1. Introduction

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^{N}\) \((N\geq1)\) with smooth boundary \(\partial\Omega\). We consider the initial-boundary value problem:

where \(p>2\) and \(\nu\) represents the unit outward normal to \(\partial\Omega\). Here, \(g(t)\) is a positive function that represents the kernel of the memory term, which will be specified in Section 2 and \begin{equation*} g\ast\Delta^{2}u(t)=\int_{0}^{t}g(t-\tau)\Delta^{2}u(\tau)d\tau. \end{equation*} The motivation of our work is due to the initial boundary problem of the double dispersive-dissipative wave equation with nonlinear damping and source terms which has been discussed by Di and Shang [1] by considering the existence of global solutions and the asymptotic behavior of global solutions with \(m\geq p\).

In the absence of the dispersive term and the nonlinear damping term, model \(2\) reduces to the following wave equation

Shang [2] studied the well-posedness, asymptotic behavior, and the finite time blow-up of the solutions under some suitable conditions on \(f\) and for \(N=1,2,3\). Zhang and Hu [3] showed the existence and the stability of global weak solutions. Xie and Zhong [4] obtained the existence of global attractors in \(H^{1}_{0}(\Omega)\times H^{1}_{0}(\Omega)\), where the nonlinear term \(f\) satisfies a critical exponential growth assumption. Xu et al., [5] used the multiplier method to investigate the asymptotic behavior of solutions for (3).

Mellah [6] considered the following initial-boundary value problem

\begin{equation*}\label{W} \left\{ \begin{array}{ll} u_{tt}-\Delta u+\Delta^{2}u-g\ast\Delta^{2}u+u_{t}=|u|^{p-1}u,& x\in\Omega,\ t>0,\\ u=0,\quad \frac{\partial u}{\partial\nu}=0,& x\in\partial\Omega,\ t>0,\\ u(x,0)=u_{0}(x),\quad u_{t}(x,0)=u_{1}(x),& x\in\Omega, \end{array} \right. \end{equation*} in a bounded domain and \(p>1\). He investigated the small data global weak solutions and general decay of solutions, respectively.

Motivated by previous works, it is interesting to prove that problem (1) has a global weak solution assuming small initial data. In addition, we show the general decay of solutions. The global solutions are constructed by means of the Galerkin approximations and the general decay is obtained by employing the technique used in [7].

2. Preliminaries

In this section, we present some materials needed in the proof of our main result. We use the following abbreviations; \(\|\cdot\|_{p}=\|\cdot\|_{L^{p}(\Omega)}\) \((1\leq p\leq+\infty)\) denotes usual \(L^{p}\) norm, \((\cdot,\cdot)\) denotes the \(L^{2}\)-inner product, and consider the Sobolev spaces \(H^{1}_{0}(\Omega)\) and \(H^{2}_{0}(\Omega)\) with their usual scalar products and norms. We also use the embedding \(H^{1}_{0}(\Omega)\hookrightarrow L^{p}(\Omega)\) for \(2< p\leq\frac{2N}{N-2}\) if \(N\geq3\) or \(2< p< \infty\) if \(N=1,2\). In this case, the embedding constant is denoted by \(C_{*}\), that is \( \|u\|_{p}\leq C_{*}\|\nabla u\|_{2}. \) We define \begin{equation*} Q(z)=\frac{1}{2}z^{2}-\frac{C_{*}^{p}}{p}z^{p}. \end{equation*} By the direct computation, we deduce that \(Q\) is increasing in \([0,z_{0}]\), where \( z_{0}=C_{*}^{\frac{p}{2-p}} \) is its unique local maximum.

Next, we give the assumptions for problem (1).

• (G1) The relaxation function \(g:\mathbb{R}_{+}\rightarrow \mathbb{R}_{+}\) is a bounded \(C^{1}\) function such that \begin{equation*} g(0)>0,\quad 0< \eta=1-\int_{0}^{\infty}g(\tau)d\tau\leq1-\int_{0}^{t}g(\tau)d\tau=\eta(t). \end{equation*}
• (G2) There exist positive constants \(\xi_{1}\) and \(\xi_{2}\) such that \begin{equation*} -\xi_{1}g(t)\leq g'(t)\leq-\xi_{2}g(t)\quad \forall t\geq0. \end{equation*}
• (G3) We also assume that \begin{equation*} 2< p\leq \frac{2N}{N-2}\ \ \mbox {if} \ \ N\geq3 \ \ \mbox {and} \ \ \ p>2 \ \ \ \mbox {if} \ \ N=1,2, \end{equation*}

where \(\lambda_{1}\) is the first eigenvalue of the following problem

Remark 1. [8] Assuming \(\lambda_{1}\) is the first eigenvalue of the problem (4), we have

The energy associated with problem (1) is given by for \(u\in H^{2}_{0}(\Omega)\), where \begin{equation*} (g\circ\Delta u)(t)=\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u(t)\|_{2}^{2}d\tau. \end{equation*} Now, we are in a position to state our main results.

3. Main results

In this section, we are going to obtain the existence of global weak solutions for problem (1) with the initial conditions \(\|\nabla u_{0}\|_{2}< z_{0}\) and \(E(0)< Q(z_{0})\).

Theorem 1. Assume that \((G1)-(G3)\) hold, and that \(\left\{u_{0},u_{1}\right\}\) belong to \(H^{2}_{0}(\Omega)\times H^{1}_{0}(\Omega)\). Further assume that \(\|\nabla u_{0}\|_{2}< z_{0}\) and \(E(0)< Q(z_{0})\). Then, problem (1) admits a global weak solution, which satisfies \[u\in L^{\infty}(0,\infty;H^{2}_{0}(\Omega)),\quad u_{t}\in L^{\infty}(0,\infty;H^{1}_{0}(\Omega)).\] Moreover, the identity

holds for \(0\leq t< \infty\). Also, for an increasing \(C^{2}\) function \(\zeta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) satisfying and, if \(\|g\|_{L^{1}(0,\infty)}\) is sufficiently small, we have for \(\kappa>0\) \begin{equation*} E(t)\leq E(0)e^{-\kappa \zeta(t)},\quad \forall t\geq0. \end{equation*}

Remark 2. From (8) and \((G2)\), we obtain

Proof of Theorem 1 (Main result)

We divide the proof into two steps. In step 1, we prove the small data global existence of weak solutions by using the Faedo-Galerkin approximation and in step 2, we establish the general decay of energy employing the method used in [7].

Step 1: Global existence of weak solutions

Let \(\left\{\omega_{j}\right\}_{j=1}^{\infty}\) be an orthogonal basis of \(H^{2}_{0}(\Omega)\) with \(\omega_{j}\) being the eigenfunction of the following problem: \begin{equation*} -\Delta \omega_{j}=\lambda_{j}\omega_{j},\quad x\in\Omega,\quad \omega_{j}=0,\quad x\in\partial\Omega. \end{equation*} Let \(V^{n}=\text{Span}\left\{\omega_{1},\omega_{2},\cdot\cdot\cdot,\omega_{n}\right\}\). By the standard method of ODE, we know that there exists only one local solution \begin{equation*} u^{n}(t)=\sum_{j=1}^{n}b^{n}_{j}(t)\omega_{j} \end{equation*} of the Cauchy problem as follows: By the standard theory of ODE system, we prove the existence of solutions of problem (10)-(11) on some interval \([0, t_{n})\), \(0< t_{n}< T\) for arbitrary \(T>0\), then, this solution can be extended to the whole interval \([0,T]\) using the first estimate given below.

A Priori Estimates

Setting \(\omega=u^{n}_{t}(t)\) in (10), we have A direct computation shows that Inserting (13) into (12) and integrating over \([0,t]\subset[0, T]\), we obtain From assumption \((G3)\) and the Sobolev embedding, we have \begin{equation*} \|u^{n}\|^{p}_{p}\leq C_{*}^{p}\|\nabla u^{n}\|^{p}_{2}, \end{equation*} and then we have By using the fact that \begin{equation*} -\int_{0}^{t}(g'\circ\Delta u^{n})(\tau)d\tau+\int_{0}^{t}g(\tau)\|\Delta u^{n}(\tau)\|^{2}_{2}d\tau\geq0, \end{equation*} estimate (15) yields From \(E(0)< \mathcal{Q}(z_{0})\) and (11), it follows that for sufficiently large \(n\). We claim that there exists an integer \(N\) such that Suppose the claim is proved, then \(\mathcal{Q}(\|\nabla u^{n}\|^{2}_{2})\geq0\) and from (16) and (17), for sufficiently large \(n\) and \(0\leq t< \infty\).

Proof of the claim

Suppose that (18) false, then for each \(n>N\), there exists \(t\in[0,t_{n})\) such that \(\|\nabla u^{n}(t)\|_{2}\geq z_{0}\). Note that from \(\|\nabla u_{0}\|_{2}< z_{0}\) and (11) there exists \(N_{0}\) such that \begin{equation*} \|\nabla u^{n}(0)\|_{2}< z_{0}\quad \forall n>N_{0}. \end{equation*} Then by continuity there exits a first \(\widetilde{t_{n}}\in[0,t_{n})\) such that from where \begin{equation*} \mathcal{Q}(\|\nabla u^{n}(t)\|_{2})\geq0 \quad \forall t\in[0,\widetilde{t_{n}}]. \end{equation*} From \(E(0)< \mathcal{Q}(z_{0})\) and (19), there exists \(N>N_{0}\) and \(\gamma\in(0,z_{0})\) such that \begin{eqnarray*} 0&\leq&\frac{1}{2}\|u^{n}_{t}(t)\|^{2}_{2}+\frac{1}{2}\|\nabla u^{n}_{t}(t)\|^{2}_{2}+\frac{\eta(t)}{2}\|\Delta u^{n}(t)\|^{2}_{2}+\frac{1}{2}(g\circ\Delta u^{n})(t)+\mathcal{Q}(\|\nabla u^{n}(t)\|^{2}_{2})\nonumber\\ &\leq&\mathcal{Q}(\gamma)\quad \forall\;\;\; t\in[0,\widetilde{t_{n}}]\quad \forall n>N. \end{eqnarray*} The monotonicity of \(\mathcal{Q}\) in \([0,z_{0}]\) implies that \begin{equation*} 0\leq\|\nabla u^{n}(t)\|^{2}_{2}\leq\gamma< z_{0}\quad \forall t\in[0,\widetilde{t_{n}}], \end{equation*} in particular, \(\|\nabla u^{n}(t)\|^{2}_{2}< z_{0}\), which is a contradiction to (20). From (19), we have Using Sobolev inequality, (5) and (21), it follows that Moreover, by (25), we get Therefore, there exist \(u\), \(\chi\) and a subsequence still denotes \(\left\{u_{n}\right\}\) such that Besides, from Lions-Aubin Lemma we also have and consequently, making use of the Lemma 1.3 in [9], we deduce Thus, we obtain that \(u\) is a global weak of problem (1). In order to prove (7), we use the mean value theorem, we see that there exists \(0< \theta_{n}< 1\) such that \begin{eqnarray*} \|u^{n}\|_{p}^{p}-\|u\|_{p}^{p} &\leq& p\left|\int_{\Omega}|u+\theta_{n}u^{n}|^{p-2}(u+\theta_{n}u^{n})(u^{n}-u)dx\right|\nonumber\\ &\leq& p\|u+\theta_{n}u^{n}\|_{p}^{p-1}\|u^{n}-u\|_{p}\nonumber\\ &\leq&c\|u^{n}-u\|_{p}\rightarrow 0 \ \ \mbox {as} \ \ \ n\rightarrow +\infty, \end{eqnarray*} and for each fixed \(t>0\), we obtain \begin{eqnarray*} |(g\circ\Delta u)(t)-(g\circ\Delta u^{n})(t)|&=&\left|\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u(t)\|^{2}_{2}d\tau-\int_{0}^{t}g(t-\tau)\|\Delta u^{n}(\tau)-\Delta u^{n}(t)\|^{2}_{2}d\tau\right|\nonumber\\ &\leq& \int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u^{n}(\tau)\|_{2}\|\Delta u(\tau)+\Delta u^{n}(\tau)\|_{2}d\tau\nonumber\\ &&+\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u^{n}(\tau)\|_{2}d\tau\|\Delta u(t)+\Delta u^{n}(t)\|_{2}\nonumber\\ &&+\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)+\Delta u^{n}(\tau)\|_{2}d\tau\|\Delta u(t)-\Delta u^{n}(t)\|_{2}\nonumber\\ &&+\int_{0}^{t}g(\tau)d\tau\|\Delta u(t)+\Delta u^{n}(t)\|_{2}\|\Delta u(t)-\Delta u^{n}(t)\|_{2}\nonumber\\ &\leq&c\int_{0}^{t}g(t-\tau)\|\Delta u(\tau)-\Delta u^{n}(\tau)\|_{2}d\tau\\ &&+c\int_{0}^{t}g(\tau)d\tau\|\Delta u(t)-\Delta u^{n}(t)\|_{2}\rightarrow 0 \ \ \mbox {as} \ \ \ n\rightarrow +\infty. \end{eqnarray*} Thus, we have \begin{equation*} \lim_{n\rightarrow +\infty}\|u^{n}\|_{p}^{p}=\|u\|_{p}^{p},\quad \lim_{n\rightarrow +\infty}(g\circ\Delta u^{n})(t)=(g\circ\Delta u)(t). \end{equation*} From (11), it follows that \(E^{n}(0)\rightarrow E(0)\) as \(n\rightarrow+\infty\). Finally, taking \(n\rightarrow +\infty\) in (14), we deduce that the energy identity (7) holds for \(0\leq t< \infty \).

Step 2: General decay of the energy

Here, we prove the energy decay estimate of the global solutions obtained in the previous section. To obtain the decay result, we use the following lemmas which are of crucial importance in the proof.

Lemma 1. Let \(u\in L^{\infty}(0,\infty;H^{2}_{0}(\Omega))\) with \(u_{t}\in L^{\infty}(0,\infty;H^{1}_{0}(\Omega))\) be the solution of (1) and \(E(0)< \mathcal{Q}(z_{0})\), \(\|\nabla u_{0}\|_{2}< z_{0}\), then we have

where \(C_{1}=\frac{1}{2}(1+B^{2})\), \(C_{2}=\frac{1}{2}(1+\lambda_{1}^{-1})\) and \(B\) is the optimal constant satisfying the Poincare inequality \(\|u_{t}\|_{2}\leq B\|\nabla u_{t}\|_{2}\).

Proof. From \(E(0)< \mathcal{Q}(z_{0})\) and \(\|\nabla u_{0}\|_{2}< z_{0}\), we can obtain \(\mathcal{Q}(\|\nabla u(t)\|_{2})\geq0\) for \(0\leq t< \infty\). Thus we have \begin{eqnarray*} E(t)&=&\frac{1}{2}\|u_{t}\|^{2}_{2}+\frac{1}{2}\|\nabla u_{t}\|^{2}_{2}+\frac{1}{2}\|\nabla u\|^{2}_{2}+\frac{1}{2}(g\circ \Delta u)(t)+\frac{1}{2}\left(1-\int_{0}^{t}g(\tau)d\tau\right)\|\Delta u\|^{2}_{2}-\frac{1}{p}\|u\|_{p}^{p}\nonumber\\ &\geq&\frac{1}{2}\|u_{t}\|^{2}_{2}+\frac{1}{2}\|\nabla u_{t}\|^{2}_{2}+\frac{\eta}{2}\|\Delta u\|^{2}_{2}+\frac{1}{2}(g\circ \Delta u)(t)+\mathcal{Q}(\|\nabla u(t)\|_{2})\\&\geq&0, \end{eqnarray*} and \begin{eqnarray*} E(t)&\leq&\frac{1}{2}\|u_{t}\|^{2}_{2}+\frac{1}{2}\|\nabla u_{t}\|^{2}_{2}+\frac{1}{2}\|\Delta u\|^{2}_{2}+\frac{1}{2}(g\circ \Delta u)(t)+\frac{1}{2}\|\nabla u\|^{2}_{2}\nonumber\\ &\leq& \frac{1}{2}B^{2}\|\nabla u_{t}\|_{2}^{2}+\frac{1}{2}\|\nabla u_{t}\|_{2}^{2}+\frac{1}{2}\lambda_{1}^{-1}\|\Delta u\|^{2}_{2}+\frac{1}{2}\|\Delta u\|^{2}_{2}+\frac{1}{2}(g\circ \Delta u)(t). \end{eqnarray*} Let \(C_{1}=\frac{1}{2}(1+B^{2})\) and \(C_{2}=\frac{1}{2}(1+\lambda_{1}^{-1})\), then we have (32).

Lemma 2. The energy \(E(t)\) satisfies

Proof. From (9), we have

From assumptions \((G2)\) and since \(\int_{0}^{t}g'(\tau)d\tau=g(t)-g(0)\), we obtain Then, Combining (34) and (35) our conclusion holds. Multiplying (33) by \(e^{\kappa\zeta(t)}\) \((\kappa>0)\) and using (32), we have Using the fact that \(\zeta_{t}\) is decreasing by (8), we conclude that Choosing \(\|g\|_{L^{1}(0,\infty)}\) sufficiently small so that \[g(0)-\xi_{1}\|g\|_{L^{1}(0,\infty)}=K>0\] and defining \[\kappa_{0}=\min\left\{\frac{1}{C_{1}\zeta_{t}(0)},\frac{\xi_{2}}{\zeta_{t}(0)},\frac{K}{2C_{2}\zeta_{t}(0)}\right\},\] we conclude by taking \(\kappa\in(0, \kappa_{0}]\) in (37) that Integrating (38) over \((0,t)\), it follows that

Example 1. For \(\zeta(t)=t+\frac{t}{t+1}\), we can get the exponential decay rate \( E(t)\leq E(0)e^{-\kappa t},\quad \forall t\geq0. \) For \(\zeta(t)=ln(1+t)\), we can get polynomial decay rate \( E(t)\leq E(0)(1+t)^{-\kappa },\quad \forall t\geq0. \)

Conflicts of Interests

''The author declares no conflict of interest.''

1. Introduction

Studies in normed spaces have been carried out with very interesting results obtained as shown in [1,2,3]. Several properties of operators in these spaces have been studied including norms, orthogonality, spectra, among others in Hilbert spaces and Banach spaces in general [4]. Regarding norm-attainable classes, a lot has been done in terms of structural properties [5,6,7,8,9,10,11,12,13]. Elementary operators have also been of considerable attention on many aspects particularly on their orthogonality, (see [14] and the references therein).

Let \(H\) be an infinite dimensional complex Hilbert space and \(B(H)\) the algebra of all bounded linear operators on \(H\). We say \(T\in B(H)\) is said to be norm-attainable if there exists a unit vector \(x_{0}\in H\) such that \(\|Tx_{0}\|=\|T\|.\) We denote by \(NA(H)\) the class of all norm-attainable operators on \(H.\) Benitez [15] gave a detailed description of several types of orthogonality which have been studied in real normed spaces namely: Robert's orthogonality, Birkhoff's orthogonality, Orthogonality in the sense of James, Isosceles, Pythagoras, Carlsson, Diminnie, Area among others as described in [16].

For \(x\in \mathcal{M}\) and \(y\in \mathcal{N},\) where \( \mathcal{M}\) and \( \mathcal{N}\) are subspaces of \(E\) which is a normed linear space, we have:

• Roberts-\(\|x-\lambda y\|=\|x+\lambda y\|,\;\forall, \lambda\in \mathbb{R};\)
• Birkhoff-\(\|x+ \lambda y\|\geq\|x\|;\)
• Isosceles-\(\|x- y\|=\|x+ y\|;\)
• Pythagorean-\(\|x- y\|^{2}=\|x\|^{2}+ \|y\|^{2};\)
• \(a\)-Pythagorean-\(\|x- ay\|^{2}=\|x\|^{2}+ a^{2}\|y\|^{2},\;a\neq 0;\)
• Diminnie-\(\sup\{f(x)g(y)-f(y)g(x): f,\;g\in S^{\prime}\}=\|x\| \|y\|\) where \(S^{\prime}\) denotes the unit sphere of the topological dual of \(E\);
• Area-\(\|x\| \|y\|=0\) or they are linearly independent and such that \(x, -x, y, -y\) divide the unit ball of their own plane (identified by \(\mathbb{R}^{2}\)) in four equal areas [4].

For characterizations of elementary operators, we consider Banach algebras and norm-attainable classes. Let \(\mathcal{A}\) be a normed space and let \(T_{A,B}:\mathcal{A}\rightarrow \mathcal{A}.\) \(T\) is called an elementary operator if it has the following representation: \[T(X)=\sum_{i=1}^{n}A_{i}XB_{i},\;\forall\;X\in \mathcal{A},\] where \(A_{i},\;B_{i}\) are fixed in \(\mathcal{A}.\)

Let \(\mathcal{A}=B(H)\). For \(A,\,B\in B(H)\) we define specific elementary operators as follows [21]:

• the left multiplication operator \(L_{A}:B(H)\rightarrow B(H)\) by \(L_{A}(X)=AX, \;\forall\;X\in B(H);\)
• the right multiplication operator \(R_{B}:B(H)\rightarrow B(H)\) by \(R_{B}(X)=XB, \;\forall\;X\in B(H);\)
• the generalized derivation (implemented by \(A,\;B\)) by \(\delta_{A,B}=L_{A}-R_{B};\)
• the basic elementary operator(implemented by \(A,\;B\)) by \(M_{A,\;B}(X)=AXB, \;\forall\;X\in B(H);\)
• the Jordan elementary operator(implemented by \(A,\;B\)) by \(\mathcal{U}_{A,\;B}(X)=AXB+BXA, \;\forall\;X\in B(H).\)

In this paper, we characterize the orthogonality of the range and the kernel of several types of important elementary operators in normed spaces. Let \(X \in B(H)\) be a compact operator, and let \(s_{1} (X) \geq s_{2} (X) \geq .. \geq 0\) denote the eigenvalues of \(|X| = (X*X)^{\frac{1}{2}}\) arranged in their decreasing order. Considering the normed classes, we call \(C_{p:_{1\leq p< \infty }}( H)\) (simply denoted by \(C_{p}( H)\)), the Von Neumann-Schatten \(p\)-class, if \[\|X\|=\left[\sum^{\infty}_{i=1}S_{i}(X)^{p}\right]^{\frac{1}{p}}=tr(|X|^{p})^{\frac{1}{p}}< +\infty,\] where \(tr\) denotes the trace functional. Hence, \(C_{1} (H)\) is the trace class, \(C_{2} (H)\) is the Hilbert-Schmidt class and the case \(p = \infty\), corresponds to the class of compact operators \(C_{\infty}(H)\) equipped with the norm \(\|X\|_{\infty}=S_{1}(X).\)

2. Preliminaries

In this section, we give the basic concepts which are useful in the sequel. We begin with the following definition;

Definition 1. Let \(W\) be a complex normed space, then for any elements \(x, y \in W,\) we say that \(x\) is orthogonal to \(y\), noted by \(x\perp y,\) if and only if for all \(\alpha, \beta \in \mathbb{C}\) there holds \(\|\alpha y + \beta x \| \geq\|\beta x \|.\)

Definition 2. Let \(W\) be a complex Banach space. If \(P\) and \(Q\) are linear subspaces in \(W\), we say that \(P\) is orthogonal to \(Q\) , denoted by \(P\perp Q\), if \(\|x + y\|\geq \|x\|\) for all \(x \in P\) and all \( y \in Q\). If \(P = {x}\), we simply write \(x\perp Q\).

Remark 1. We note that the orthogonality in the definition above is not symmetric [17] and if \(W\) is a Hilbert space with its inner product then it follows from [18] that \(\langle x, y\rangle= 0\) which means that Birkhoff-James's orthogonality generalizes the usual sense of orthogonality in a Hilbert space [19].

Definition 3. Let \(F: X \rightarrow Y\) and \(J: Y \rightarrow Z\) be operators between norm-attainable classes. We say \(J\) is orthogonal to \(F\) if \(s\in Ker T\Rightarrow \|s+E(x)\|\geq \|s\|,\, \forall\, x\in X\). Moreover, if \(F=J\), we shall say that \(F\) is orthogonal.

Remark 2. The authors in [10] proved that if \(A\) and \(B\) are normal operators then for all \(X, S \in B (H),\, S \in ker \delta _{A,B} \Rightarrow \|\delta _{A,B} (X) + S\| = \|S\|,\) where the \( ker \delta _{A,B}\) denotes the kernel of \(\delta _{A,B}.\) This means that the kernel of \(\delta _{A,B}\) is orthogonal to its range. This result has been generalized in different directions, to non-normal operators [20], to \(C_{p}(H) \), and to some elementary operators [19].

3. Conditions for orthogonality in norm-attainable class and \(C_{p}\)-classes

Let \( X\) be a normed linear space over the field \(\mathbb{K}\) and \(X^{\dagger}\) its topological dual. For all \( x\in X,\) \[D(X)=\left\{ \varphi\in X^{\dagger}:\varphi(x)=\|x\|^{2};\|\varphi \|=\|x\|\right\}\] is called the duality mapping. The Hahn-Banach's theorem [12] ensures that there always exists at least one support functional (a support functional \(\varphi\) at \(x \in X\) is a norm-one linear functional in \(X^{\dagger}\) such that \( (\varphi(x) = \|x\|)\) at each vector \(x \in X )\) and therefore \(D(x)\) is non-empty for every \(x \in X\). Moreover, it is well known that \(D(x)\) is convex and weak*-compact subset of \(X^{\dagger}\). Hence, \(D \) is not linear in general but it is homogeneous, that is, for all \( \alpha \in \mathbb{R}, \; D (\alpha x) = \alpha D(x)\).

Proposition 1. Let \(K\) be a norm-attainable subclass of a norm-attainable set \(X\) and \(x \not\in K,\) then \(x \perp K\) if and only if there exists \( \tilde{\varphi} \in D (x)\) such that \( K \subseteq ker \tilde{\varphi}\).

Proof. Since \(K\) is a norm-attainable class, let \(\varphi \in D(x)\) be such that \(\langle \varphi,y \rangle=0,\) for all \(y\in K\). Then \(\varphi(x+y)=\varphi(x)=\varphi\|x\|^{2}\) and \( \|x\| ^{2}= \|\varphi\| \|x + y\| = \|x\| \|x + y\| , \) that is, \(\|x\| = \|x + y\|,\) for all \(y\in K\). Hence, \(x\perp K.\)

Conversely, let \(x \not\in K\) such that \(x \perp K\). Then, for all \(y \in K\), \(x\) and \(y\) are linearly independent vectors. Let \(L\) be the closed norm-attainable subclass spanned by \(K\) and \({x}, L = [K, {x}]\). Define the function \(\varphi\) on \(L\) by \(\varphi (\alpha x + \beta y) = \alpha \|x\|^{2},\) for all \(y \in K\) and \(\alpha\, , \beta \in \mathbb{C}\). Clearly, \(\varphi\) is linear (by the assumption that \(K\) is a linear subset of \(X\) ). To prove the continuity of \(\varphi\), let \(z \in L\), then \(z = \alpha x + \beta y\) and \(\varphi (z) = \alpha \|x\|^{ 2}\). By the definition of \(\varphi\) and the assumption that \(x \perp K,\) we derive that \(\|z\| = \|\alpha x\|.\) If \(\alpha \neq 0\), it is easy to see from known inequalities that \(|\varphi(z)|=|\alpha|\|x\|^{2}.\frac{\|z\|}{\|z\|}\leq \frac{|\alpha|\|x\|^{2}}{\|\alpha x\|}\|z\|=\|x\|\|z\|. \) If \(\alpha =0\), then \(|\varphi(z)|=|\varphi (\beta y)|=0\). Hence, \( |\varphi (z)|\leq \|x\|\|z\|, \) for all \(z\in L\). Therefore, \(\varphi\) is continuous on \(L\) and \(\|\varphi \| = \|x\|.\) By Hahn-Banach theorem there is a continuous linear functional \( \tilde{\varphi}\) on \(X\) such that \( \tilde{\varphi}_{|L}=\varphi\) and \(\| \tilde{\varphi}\|=\|\varphi\|,\) where \(\tilde{\varphi}_{|L}\) is the restriction of \(\varphi\) on \(L\). It follows, by the definition of \(\varphi\) and \(\tilde{\varphi}_{|L}=\varphi\) that \(K \subseteq ker \tilde{\varphi} \) and \(\tilde{\varphi}\in D (x).\)

Theorem 1. Let \(K\) be a norm-attainable subclass of a norm-attainable set \(X\). For all \(x, y \in X\,, \,x \perp y\) if and only if there exists \(\varphi\in D (x)\) such that \(\varphi (y) = 0.\) Moreover, for all \(x \in X \) and for all \( \varphi \in D (x)\), \( x \perp ker \varphi.\)

Proof. From Proposition 1 and an analogous computation from [20], the proof is clear.

Remark 3. We can consider general normed spaces as follows: Let \(K\) be a nonempty subset of a Banach space \(X\) and \(T \in NA(X )\), we denote the duality adjoint of \(T\) by \(T^{\dagger}\) and set \(K^{\perp r}=\{x \in X : x\perp y; \forall y \in K\}.\) It is clear that if \(\{x_{n}\}_{n}\) is a sequence in a subset \(K\) converging to \(y\) and \(x \perp x_{n},\) for all \(n\), then \(x \perp y.\) Hence, \(x \perp K \rightarrow x \perp \overline{K}.\)

Proposition 2. Let \(X\) be a norm-attainable subclass of \(NA(H)\), then

• (1). If \(K\) and \(L\) are closed subclasses of \(X\) and \(K \oplus L\).
• (2). Let \(T \in NA(H)\) and \(s \in X.\) Then
  • (i). \(s\perp ran T \Leftrightarrow \exists\varphi\in D (s)\) such that \(\varphi \in ker T^{\dagger}.\)
  • (ii). If \(T\) is orthogonal, then \(ker T \oplus \overline{ran (T )}\) is a closed subclass of \(X.\)

Proof.

• (1). Let \( z \in X : z = \lim_{n}(x_{n} + y_{n})\; \text{and}\; z_{n}= (x_{n} + y_{n})\); for all \(n \geq 1,\) where \(x_{n} \in K\; \text{and}\; y_{n} \in L\). From \(K \in L^{\perp r}\), we obtain, \[\|z_{n} - z_{n+p}\| = k\|y_{n} - y_{n+p} + x_{n} - x_{n+p}\| = \|x_{n} - x_{n+p}\|,\,\,\text{ for all}\,\,n, p.\] So, \(\{x_{n}\}_{n}\) is a Cauchy sequence, hence \(\lim_{n}x_{n} \in K\). Setting \(x = \lim_{n}x_{n}\), we get \(\lim_{n}y_{n} = z - x \in L\) and therefore, \(z \in K \oplus L.\)
• (2).
  • (i). By Proposition 1, \(s\perp ran T\Leftrightarrow \exists \varphi D(s), ran (T)\subseteq ker \varphi\) that is \(\varphi(Tx)=(T^{\dagger}\varphi)x=0,\) for all \(x\in X.\) Hence \(s\perp ran\, T\Leftrightarrow \exists \varphi\in D(s):\varphi\in Ker T^{\dagger}.\)
  • (ii). It is a direct consequence of the assertions \((1)\) and \((2)(i).\)

Remark 4. Let \(f : X \rightarrow X\) be a map on \(X\), not necessarily linear or additive, and \(F_{f} : X \rightarrow \mathbb{R}^{+}\) be a map defined by \( F_{f}(x)=\|f(x)\|,\,\forall\, x\in X.\) We say that \(F_{f}\) has a global minima at \(a \in X\) if \(\|f (a)\| = \|f (x)\|\), for all \(x \in X \).

As an application of the previous results, the following result gives us a necessary and sufficient conditions in term of Birkhoff-James orthogonality for minimizing the map \(F_{f}.\)

Lemma 1. Let \(T\) and \(f : X \rightarrow X\) be norm-attainable maps and \(a \in X.\) Suppose that the relation \(f (x) + T (y) = f (y) + T (x)\) holds for all \(x, y \in X\), if \(f (a)\perp T (x),\, \forall x \in X\) then the map \(F_{f}\) has a global minima at \(a\). Moreover, if we suppose that \(T\) is linear and \(f (x) = T (x) + f (0), \forall x \in X \), then \(F_{f}\) has a global minima at \(a\) if and only if \(f (a)\perp ran T\) if and only if there is \(\varphi \in D (f (a))\) such that \(\varphi \in ker T^{\dagger}\), where \(T^{\dagger}\) is the duality map of \(T.\) Lastly, if \(f(a)\) is a smooth point, then the existence of \(\varphi\) is unique.

Proof. It follows from Theorem 1, that for all \(x \in X , f (a)\perp T (x) \Leftrightarrow \exists \varphi \in X^{\dagger}: \varphi (f (a)) = \|f (a)\|_{2}= \|\varphi\|^{2}\) and \(\varphi (T (x)) = 0\). Then by the relation defined in (i), we get \(\varphi (f (a)) = \varphi (f (a) + T (x)) = \varphi (f (x) + T (a)) =\varphi (f (x))\), so that \(\|\varphi (f (a))\| = \|f (a)\| \|\varphi\| \leq \|\varphi\| \|f (x)\|\). Hence, \(\|f (a)\| \leq \|f (x)\|\), for all \(x \in X \). Since the maps \(f, T\) satisfy the relation cited in (i), the sufficient condition follows from the first part of the proof. By linearity of \(T\), we get \(f (a) + \lambda T (x) = f (a + \lambda x),\) for all \(x \in X , \lambda \in \mathbb{C}\). Hence, \(F_{f}\) has a global minima at a implies \(\|f (a) + \lambda g(x)\| \geq \|f (a + \lambda x)\| = \|f (a)\|\). The other equivalence follows immediately. To complete the proof, if \(f (a)\) is a smooth point, then \(f (a)\) has only one functional support and therefore \(D (f (a))\) has one element.

Lemma 2. If \(\mathcal{J}\) is a separable ideal of norm-attainable operators in \(NA(H)\) equipped with unitary invariant norm, then its dual \(\mathcal{I}\) is isometrically isomorphic to an ideal of compact operator \(\mathcal{Q}\) not necessarily separable, i.e., \[ \phi:\mathcal{Q}\rightarrow \mathcal{J^{\dagger}},\, R\mapsto \phi_{R}(X)=tr (XR).\]

Proof. Following the argument in Lemma 1, the proof is trivial.

Theorem 2. Let \(A \in C_{p}(H), T \in B(C_{p}(H))\) and \(f, F_{f}\) are defined as in Lemma 1, where \(f (A)\) is given by its polar decomposition \(f (A) = u |f (A)|.\) If \(A \in C_{p}(H)\), then \(F_{f}\) has a global minimizer at A if and only if \(f(A)\perp ran T\) if and only if \(|f (A)|^{p-1}u^{*}\in ker T^{\dagger}\). Moreover, if \(A \in C_{1} (H)\) and \(f (A)\) is a smooth point, then \(F_{f}\) has a global minimizer at \(A\) if and only if \(f (A)\perp ran T\) if and only if \(u^{*}\in ker T^{\dagger}\) when \(f (A)\) is injective (or \(u \in kerT^{\dagger}\) when \(f (A)^{*}\)is injective).

Proof. From Lemma 1, we have that \(F_{f}\) has a global minimizer at \(A\) if and only if there exists \(\varphi\in D (f (A))\) such that \(\varphi \in ker T^{\dagger}\). If \(A \in C_{p}(H)\), then by the properties of the isomorphism, it follows that \( \varphi \in C_{p}(H)^{\dagger}\) if and only if there exists \( R\in C_{p}(H)\) such that \(\phi_{R}=\varphi, \) \( \|\varphi\|=\|R\| \) and \(\varphi(X)=tr\, RX,\) for all \(X\in C_{p}(H).\) Hence, the smoothness of \( C_{p}(H),\) \(F_{f}\) has a global minimizer at \(A\) if and only if there is a unique operator \(R\) such that \(\varphi (f (A)) = tr (f (A)R) =\|f (A)\|^{2}_{p}= \|R\|^{2}_{q}\) and \(tr (T^{\dagger}(R) X)= 0\), for all \(X \in C_{p}(H)\). To complete the proof, it is well known that \(C_{1} (H)\) is neither reflexive, nor smooth and its dual \(C_{1} (H)^{\dagger}\) is isometrically isomorphic to \(NA(H).\) This isomorphism is given by \( \phi\in NA(H)\) if and only if \( C_{1} (H)^{\dagger}\) contains \(R\mapsto\phi_{R}\) such that \(\phi_{R}(X)=tr(XR)\) so we have that \( \varphi\in C_{1} (cH)^{\dagger}\Leftrightarrow\exists R \in NA(H):\phi_{R}=\varphi,\) \(\|\varphi\|=\|R\|\) and \(\varphi(X)=tr\, RX\, \forall\, X\in C_{1} (H)\), so that if \(f (A)\) is a smooth point then \(F_{f}\) has a global minimizer at \(A\) if and only if there is a unique operator \(R\) such that \( \varphi (f (A)) = tr (f (A)R) = \|f (A)\|^{2}_{1}= \|R\|^{2}\) and \( tr (T^{\dagger}(R) X)= 0, \forall\, X\in C_{1} (H).\) Since \(f (A)\) is smooth then by [7], either \(f (A)\) or \(f (A)^{*}\) is injective, thus either \(u\) or \(u^{*}\) is an isometry i.e., \(uu^{*}= I\) or \(u^{*}u = I\). So it suffices to take \(R = \|f (A)\|_{1}u^{*} \)or \(R = \|f (A)\|_{1} u\), which is the unique operator required in both cases. So, \(f (A)\) or \(f (A)^{*}\) is injective.

Proposition 3. Let \(K\) be a closed subclass of a norm-attainable class \(X\). If \(X\) is separable and \(K^{\perp r}= {0}\), then \(K = X.\)

Proof. If \(K \neq X\), then there exists \(\varphi \in X^{\dagger} :K\subseteq ker\varphi.\) Since \(D (\varphi)\) is not empty, then there is \(f \in X^{\dagger\dagger}\) with \(f (\varphi) = \|\varphi\|^{2}= \|f \|^{2}\). Let \(J\) be the natural injection between \(X\) and \( X^{\dagger\dagger}\) i.e., \(J : X \rightarrow X^{\dagger\dagger}, \forall x \in X , \forall \psi \in X^{\dagger},\) such that \( J (x)\psi = \psi (x),\; \|J (x)\|= \|x\|.\) So, by the separability of \(X , J\) is a bijection and, then there is \(0 \neq x \in X\) such that \(J (x) = f\). Hence, \(\varphi (x) = \|x\|^{2}= \|\varphi\|^{2}\). Thus, \(\varphi \in D (x)\) and by application of Proposition 1, we get \(0 \neq x \in K^{\perp r}= {0}\), a contradiction.

Proposition 4. Let \(X\) be a separable, smooth and strictly convex norm-attainable class and \(T \in NA(X )\). If \(T^{\dagger}\)is orthogonal, then \(\forall s\in X,\; s\perp ran T\Rightarrow s\in Ker T.\)

Proof. Let \(s \in X\) such that \(\perp ran T\). Then by [7] and the smoothness of \(X\), there is a unique \(\varphi_{s} \in D (s)\) such that \(\varphi_{s}\in ker T^{\dagger}\). Again, by assumptions of the proposition and arguments in [21], there is \(\psi_{\varphi_{s}}\in D (\varphi_{s})\) such that \(\psi_{\varphi_{s}}\in ker T^{\dagger\dagger}\). Let \(J\) be the natural injection between \(X\) and \(X^{*}\) as defined in [22]. We see that \(J (s) \varphi_{s} = \|\varphi_{s}k\|\) and \(\|J (s)\| = \|\varphi_{s}\|,\) which means that \(J (s) \in D (\varphi_{s}).\) By the separability of \(X , J\) is a bijection. Hence, there is \(c \in X\) such that \( \psi_{\varphi_{s}}= J (c)\) and \(\|J (c)\|=\|c\|=\|\varphi\|, J (c) \varphi_{s} =\varphi_{s} (c) = \psi_{\varphi_{s}}(\varphi_{s}) = \|\varphi_{s}\|^{2}\). Then \(\varphi_{s} \in D (s) \bigcap D (c)\) and since \(X\) is strictly convex, we get \(c = s\). Thus, \(J (c) = J (S) \in ker T^{\dagger\dagger}.\) Therefore, it immediately follows that \( (T^{\dagger\dagger} J (s))\varphi =J (s)( T^{\dagger})=(T^{\dagger}\varphi)s = \varphi (T s) = 0,\) for all \(\varphi \in X^{\dagger}\). That is \(s \in ker T\).

In the next result we consider general Banach spaces. If \(X\) is a reflexive separable Banach space and \(T^{\dagger}\) is orthogonal then the implication of orthogonality holds with respect to a suitable norm in \(X\). Indeed, if \(X\) is separable, then there is an equivalent norm which is smooth and strictly convex in \(X\). This can be seen in the next theorem.

Theorem 3. Let \(X\) be a reflexive, smooth and strictly convex Banach space and \(T \in B (X ).\) If \(T\) and \(^{\dagger}\) are orthogonal. Then \( \forall s \in X : s\perp ran T \Leftrightarrow s \in ker T ,\) where \( X = ker T \oplus \overline{ran (T )}. \)

Proof. If \(T\) is orthogonal then, by Definition 2, it follows that \( \forall s \in X \) such that \( s\perp ran T\), it implies that \( s \in kerT \) and the reverse implication follows by Proposition 1. Let us prove the decomposition. Let \( y \in X\) such that \(y \in (ker T \oplus ran (T ))^{\perp r}\), then there is \(\varphi _{y} \in D (y)\) such that \(\varphi _{y}(s \oplus T X ) = 0\), for all \(s \in ker T\) and all \(x \in X\). For \(s = 0\), it follows, by [12], that \(Y \perp ran T \), and by [13], \(ran T \subseteq ker T\). So, we can choose \(x = 0\) and \(s = y\), such that this yields \(\varphi_{y}(y) = 0\). This means \(y\perp y,\) and hence \(y = 0\). Finally, the decomposition follows immediately.

4. Orthogonality conditions for elementary operators

In this section, we consider the important case, when the operator \(T\), cited in the previous section, is replaced by the elementary operators defined as follows: \[ E (X) =\sum X _{i=1}^{n} A_{i} X B_{i}\,\text{on}\,C_{p}(H)\] where \(A = (A_{1}, A_{2}, ..., A_{n})\) and \(B = (B_{1}, B_{2}, ..., B_{n})\) are \(n\)-tuples in \((NA(H))_{n}\). The duality adjoint of \(E\) on \(C_{p}(H) \) has the form \(E^{\dagger}(X) =\sum_{i=1}^{n}B_{i} XA_{i}.\) Indeed, let \(X \in C_{p}(H)\) and \(R \in B (H) (or R \in C_{q} (H)\) if \(X \in C_{p}(H),\) where we have \(\frac{1}{p}+\frac{1}{q}=1\) and that \(1< p,q< \infty\) then the following form suffices, that is, \[\phi_{R} (E (X)) = tr\left(\sum_{i=1}^{n}A_{i}XB_{i}R\right) =tr\left(X\sum_{i=1}^{n}A_{i}RB_{i}\right) = tr (XE^{\dagger}(R)) = \phi _{E^{\dagger}(R)}(X).\] We denote the formal adjoint of \(E\) by \(\tilde{E}=\sum_{i=1}^{n}A_{i}^{*}XB^{*}_{i}\), where \((A_{1}^{*},A_{2}^{*},...,A_{n}^{*}) \) and \((B_{1}^{*},B_{2}^{*},...,B_{n}^{*})\) are \(n\)-tuples of operators in \((B(H))^{n}\).

Proposition 5. Let \(H, K\) be Hilbert spaces, \(A \in NA(H),\, B \in NA(K)\) and \(E \in B (NA(K, H))\) such that \(E (X) = AXB+BXA\). If \(A \) and \(B^{*}\) are injective operators then \(E\) is injective.

Proof. If either \(AXB = 0\) or \(BXA = 0\) with \(A\) injective, then we have that \(XB = 0 = B^{*}X= 0\) implies \(X^{*}= 0 =X\) since \(B^{*}\) is injective. Thus, \(E\) is injective.

Proposition 6. Let \(A = (A_{1}, A_{2}, ..A_{n})\) and \(B = (B_{1}, B_{2}, ..B_{n})\) with \(A_{i} , B_{i}\) be operators in \(B (H)\) such that \( \sum\limits_{i=1}^{n}A_{i}A_{i}^{*}\leq 1, \sum\limits_{i=1}^{n}A_{i}A_{i}^{*}\leq 1, \sum\limits_{i=1}^{n}B_{i}B_{i}^{*}\leq 1,\,\text{and}\ \ \sum\limits_{i=1}^{n}B_{i}B_{i}^{*}\leq 1. \) If \(E\) is the elementary operator defined on \(C_{p:1\leq p< \infty}\) by \( E(X)=\sum_{i=1}^{n}A_{i}XB_{i}-X \) then \(ker E = ker \tilde{E} \Rightarrow ker E^{\dagger}= ker \tilde{E^{\dagger}}\). Moreover, if \( E (S) = 0 =\tilde{E} (S)\) for some compact operator \(S\), then \([|S| , B_{i}] = 0 \) \(\forall 1 \leq i < n\).

Proof. We have that \(E^{\dagger}(S)=0\Leftrightarrow \tilde{E}S^{*}=0.\) Then from the equality, \(Ker E=Ker \tilde{E}. \) It follows that \(E^{\dagger}(S)=0\Leftrightarrow E(S^{*})=0\Leftrightarrow \tilde{E^{\dagger}}(S)=0\). The rest is trivial.

At this point we give certain necessary conditions and characterization of the operators in \(C_{p}\)-classes whose kernels are orthogonal to the ranges of certain kinds of elementary operators, in particular, we consider the Jordan elementary operator.

Proposition 7. Let \(E\) be an elementary operator defined on \(C_{p}\) then \(\forall S, X \in C_{p}(H),\; \|E (X) + S\|_{p}\geq \|S\|_{p}\Rightarrow S\in ker E\) then \(E (X) = AXB\) with \(A^{*}\) and \(B\) injective operators \(E (X) = AXB - CXD,\) where \(A, B\) normal operators, \(D, C^{*}\) hyponormal operators with \([A, C] = [B, D] = 0\) and \(ker A^{*}\bigcap ker C^{*}= {0} = ker B \bigcap ker D\).

Proof. The duality adjoint \(E^{\dagger}\) is defined by \(E^{\dagger}(X) = BXA\) and using a result of [19], we get \(E^{\dagger}\) is injective and hence orthogonal. So, the result follows by Proposition 1. Next, we have \(E^{\dagger}(X) = BXA - DXC\) and applying the result of [20], we get \(E^{\dagger}\) is orthogonal, and so by [16], the proof is complete.

Theorem 4. Let \(A,B\in NA(H)\) be hyponormal operators, such that \(AB=BA\), and let \(\mathcal{U}(X)=AXB+BXA.\) Furthermore, suppose that \(A^{*}A+B^{*}B>0.\) If \(S\in Ker\mathcal{U},\) then \(|\|\mathcal{U}(X)+S\||\geq|\|S\||.\)

Proof. Follows trivially from the proof of the sum of two basic elementary operators as shown in [5] and from the fact that \(|\|.|\|\) is a unitarily invariant norm.

We extend Theorem 4 to distinct hyponormal operators \(A,B,C,D\in NA(H)\) in the theorem below:

Theorem 5. Consider \(A,B,C,D\in NA(H)\) as hyponormal operators, such that \(AC=CA\), \(BD=DB,\) \(AA^{*}\leq CC^{*}\) , \(B^{*}B\leq D^{*}D\). Let the Jordan elementary operator be given as \(\mathcal{U}(X)=AXB+CXD\) and \(S\in NA(H)\) satisfying \(ASB=CSD,\) then \(\|\mathcal{U}(X)+S\|\geq \|S\|,\) for all \(X\in NA(H).\)

Proof. Since \(\mathcal{U}(X)=AXB+CXD\) and the fact that \(AC=CA\), \(BD=DB,\) \(AA^{*}\leq CC^{*}\) , \(B^{*}B\leq D^{*}D\), then it is easy to see that the operator is injective. So, with \(S\in NA(H)\) satisfying \(ASB=CSD,\) then \(\|\mathcal{U}(X)+S\|\geq \|S\|\) follows analogously from the proof of the sum of two basic elementary operators as shown in [8].

Theorem 6. Consider \(A,B,C,D\in NA(H)\) as hyponormal operators, such that \(AC=CA\), \(BD=DB,\) \(AA^{*}\leq CC^{*}\) , \(B^{*}B\leq D^{*}D\). Let the Jordan elementary operator be give as \(\mathcal{U}(X)=AXB-CXD\) and \(S\in NA(H)\) satisfying \(ASB=CSD,\) then \(\|\mathcal{U}(X)+S\|\geq \|S\|,\) for all \(X\in NA(H).\)

Proof. From \(AA^{*}\leq CC^{*}\) and \(B^{*}B\leq D^{*}D,\) let \(A=CU,\) and \(B=VD,\) where \(U,V\) are unitaries. So we have \(AXB-CXD=CUXVD-CXD=C(UXV-X)D.\) Assume \(C\) and \(D^{*}\) are injective, \(ASB=CSD\) if and only if \(USV=S.\) Moreover, \(C\) and \(U \) commute. Indeed, from \(A=CU\) we obtain \(AC=CUC.\) Therefore, \(C(A-UC)=0.\) Thus since \(C\) is injective \(A=CU.\) Similarly, \(D\) and \(V \) commute. So, \[ \|\mathcal{U}(X)+S\| = \|[AXB-CXD]+S\| = \|[U(CXD)V-CXD] +S\| \geq \|S\|, \forall X\in NA(H). \] The rest is clear from the analogous assertions in [10] for hyponormal operators.

5. Conclusion

In this paper, we have given necessary conditions and characterize the elements that are orthogonal to the range of an operator defined in certain classes of normed spaces. We have also given range-kernel orthogonality conditions for elementary operators defined on \(C_{p}\)-classes and on norm-attainable class, particularly, the Jordan elementary operators.

Acknowledgments

The first author is grateful to National Research Fund(NRF)-Kenya for funding this research.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

In recent years, several extensions of the well known special functions have been considered by several authors [1,2,3,4,5]. Extensions of Euler's Gamma function together with the set of related higher transcendental special functions were introduced by Chaudhry and Zubair [1] as:

Clearly, we have \(\Gamma_0(x)=\Gamma(x)\) where \(\Gamma(x)\) is the well known classical Gamma function defined by [6]: The integral (1) can be presented as (see [1, p.101] for \(x=0\)) also [2 p,.79]: where \(a+p>0,\ \ Re(a)>0,\ \ p=0,\ \ Re(x)>0,\) which for \(a=1\) reduces to (1).

Here in this paper, we denote the right hand side of (3) as \(\Gamma_{(a,p)}(x)\), i.e.,

where \(a+p>0,\ \ Re(a)>0,\ \ p=0,\ \ Re(x)>0\).

Note that,

Recently, Dattoli et al., [7] introduced the extended Gegenbauer polynomials of two variable \(C_n^\alpha(x,y;a)\) as follows: which specified by the following generating function and integral representation: where \(H_n(x,y)\) denotes the 2-variable Hermite-Kamp\({\acute e}\) de F\({\acute e}\)riet polynomials defined by [8]: and specified by the following generating function: In many recent works (see for example [9,10,11,12]), the extended Beta function and its systemic generalizations are used to introduce new extended special functions such as hypergeometric function, Appell's and Lauricella's hypergeometric functions, Mittag Leffler function and Zeta function. Very recently, in terms of the extended Gamma function defined in (1) Atash and Al-Gonah [13] introduced the extended Gegenbauer polynomials of two variables \(C_n^\alpha(x,y;p)\) as follows: where \(Re(p)>0,\ \ p=0,\ \ Re(\alpha+n-k)>0,\) which specified by the following generating function and integral representation: From Equations (9) and (14), we have where \(C_n^\alpha(x)\) is the classical Gegenbauer polynomials [6]. Further, from Equations (9) and (14), we have where \(C_n^\alpha(x;p)\) is the extended Gegenbauer polynomials given in [13].

This paper is a further attempt in this direction to stress the importance of the use of extended Gamma function in introducing new extended special polynomials. The main object of this paper is to introduce a new generalization for the extended Gegenbauer polynomials defined in Equation (14) by using the extended Gamma function defined in Equation (4).

2. A generalization of extended Gegenbauer polynomials

In terms of the extended Gamma function \(\Gamma_{(a,p)}(x)\) defined in (4), we introduce a new generalization of extended Gegenbauer polynomials of two variables , denoted by \(C_n^\alpha(x,y;a,p)\), as follows: where \(a+p>0,\ \ Re(p)>0, p=0,\ \ Re(\alpha+n-k)>0.\)

Remark 1. From Equation (22), we note that:

• 1. For \(y=1\), Equation (22) reduces to the following new extended Gegenbauer polynomials \(C_n^\alpha(x;a,p)\):
  \begin{equation} \label{e22} C_n^\alpha(x;a,p)=\frac{1}{\Gamma(\alpha)} \sum^{[\frac{n}{2}]}_{k=0}\frac{(2x)^{n-2k}(-1)^k \Gamma_{(a,p)}(\alpha+n-k)}{k!(n-2k)!}. \end{equation}

  (23)
• 2. For \(p=0\) and using relation (5), Equation (22) reduces to the extended Gegenbauer polynomials of two variables \(C_n^\alpha(x,y;a)\) defined in (9).
• 3. For \(a=1\) and using relation (6), Equation (22) reduces to the extended Gegenbauer polynomials of two variables \(C_n^\alpha(x,y;p)\) defined in (14).
• 4. For \(y=a=1\) and using relation (6), Equation (22) reduces to the extended Gegenbauer polynomials \(C_n^\alpha(x;p)\) defined in (20).

Also, note that from Equations (22), (14), (23) and using relation (5), we have the following relations: Now, we establish some properties for the generalization of extended Gegenbauer polynomials of two variables \(C_n^\alpha(x,y;a,p)\) in the form of the following theorems:

Theorem 1. The following integral representation for the new extended Gegenbauer polynomials \(C_n^\alpha(x,y;a,p)\) holds true:

Proof. Using Equations (22) and (17) in the left hand side of Equation (26), we get

Making use of Equation (12) in the right hand side of Equation (27), we get assertion (26) of Theorem 1.

For \(y=1\) in assertion (26) of Theorem 1, we get the following result:

Corollary 1. The following integral representation for the new extended Gegenbauer polynomials \(C_n^\alpha(x;a,p)\) holds true:

Remark 2. From results (26) and (28), we note that:

• 1. For \(p=0\), result (26) reduces to a known result (21) given in [7].
• 2. For \(a=1\), result (26) reduces to a known result (16) given in [13].
• 3. For \(a=1\), result (28) reduces to a known result given in [13].

Further, by making use of result (26), we get the following results for \(C_n^\alpha(x,y;a,p)\):

Theorem 2. The following recurrence relation for the new extended Gegenbauer polynomials \(C_n^\alpha(x,y;a,p)\) holds true:

Proof. Consider the following recurrence relation [14]:

Replacing \(x\) by \(2xt\) and \(y\) by \(-yt\) in relation (30) and then multiplying both sides by \(\frac{t^{\alpha-1}\exp(-at-pt^{-1})}{\Gamma(\alpha)n!}\) and integrating the resultant equation with respect to \(t\) between the limits \(0\) to \(\infty\), we get which on using relation (26) yields assertion (29) of Theorem 2.

For \(y=1\) in assertion (26) of Theorem 2, we get the following result:

Corollary 2. The following recurrence relation for the new extended Gegenbauer polynomials \(C_n^\alpha(x;a,p)\) holds true:

Remark 3.

• 1. Setting \(p=0\) in result (30), we obtain a known result given in [7].
• 2. Setting \(a=1\) in results (30) and (32), we obtain a known results given in [13].

Theorem 3. The following differential equation of the new extended Gegenbauer polynomials \(C_n^\alpha(x,y;a,p)\) holds true:

Proof. Consider the following differential equation [14]:

Replacing \(x\) by \(2xt\) and \(y\) by \(-yt\) in relation (34) and using the relation and then multiplying both sides by \(\frac{t^{\alpha-1}\exp(-at-pt^{-1})}{\Gamma(\alpha)n!}\) and integrating the resultant equation with respect to \(t\) between the limits \(0\) to \(\infty\), we get Using relation (26) in the above equation and then using the following relation: in the first term of the resultant equation, we get the desired result.

For \(y=1\) in assertion (33) of Theorem 3, we get the following result:

Corollary 3. The following differential equation of the new extended Gegenbauer polynomials \(C_n^\alpha(x;a,p)\) holds true:

3. Generating functions and other properties of \(C_n^\alpha(x,y;a,p)\)

Very recently, many generating functions for Gegenbauer polynomials and its extension are obtained (see for example [13,15,16]). Here we prove some generating functions for the new extended Gegenbauer polynomials \(C_n^\alpha(x,y;a,p)\) in the form of the following theorems:

Theorem 4. The following generating function for the new extended Gegenbauer polynomials \(C_n^\alpha(x,y;a,p)\) holds true:

where \(Re(a-2xu+yu^2)>0\).

Proof. Using Equations (22) in the left hand side of Equation (39) and then putting \(n=n+2k\) in the resultant equation, we get

Now, using Equation (17) in the right hand side of the above equation, we obtain which in view of Equation (17) yields assertion (39) of Theorem 4.

For \(y=1\) in assertion (39) of Theorem 4, we get the following result:

Corollary 4. The following generating function for the new extended Gegenbauer polynomials \(C_n^\alpha(x;a,p)\) holds true:

where \(Re(a-2xu+u^2)>0\).

Remark 4.

• 1. Setting \(p=0\) in result (39) and using relation (7), we obtain a known result (20) given in [7].
• 2. Setting \(a=1\) in result (39) and then using relation (5), we obtain a known result (15) given in [13].
• 3. Setting \(a=1\) in result (42) and then using relation (5), we obtain a known result given in [13].

Theorem 5. The following generating function for the new extended Gegenbauer polynomials \(C_n^\alpha(x,y;a,p)\) holds true:

Proof. Consider the following generating function [17, p.452]:

Replacing \(x\) by \(xt\) and \(y\) by \(-yt\) in above equation and multiplying both sides by \(\frac{t^{\alpha-1}\exp(-at-pt^{-1})}{\Gamma(\alpha) k!}\) and integrating the resultant equation with respect to \(t\) from \(0\) to \(\infty\), we get which on using relation (26) yields assertion (43) of Theorem 5.

For \(y=1\) in assertion (43) of Theorem 5, we get the following result:

Corollary 5. The following generating function for the new extended Gegenbauer polynomials \(C_n^\alpha(x;a,p)\) holds true:

Remark 5.

• 1. Setting \(p=0\) in result (43) and using relation (7), we obtain a known result given in [7].
• 2. Setting \(a=1\) in results (43) and (46), we have
  \begin{align} \label{e39} &\sum^{\infty}_{n=0}(1+k)_nC_{n+k}^\alpha(x,y;p)\frac{u^n}{n!}=C_{k}^\alpha(x-yu,y;1-2xu+yu^2,p),\\ \end{align}

  (47)
  \begin{align} \label{e310} \sum^{\infty}_{n=0}(1+k)_nC_{n+k}^\alpha(x;p)\frac{u^n}{n!}=C_{k}^\alpha(x-u;1-2xu+u^2,p), \end{align}

  (48)
  which in view of relations (24) and (25) gives a known results given in [13].

Theorem 6. The following generating function for the new extended Gegenbauer polynomials \(C_n^\alpha(x,y;a,p)\) holds true:

where \(\Delta=a-2xu+yu^2-2xv+2yuv+yv^2\).

Proof. Using relation (26) in the left hand side of Equation (49) and interchanging the order of the summation and integration, we get

Now, using relation (44) in the right hand side of the above equation, we get which on using relation (13) gives Making use of relation (17) in the right hand side of Equation (52), we get assertion (49) of Theorem 6.

For \(y=1\) in assertion (49) of Theorem 6, we get the following result:

Corollary 6. The following generating function for the new extended Gegenbauer polynomials \(C_n^\alpha(x;a,p)\) holds true:

where \(\Delta=a-2xu+u^2-2xv+2uv+v^2\).

Remark 6. Setting \(a=1\) in results (49) and (53) and using relation (6), we obtain a known results given in [13].

Further properties for the new extended Gegenbauer polynomials of two variables \(C_n^\alpha(x,y;a,p)\) can be obtained in the form of the following theorems:

Theorem 7. The following Mellin transform representation of the new extended Gegenbauer polynomials \(C_n^\alpha(x,y;a,p)\) holds true:

Proof. Multiplying both sides of Equation (26) by \(p^{s-1}\) and integrating with respect to \(p\) between the limits \(0\) to \(\infty\), we get

Now, using the following relation [18]: in the R.H.S. of Equation (55), we get which on using relation (21) yields assertion (54) of Theorem 7.

For \(y=1\) in assertion (54) of Theorem 7, we get the following result:

Corollary 7. The following Mellin transform representation of the new extended Gegenbauer polynomials \(C_n^\alpha(x;a,p)\) holds true:

Theorem 8. The following series representations for the new extended Gegenbauer polynomials \(C_n^\alpha(x,y;a,p)\) and \(C_{2n}^\alpha(x,y;a,p)\) hold true:

Proof of (59). From relation (26), we have

which on using relation (21) yields assertion (59) of Theorem 8.

Proof of (60). Consider the following relation [19]:

Replacing \(x\) by \(2xt\) and \(y\) by \(-yt\) in relation (62) and then multiplying both sides by \(t^{\alpha-1}\exp(-at-pt^{-1})\) and integrating the resultant equation with respect to \(t\) between the limits \(0\) to \(\infty\), we get Next, using the following relation [19]: in the right hand side of Equation (63) and interchanging the order of summation and integration, we obtain which on using relation (26) yields assertion (60) of Theorem 8. Thus the proof of Theorem 8 is completed.

For \(y=1\) in assertions (59) and (60) of Theorem 8, we get the following results:

Corollary 8. The following series representations for the new extended Gegenbauer polynomials \(C_n^\alpha(x;a,p)\) hold true:

Remark 7. If we take \(a=1\) in relations (54),(58), (59), (60) (66) and (30), we obtain a known corresponding results given in [13].

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

Vito Volterra in 1926 introduced integro-differential for the first time when he investigated the population growth, focussing his study on the hereditary influences, whereby through his research work the topic of integro-differential equations was established [1]. Mathematical modeling of real life problems often result in functional equations such as differential, integral and integro-differential equations. Many mathematical formulation of physical phenomena reduced to integro-differential equations like fluid dynamics, control theory, biological models and chemical kinetics [1,2,3,4,5,6,7,8].

Linear Integro-Differential Equation (LIDE) is an important branch of modern mathematics and arises often in many applied areas which include engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory and electrostatics [9].

A variational iteration method and trapezoidal rule by Saadati et al., [10] was used for solving LIDEs. Manafianheris [11] applied modified laplace Adomian decomposition method. Bashir and Sirajo [12] used finite difference Simpson's approach on Fredholm Integro-differential equation and proved the error estimation of the method. In the work of Aruchunan and Sulaiman [2], a numerical solution of first order linear Fredholm integro-differential equations was obtained using conjugate gradient method. A reliable algorithm with application was presented by Alwaneh et al., [13]. Consider a linear Volterra integro-differential equation (LVIDE) of the form:

where \(a\) and \(\lambda\) are constants and \(x\) is a variable, \( f(x)\) and \(k(x,t)\) are known function with \(k(x,t)\) is the kernel and \(u\) is the unknown function to be determined.

The main objective of this paper is to propose a combination of finite difference-Simpson's approach (FDSM) to solve (LVIDE) (1) by transforming the problem into a system of linear algebraic equations. We used Maple software to obtained the numerical solution. Some examples are given to test the accuracy and efficiency of the method.

The paper has been organized as follows; In Section 2, we presented the derivation of the method. Error estimation of the scheme is proved in Section 3 and numerical results and discussions are provided in Section 4. Finally Section 5 conclude the paper.

2. Derivation of the method

Consider a linear Volterra integro-differential equation LVIDE of (1) and partition the domain \([a,x]\) into \(N\) finite parts of uniform step length \({\displaystyle{h= \frac{x_{N}-a}{N}}}\), such that \(x_{i}=a+ih, \quad i=1,2,...,N\).

Now using composite Simpson's with \(N\) subintervals. The integral part of (1) is approximated as:

\begin{align*}%\label{eqn2} \nonumber\displaystyle{\int_{a}^{x}k(x,t)u(t)dt} \approx& \displaystyle{\frac{h}{3}}[k(x,t_{0})u(t_{0})+4(k(x,t_{1})u(t_{1})+...+k(x,t_{N-1})u(t_{N-1})) \\ &+2(k(x,t_{2})u(t_{2})+...+k(x,t_{N-2})u(t_{N-2}))+k(x,t_{N})u(t_{N})]\\%\label{eqn3} % \nonumber to remove numbering (before each equation) \approx&\displaystyle{\frac{h}{3}}[k(x,t_{0})u(t_{0})+4k(x,t_{1})u(t_{1})+2k(x,t_{2})u(t_{2})+...+2k(x,t_{N-2})u(t_{N-2})) \nonumber \\ &+4k(x,t_{N-1})u(t_{N-1})+k(x,t_{N})u(t_{N})]. \end{align*} By discritizing along \(x\) and taking \(u'(x_{i})=u'_{i}\), \(f(x_{i})=f_{i}\), \(k(x_{i},t_{i})=k_{ij},\) \(k(x_{i},t_{i})\) vanishes for \(t_{j}>x_{i}\) we have By central difference we can approximate the derivative part of (2) as \[u'_{i}= \displaystyle{\frac{u_{i+1}-u_{i-1}}{2h}},\quad i=1,2,...,N-1,\] and at the end point \(N\) we use second Backward Finite difference \[u'_{i}=\displaystyle{\frac{3u_{N}-4u_{N-1}+u_{N-2}}{2h}},\quad i=N.\] By replacing \(u'_{i}\) in (2) we have for \(i=1,2,...,N-1,\) and for \(i=N\).

Using Equations (3) for \(i=1,2,...,N-1\) and (4) for \(i=N\), we can generate a systems of linear equations for \(u_{1},u_{2},...,u_{N}\), which can be represented in a matrix form as \( KU=W, \) where

\[K=\left( \begin{array}{cccccccccccc} A_{11} & 1 & 0 & 0 & ... & 0 & 0 & 0 \\ A_{21}-1 & B_{22} & 1 & 0 & ... & 0 & 0 & 0 \\ A_{31}-1 & B_{32}-1 & A_{33} & 1 & ... & 0 & 0 & 0 \\ % A_{41}-1 & B_{42} & A_{43}-1 & 0 & ... & 0 & 0& 0 \\ . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . \\ . & . & . & . & . & . & . & . \\ % A_{N-31} & B_{N-32} & A_{N-33} & B_{N-34} & ... & B_{N-3N-2}-1 & A_{N-3N-1} & 0 \\ A_{N-21} & B_{N-22} & A_{N-23} & B_{N-24} & ... & B_{N-2N-2} & 1 & 0 \\ A_{N-11} & B_{N-12} & A_{N-13} & B_{N-14} & ... & B_{N-1N-2}-1 & A_{N-1N-1} & 1 \\ A_{N1} & B_{N2} & A_{N3} & B_{N4} & ... & B_{NN-2}+1 & A_{NN-1}-4 & C_{NN}+3 \\ \end{array} \right),\] \(U=\left( \begin{array}{c} u_{1} \\ u_{2} \\ . \\ . \\ . \\ u_{N} \\ \end{array} \right),\) and \(W=\left( \begin{array}{c} 2hf_{1}+(\frac{2}{3}h^2k_{10})u_{0} \\ 2hf_{2}+\frac{2}{3}h^2k_{20}u_{0} \\ . \\ . \\ 2hf_{N-1}+\frac{2}{3}h^2k_{N-10}u_{0} \\ 2hf_{N}+\frac{2}{3}h^2k_{N0}u_{0} \\ \end{array} \right).\)

3. Error estimation

Theorem 1. Suppose that \(\mu_{1}, \mu_{2}, \mu_{3}\in(a,b)\) such that the errors \(e_{1}\) of central difference, \(e_{2}\) of second backward difference approximation and \(e_{3}\) of Simpson's rule respectively are given by \(\frac{h^{2}}{6}u^{(3)}(\mu_{1})\), \(\frac{h^{2}}{4}u^{(4)}(\mu_{3})\) and \(\frac{(x_{N}-a)}{180}h^{4}u^{(4)}(\mu_{2})\). Then the error estimation of approximate solution of linear Volterra integro-differential Equation (1) by the scheme (3) and (4) is \( {\displaystyle e^{*}}\leq\frac{5(x_{N}-a)^{2}}{12N^{2}}G.\)

Proof. From the problems of LVIDES (3) and (4), the exact solution for \(i=1,2,...,N-1\)

and for \(i=N\) where \(\mu_{1}, \mu_{2}, \mu_{3}\in(a,b).\)

Subtracting (3) and (4) from (5) and (6), we obtained the error term as follow:

\begin{align*} e&=\left|\frac{h^{2}}{6}u^{(3)}(\mu_{1})+\frac{h^{2}}{4}u^{(4)}(\mu_{3})-\left(\frac{(x_{N}-a)}{180}h^{4}u^{(4)}(\mu_{2})+\frac{(x_{N}-a)}{180}h^{4}u^{(4)}(\mu_{2})\right)\right|,\\ &= \left|\frac{h^{2}}{6}u^{(3)}(\mu_{1})+\frac{h^{2}}{4}u^{(4)}(\mu_{3})-\frac{(x_{N}-a)}{90}h^{4}u^{(4)}(\mu_{2})\right|,\\ &\leq \left|\frac{h^{2}}{6}u^{(3)}(\mu_{1})+\frac{h^{2}}{4}u^{(4)}(\mu_{3})\right|,\\ &\leq \left|\frac{h^{2}}{6}G_{1}+\frac{h^{2}}{4}G_{2}\right|, \end{align*} after taking \(G_{1}=u^{(3)}(\mu_{1})\) and \(G_{2}=u^{(4)}(\mu_{3}).\)

Now, if we let \(G=max\{G_{1}, G_{2}\}\) we have

\begin{eqnarray*} e&\leq \left|\frac{h^{2}}{6}G+\frac{h^{2}}{4}G\right|,\leq \left|\frac{5h^{2}}{12}G\right|. \end{eqnarray*} Substituting \({\displaystyle h=\frac{x_{N}-a}{N}}\), we get \begin{equation*} e^{*}\leq \left|\frac{5(x_{N}-a)^{2}}{12N^{2}}G\right|. \end{equation*} Hence the proposed scheme is of second order convergence.

4. Results and discussion

In this section the proposed method of finite difference-Simpson's approach is used to obtain the numerical solutions of problems of LVIDES in order to study the performance of the method.

Example 1. Consider LVIDE equation:

with exact solution \(u(x)=\frac{e^{x}}{4}-\frac{3e^{-x}}{4}-\frac{cosx}{2}\). We obtain the numerical results at different values of \(x\) which are represented in Table 1:

Table 1. Numerical result of Example 1.

\(x\)	Exact solution	FDSM	REA\cite{15}
0.0625	-0.9374599377	-0.970285632	-0.939313
0.125	-0.8746843974	-0.881205977	-0.877167
0.1875	-0.8114509330	-0.848304304	-0.816836
0.250	-0.7475504439	-0.760345075	-0.752159
0.3125	-0.6827862105	-0.723778451	-0.691295
0.375	-0.6169729163	-0.635854259	-0.623507
0.43750	-0.5499356617	-0.595245125	-0.561194
0.500	-0.4815089580	-0.506345938	-0.489657
0.5625	-0.4115357039	-0.461405361	-0.425194
0.6250	-0.3398661420	-0.370577838	-0.349485
0.6875	-0.2663567891	-0.316674778	-0.282077
0.750	-0.1908693447	-0.231335606	-0.201611
0.8125	-0.1132695669	-0.165028649	-0.130719
0.8750	-0.0334261204	-0.08294371	-0.0451471
0.93750	0.0487906069	-0.005464919	-0.0299532
1	0.1335097231	0.075642841	-0.121113

Table 1 shows the numerical results of Example 1. We compare the results obtained by using our method and numerical method of Romberg extrapolation algorithm (REA) given in [13]. Figure 1 indicate that our method of FDSM coincide with the exact solution and gives a better result than REA. This prove that our method is a good tool of approximating LVIDES problems of second kind.

Example 2. Consider LVIDE equation:

with exact solution \( u(x)=e^{-x}\). The numerical results are represented in Table 2 below:

Table 2. Numerical result of Example 2.

\(x\)	Exact solution	FDSM	BVMs\cite{17}
0.1	0.7788007831	0.6913562245	0.5627307831
0.2	0.6065306597	0.5986143128	0.5781196597
0.3	0.4723665527	0.0467629062	0.4687287527
0.4	0.3678794412	0.3678699844	0.3674183412

Table 2 shows the exact and the approximate solution obtained by our method at different values of \(x\) with the results of Boundary value methods (BVMs) given in [14]. Figure 2 provided a graphical presentation of the result which shows that our method can converge to the exact solution and gives a better result than BVMs.

Example 3. Consider LVIDE equation:

with exact solution \( u(x)=sinx\). The numerical results are represented in Table 3.

Table 3. The exact and approximate solution of Example 3.

\(x\)	Exact solution	FDSM	Absolute error
0.1	0.09983341665	0.0966907300	3.143\(e-3\)
0.2	0.1986693308	0.2009669073	2.298\(e-3\)
0.3	0.2955202067	0.3006342060	5.114\(e-3\)
0.4	0.3894183423	0.4099264019	2.051\(e-2\)
0.5	0.4794255386	0.5166993051	3.727\(e-2\)
0.6	0.5646424734	0.6352577593	7.061\(e-2\)
0.7	0.6442176872	0.7535502344	1.093\(e-1\)
0.8	0.7173560909	0.8859967676	1.686\(e-1\)
0.9	0.7833269096	1.020684713	2.374\(e-1\)
1	0.8414709848	1.172198061	3.307\(e-1\)

Table 3 shows the exact solution of the problem in Example 3 and the approximate solution obtained by our method. The absolute error obtained indicated that our method can give good approximation to LVIDE problems.

5. Conclusion

In this paper, we presented a numerical method to solve problems of linear Volterra integro-differential equations using a finite difference-Simpson’s approach by transforming the problem into a system of linear algebraic equations. Error estimation of the method shows that the method has second order convergence. Some problems and comparison with two method of REA and BVMs is given in order to test the applicability and efficiency of the derived method which prove that our method converges to the exact solution.

Acknowledgments

The authors are grateful to the referees for their useful comments.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction

Let \(\Sigma\) be the class of functions of the form

\[f(z)=\frac{1}{z}+\sum_0^\infty a_nz^n,\] which are analytic in the punctured unit disc \( \mathbb E_0=\mathbb E\setminus\{0\},\) where \(\mathbb E = \{z:|z|< 1\}. \) A function \(f \in \Sigma \) is said to be meromorphic starlike of order \(\alpha \) if \(f(z) \neq 0 \) for \(z \in \mathbb E_0 \) and \[-\Re\left(\frac{zf'(z)}{f(z)}\right)>\alpha,\hspace{1.7cm}(\alpha< 1;z \in \mathbb E).\] The class of such functions is denoted by \(\mathcal {MS}^*(\alpha)\) and the class of meromorphic starlike functions is denoted by \(\mathcal {MS}^*=\mathcal {MS}^*(0)\).

In the theory of meromorphic functions, many authors have obtained different sufficient conditions for meromorphically starlike functions. Some of them are stated below:

Kargar et al., [1] proved the following results:

Theorem 1. Assume that \(f(z)\neq 0\) for \(\mathbb E_0.\) If \(f\in\Sigma(p)\) satisfies \[\left|\frac{1}{\sqrt[p]{ f(z)}} \left(\frac{f'(z)}{f(z)}+\right)+p\right| < p\lambda(\beta)|b(z)|, ~z\in\mathbb E_0,\] then \(f\) is a \(p\)-valently meromorphic strongly-starlike of order \(\beta\).

Theorem 2. Assume that \(f(z)\neq 0\) for \(\mathbb E_0.\) If \(f\in\Sigma\) satisfies \[\left|\left(\frac{f(z)}{z^{-\alpha}}\right)^\frac{1}{\alpha-1} \left(\frac{f'(z)}{f(z)}+\frac{\alpha}{z}\right)+1-\alpha\right| < \frac{2}{\sqrt 5}, ~z\in\mathbb E_0,\] then \(f\) is meromorphic starlike function of order \(\alpha\).

Goswami et al., [2] proved the following results:

Theorem 3. If \(f(z) \in \Sigma_p,n\) with \(f(z)\neq 0\) for all \(z\in\mathbb E_0\), satisfies the following inequality \[\left|\displaystyle [z^p f(z)]^{\frac{1}{\alpha-p}}\left(\frac{z f'(z)}{f(z)}+\alpha\right)+p-\alpha\right|< \frac{(n+1)(p-\alpha)}{\sqrt{(n+1)^2+1}}, z\in\mathbb E, \] for some real values of \(\alpha~(0\leq\alpha< p)\), then \(f\in\mathcal{MS}_{p,n}^*(\alpha).\)

Theorem 4.If \(f(z) \in \Sigma_p,n\) with \(f(z)\neq 0\) for all \(z\in\mathbb E_0\) satisfies the following inequality \[\left|\frac{ \gamma [z^p f(z)]^\gamma}{z}\left(\frac{z f'(z)}{f(z)}+p\right)\right| \leq\frac{(n+1)}{2\sqrt{(n+1)^2+1}}, z\in\mathbb E, \] for \(\gamma\leq-\displaystyle\frac{1}{p}\), then \(f\in\mathcal{MS}_{p,n}^*\left(p+\displaystyle\frac{1}{\gamma}\right).\)

In [3], Sahoo et al., investigated a new class \(\mathcal{U}_n(\alpha,\lambda,\mu),\) of non-Bazilevic analytic functions by

\[\mathcal{U}_n(\alpha,\lambda,\mu)=\left\{f\in\mathcal {A}_n:\left|(1-\alpha)\left(\frac{z}{f(z)}\right)^\mu+\alpha f'(z)\left(\frac{z}{f(z)}\right)^{\mu+1}-1\right|< \lambda, ~z\in\mathbb E\right\}.\] For different choices of \(\mu\) with \(\alpha=1\), many authors has studied this class which are included in [4,5,6]. In this paper, we define above class of non-Bazilevic functions in punctured unit disk and study a differential inequality to obtain certain new criteria for starlikeness of meromorphic functions.

2. Main results

To prove our main result, we shall make use of following lemma of Hallenback and Ruscheweyh [7].

Lemma 1. Let G be a convex function in \(\mathbb E\), with \(G(0)=a\) and let \(\gamma\) be a complex number, with \(\Re(\gamma)>0\). If \(F(z)=a+a_nz^n+a_{n+1}z^{n+1}+\ldots\) , is analytic in \(\mathbb E\) nd \(F\prec G\), then \[\frac{1}{z^\gamma} \int^z_0 F(w)w^{\gamma-1}dw\prec \frac{1}{nz^{\frac{\gamma}{n}}}\int^z_0 G(w)w^{\frac{\gamma}{n}-1}dw .\]

Theorem 5. Let \(\alpha, \beta, \delta\) be real numbers such that \(\displaystyle \alpha< \frac{2}{\delta-1},\) \(\beta>0\), \(0\leq\delta< 1\) and let

If \(f\in\Sigma\) satisfies the differential inequality then \[-\Re\left(\frac{zf'(z)}{f(z)}\right)>\delta, ~z\in\mathbb E.\]

Proof. Let us define \[\left(\frac{1}{zf(z)}\right)^\beta=u(z),~z\in\mathbb E.\] Differentiate logarithmically, we obtain

Therefore, in view of (3), we have The use of Lemma 1 \(\left(taking~ \displaystyle\gamma =-\frac{\beta}{\alpha}\right)\) in (4) gives \[u(z)\prec1+\frac{\gamma Mz}{\gamma+1},\] or \[ |u(z)-1|< \frac{\beta M}{\beta-\alpha}< 1,\] therefore, we obtain Write \(\displaystyle -\frac{zf'(z)}{f(z)}=(1-\delta)w(z)+\delta,\) \(0\leq\delta< 1\) and therefore (2) reduces to \[|(1+\alpha) u(z)-\alpha u(z) [(1-\delta)w(z)+\delta]-1|< M.\] We need to show that \(\Re (w(z))>0, ~z\in\mathbb E.\) If possible, suppose that \(\Re (w(z))\ngtr 0, ~z\in\mathbb E,\) then there must exist a point \(z_0\in\mathbb E\) such that \(w(z_0)=ix,x\in\mathbb R.\) To prove the required result, it is now sufficient to prove that By making use of (3), we have Now (7) is true in view of (1) and therefore, (6) holds. Hence \(\Re(w(z))>0\), i.e., \[\displaystyle -\Re\left(\frac{zf'(z)}{f(z)}\right)>\delta, ~0\leq\delta< 1, ~z\in\mathbb E.\]

Remark 1. Let \(\alpha, \beta, \delta\) be real numbers such that \(\displaystyle \alpha< \frac{2}{\delta-1},\) \(0\leq\delta< 1\), \(\beta>0\) and if \( f(z)\in\Sigma\) satisfies \[\left|\left(\frac{1}{zf(z)}\right)^\beta \left(\frac{1}{\alpha}+1+\frac{zf'(z)}{f(z)}\right)-\frac{1}{\alpha}\right|< \displaystyle \frac{(\beta-\alpha)[2+\alpha(1-\delta)]}{\alpha^2[1+\beta(1-\delta)]},\] then \[-\Re\left(\frac{zf'(z)}{f(z)}\right)>\delta, ~z\in\mathbb E.\]

Letting \(\alpha\rightarrow\infty\) in above remark, we get the following result:

Theorem 6. Let \(\beta, \delta\) be real numbers such that \(\beta>0, 0\leq\delta< 1\) and let \(f(z)\in\Sigma\) satisfy \[\left|\left(\frac{1}{zf(z)}\right)^\beta \left(1+\frac{zf'(z)}{f(z)}\right)\right|< \displaystyle \frac{1-\delta}{1+\beta(1-\delta)}\] then \[-\Re\left(\frac{zf'(z)}{f(z)}\right)>\delta, ~z\in\mathbb E.\]

3. Deductions

Setting \(\beta=1\) in Theorem 5, we obtain

Corollary 1. Let \(\alpha\) and \(\delta\) be real numbers such that \(\displaystyle \alpha< \frac{2}{\delta-1},\) \(0\leq\delta< 1\) and suppose that \(f\in\Sigma\) satisfies \[\displaystyle \left|\frac{1}{zf(z)} \left(1+\alpha+\alpha\frac{zf'(z)}{f(z)}\right)-1\right|< \frac{(1-\alpha)(2+\alpha(1-\delta))}{\alpha(2-\delta)},~z\in\mathbb E,\] then \[-\Re\left(\frac{zf'(z)}{f(z)}\right)>\delta~, z\in\mathbb E,\] i.e., \(f\in\mathcal{MS}^*(\delta),~z\in\mathbb E.\)

Writing \(\delta=0\) in above corollary, we get the following result:

Corollary 2. Let \(f\in\Sigma\) satisfy \[\displaystyle\left|\frac{1}{zf(z)} \left(1+\alpha+\alpha\frac{zf'(z)}{f(z)}\right)-1\right|< \frac{(1-\alpha)(2+\alpha)}{2\alpha},~z\in\mathbb E,\] then \(f\in\mathcal{MS}^*,~z\in\mathbb E\).

Setting \(\beta=1\) in Theorem 6, we get the following result:

Corollary 3. Let \(\delta\) be a real number such that \( 0\leq\delta< 1\) and let \(f(z)\in\Sigma\) satisfy \[\left|\frac{1}{zf(z)} \left(1+\frac{zf'(z)}{f(z)}\right)\right|< \displaystyle \frac{1-\delta}{2-\delta}\] then \[-\Re\left(\frac{zf'(z)}{f(z)}\right)>\delta, ~z\in\mathbb E.\]

Setting \(\delta=0\) in above corollary, we get the following result:

Corollary 4. Let \(f(z)\in\Sigma\) satisfy \[\left|\frac{1}{zf(z)} \left(1+\frac{zf'(z)}{f(z)}\right)\right|< \displaystyle \frac{1}{2}\] then \(f\in\mathcal{MS}^*,~z\in\mathbb E\).

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

The fuzzy sets were introduced for the first time by Zadeh in [1]. Hundreds of examples have been supplied where the nature of uncertainty in the behavior of given system processes is fuzzy rather than stochastic nature. Recently, many authors showed interest in the study of the theoretical framework of fuzzy initial value problems. Chang and Zadeh [2] introduced the concept of fuzzy derivative. Dubosi and Prade [3] presented the extension principle. The differential and integral calculus for fuzzy-set-valued functions, shortly fuzzy-valued functions was developed in resent work, see [4,5,6,7,8,9].

It is known that phenomena of nature or physical systems can be modeled using partial differential equations (PDEs) such as wave equations, heat equations, Poisson's equation and so on. Hence, studies of PDEs become one of the main topics of modern mathematical analysis and attracted much attention. Many authors developed different methods for solving different kinds of PDEs, see [10,11,12,13,14,15,16,17].

The differential transform method (DTM) was introduced by Zhou [18] and he applied it to solve initial value problems for electric circuit analysis. The DTM is based on Taylor,s series expansion and can be applied to solve both linear and nonlinear ordinary differential equations as well as PDEs. Keskin and Oturanc [19] proposed the RDTM, defining a set of transformation rules to overcome the complicated complex calculations of traditional DTM. Recently some researchers used RDTM for solving different equations, see [20,21,22,23,24,25,26].

This paper is structured as follows: In Section 2, we call some definition on a fuzzy number, fuzzy-valued function and strongly generalized Hukuhara differentiability. In Section 3, Taylor's formula, one-dimensional DTM, and two-and three-dimensional RDTM is introduced. In Section 4, we provide some examples to show the efficiency and simplicity of RDTM. Finally, Section 5 consists of some brief conclusions.

2. Basic concepts

The fuzzy set \(\tilde{u}\in E^{1}\) is called a fuzzy number if \(\tilde{u}\) is a normal, convex fuzzy set, upper semi-continuous and supp\(u=\{x\in R|u(x)>0\}\) is compact. Here \(\overline{A}\) denotes the closure of \(A.\) We use \(E^{1}\) to denote the fuzzy number space [27,28].

For \(\tilde{u},\tilde{v}\in E^{1}\), \(k\in R\), the addition and scalar multiplication are defined by

\begin{align} [\tilde{u}+\tilde{v}]_{r}&=[\tilde{u}]_{r}+[\tilde{v}]_{r},\nonumber\\ [k\tilde{u}]_{r}&=k[\tilde{u}]_{r},\nonumber \end{align} respectively, where \([\tilde{u}]_{r}=\{x:u(x)\geq r\}=[\underline{u}_{r},\overline{u}_{r}]\) for any \(r\in[0,1]\).

We use the Hausdorff distance between fuzzy numbers [28] \(D:E^{1}\times E^{1}\rightarrow[0,+\infty)\) defined by

\begin{align} D(\tilde{u}+\tilde{v})&=\sup_{r\in[0,1]}d([\tilde{u}]_{r}[\tilde{v}]_{r})\nonumber\\ &=\sup_{r\in[0,1]}\max\{|\underline{u}_{r}-\underline{v}_{r}|,|\overline{u}_{r} -\overline{v}_{r}|\},\nonumber \end{align} where \(d\) is the Hausdorff metric. \(D(\tilde{u},\tilde{v})\) is called the distance between \(\tilde{u}\) and \(\tilde{v}\).

Definition 1. [29,30] Let \(\tilde{u}\) be a fuzzy number defined in \({F(R)}\). The \(r\)-level set of \(\tilde{u}\), for any \(r\in[0,1]\) denoted by \(\tilde{u}_{r}\) is a crisp set that contains all elements in \(R\), such that the membership value of \(\tilde{u}\) is greater or equal to \(r\), that is \[ \tilde{u}_{r}=\{{x\in R|\tilde{u}(x)\geq r}\}. \] Whenever we represent the fuzzy number with \(r\)-level set, we mean that it is closed and bounded and it is denoted by \([\underline{u}_{r},\overline{u}_{r}]\), where they represent the lower and upper bound \(r\)-level set of a fuzzy number.

The researchers [31,32] defined the parametrical representation of the fuzzy numbers as in the following definition:

Definition 2. [33] A fuzzy number \(\tilde{u}\) in parametric form is a pair \([\underline{u}_{r},\overline{u}_{r}]\) of functions \(\underline{u}_{r}\) and \(\overline{u}_{r}\) for any \(r\in[0,1]\), which satisfies the following requirements

• \(\underline{u}_{r}\) is a bounded non-decreasing left continuous function in (0,1];
• \(\overline{u}_{r}\) is a bounded non-increasing left continuous function in (0,1];
• \(\underline{u}_{r}\leq\overline{u}_{r}.\)

Some researchers classified the fuzzy numbers into several types of the fuzzy membership function and the triangular fuzzy membership function or also often referred to as triangular fuzzy number is the most widely used membership function.

In order to avoid the inconvenience, in the whole paper, the fuzzy numbers and fuzzy-valued functions are represented with a tilde sign at the top, while the real-value function and interval-valued functions are written directly.

Definition 3. [34] A fuzzy valued function \(\tilde{f}\) of two variable is a rule that assigns to each ordered pair of real numbers, \((x,t)\), in a set \(D\), a unique fuzzy number denoted by \(\tilde{f}(x,t)\). The set \(D\) is the domain of \(\tilde{f}\) and its range is the set of values taken by \(\tilde{f}\), i.e., \(\{\tilde{f}(x,t)|(x,t)\in D\}\).

The parametric representation of the fuzzy valued function \(f:D\rightarrow E^{1}\) is expressed by \(\tilde{f}(x,t;r)=[\underline{f}(x,t;r),\overline{f}(x,t;r)]\), for all \((x,t)\in D\) and \(\alpha\in[0,1]\).

Definition 4. [34,35] A fuzzy valued function \(f:D\rightarrow E^{1}\) is said to be fuzzy continuous at \((x_{0},t_{0})\in D\) if \(\lim_{(x,t)\rightarrow(x_{0},t_{0})}\tilde{f}(x,t)=f(x_{0},t_{0})\). We say that \(\tilde{f}\) is fuzzy continuous on \(D\) if \(\tilde{f}\) is fuzzy continuous at every point \((x_{0},t_{0})\) in \(D\).

Definition 5. [36,37] The generalized Hukuhara difference of two fuzzy numbers \(\tilde{u},\tilde{v}\in E^{1}\) is defined as follows:

In terms of the \(r-\)levels, we get \([\tilde{u}\ominus_{gH}\tilde{v}]=[\min\{\underline{u}_{r}- \underline{v}_{r},\overline{u}_{r}-\overline{v}_{r}\},\max\{\underline{u}_{r} -\underline{v}_{r},\overline{u}_{r}-\overline{v}_{r}\}]\) and if the H-difference exists, then \(\tilde{u}\ominus\tilde{v}=\tilde{u}\ominus_{gH}\tilde{v}\); the conditions for existence of \(\tilde{w}=\tilde{u}\ominus_{gH}\tilde{v}\in E^{1}\) are It is easy to show that (i) and (ii) are both valid if and only if \(\tilde{w}\) is a crisp number. In this case, it is possible that the gH-difference of two fuzzy numbers does not exist. To address this shortcoming, a new difference of fuzzy numbers was presented in [37]

Definition 6. [38] Let \(\tilde{u}(x,t): D\rightarrow E^{1}\) and \((x_{0},t)\in D\). We say that \(\tilde{u}\) is strongly generalized Hukuhara differentiable on \((x_{0},t)\) (gH-differentiable for short) if there exists an element \(\frac{\partial \tilde{u}}{\partial x}|_{(x_{0},t)}\in E^{1}\) such that

• (i)   for all \(h>0\) sufficiently small, \(\exists\; \tilde{u}(x_{0}+h,t)\ominus_{gH} \tilde{u}(x_{0},t), \tilde{u}(x_{0},t) \ominus_{gH} \tilde{u}(x_{0}-h,t)\) and the limits (in the metric D) \begin{equation} \lim_{h\rightarrow0+}\frac{\tilde{u}(x_{0}+h,t)\ominus_{gH} \tilde{u}(x_{0},t)}{h}=\lim_{h\rightarrow0+}=\frac{ \tilde{u}(x_{0},t) \ominus_{gH} \tilde{u}(x_{0}-h,t)}{h}=\left.\frac{\partial \tilde{u}}{\partial x}_{_{gH}}\right|_{(x_{0},t)},\nonumber \end{equation} or
•   (ii) for all \(h>0\) sufficiently small, \(\exists\; \tilde{u}(x_{0},t) \ominus_{gH} \tilde{u}(x_{0}+h,t), \tilde{u}(x_{0}-h,t)\ominus_{gH} \tilde{u}(x_{0},t)\) and the limits \begin{equation} \lim_{h\rightarrow0+}\frac{\tilde{u}(x_{0},t)\ominus_{gH} \tilde{u}(x_{0}+h,t)}{-h} =\lim_{h\rightarrow0+}\frac{\tilde{u}(x_{0}-h,t)\ominus_{gH} \tilde{u}(x_{0},t)}{-h}=\left.\frac{\partial \tilde{u}}{\partial x}_{gH}\right|_{(x_{0},t)},\nonumber \end{equation} or
• (iii)   for all \(h>0\) sufficiently small, \(\exists\; \tilde{u}(x_{0}+h,t)\ominus_{gH} \tilde{u}(x_{0},t), \tilde{u}(x_{0}-h,t)\ominus_{gH} \tilde{u}(x_{0},t)\) and the limits \begin{equation} \lim_{h\rightarrow0+}\frac{\tilde{u}(x_{0}+h,t)\ominus_{gH} \tilde{u}(x_{0},t)}{h}=\lim_{h\rightarrow0+}\frac{\tilde{u}(x_{0}-h,t)\ominus_{gH} \tilde{u}(x_{0},t)}{-h}=\left.\frac{\partial \tilde{u}}{\partial x}_{gH}\right|_{(x_{0},t)},\nonumber \end{equation} or
• (iv)   for all \(h>0\) sufficiently small, \(\exists\; \tilde{u}(x_{0},t) \ominus_{gH} \tilde{u}(x_{0}+h,t), \tilde{u}(x_{0},t) \ominus_{gH} \tilde{u}(x_{0}-h,t)\) and the limits \begin{equation} \lim_{h\rightarrow0+}\frac{\tilde{u}(x_{0},t)\ominus_{gH} \tilde{u}(x_{0}+h,t)}{-h}=\lim_{h\rightarrow0+}\frac{ \tilde{u}(x_{0},t) \ominus_{gH} \tilde{u}(x_{0}-h,t)}{h}=\left.\frac{\partial \tilde{u}}{\partial x}_{gH}\right|_{(x_{0},t)}.\nonumber \end{equation}

Lemma 1. [39] Let \(\tilde{u}(x,t): D \rightarrow E^1.\) Then the following statements hold:

• (a)   If \(\tilde{u}(x,t)\) is \((i)\)-partial differentiable for \(x\) (i.e., \(\tilde{u}\) is partial differentiable for \(x\) under the meaning of Definition 5 \((i)\), similarly to \(t\)), then
  \begin{equation} \left[\frac{\partial \tilde{u}}{\partial x}\right]_r=\left[\frac{\partial \underline{u}(x,t)(r)}{\partial x}, \frac{\partial \bar{u}(x,t)(r)}{\partial x}\right], \end{equation}

  (4)
• (b)   If \(\tilde{u}(x,t)\) is \((ii)\)-partial differentiable for \(x\) (i.e., \(\tilde{u}\) is partial differentiable for \(x\) under the meaning of Definition 5 \((ii)\), similarly to \(t\)), then
  \begin{equation} \left[\frac{\partial \tilde{u}}{\partial x}\right]_r=\left[\frac{\partial \bar{u}(x,t)(r)}{\partial x}, \frac{\partial \underline{u}(x,t)(r)}{\partial x}\right]. \end{equation}

  (5)

Remark 1. For \(\tilde{u}(x,t): D \rightarrow E^1,\) the following results hold:

if \((i,i), (ii,ii)\)-\(\frac{\partial^{2} \tilde{u}}{\partial x^{2}}\) exist, and if \((i,ii), (ii,i)\)-\(\frac{\partial^{2}\tilde{u}}{\partial t^{2}}\) exist.

3. Analysis of the method

In this section, we shall give some definitions and theorems of the Taylor series, one-dimensional DTM, and two-and three-dimensional RDTM.

Definition 7. [40] A Taylor series for the polynomial of degree \(n\) is defined as

Theorem 1. If the function \(f(x)\) has \((n+1)\) derivatives on an interval \((c-r,c+r)\) for some \(r>0\), and \(\lim_{n\rightarrow\infty}R_{n}(x)=0\), for all \(x\in(c-r,c+r)\), where \(R_{n}(x)\) is the error between \(F_{n}(x)\) and the polynomial function \(f(x)\) then the Taylor series expanded about \(x=c\) converges to \(f(x).\) Thus

3.1. Differential transform method

We consider the following one-dimensional DTM:

Definition 8. [26] The differential transform \(F(j)\) of the function \(f(x)\) for \(j^{th}\) order derivative is defined as

Definition 9. [26] The inverse differential transform of \(F(j)\) is defined as

The Equation (9) is the Taylor series expansion of \(f(x)\) at \(x=x_{0}\). From Equations (10) and (11), the following basic operations of DTM can be deduced

•   If \(f(x)=y_{1}(x)\pm y_{2}(x),~~\mbox {then}~~ F(j)=Y_{1}(j)\pm Y_{2}(j).\)
•   If \(f(x)=ay_{1}(x), ~\mbox {then}~~ F(j)=aY_{1}(j), \mbox{where a is a constant}.\)
•   If \(f(x)=\frac{dy_{1}(x)}{dx}~~\mbox {then}~~F(j)=(j+1)Y_{1}(j+1)\).
•   If \(f(x)=\frac{d^{2}y_{1}(x)}{dx^{2}}~~\mbox {then}~~F(j)=(j+1)(j+2)Y_{1}(j+2)\).
•   If \(f(x)=\frac{d^{n}y_{1}(x)}{dx^{n}},~~\mbox {then}~~ F(j)=\frac{(j+1)!}{j!}Y_{1}(j+1).\)
•   If \(f(x)=y_{1}(x)y_{2}(x),~~\mbox {then}~~ F(x)=\sum^{j}_{i=0}Y_{1}(i)Y_{2}(j-i).\)
•   If \(f(x)=x^{n},~~\mbox {then}~~ F(j)=\mu(j-n)~~\mbox where ~\mu(j-n) =\left\{\begin{aligned} &1, ~~~j=n,\\ &0, ~~~j\neq n. \end{aligned}\right.\)
•   If \(f(x)=(1+x)^{n}~~\mbox {then}~~F(j)=\frac{n(n-1)......(n-j+1)}{j!}\).
•   If \(f(x)=e^{a x},~~\mbox {then}~~ F(j)=\frac{a^{j}}{j!},\mbox {where \(a\) is a constants}.\)
•   If \(f(x)=\sin(\omega x+\alpha),\mbox {then}~ F(j)=\frac{\omega^{j}}{j!}\sin(\frac{j\pi}{2}+\alpha),~\mbox{ where \(\omega\) and \(\alpha\) are constants}.\)
•   If \(f(x)=cos(\omega x+\alpha), \mbox {then} F(j)=\frac{\omega^{j}}{j!}\cos(\frac{j\pi}{2}+\alpha),~\mbox{ where \(\omega\) and \(\alpha\) are constants}.\)

3.2. Reduced differential transform method

We consider the following two-dimensional RDTM:

Definition 10. [41] If the function \(w(x,t)\) is analytical and differentiable continuously with respect to time \(t\) and space \(x\) in the domain of interest, then we get

where the t-dimensional spectrum function \(W_{j}(x)\) is the transformed function of \(w(x,t)\). Here the lower case function \(w(x,t)\) represents the original function while the upper case \(W_{j}(x)\) stands for the transformed function.

Definition 11. [41] The inverse differential transform of \(W_{j}(x)\) is defined as

Thus combining (12) and (13), we can express the solution as The basic concept of RDTM mainly comes from the power series expansion. For two-dimensional function

• \(w(x,t)~~~\text{then}~~~W_{j}(x)=\frac{1}{j!}\left[\frac{\partial^{j}}{\partial x^{j} }u(x,t)\right]_{t=0}\).
• \(u(x,t)=w(x,t)\pm v(x,y),~~~\mbox {then}~~~U_{j}(x)=W_{j}(x)\pm V_{j}(x).\)
• \(u(x,t)=\alpha w(x,t),~~~\mbox {then}~~~ U_{j}(x)=\alpha W_{j}(x),\)   where \(\alpha\) a is constant.
• \(u(x,t)=x^{m} t^{n},~~~\mbox {then}~~~ U_{j}(x)=x^{m}\delta(j-n), \mbox{where}   \delta(j-n)= \left\{\begin{aligned} &1~~~~~~~\mbox {for}~~~j=n,\\ &0~~~~~~~\mbox {for}~~j\neq n. \end{aligned}\right.\)
• \(u(x,t)=x^{m} t^{n}w(x,t),~~~\mbox {then}~~~U_{j}(x)=x^{m}W_{j-n}(x)\).
• \(u(x,t)=w(x,t)v(x,t),~~~\mbox {then}~~~ U_{j}(x)=\sum^{j}_{r=0}W_{r}(x)V_{j-r}(x)=\sum^{j}_{r=0}V_{r}(x)W_{j-r}(x)\).
• \(u(x,t)=\frac{\partial^{r}}{\partial t^{r}}w(x,t),~~~\mbox {then}~~~ U_{j}(x)=\frac{(j+r)!}{j!}W_{j+r}(x)\).
• \(u(x,t)=\frac{\partial}{\partial x}w(x,t),~~~\mbox {then}~~~ U_{j}(x)=\frac{\partial}{\partial x}W_{j}(x).\)
• \(u(x,t)=\frac{\partial^{2}}{\partial x^{2}}w(x,t),~~~\mbox {then}~~~ U_{j}(x)=\frac{\partial^{2}}{\partial x^{2}}W_{j}(x).\)

For three-dimensional function

• \(w(x,y,t)~~~ \mbox{then}~~~ W_{j}(x,y)=\frac{1}{j!}\left[\frac{\partial^{j}}{\partial x^{j}}u(x,y,t)\right]_{t=0}\).
• \(u(x,y,t)=w(x,y,t)\pm v(x,y,t)~~~\mbox{then}~~~ U_{j}(x,y)=W_{j}(x,y)\pm V_{j}(x,y)\).
• \(u(x,y,t)=\lambda w(x,y,t)~~~\mbox{then}~~~U_{j}(x,y)=\lambda W_{j}(x,y), \mbox{where \(\lambda\) a is constant}\).
• \(u(x,y,t)=x^{m} y^{n} t^{p}~~~\mbox{then}~~~U_{j}(x)=x^{m} y^{n}\delta(j-p),~~\mbox{where}~~ \delta(j-p)= \left\{\begin{aligned} &1~~~~~~~\mbox {for}~~~j=p,\\ &0~~~~~~~\mbox {for}~~j\neq p. \end{aligned}\right.\)
• \(u(x,y,t)=x^{m} y^{n} t^{p}w(x,y,t)~~~\mbox{then}~~~U_{j}(x,y)=x^{m} y^{n}W_{j-p}(x,y)\).
• \(u(x,y,t)=w(x,y,t)v(x,y,t)~~~\mbox{then}~~~U_{j}(x,y)=\sum^{j}_{r=0}W_{r}(x,y)V_{j-r}(x,y)= \sum^{j}_{r=0}V_{r}(x,y)W_{j-r}(x,y)\).
• \(u(x,y,t)=\frac{\partial^{r}}{\partial t^{r}}w(x,y,t)~~~\mbox{then}~~~ ~U_{j}(x,y)=\frac{(j+r)!}{j!}W_{j+1}(x,y)\).
• \(u(x,y,t)=\frac{\partial}{\partial x}w(x,y,t)~~~\mbox{then}~~~U_{j}(x,y)=\frac{\partial }{\partial x}W_{j}(x,y)\).
• \(u(x,y,t)=\frac{\partial^{2}}{\partial x^{2}}w(x,y,t)~~~\mbox{then}~~~U_{j}(x,y)=\frac{\partial^{2}}{\partial x^{2}}W_{j}(x,y)\).
• \(u(x,y,t)=\frac{\partial^{2}}{\partial y^{2}}w(x,y,t)~~~\mbox{then}~~~U_{j}(x,y)=\frac{\partial^{2}}{\partial y^{2}}W_{j}(x,y)\).

4. Examples

In this section, we demonstrate how RDTM can be easily applied to obtain the exact solutions of the fuzzy partial differential equations.

Example 1. Consider the following one-dimensional initial value problem describing fuzzy heat-like equations

subject to the initial condition where \((n = 1,2,3,\ldots).\) Now \([\underline{\tilde{k}^{n}}](r)=r^{n}\) and \( [\overline{\tilde{k}^{n}}](\alpha)=(2-r)^{n}.\) The parametric form of (15) is for \(r\in [0,1],\) where \(\underline{u}\) stands for \(\underline{u}(x,t;r),\) and \(\overline{u}\) stands for \(\overline{u}(x,t;r).\) Applying the RDTM on Equations (18) and (19), we get the recurrence relation as where \(\tilde{U}_{j}(x;r) = [\underline{U}_{j}(x;r),\overline{U}_{j}(x;r)]\) is the transform function. From the initial condition (16), we get Substituting \(\tilde{U}_{0}(x;r) = [\underline{U}_{0}(x;r), \overline{U}_{0}(x;r)]\) into the recurrence relation (20), we get the following \(\tilde{U}_{j}(x;r)\) values successively \begin{align} \left.\begin{aligned} \underline{U}_{1}(x;r)&=2(x^{2}+5)r^{n}\nonumber\\ \underline{U}_{2}(x;r)&=\frac{2^{2}(x^{2}+5)r^{n}}{2!}\nonumber\\ \overline{U}_{3}(x;r)&=\frac{2^{3}(x^{2}+5)r^{n}}{3!}\nonumber\\ \overline{U}_{4}(x;r)&=\frac{2^{4}(x^{2}+5)r^{n}}{4!}\nonumber\\ \vdots\nonumber \end{aligned}\right\} \end{align} and \begin{align} \left.\begin{aligned} \underline{U}_{1}(x;r)&=2(x^{2}+5)(2-r)^{n}\nonumber\\ \underline{U}_{2}(x;r)&=\frac{2^{2}(x^{2}+5)(2-r)^{n}}{2!}\nonumber\\ \overline{U}_{3}(x;r)&=\frac{2^{3}(x^{2}+5)(2-r)^{n}}{3!}\nonumber\\ \overline{U}_{4}(x;r)&=\frac{2^{4}(x^{2}+5)(2-r)^{n}}{4!}\nonumber\\ \vdots\nonumber \end{aligned}\right\} \end{align} The inverse differential transform of \(\tilde{U}_{j}(x;r)\) is obtained from the relations \begin{align} \underline{u}(x,t;r)&=\sum^{\infty}_{j=0}\underline{U}_{j}(x;r)t^{j},\nonumber\\ \overline{u}(x,t;r)&=\sum^{\infty}_{j=0}\overline{U}_{j}(x;r)t^{j},\nonumber \end{align} and the exact solution is \begin{align} \tilde{u}(x,t;r)&=\left[r^{n},(2-r)^{n}\right]\odot\left((x^{2}+5)e^{2t}+x-5\right), \;\;\;\; 0\leq r\leq1.\nonumber \end{align}

Example 2. Consider the following two-dimensional initial value problem describing fuzzy heat-like equations

subject to the initial condition where \((n = 1,2,3,\ldots).\) Now \([\underline{\tilde{k}^{n}}](r)=(0.2+0.2r)^{n}\) and \([\overline{\tilde{k}^{n}}](\alpha)=(0.6-0.2r)^{n}.\) The parametric form of (23) is for \(r\in [0,1],\) where \(\underline{u}\) stands for \(\underline{u}(x,t)(r)\) and \(\overline{u}\) stands for \(\overline{u}(x,t)(r),.\) Applying the RDTM on Equations (26) and (27), we get the recurrence relation as where \(\tilde{U}_{j}(x,y;r)\) is the transform function. From the initial condition (24), we get \begin{align*} \underline{U}_{0}(x,y;r)&=(0.2+0.2r)^{n}+(x^{2}+y^{2}),\\ \overline{U}_{0}(x,y;r) &=(0.6-0.2r)^{n}+(x^{2}+y^{2}). \end{align*} Substituting \(\tilde{U}_{0}(x,y;r)\) into the recurrence relation (28), we get the following \(\tilde{U}_{j}(x,y;r)\) values successively \begin{align} \left.\begin{aligned} \underline{U}_{1}(x,y;r)&=(0.2+0.2r)^{n}+(x^{2}+y^{2})\nonumber\\ \underline{U}_{2}(x,y;r)&=\frac{(0.2+0.2r)^{n}+(x^{2}+y^{2})}{2!}\nonumber\\ \underline{U}_{3}(x,y;r)&=\frac{(0.2+0.2r)^{n}+(x^{2}+y^{2})}{3!}\nonumber\\ \underline{U}_{4}(x,y;r)&=\frac{(0.2+0.2r)^{n}+(x^{2}+y^{2})}{4!}\nonumber\\ \vdots\nonumber \end{aligned}\right\} \end{align} and \begin{align} \left.\begin{aligned} \overline{U}_{1}(x,y;r)&=(0.6-0.2r)^{n}+(x^{2}+y^{2})\nonumber\\ \overline{U}_{2}(x,y;r)&=\frac{(0.6-0.2r)^{n}+(x^{2}+y^{2})}{2!}\nonumber\\ \overline{U}_{3}(x,y;r)&=\frac{(0.6-0.2r)^{n}+(x^{2}+y^{2})}{3!}\nonumber\\ \overline{U}_{4}(x,y;r)&=\frac{(0.6-0.2r)^{n}+(x^{2}+y^{2})}{4!}\nonumber\\ \vdots\nonumber \end{aligned}\right\} \end{align} The solution for \(\tilde{u}(x,t;r)\) is \begin{align} \underline{u}(x,y,t;r)&=\sum^{\infty}_{j=0}\underline{U}_{j}(x,y;r)t^{j}=(0.2+0.2r)^{n}+\left[(x^{2}+y^{2})\left(1+t+\frac{t^{2}}{2!}+\frac{t^{3}}{3!} +\frac{t^{4}}{4!}+\cdot\cdot\cdot\infty\right)\right],\nonumber\\ \overline{u}(x,y,t;r)&=\sum^{\infty}_{j=0}\overline{U}_{j}(x,y;r)t^{j}=(0.6-0.2r)^{n}+\left[(x^{2}+y^{2})\left(1+t+\frac{t^{2}}{2!}+\frac{t^{3}}{3!} +\frac{t^{4}}{4!}+\cdot\cdot\cdot\infty\right)\right],\nonumber \end{align} and the exact solution is \begin{align} \tilde{u}(x,y,t;r)=\left[(0.2+0.2r)^{n},(0.6-0.2r)^{n}\right]\oplus\left((x^{2}+y^{2})\exp(t)\right), 0\leq r\leq 1.\nonumber \end{align}

Example 3. We consider following two-dimensional initial value problem describing heat-like equations

subject to the initial condition where \begin{align} \label{e4.21} \tilde{\nu}(x,y,t;r) & = (-1,0,1)\odot(xy)^{2}\nonumber\\ & = \left[(r-1)^{n},(1-r)^{n}\right]\odot(xy)^{2}, 0\leq r\leq1, (n = 1,2,3,...), \tilde{0}\in E^{1}.\nonumber \end{align} The parametric form of (29) is \begin{align*} \frac{\partial \underline{u}}{\partial t} &= (r-1)^{n}(xy)^{2}+\frac{1}{4}\left[x^{2}\frac{\partial^{2} \underline{u}}{\partial x^{2}}+ y^{2}\frac{\partial^{2} \underline{u}}{\partial y^{2}}\right], 0< x,y< 1, t>0,\\ \frac{\partial\overline{ u}}{\partial t} &= (1-r)^{n}(xy)^{2}+\frac{1}{4}\left[x^{2}\frac{\partial^{2} \overline{u}}{\partial x^{2}}+ y^{2}\frac{\partial^{2} \overline{u}}{\partial y^{2}}\right], 0< x,y< 1, t>0. \end{align*} Applying the RDTM, we get the recurrence relation as where \(\delta(j) = 1\) when \(j = 0\), and \(\delta(j) = 0\) when \(j\neq0\). Moreover \(\tilde{U}_{j}(x,y;r) = \left[\underline{U}_{j}(x,y;r), \overline{U}_{j}(x,y;r)\right]\) is the transform function. From the initial conditions, we obtain Substituting \(\tilde{U}_{0}(x,y;r)\) into the recurrence relation (31), we get the following \(\tilde{U}_{j}(x,y;r)\) values successively \begin{align} \left.\begin{aligned} \underline{U}_{1}(x,y;r) &= (r-1)^{n}x^{2}y^{2}\\ \underline{U}_{2}(x,y;r) &= \frac{(r-1)^{n}x^{2}y^{2}}{2!}\nonumber\\ \underline{U}_{3}(x,y;r) &= \frac{(r-1)^{n}x^{2}y^{2}}{3!}\\ \underline{U}_{4}(x,y;r) &= \frac{(r-1)^{n}x^{2}y^{2}}{4!}\\ &\vdots \end{aligned}\right\} \end{align} and \begin{align} \left.\begin{aligned} \overline{U}_{1}(x,y;r) &= (1-r)^{n}x^{2}y^{2}\\ \overline{U}_{2}(x,y;r) &= \frac{(1-r)^{n}x^{2}y^{2}}{2!}\nonumber\\ \overline{U}_{3}(x,y;r) &= \frac{(1-r)^{n}x^{2}y^{2}}{3!}\\ \overline{U}_{4}(x,y;r) &= \frac{(1-r)^{n}x^{2}y^{2}}{4!}\\ \vdots \end{aligned}\right\} \end{align} The solution for \(\tilde{u}(x,t;r)\) is \begin{align} \underline{u}(x,y,t;r) &= \sum^{\infty}_{j=0}\underline{U}_{j}(x,y,t;r)t^{j}= (r-1)^{n}\left[(x^{2}y^{2})\left(1+t+\frac{t^{2}}{2!}+\frac{t^{3}}{3!}+\frac{t^{4}}{4!}+...\infty\right)\right], \nonumber\\ \overline{u}(x,y,t;r) &= \sum^{\infty}_{j=0}\overline{U}_{j}(x,y,t;r)t^{j}= (1-r)^{n}\left[(x^{2}y^{2})\left(1+t+\frac{t^{2}}{2!}+\frac{t^{3}}{3!}+\frac{t^{4}}{4!}+...\infty\right)\right], \nonumber \end{align} and the exact solution is \begin{align} \tilde{u}(x,y,t;r) = \left[(r-1)^{n},(1-r)^{n}\right]\odot\left((x^{2}y^{2})\exp(t)\right),\;\;\; 0\leq r\leq 1. \nonumber \end{align}

Example 4. We consider the following fuzzy partial differential equation

subject to the initial condition where (n = 1,2,3,...). Now \([\underline{\tilde{k}^{n}}](r)=(0.5+0.5r)^{n}\) and \([\overline{\tilde{k}^{n}}](\alpha)=(1.5-0.5r)^{n}.\) The parametric form of (34) is for \(r\in [0,1],\) where \(\underline{u}\) stands for \(\underline{u}(x,t)(r)\) and \(\overline{u}\) stands for \(\overline{u}(x,t)(r)\). Applying the RDTM on Equations (37) and (38), we get the recurrence relation as The transformed initial condition (35) becomes For different values of \(j\), we get the following results \begin{align} \left.\begin{aligned} \underline{U}_{1}(x;r)&=(0.5+0.5r)-1\nonumber\\ \underline{U}_{2}(x;r)&=(0.5+0.5r)+1\nonumber\\ \underline{U}_{3}(x;r)&=(0.5+0.5r)-1\nonumber\\ \underline{U}_{4}(x;r)&=(0.5+0.5r)+1\nonumber\\ \vdots\nonumber \end{aligned}\right\} \end{align} and \begin{align} \left.\begin{aligned} \overline{U}_{1}(x;r)&=(1.5-0.5r)-1\nonumber\\ \overline{U}_{2}(x;r)&=(1.5-0.5r)+1\nonumber\\ \overline{U}_{3}(x;r)&=(1.5-0.5r)-1\nonumber\\ \overline{U}_{4}(x;r)&=(1.5-0.5r)+1\nonumber\\ \vdots\nonumber \end{aligned}\right\} \end{align} The solution for \(\tilde{u}(x,t;r)\) is \begin{align} \underline{u}(x,t;r)&=\sum^{\infty}_{j=0}\underline{U}_{j}(x;r)t^{j} = \left(\underline{U}_{0}(x;r)+\underline{U}_{1}(x;r)t+\underline{U}_{2}(x;r)t^{2}+ \underline{U}_{3}(x;r)t^{3}+\cdot\cdot\cdot\right)\nonumber\\ &=(0.5+0.5r)+(1-t+t^{2}-t^{3}+\cdot\cdot\cdot),\nonumber\\ \overline{u}(x,t;r)&=\sum^{\infty}_{j=0}\overline{U}_{j}(x;r)t^{j} = \left(\overline{U}_{0}(x;r)+\overline{U}_{1}(x;r)t+\overline{U}_{2}(x;r)t^{2} +\overline{U}_{3}(x;r)t^{3}+\cdot\cdot\cdot\right)\nonumber\\ &=(1.5-0.5r)+(1-t+t^{2}-t^{3}+\cdot\cdot\cdot),\nonumber \end{align} and the exact solution is \begin{align} \tilde{u}(x,t;r) = \left[(0.5+0.5r)^{n},(1.5-0.5r)^{n}\right]\oplus\left[\frac{1}{1+t}\right],\;\;\; 0\leq r\leq 1.\nonumber \end{align} Figure 1 illustrate that the left-hand functions of the r-level set of \(\tilde{u}\) (u lower) are always increasing functions of \(r\) and the right-hand functions of the r-level set of \(\tilde{u}\) (u upper) are always decreasing functions of \(r\) in the above examples.

5. Conclusion

In this paper, the reduced differential transform method (RDTM) has been successfully applied for solving fuzzy nonlinear partial differential equations under gH-differentiability. The solutions are considered as infinite series expansions that converge rapidly to the exact solutions. We solved some examples to illustrate the proposed method. The results reveal that the proposed method is a powerful and efficient technique for solving fuzzy nonlinear partial differential equations.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

1. Introduction

Let \(\mathcal{A}\) be the family of holomorphic functions, normalized by the conditions \(f(0)=f'(0)-1=0\) which is of the form

in the open unit disk \(\varOmega=\{z;z\in \mathbb{C}\;\text{and}\;|z|< 1\}\). We denote by \(\mathcal{G}\) the subclass of functions in \(\mathcal{A} \) which are univalent in \(\varOmega\) (for more details see [1]).

The Keobe-One Quarter Theorem [1] state that the image of \(\varOmega\) under all univalent function \(f\in \mathcal{A}\) contains a disk of radius \(\frac{1}{4}\). Hence all univalent function \(f\in \mathcal{A}\) has an inverse \(f^{-1}\) satisfy \(f^{-1}(f(z))\) and \(f(f^{-1}(\upsilon))=\upsilon\) \((|\upsilon|< r_0(f),\;r_0(f)\ge\frac{1}{4})\), where

A function \(f\in \mathcal{A}\) denoted by \(\varSigma\) is said to be bi-univalent in \(\varOmega\) if both \(f^{-1}(z)\) ans \(f(z)\) are univalent in \(\varOmega\) (see for details [2,3,4,5,6,7,8,9,10,11]).

A domain \(\varPsi\) is said to be \(m\)-fold symmetric if a rotation of \(\varPsi\) about the origin through an angle \(2\pi/m\) carries \(\varPsi\) on itself. Therefore, a function \(f(z)\) holomorphic in \(\varOmega\) is said to be \(m\)-fold symmetric if

\begin{equation*} f\left(e^{\frac{2\pi i}{m}}z\right) =e^{\frac{2\pi i}{m}}f(z).\end{equation*} A function is said to be \(m\)-fold symmetric if it has the following normalized form Let \(\mathfrak{S}_m\) the class of \(m\)-fold symmetric univalent functions in \(\varOmega\), that are normalized by (3), in which, the functions in the class \(\mathfrak{S}\) are \(one\)-fold symmetric. Similar to the concept of \(m\)-fold symmetric univalent functions, we introduced the concept of \(m\)-fold symmetric bi-univalent functions which is denoted by \(\varSigma_m\). Each of the function \(f\in \varSigma\) produces \(m\)-fold symmetric bi-univalent function for each integer \(m\in\mathcal{N}\).

The normalized form of \(f(z)\) is given as in (3) and the series expansion for \(f^{-1}(z)\), which has been investigated by Srivastava et al., [12], is given below:

Some of the examples of \(m\)-fold symmetric bi-univalent functions are \[\Biggl\{\frac{z^m}{1-z^m}\Biggr\}^{\frac{1}{m}},\] \[\left[-\log(1-z^{m})\right]^{\frac{1}{m}},\] and \[\Biggl\{\frac{1}{2}\log \left(\frac{1+z^m}{1-z^m}\right)^{\frac{1}{m}}\Biggr\}.\] For more details on \(m\)-fold symmetric analytic bi-univalent functions (see [5,12,13,14,15,16,17]).

Jackson [18,19] introduced the \(q\)-derivative operator \(\mathcal{D}_q\) of a function as follows;

and \(\mathcal{D}_qf(0)=f'(0)\). In case of \(g(z)=z^{k}\) for \(k\) is a positive integer, the \(q\)-derivative of \(f(z)\) is given by \begin{equation*} \mathcal{D}_{q}z^{k}=\frac{z^{k}-(zq)^{k}}{(q-1)z}=[k]_qz^{k-1}. \end{equation*} As \(q\longrightarrow1^{-}\) and \(k\in \mathcal{N}\), we get where \((z\neq 0,\;q\neq0)\). For more details on the concepts of \(q\)-derivative (see [5,20,21,22,23,24,25,26,27]).

Definition 1. [28] Let \(f(z)\in \mathcal{A}\), \(0\leq\chi< 1\) and \(\sigma\ge 1\) is real. Then \(f(z)\in L_{\sigma}(\chi)\) of \(\sigma\)-pseodu-starlike function of order \(\chi\) in \(\varOmega\) if and only if

Babalola [28] verified that, all pseodu-starlike function are Bazilevic of type \(\left(1-\frac{1}{\sigma}\right)\), order \(\chi^{\frac{1}{\sigma}}\) and univalent in \(\varOmega\).

Lemma 1. [1] Let the function \(\omega\in \mathcal{P}\) be given by the following series \(\omega(z)=1+\omega_1z+\omega_2z^2+\cdots\quad(z\in \varOmega).\) The sharp estimate given by \(|\omega_n|\leq2\quad(n\in \mathcal{N})\) holds true.

In [29] Girgaonkar et al., introduced a new subclasses of holomorphic and bi-univalent functions as follows:

Definition 2. A function \(f(z)\) given by (1) is said to be in the class \(\mathcal{M}_{\varSigma}(\chi)\;(0< \chi\leq1,(z,\upsilon)\in\varOmega)\) if \( f\in\mathcal{E}\), \(|\arg(f'(z))^{\sigma}|< \frac{\chi\pi}{2} \) and \( |\arg(g'(\upsilon))^{\sigma}|< \frac{\chi\pi}{2}, \) where \(g(\upsilon)\) is given by (2).

Definition 3. A function \(f(z)\) given by (1) is said to be in the class \(\mathcal{M}_{\varSigma}(\psi)\;(0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) if \( \vartheta\in\varSigma\), \( \Re[(f'(z))^{\sigma}]>\psi \) and \( \Re[(g'(\upsilon))^{\sigma}]>\psi,\) where \(g(\upsilon)\) is given by (2).

In this current research, we introduced two new subclasses denoted by \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\) and \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\) of the function class \(\varSigma_m\) and obtain estimates coefficient \(|\rho_{m+1}|\) and \(|\rho_{2m+1}|\) for functions in these two new subclasses.

2. Main 4esults

Definition 4. A function \(f(z)\) given by (3) is said to be in the class \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\;(m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) if

and where \(g(\upsilon)\) is given by (2).

Remark 1. We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{\sigma}_{\varSigma,1}(\chi)=\mathcal{M}^{\sigma}_{\varSigma}(\chi)\) which was introduced and studied by Girgaonkar et al., [29].

Remark 2. We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{1}_{\varSigma,1}(\chi)=\mathcal{M}_{\varSigma}(\chi)\) which was introduced and studied by Srivastava et al., [11].

Theorem 1. Let \(f(z)\in \mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\), \((m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (3). Then

and

Proof. Using inequalities (1) and (9), we get

and respectively, where \(\tau(z)\) and \(\varsigma(\upsilon)\) in \(\mathcal{P}\) are given by the following series and Clearly, \begin{equation*} [\tau(z)]^{\chi}=1+\chi\tau_{m}z^{m}+\left(\chi\tau_{2m}+\frac{\chi(\chi-1)}{2}\tau_{m}^2\right)z^{2m}+\cdots, \end{equation*} and \begin{equation*} [\varsigma(\upsilon)]^{\chi}=1+\chi\varsigma_{m}\upsilon^{m}+\left(\chi\varsigma_{2m}+\frac{\chi(\chi-1)}{2}\varsigma_{m}^2\right)\upsilon^{2m}+\cdots. \end{equation*} Also \begin{equation*} (\mathcal{D}_{q}f(z))^{\sigma}=1+\sigma[m+1]_q\rho_{m+1}z^{m}+\left(\sigma[2m+1]_q\rho_{2m+1}+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\right)z^{2m}+\cdots, \end{equation*} and \begin{align*} (\mathcal{D}_{q}g(\upsilon))^{\sigma}=&1-\sigma[m+1]_q\rho_{m+1}\upsilon^{m}-\sigma[2m+1]_q\rho_{2m+1}\upsilon^{2m}\\&+\Biggl(\sigma(m+1)[2m+1]_q\rho_{m+1}^2+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\Biggr)\upsilon^{2m}+\cdots \end{align*} Comparing the coefficients in (12) and (13), we have From (16) and (18), we obtain and Further from (17), (19) and (21), we obtain that \begin{equation*} \sigma(\sigma-1)\chi[m+1]_q^2\rho_{m+1}^2+(m+1)\sigma\chi[2m+1]_q\rho_{m+1}^2-(\chi-1)\sigma^2[m+1]_q^2\rho_{m+1}^2=\chi^2(\tau_{2m}+\varsigma_{2m}). \end{equation*} Therefore, we have By applying Lemma 1 for the coefficients \(\tau_{2m}\) and \(\varsigma_{2m}\), then we have \begin{equation*} |\rho_{m+1}|\leq\frac{2\chi}{\sqrt{(m+1)\sigma\chi[2m+1]_q-(\chi-\sigma)\sigma[m+1]_q^2}}. \end{equation*} Also, to find the bound on \(|\rho_{2m+1}|\), using the relation (19) and (17), we obtain It follows from (20), (21) and (23), Applying Lemma 1 for the coefficients \(\tau_{m},\tau_{2m},\varsigma_{m},\varsigma_{2m}\), then we have \begin{equation*} |\rho_{2m+1}|\leq\frac{2\chi}{\sigma[2m+1]_q}+\frac{2(m+1)\chi^2}{\sigma^2[m+1]_q^2}. \end{equation*} Choosing \(q\longrightarrow1^{-1}\) in Theorem 1, we get the following result:

Corollary 1. Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma,m}(\chi)\), \((m\in \mathcal{N}, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (3). Then

and Choosing \(m=1\) (0ne-fold case) in Theorem 1, we get the following result:

Corollary 2. Let \(f(z)\in \mathcal{M}^{q,\sigma}_{\varSigma}(\chi)\), \((0< q< 1, \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (1). Then

and Choosing \(q\longrightarrow1^{-1}\) in Corollary 2, we get the following result:

Corollary 3. [29] Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\chi)\), \(( \sigma\ge1,0< \chi\leq1,(z,\upsilon)\in\varOmega)\) be given (1). Then

and

Remark 3. For one-fold case, we have \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{q,1}_{\varSigma,1}(\chi)=\mathcal{M}_{\varSigma}(\chi)\), and we can get the results of Srivastava et al., [11].

Definition 5. A function \(f(z)\) given by (3) is said to be in the class \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\;(m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) if

and where \(g(\upsilon)\) is given by (2).

Remark 4. We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{\sigma}_{\varSigma,1}(\psi)=\mathcal{M}^{\sigma}_{\varSigma}(\chi)\) which was introduced and studied by Girgaonkar et al., [29].

Remark 5. We have the class \(\lim_{q\longrightarrow1^{-1}}\mathcal{M}^{1}_{\varSigma,1}(\psi)=\mathcal{M}_{\varSigma}(\chi)\) which was introduced and studied by Srivastava et al., [11].

Theorem 2. Let \(f(z)\in \mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\), \((m\in \mathcal{N}, 0< q< 1, \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (3). Then

and

Proof. Using inequalities (31) and (32), we get

and here \(\tau(z)\) and \(\varsigma(\upsilon)\) in \(\mathcal{P}\) are given by the following series \begin{equation*} \tau(z)=1+\tau_mz^m+\tau_{2m}z^{2m}+\tau_{3m}z^{3m}+\cdots, \end{equation*} and \begin{equation*} \varsigma(\upsilon)=1+\varsigma_m\upsilon^m+\varsigma_{2m}\upsilon^{2m}+\varsigma_{3m}\upsilon^{3m}+\cdots. \end{equation*} Clearly, \begin{equation*} \psi+(1-\psi)\tau(z)=1+(1-\psi)\tau_{m}z^{m}+(1-\psi)\tau_{2m}z^{2m}+\cdots, \end{equation*} and \begin{equation*} \psi+(1-\psi)\varsigma(\upsilon)=1+(1-\psi)\varsigma_{m}\upsilon^{m}+(1-\psi)\varsigma_{2m}\upsilon^{2m}+\cdots. \end{equation*} Also \begin{equation*} (\mathcal{D}_{q}f(z))^{\sigma}=1+\sigma[m+1]_q\rho_{m+1}z^{m}+\left(\sigma[2m+1]_q\rho_{2m+1}+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\right)z^{2m}+\cdots, \end{equation*} and \begin{align*} (\mathcal{D}_{q}g(\upsilon))^{\sigma}=&1-\sigma[m+1]_q\rho_{m+1}\upsilon^{m}-\sigma[2m+1]_q\rho_{2m+1}\upsilon^{2m}\\&+\Biggl(\sigma(m+1)[2m+1]_q\rho_{m+1}^2+\frac{\sigma(\sigma-1)}{2}[m+1]^2_q\rho_{m+1}^2\Biggr)\upsilon^{2m}+\cdots. \end{align*} Now comparing the coefficients in (35) and (36), we get From (37) and (39), we obtain and Also, from (38) and (40), we get Applying the Lemma 1 for the coefficients \(\tau_{m},\tau_{2m},\varsigma_{m},\varsigma_{2m}\), we find that \begin{equation*} |\rho_{m+1}|\leq2\sqrt{\frac{(1-\psi)}{\sigma(\sigma-1)[m+1]_q^2+(m+1)\sigma[2m+1]_q}}. \end{equation*} Also, to find the bound on \(|\rho_{2m+1}|\), using the relation (40) and (38), we obtain or equivalently By substituting the value of \(\rho_{m+1}^2\) from (42), we have Applying the Lemma 1 for the coefficients \(\tau_{m},\tau_{2m},\varsigma_{m},\varsigma_{2m}\), we get \begin{equation*} |\rho_{2m+1}|\leq\frac{2(1-\psi)}{\sigma[2m+1]_q}+\frac{2(m+1)(1-\psi)^2}{2\sigma^2[m+1]_q^2}. \end{equation*} Also, by using (43) and (45), and applying Lemma 1 we obtain \begin{equation*} |\rho_{2m+1}|\leq\frac{2(m+1)(1-\psi)}{\sigma(\sigma-1)[m+1]_q^2+(m+1)\sigma[2m+1]_q}+\frac{2(1-\psi)}{\sigma[2m+1]_q}. \end{equation*} This complete the proof.

Choosing \(q\longrightarrow1^{-1}\) in Theorem 2, we get the following result:

Corollary 4. Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma,m}(\psi)\), \((m\in \mathcal{N}, \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (3). Then \begin{equation*} |\rho_{m+1}|\leq\left \{ \begin{array}{cc} 2\sqrt{\frac{(1-\psi)}{\sigma(\sigma-1)[m+1]^2+(m+1)\sigma[2m+1]}} & 0\leq\psi\leq\frac{m}{1+2m},\\ \frac{2(1-\psi)}{\sigma[m+1]} & \frac{m}{1+2m}\leq\psi< 1, \end{array} \right. \end{equation*} and \begin{equation*} |\rho_{2m+1}|\leq\frac{2(m+1)(1-\psi)}{\sigma(\sigma-1)[m+1]^2+(m+1)\sigma[2m+1]}+\frac{2(1-\psi)}{\sigma[2m+1]}. \end{equation*} For one fold case, Corollary 4, yields the following Corollary:

Corollary 5. Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\psi)\), \(( \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (1). Then \begin{equation*} |\rho_{2}|\leq\left \{ \begin{array}{cc} \sqrt{\frac{2(1-\psi)}{\sigma(2\sigma+1)}} & 0\leq\psi\leq\frac{1}{3},\\ \frac{(1-\psi)}{\sigma} & \frac{1}{3}\leq\psi< 1, \end{array} \right. \end{equation*} and \begin{equation*} |\rho_{3}|\leq\frac{(1-\psi)(2\sigma-3\psi+3)}{3\sigma^2}. \end{equation*}

Remark 6. Corollary 5 gives above is the improvement of the estimates for coefficients on \(|\rho_{2}|\) and \(|\rho_{3}|\) investigated by Girgaonkar et al., [29].

Corollary 6. [29] Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\psi)\), \(( \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (1). Then \begin{equation*} |\rho_{2}|\leq\sqrt{\frac{2(1-\psi)}{\sigma(2\sigma+1)}}, \end{equation*} and \begin{equation*} |\rho_{3}|\leq\frac{(1-\psi)(2\sigma-3\psi+3)}{3\sigma^2}. \end{equation*} Taking \(\sigma=1\) in Corollary 7, we get the following result:

Corollary 7. [11] Let \(f(z)\in \mathcal{M}^{\sigma}_{\varSigma}(\psi)\), \(( \sigma\ge1,0\leq\psi< 1,(z,\upsilon)\in\varOmega)\) be given (1). Then \begin{equation*} |\rho_{2}|\leq\sqrt{\frac{2(1-\psi)}{3}}, \end{equation*} and \begin{equation*} |\rho_{3}|\leq\frac{(1-\psi)(5-3\psi)}{3}. \end{equation*}

3. Conclusion

In this present paper, two new subclasses indicated by \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\chi)\) and \(\mathcal{M}^{q,\sigma}_{\varSigma,m}(\psi)\) of function class of \(\mathcal{E}_m\) was obtained and worked on. Also, the estimates coefficients for \(|\rho_{m+1}|\) and \(|\rho_{2m+1}|\) of functions in these classes are determined.

Conflicts of Interest

The author declares no conflict of interest.

1. Introduction

The concept of \(b\)-metric spaces was first introduced by Bakhtin [1] and Czerwik [2] and utilized for \(s=2\) and for an arbitrary \(s\geq 1\) to prove some generalizations of Banach's fixed point theorem [3]. In 2010, Khamsi and Hussain [4] reintroduced the notion of \(b\) -metric and called it a metric-type. Afterwards, several authors proved fixed and common fixed point theorems for single-valued mappings in \(b\)-metric spaces, see [5,6,7,8,9,10,11,12,13,14,15,16].

In this paper, we introduce the definition of \(C\)-class functions and \(% \left( \psi ,\varphi ,F\right) \)-contraction type mappings where \(\psi \) is the altering distance function and \(\varphi \) is the ultra altering distance function. The unique fixed point theorem for self mapping in the setting of \(% b\)-complete metric spaces is proven. In the end of paper, we apply our main result to approximating the solution of the Fredholm integral equation.

In the sequel, we always denote by \(\mathbb{N}\), \(\mathbb{R}\), and \(\mathbb{R}_{+}\) the set of positive integers, real numbers, and nonnegative real numbers, respectively. The following definitions, notations, basic lemma and remarks will be needed in the sequel.

Definition 1. [1] Let \(X\) be a nonempty set and \(s\geq 1\) a given real number. A function \( d:X\times X\rightarrow\mathbb{R}_{+}\) is called a \(b\)-metric on \(X\) if for all \(x,y,z\in X\), the following conditions are satisfied;

• (bm-1)   \(d(x,y)=0\iff x=y,\)
• (bm-2)   \(d(x,y)=d(y,x),\)
• (bm-3)   \(d(x,y)\leq s(d(x,z)+d(z,y))\).

The pair \((X,d)\) is called a \(b\)-metric space with a coefficient \(s\).

Every metric space is a \(b\)-metric space with \(s=1\), but the converse is not true in general as it is shown by the following example.

Example 1. [17] Let \(X=\{0,1,2\}\) and \(d:X\times X\rightarrow\mathbb{R}_{+}\) defined by \begin{align*} d(0,0) &=d(1,1)=d(2,2)=0\text{,} \\ d(1,0) &=d(0,1)=d(2,1)=d(1,2)=1\text{,} \\ d(0,2) &=d(2,0)=m\text{,} \end{align*} where, \(m\) is given real number such that \(m\geq 2\). It is easy to check that for all \(x,y,z\in X\) \begin{equation*} d(x,y)\leq \frac{m}{2}(d(x,z)+d(z,y))\text{.} \end{equation*} Therefore, \((X,d)\) is a \(b\)-metric space with a coefficient \(s=\dfrac{m}{2}\). The ordinary triangle inequality does not hold if \(m>2\) and so \((X,d)\) is not a metric space.

Example 2. [13] Let \((X,d)\) be a metric space and \(\rho (x,y)=(d(x,y))^{p}\), where \(p>1\) is a real number. Then \(\rho \) is a \(b\)-metric with \(s=2^{p-1}\).

For other examples of a \(b\)-metric, see [1].

Definition 2. [18] Let \((X,d)\) be a \(b\)-metric space and \(\left\{ x_{n}\right\} \) a sequence in \(X\). The sequence \(\left\{ x_{n}\right\} \) is said to be

• (i)   Convergent to \(x\in X\) if \(\lim_{n\rightarrow +\infty }d(x_{n},x)=0\). In this case, we write \(\lim_{n\rightarrow +\infty }x_{n}=x\).
• (ii)   A Cauchy sequence if \(\lim_{n,m\rightarrow +\infty }d(x_{n},x_{m})=0\).
• (iii)   \((X,d)\) is complete if every Cauchy sequence in \(X\) is convergent.

Remark 1. In general, a \(b\)-metric need not be continuous in each variable [13].

The following lemma was established by [12].

Lemma 1. Let \((X,d)\) be a \(b\)-metric space with a coefficient \(s\geq 1\) and \(\left\{ x_{n}\right\} \) a sequence in \(X\) such that \begin{equation*} d(x_{n},x_{n+1})\leq \lambda d(x_{n-1},x_{n}),\text{ }n=1,2,..., \end{equation*} where \(0\leq \lambda < 1\). Then \(\{x_{n}\}\) is a Cauchy sequence.

Recently, Ansari [19] introduced the concept of following \(C\)-class functions which covers a large class of contractive conditions.

Definition 3. [19] A continuous function \(F:[0,+\infty )\rightarrow\mathbb{R}\) is called \(C\)-class function if for any \(s,t\in \lbrack 0,+\infty )\); the following conditions hold

• (c1)   \(\ F(s,t)\leq s\),
• (c2)   \(\ F(s,t)=s\) implies that either \(s=0\) or \(t=0\).

An extra condition on \(F\) that \(F(0,0)=0\) could be imposed in some cases if required. The letter \(C\) will denote the class of all \(C\)- functions.

Example 3. The following examples show that the class \(C\) is nonempty;

• 1.   \(F(s,t)=s-t\),
• 2.   \(F(s,t)=ms\); for some \(m\in (0,1)\),
• 3.   \(F(s,t)=\frac{s}{(1+t)^{r}}\) for some \(r\in (0,1)\),
• 4.   \(F(s,t)=\frac{\log (t+a^{s})}{(1+t)}\), for some \(a>1\),
• 5.   \(F(s,t)=s-(\frac{1+s}{2+s})(\frac{t}{1+t})\),
• 6.   \(F(s,t)=s\beta (s)\), \(\beta :[0,+\infty )\rightarrow (0,1)\) is continuous,
• 7.   \(F(s,t)=s-\frac{t}{k+t}.\)
• 8.   \(F(s,t)=s-(\frac{2+t}{1+t})t,\)
• 9.   \(F(s,t)=\sqrt[n]{\ln \left( 1+s^{n}\right) }.\)

Let \(\Phi _{u}\) denote the class of the functions \(\varphi :[0,+\infty )\rightarrow \lbrack 0,+\infty )\) which satisfy the following conditions;

• (a)   \(\varphi \) is continuous,
• (b)   \(\varphi (t)>0,\) \(t>0\) and \(\varphi (0)\geq 0\).

In 1984, Khan et al., [20] introduced altering distance function as follows;

Definition 4. [20] A function \(\psi :[0,+\infty )\rightarrow \lbrack 0,+\infty )\) is called an altering distance function if the following properties are satisfied;

• (i)   \(\psi \) is non-decreasing and continuous,
• (ii)   \(\psi (t)=0\) if and only if \(t=0\).

Let us suppose that \(\Psi \) denote the class of the altering distance functions.

Definition 5. A tripled \((\psi ,\varphi ,F)\) where \(\psi \in \) \(\Psi \); \(\varphi \) \(\in \Phi _{u}\) and \(F\in C\) is said to be a monotone if for any \(x,y\in \lbrack 0,+\infty )\), we have \begin{equation*} x\leq y\Rightarrow F(\psi (x),\varphi (x))\leq F(\psi (y),\varphi (y)). \end{equation*}

Example 4. Let \(F(s,t)=s-t\), \(\varphi (x)=\sqrt{x}\) and \( \psi (x)=\left\{ \begin{array}{c} \sqrt{x} \;\;\;\;\;\; \text{if}\;\;0\leq x\leq 1 \\ x^{2}\;\;\text{if}\;\;x>1 \end{array} \right., \) then \((\psi ,\varphi ,F)\) is monotone.

2. Main result

In this section we assume \(\psi \) is altering distance function, \(% \varphi \) is ultra altering distance function and \(F\) is a \(C\)-class function.

Theorem 1. Let \((X,d)\) be a \(b\)-complete metric space and \(T\) be a self-mapping on \(X\) that satisfies the following contractive condition;

for all \(x,y\in X\), where \(\psi \in \Psi \), \(\varphi \in \Phi _{u}\) and \(F\in C\) such that \((\psi ,\varphi ,F)\) is monotone and Then, \(T\) has a unique fixed point in \(X\).

Proof. Let \(x\) in \(X\) and \(\left\{ x_{n}\right\} _{n}\) be a sequence in \(X\) defined as \begin{equation*} Tx_{n}=x_{n+1},y_{n}=x_{n-1}\text{ }n=0,1,2\ldots . \end{equation*} Applying the inequality (1), we obtain \begin{equation*} \psi \left( d(Tx_{n},Tx_{n-1})\right) \leq F\left( \psi (M(x_{n},x_{n-1})),\varphi (M(x_{n},x_{n-1}))\right) , \end{equation*} where \begin{align*} M(x_{n},x_{n-1}) &=\max \left\{ d(x_{n},x_{n-1}),\frac{d^{2}(x_{n},x_{n-1})% }{1+d(x_{n-1},Tx_{n-1})},\frac{d^{2}(x_{n-1},Tx_{n-1})}{1+d(x_{n},x_{n-1})},% \frac{d(x_{n},Tx_{n})d(x_{n-1},Tx_{n-1})}{1+d(Tx_{n-1},Tx_{n})}\right\} \\ &\leq d(x_{n-1},x_{n}). \end{align*} Thus \begin{align*} \psi \left( d(Tx_{n},Tx_{n-1})\right) &\leq F\left( \psi (d(x_{n},x_{n-1})),\varphi (d(x_{n},x_{n-1}))\right) \\ &\leq \psi (d(x_{n},x_{n-1})). \end{align*} Since \(\psi \) is non-decreasing, then \(d(Tx_{n},Tx_{n+1})\leq d(x_{n},x_{n-1}).\) This means \(\left\{ d(x_{n},x_{n+1})\right\} \) is a decreasing sequence. Thus it converges and there exists \(r\geq 0\) such that \(\lim\limits_{n% \rightarrow +\infty }d(x_{n},x_{n+1})=r\). Taking \(n\rightarrow +\infty \), then contractive condition implies \( \psi \left( r\right) \leq F\left( \psi (r),\varphi (r)\right) \leq \psi \left( r\right) . \) So, \(\psi (r)=0\). Therefore \(r=0\), that is \( \lim\limits_{n\rightarrow +\infty }d(x_{n},x_{n+1})=0. \)

Now, we prove that the sequence \(\left\{ x_{n}\right\} \) is a Cauchy sequence. Suppose that \(\left\{ x_{n}\right\} \) is not a Cauchy sequence, then there exists an \(\varepsilon >0\) for which we can nd two sequences of positive integers \(m(k)\) and \(n(k)\) such that for all positive integers \(k\), \(n(k)>m(k)>k\) and \(d(x_{m(k)},x_{n(k)})\geq \varepsilon .\) Let \(n(k)\) be the smallest positive integer \(n(k)>m(k)>k\), such that

\begin{equation*} d(x_{m(k)},x_{n(k)})\geq \varepsilon ,\text{ }d(x_{m(k)},x_{n(k)-1})\leq \varepsilon . \end{equation*} Then, we find \(\psi (\varepsilon )=0\) which is a contradiction. Thus \(% \left\{ x_{n}\right\} \) is a \(b\)-Cauchy sequence in \(X\). Since \((X,d)\) is a complete \(b\)-metric space, so there exists \(u\in X,\) such that \(% \lim\limits_{n\rightarrow +\infty }x_{n}=u.\)

Uniqueness of fixed point

Let \(v\neq u\) be another fixed point of \(f\), then from the contraction condition, we have \begin{equation*} \psi \left( d(u,v)\right) \leq \psi \left( sd(u,v)\right) =\psi \left( sd(Tu,Tv)\right) \leq F\left( \psi (M(u,v)),\varphi (M(u,v))\right) , \end{equation*} where \begin{equation*} M(u,v)=\max \left\{ d(u,v),\frac{d^{2}(u,v)}{1+d(v,Tv)},\frac{d^{2}(v,Tv)}{% 1+d(u,v)},\frac{d(u,Tu)d(v,Tv)}{1+d(Tv,Tu)}\right\} . \end{equation*} Then \(\psi \left( d(u,v)\right) =0,\) thus \(d(u,v)=0\). This shows \(T\) has a unique fixed point.

The following example supports our Theorem 1.

Example 5. Let the complete \(b\)-metric space \((X,d)\) with \(X=\left[0,\frac{1}{2% }\right]\) and \begin{equation*} d(x,y)=|x-y|\text{ }for\text{ }all\text{ }x,y\in X. \end{equation*} Consider \(T:X\rightarrow X\) be given by \(Tx=\frac{x}{4}\) for all \(x\in X\). Then, for \(\psi \left( t\right) =t\) and \(F(s,t)=ms\) for some \(m\in (0,1),\) we have \begin{align*} d(Tx,Ty) &=\frac{1}{4}\left\vert x-y\right\vert \leq \frac{1}{2}\left\vert x-y\right\vert \\ &\leq \frac{1}{2}d(x,y)\leq \frac{1}{2}M(x,y). \end{align*} Thus, \(T\) is satisfying all the conditions of Theorem 1 and \(0\) is its fixed point, which is unique.

The following results can be obtained immediately from Theorem 1.

Corollary 1. Let \((X,d)\) be a complete \(b\)-metric space and \(T\) be a self-mapping on \(% X\) that satisfies the following contractive condition; \begin{equation*} \psi \left( d(Tx,Ty)\right) \leq \psi (M(x,y))-\varphi (M(x,y)), \end{equation*} for all \(x,y\in X\) where \(\psi \in \Psi \); \(\varphi \in \Phi _{u}\) and \(F\in C\) such that \((\psi ,\varphi ,F)\) is monotone and \begin{equation*} M(x,y)=\max \left\{ d(x,y),\frac{d^{2}(x,y)}{1+d(y,Ty)},\frac{d^{2}(y,Ty)}{% 1+d(x,y)},\frac{d(x,Tx)d(y,Ty)}{1+d(Ty,Tx)}\right\} . \end{equation*} Then, \(T\) has a unique fixed point in \(X.\)

Proof. Taking \(F(s,t)=s-t\), in Theorem 1, we obtain the desired result.

Corollary 2. Let \((X,d)\) be a complete \(b\)-metric space and \(T\) be a self-mapping on \(% X\) that satisfies the following contractive condition; \begin{equation*} \psi \left( d(Tx,Ty)\right) \leq M(x,y)\Phi \left( \varphi (M(x,y)\right) , \end{equation*} for all \(x,y\in X\) where \(\psi \in \Psi \); \(\varphi \in \Phi _{u}\) and \(F\in C\) such that \((\psi ,\varphi ,F)\) is monotone and \begin{equation*} M(x,y)=\max \left\{ d(x,y),\frac{d^{2}(x,y)}{1+d(y,Ty)},\frac{d^{2}(y,Ty)}{% 1+d(x,y)},\frac{d(x,Tx)d(y,Ty)}{1+d(Ty,Tx)}\right\} . \end{equation*} Then, \(T\) has a unique fixed point in \(X\)

Proof. Taking \(\psi \left( t\right) =t\) and \(F(s,t)=\frac{s}{(1+t)^{r}}\) for some \(% r\in (0,1)\) in Theorem 1, we obtain the desired result.

Corollary 3. Let \((X,d)\) be a complete \(b\)-metric space and \(T\) be a self-mapping on \(% X\) that satisfies the following contractive condition; \begin{equation*} d(Tx,Ty)\leq \frac{M(x,y)}{\left( 1+M(x,y)\right) ^{r}}, \end{equation*} for all \(x,y\in X\) where \(\psi \in \Psi \); \(\varphi \in \Phi _{u}\) ,\(% r\in (0,1)\) and \(F\in C\) such that \((\psi ,\varphi ,F)\) is monotone and \begin{equation*} M(x,y)=\max \left\{ d(x,y),\frac{d^{2}(x,y)}{1+d(y,Ty)},\frac{d^{2}(y,Ty)}{% 1+d(x,y)},\frac{d(x,Tx)d(y,Ty)}{1+d(Ty,Tx)}\right\} . \end{equation*} Then, \(T\) has a unique fixed point in \(X\)

Proof. Taking \(F(s,t)=s\Phi \left( t\right) \), (\(s,t>0\)) in Theorem 1, we obtain the desired result.

3. Application to integral equations

Let \(X=C[a,b]\) be a set of all real valued continuous functions on \([a,b]\), where \([a,b]\) is closed and bounded interval in \(\mathbb{R}\). For a real number \(p>1\), define \(d:X\times X\rightarrow\mathbb{R}_{+}\) by \begin{equation*} d(x,y)=\max_{t\in \lbrack a,b]}\left\vert x(t)-y(t)\right\vert ^{p}, \end{equation*} for all \(x,y\in X\). Therefore, \(\left( X,d\right) \) is a complete \(b\)-metric space with \(s=2^{p-1}\). We apply Theorem 1 to establish the existence of solution of Fredholm type defined by where \(x\in C[a,b]\) is the unknown function, \(\lambda \in\mathbb{R},t,\tau \in \lbrack a,b]\), \(K:[a,b]\times \lbrack a,b]\times\mathbb{R}\rightarrow\mathbb{R}\) and \(f:[a,b]\rightarrow\mathbb{R}\) are given continuous functions.

Theorem 2. We assume the following conditions;

• (i)   There exists a continuous function \(\psi :[a,b]\times \lbrack a,b]\rightarrow\mathbb{R}_{+}\) such that for all \(x,y \in X\), \(\lambda \in\mathbb{R}\) and \(t,\tau \in \lbrack a,b]\), we get \( \left\vert K(t,\tau ,x)-K(t,\tau ,y)\right\vert ^{p}\leq \psi (t,\tau ).\left\vert x-y\right\vert ^{p}\text{,} \)
• (ii)   \(\left\vert \lambda \right\vert \leq 1\),
• (iii)  \( \max_{t\in \lbrack a,b]} \int\limits_{a}^{b}\psi (t,\tau )d\tau \leq \dfrac{1% }{(b-a)^{p-1}}\text{,}\) where \(s=\dfrac{1}{2^{p-1}}\).

Then, the Equation (3) has a solution \(z\in C[a,b]\).

Proof. Define the mapping \(T:X\rightarrow X\) by \begin{equation*} Tx(t)=f(t)+\lambda \int\limits_{a}^{b}K(t,\tau ,x(\tau ))d\tau , \end{equation*} for all \(t\in \lbrack a,b]\). So, the existence of a solution of (3) is equivalent to the existence of fixed point \(T\). Let \(q\in\mathbb{R}\) such that \(\dfrac{1}{p}+\dfrac{1}{q}=1\). Using the Hölder inequality, and conditions (i)-(iii), we have \begin{align*} d(Tx,Ty) &=\max\limits_{t\in \lbrack a,b]}|Tx(t)-Ty(t)|^{p} \\ &\leq \left\vert \lambda \right\vert ^{p}\max\limits_{t\in \lbrack a,b]}\left( \int\limits_{a}^{b}\left\vert K(t,\tau ,x)-K(t,\tau ,y\right\vert ^{p}d\tau \right) \\ &\leq \left[ \max\limits_{t\in \lbrack a,b]}\left( \int\limits_{a}^{b}1^{q}dz\right) ^{\frac{1}{q}}\left( \int\limits_{a}^{b}\left\vert \left( K(t,\tau ,x)-K(t,\tau ,y\right) )\right\vert ^{p}d\tau \right) ^{\frac{1}{p}}\right] ^{p} \\ &\leq \left( b-a\right) ^{\frac{p}{q}}\left[ \max\limits_{t\in \lbrack a,b]}\left( \int\limits_{a}^{b}\psi (t,\tau )\left\vert x-y\right\vert ^{p}d\tau \right) \right] \\ &\leq \left( b-a\right) ^{p-1}\max\limits_{t\in \lbrack a,b]}\left( \int\limits_{a}^{b}\psi (t,\tau )d\tau \right) d(x,y) \\ &\leq \left( b-a\right) ^{p-1}\dfrac{1}{(b-a)^{p-1}}.M\left( x,y\right). \end{align*} Thus \begin{equation*} d(Tx,Ty)\leq M\left( x,y\right). \end{equation*} Hence, all the conditions of Theorem 1 hold. Consequently, the Equation (3) has a solution \(z\in C[a,b]\).

Acknowledgments:

The author is grateful to the referees for the useful comments.

Conflicts of Interest:

The author declares no conflict of interest.

1. Introduction

In harmonic analysis, uncertainty principles play an important role. It states that a non-zero function and its Fourier transform cannot be simultaneously sharply concentrated. many of them have already been studied from several points of view for the Fourier transform, Heisenberg-Pauli-Weyl inequality [1] and local uncertainty inequality [2]. As a classical uncertainty principle, the Heisenberg uncertainty principle has been extended to transforms such as the spherical mean transforms [3,4], the Dunkl transform [5] and so forth.

The Hankel transform \(\mathcal{H}_\alpha\) is defined for every integrable function \(f\) on \(\mathbb{R}_+=[0,+\infty[\) with respect to the measure \(d\nu_\alpha\), by

\begin{equation*} \mathcal{H}_\alpha(f)(\lambda)=\int_0^{+\infty}f(x)j_\alpha(\lambda x)d\nu_\alpha(x),\end{equation*} where \(d\nu_\alpha\) is the measure defined on \(\mathbb{R}_+ \) by \[d\nu_\alpha(x)=\frac{x^{2\alpha+1}}{2^{\alpha}\Gamma(\alpha+1)}dx,\] and \(j_\alpha\) is the modified Bessel function given in the next section.

The Hankel transform is found as a very useful mathematical tool in many fields of physics, signal processing and other [6,7]. Also, many uncertainty principles related to this transform \(\mathcal{H}_\alpha\) have been proved [8,9,10].

Time-frequency analysis plays an important role in harmonic analysis, in particular in signal theory. With the development of time-frequency analysis, the study of uncertainty principles have gained considerable attention and have been extended to a wide class of integral transforms such as Weinstein transforms [11,12], Dunkl transforms [13], Hankel-Stockwell transforms [14] and so on.

Based on the ideas of Faris [15] and Price [2,16], we show a general form of the local uncertainty principles for the Hankel-Stockwell transform and we deduce \(L^p\) version of Heisenberg-Pauli-Weyl uncertainty principle. We shall use also the Heisenberg uncertainty principle, the properties of the Hankel-Stockwell transform and the techniques of Donoho-Stark [17,18], we show a continuous-time principle for the \(L^p\) theory, when \(1 < p \leqslant 2\). Finally, Pitt's inequality and Beckner's uncertainty principle are proved for this transform.

This work is organized as follows; in Section 2 we recall some harmonic analysis results related to the Hankel transform. In Section 3, we present some elements of harmonic analysis related to the Hankel-Stockwell transform. In Section 4, we introduce some uncertainty principles for this transform.

2. The Hankel transform

In this section, we summarize some harmonic analysis tools related to the Hankel transform that will be used hereafter, (see [19]). The modified Bessel function \(x\longmapsto j_{\alpha}(x)\) has the following integral representation [20,21]; \begin{alignat*}{2} j_{\alpha}(x)=\left\{ \begin{array}{ll} \frac{2\Gamma(\alpha+1)}{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})}\int_0^1(1-t^2)^{\alpha-\frac{1}{2}}\cos(x t)dt, & \hbox{if \(\alpha>\frac{-1}{2}\);} \\ \cos x, & \hbox{if \(\alpha=\frac{-1}{2}\).} \end{array} \right. \end{alignat*} In particular, for every \(x\in\mathbb{R}\) and \(k\in\mathbb{N}\), we have \begin{equation*} \left|j_{\alpha}^{(k)}(x)\right|\leqslant 1.\end{equation*} We define the Hankel translation operators \(\tau_x\), \(x\in[0,+\infty[\) by \begin{eqnarray*} \tau_x(f)(y)=\left \{ \begin{array}{ll} \frac{\Gamma(\alpha+1)}{{\sqrt{\pi}\Gamma(\alpha+\frac{1}{2})}}\int_{0}^{\pi}f(\sqrt{x^{2}+y^{2}+2xy\cos\theta},x+y)\sin^{2\alpha}(\theta) d\theta , &\hbox{if \(\alpha >\frac{-1}{2}\),}\\ \frac{f(x+y)+f(|x-y|)}{2} ,& \hbox{if \(\alpha=\frac{-1}{2},\)} \end{array} \right. \end{eqnarray*} whenever the integral in the right-hand side is well defined. In the following, we denote by;

• \(S_e(\mathbb{R})\) the Schwartz space, constituted by the even infinitely differentiable functions on the real line, rapidly decreasing together with all their derivatives,
• \(L^p(d\nu_\alpha)\) the Lebesgue space of measurable functions \(f\) on \(\mathbb{R}_+\), such that \(\|f\|_{p,\nu_\alpha}< +\infty.\)

For every \(f\in L^p(d\nu_\alpha),\, p\in[1,+\infty]\), and for every \(x\in\mathbb{R}_+\), the function \(\tau_x(f)\) belongs to the space \(L^p(d\nu_\alpha)\) and \[\|\tau_x(f)\|_{p,\nu_\alpha}\leqslant\|f\|_{p,\nu_\alpha}.\] In particular, for every \(x, y \in\mathbb{R}_+\), we have \[\tau_x(f)(y)=\tau_y(f)(x).\] If \(f\in L^1(d\nu_\alpha)\), then \[\int_0^{+\infty}\tau_x(f)(y)d\nu_\alpha(y)=\int_0^{+\infty}f(y)d\nu_\alpha(y).\] The convolution product of \(f,g\in L^1(d\nu_\alpha)\) is defined by \[f\ast g(x)=\int_0^{+\infty}\tau_x(f)(y)g(y)d\nu_\alpha(y).\] Let \(p,q,r\in[1,+\infty]\) such that \(\displaystyle\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r}\). Then for every \(f\in L^p(d\nu_{\alpha})\) and \(g\in L^q(d\nu_{\alpha})\), the function \(f\ast g\) belongs to the space \(L^r(d\nu_{\alpha})\), and we have the following Young's inequality \begin{equation*} \|f\ast g\|_{r,\nu_{\alpha}}\leqslant\|f\|_{p,\nu_{\alpha}}\|g\|_{q,\nu_{\alpha}}.\end{equation*} The Hankel transform \(\mathcal{H}_\alpha\) is defined on \(L^1(d\nu_\alpha)\) by \begin{equation*}\forall \lambda\in\mathbb{R}~;~\mathcal{H}_\alpha(f)(\lambda)=\int_0^{+\infty}f(x)j_\alpha(\lambda x)d\nu_\alpha(x).\end{equation*}

Theorem 1.

• (1) [Inversion formula] Let \(f\in L^1(d\nu_{\alpha})\) such that \(\mathcal{H}_{\alpha}(f)\in L^1 (d\nu_{\alpha})\), then we have \begin{alignat*}{2} f(x)&=\int_0^{+\infty}\mathcal{H}_{\alpha}(f)(\lambda)j_\alpha(\lambda x)d\nu_\alpha(\lambda),\,\,a.e. \end{alignat*}
• (2) [Plancherel theorem] The Fourier transform \(\mathcal{H}_{\alpha}\) can be extended to an isometric isomorphism from \(L^2(d\nu_{\alpha})\) onto itself and we have \begin{equation*} \|\mathcal{H}_{\alpha}(f)\|_{2,\nu_{\alpha}}=\|f\|_{2,\nu_{\alpha}} .\end{equation*}
• (3) [Parseval's formula] For all functions \(f\) and \(g\) in \(L^2(d\nu_{\alpha})\) , we have \begin{alignat*}{2} \int_{0}^{+\infty}f(x)\overline{g(x)}d\nu_{\alpha}(x)&=\int_{0}^{+\infty}\mathcal{H}_\alpha(f)(\lambda)\overline{\mathcal{H}_\alpha(g)(\lambda)}d\nu_{\alpha}(\lambda). \end{alignat*}

The Hankel transform \(\mathcal{H}_\alpha\) satisfies the following properties;

For every \(f\in L^1(d\nu_{\alpha})\) and \(g\in L^p(d\nu_{\alpha}), p = 1, 2 \), the function \(f \ast g\) belongs to \(L^p(d\nu_{\alpha}), p = 1, 2\), and we have

\begin{equation*}\mathcal{H}_{\alpha}(f\ast g)=\mathcal{H}_{\alpha}(f)\mathcal{H}_{\alpha}(g).\end{equation*} Let \(f, g \in L^2(d\nu_\alpha)\). Then \(f\ast g\in L^2(d\nu_\alpha)\), if and only if \(\mathcal{H}_{\alpha}(f)\mathcal{H}_{\alpha}(g)\in L^2(d\nu_\alpha)\) and we have moreover, \begin{equation*} \int_{0}^{+\infty}|f\ast g(x)|^2d\nu_\alpha(x)=\int_{0}^{+\infty}|\mathcal{H}_{\alpha}(f)(\lambda)|^2|\mathcal{H}_{\alpha}(g)(\lambda)|^2d\nu_\alpha(\lambda), \end{equation*} where both integrals are finite or infinite.

3. Hankel-Stockwell transform

We recall some results introduced and proved in [14]. The modulation operator is defined for every function \(\psi\) in \(L^2(d\nu_\alpha)\) by \begin{equation*}\label{modulation} M_a(\psi)=\mathcal{H}_\alpha\left(\sqrt{\tau_a(|\mathcal{H}_\alpha(\psi)|^2)}\right),\quad a>0. \end{equation*} Then for every \(\psi\in L^2(d\nu_\alpha)\), \(M_a(\psi)\) belongs to \(L^2(d\nu_\alpha)\) and we have \begin{equation*} \|M_a(\psi)\|_{2,\nu_\alpha}=\|\psi\|_{2,\nu_\alpha}. \end{equation*} Now, for every \(\psi\in L^2(d\nu_\alpha)\), we consider the family \(\psi^{a,r}, (a,r)\in\mathbb{R}^{*}_+\times\mathbb{R}_+\) defined by \[\forall x\in\mathbb{R}_+,\quad \psi^{a,r}(x)=\tau_rM_aD_a\psi(x),\] where \(D_a\) is the dilatation operator given by \[D_a(\psi)(x)=a^{\alpha+1}\psi(ax).\] Then, we have the following properties;

• (i) For every \(\psi\in L^2(d\nu_\alpha)\)
  \begin{equation} \label{dilatatio de trans} \tau_xD_a(\psi)=D_a\tau_ax(\psi). \end{equation}

  (2)
• (ii) For every \(\psi\in L^2(d\nu_\alpha)\)
  \begin{equation} \label{dila de fourie} \mathcal{H}_\alpha(D_a(\psi))=D_{\frac{1}{a}}(\mathcal{H}_\alpha(\psi)). \end{equation}

  (3)

Definition 1. A nonzero function \(\psi\in L^2(d\nu_\alpha) \) is said to be an admissible window function if \[0< C_\psi=\frac{1}{2^\alpha\Gamma(\alpha+1)}\int_0^{+\infty}\tau_1(|\mathcal{H}_\alpha(\psi)|^2)(a)\frac{da}{a}< +\infty.\] In the following we denote by \(\mu_\alpha\) the measure defined on \(\mathbb{R}^*_+\times\mathbb{R}_+\) by \[d\mu_\alpha(a,r)=d\nu_\alpha(a)d\nu_\alpha(r),\] and \(L^p(d\mu_\alpha), 1\leqslant p\leqslant+\infty\), the Lebesgue space on \(\mathbb{R}^*_+\times\mathbb{R}_+\) with respect to the measure \(\mu_\alpha\) with the \(L^p\)-norm denoted by \(\|.\|_{p,\mu_\alpha}\).

Definition 2. Let \(\psi\) be an admissible window function. The continuous Hankel-Stockwell transform \(S^{\alpha}_\psi\) is defined in \(L^2(d\nu_\alpha)\) by

where \(\langle ,\rangle_{\nu_\alpha}\) is the usual inner product in the Hilbert space \(L^2(d\nu_\alpha).\)

Proposition 1. Let \(\psi\) be an admissible window function. For every \(f\in L^2(d\nu_\alpha)\), we have

Proposition 2. Let \(\psi\) be an admissible window function.

• (i) [Plancherel formula] For every \(f\in L^2(d\nu_\alpha)\), we have
  \begin{equation} \label{formule de planch fo VG} \|S^{\alpha}_\psi(f)\|_{2,\mu_\alpha}=\sqrt{C_\psi}\|f\|_{2,\nu_\alpha}. \end{equation}

  (6)
• (ii) [Parseval formula] For all \(f, h\in L^2(d\nu_\alpha)\), we have \begin{equation*}\label{parseval for gabor} \int_0^{+\infty}\int_0^{+\infty}S^{\alpha}_\psi(f)(a,r)\overline{S^{\alpha}_\psi(h)(a,r)}d\mu_\alpha(a,r)=C_\psi\int_0^{+\infty}f(s)\overline{h(s)}d\nu_\alpha(s). \end{equation*}
• (iii) [Inversion formula] For all \(f\in L^1(d\nu_\alpha)\cap L^2(d\nu_\alpha)\), such that \(\mathcal{H}_\alpha(f)\) belongs to \(L^1(d\nu_\alpha)\), we have \[f(u)=\frac{1}{C_\psi}\int_0^{+\infty}\left(\int_0^{+\infty}S^\alpha_\psi(f)(a,r)\psi^{a,r}(u)d\nu_\alpha(r)\right)d\nu_\alpha(a), a.e.,\] where for each \(u\in\mathbb{R}_+\), both the inner integral and the outer integral are absolutely convergent, but possible not the double integral.

By Riesz-Thorin's interpolation theorem we obtain the following.

Proposition 3. Let \(\psi\) be an admissible window function, \(f\in L^2(d\nu_\alpha)\) and \(2\leqslant p\leqslant+\infty\), then we have

4. Uncertainty principle for the Hankel-Stockwell transform

In this section, we obtain some uncertainty principles for the Hankel-Stockwell transform.

Theorem 2. [\(L^p\) local uncertainty principle for \(S^{\alpha}_\psi\)] Let \(\psi\) be an admissible window function and \(\Sigma\) be measurable subset of \(\mathbb{R}^*_+\times\mathbb{R}_+\) such that \(0< \mu_{\alpha}(\Sigma)< +\infty\). Let \(p\in]1,2], q=\frac{p}{p-1}\), then or every \(f\in L^p(d\nu_{\alpha})\), we have \begin{alignat*}{2} &\|\chi_\Sigma S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}\leqslant\left \{ \begin{array}{ll} C_1(b,\psi)(\mu_{\alpha}(\Sigma))^{\frac{b}{\alpha+1}}\left(\|r^{b}f\|_{2p,\nu_{\alpha}}+\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{b}f\|_{2,\nu_{\alpha}}\right),&\hbox{if \(0< b< \frac{\alpha+1}{q}\),}\\ \displaystyle C_2(b,\psi)(\mu_{\alpha}(\Sigma))^{\frac{1}{q}}\|f\|^{1-\frac{\alpha+1}{qb}}_{2p,\nu_{\alpha}} \|r^bf\|^{\frac{\alpha+1}{qb}}_{2p,\nu_{\alpha}} ,&\hbox{if \(b>\frac{\alpha+1}{q}\)},\\ C_3(b,\psi)(\mu_{\alpha}(\Sigma))^{\frac{1}{2q}}\Big(\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}+\|f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\Big),&\hbox{if \(b=\frac{\alpha+1}{q}\)}. \end{array} \right. \end{alignat*} where \begin{align*}C_1(b,\psi)&=\left(\frac{1}{2^{\alpha+1}\Gamma(\alpha+1)(\alpha+1-bq)}\right)^{\frac{b}{2(\alpha+1)}}C^{\frac{1}{q}-\frac{b}{\alpha+1}}_\psi\|\psi\|_{2,\nu_{\alpha}},\\ C_2(b,\psi)&=\left(\frac{\Gamma(\frac{\alpha+1}{bp})\Gamma\left(\frac{qb-(\alpha+1)}{bp}\right)}{bp2^{\alpha+1}\Gamma(\alpha+1)\Gamma\left(\frac{q}{p}\right)}\right)^{\frac{1}{2q}} \left(\frac{qb}{qb-(\alpha+1)}\right)^{\frac{1}{2p}}\left(\frac{qb}{\alpha+1}-1\right)^{\frac{\alpha+1}{2qbp}}\|\psi\|_{2,\nu_{\alpha}},\ \ \ \text{and}\\ C_3(b,\psi)&=2C_1\left(\frac{b}{2},\psi\right).\end{align*}

Proof.

• (i) It is clear that the first inequality holds if \[\|r^{b}f\|_{2p,\nu_{\alpha}}+\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{b}f\|_{2,\nu_{\alpha}}=+\infty.\] Let \(f\in L^p(d\nu_{\alpha}), 1< p\leqslant2, q=\frac{p}{p-1}\) such that \[\displaystyle \|r^{b}f\|_{2p,\nu_{\alpha}}+\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{b}f\|_{2,\nu_{\alpha}}< +\infty.\] Denote by \(\chi_\Sigma\) the characteristic function associated to \(\Sigma\). Using Minkowski's inequality, relations (5) and (7), we obtain for every \(\rho>0\) \begin{alignat*}{2} \|\chi_\Sigma S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}&\leqslant\|\chi_\Sigma S^{\alpha}_\psi(\chi_{[0,\rho[}f)\|_{q,\mu_{\alpha}}+\|\chi_\Sigma S^{\alpha}_\psi(f)(\chi_{[\rho,+\infty[}f)\|_{q,\mu_{\alpha}}\\ &\leqslant(\mu_{\alpha}(\Sigma))^{\frac{1}{q}}\|S^{\alpha}_\psi(\chi_{[0,\rho[}f)\|_{\infty,\mu_{\alpha}}+\|S^{\alpha}_\psi(f)(\chi_{[\rho,+\infty[}f)\|_{q,\mu_{\alpha}}\\ &\leqslant(\mu_{\alpha}(\Sigma))^{\frac{1}{q}}\|\psi\|_{2,\nu_{\alpha}}\|\chi_{[0,\rho[} f\|_{2,\nu_{\alpha}}+C^{\frac{1}{q}}_\psi\|\psi\|^{1-\frac{2}{q}}_{2,\nu_{\alpha}}\|\chi_{[\rho,+\infty[}f\|_{2,\nu_{\alpha}}. \end{alignat*} On the other hand, by Hölder's inequality \[\|\chi_{[0,\rho[} f\|_{2,\nu_{\alpha}}\leqslant\|r^{-b}\chi_{[0,\rho[}\|_{2q,\nu_{\alpha}}\|r^{b}f\|_{2p,\nu_{\alpha}}.\] By simple calculus and the hypothesis \(0< b< \frac{\alpha+1}{q}\), we obtain
  \begin{equation} \label{inega 1}\|\chi_{[0,\rho[} f\|_{2,\nu_{\alpha}}\leqslant C_{b,\alpha,q}\rho^{\frac{\alpha+1}{q}-b}\|r^{b}f\|_{2p,\nu_{\alpha}},\end{equation}

  (8)
  where \(\displaystyle C_{b,\alpha,q}=\left(\frac{1}{2^{\alpha+1}\Gamma(\alpha+1)(\alpha+1-bq)}\right)^{\frac{1}{2q}}.\) Moreover,
  \begin{equation} \label{inega 2} \|\chi_{[\rho,+\infty[}f\|_{2,\nu_{\alpha}}\leqslant\rho^{-b}\|r^{b}f\|_{2,\nu_{\alpha}}. \end{equation}

  (9)
  From (8) and (9), we get \begin{alignat*}{2}\|\chi_\Sigma S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}&\leqslant C_{b,\alpha,q}(\mu_{\alpha}(\Sigma))^{\frac{1}{q}}\|\psi\|_{2,\nu_{\alpha}}\rho^{\frac{\alpha+1}{q}-b}\|r^{b}f\|_{2p,\nu_{\alpha}}+\rho^{-b}C^{\frac{1}{q}}_\psi\|\psi\|^{1-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{b}f\|_{2,\nu_{\alpha}}. \end{alignat*} We choose \[\rho=(C_{b,\alpha,q})^{\frac{-q}{\alpha+1}}(\mu_{\alpha}(\Sigma))^{\frac{-1}{\alpha+1}}C^{\frac{1}{\alpha+1}}_\psi,\] hence, we obtain the first inequality.
• (ii) It is clear that the second inequality holds if \(\displaystyle \|f\|_{2p,\nu_{\alpha}}\) or \(\displaystyle \|r^bf\|_{2p,\nu_{\alpha}}=+\infty\). Assume that \[\displaystyle \|f\|_{2p,\nu_{\alpha}}+\|r^bf\|_{2p,\nu_{\alpha}}< +\infty.\] From hypothesis \(b>\frac{\alpha+1}{q}\), we deduce that the function \(\displaystyle r\longrightarrow(1+r^{2bp})^{\frac{-1}{p}}\) belongs to \(L^q(d\nu_{\alpha})\) and By Hölder's inequality, we have
  \begin{alignat} {2}\label{norme 1 de f < cnn} \|f\|^{2p}_{2,\nu_{\alpha}}&=\left(\int_0^{+\infty}(1+r^{2bp})^{\frac{-1}{p}}(1+r^{2bp})^{\frac{1}{p}}|f(r)|^2d\nu_{\alpha}(r)\right)^p\nonumber\\ &\leqslant\left(\int_0^{+\infty}\frac{d\nu_{\alpha}(r)}{(1+r^{2bp})^{\frac{q}{p}}}\right)^{\frac{p}{q}}\left(\|f\|^{2p}_{2p,\nu_{\alpha}}+\|r^bf\|^{2p}_{2p,\nu_{\alpha}}\right). \end{alignat}

  (10)
  However, with a standard computation, we obtain \[\left(\int_0^{+\infty}\frac{d\nu_{\alpha}(r)}{(1+r^{2bp})^{\frac{q}{p}}}\right)^{\frac{p}{q}}=\left(\frac{\Gamma(\frac{\alpha+1}{bp})\Gamma(\frac{qb-(\alpha+1)}{bp})}{bp2^{\alpha+1}\Gamma(\alpha+1)\Gamma(\frac{q}{p})}\right)^{\frac{p}{q}}=M^{\frac{p}{q}}_{b,\alpha,q}.\] Replacing \(f(r)\) by \(f_t(r)=f(rt), t>0\) in the relation (10), we deduce that for all \(t>0\) \begin{alignat*}{2} \|f\|^{2p}_{2,\nu_{\alpha}}&\leqslant M^{\frac{p}{q}}_{b,\alpha,q}\left(t^{(2\alpha+2)(p-1)}\|f\|^{2p}_{2p,\nu_{\alpha}}+t^{(2\alpha+2)(p-1)-2pb}\|r^bf\|^{2p}_{2p,\nu_{\alpha}}\right). \end{alignat*} In particular for \(t=\left(\frac{(2bp-(2\alpha+2)(p-1))\|r^bf\|^{2p}_{2p,\nu_{\alpha}}}{(2\alpha+2)(p-1)\|f\|^{2p}_{2p,\nu_{\alpha}}}\right)^{\frac{1}{2bp}}\), we obtain \[\|f\|_{2,\nu_{\alpha}}\leqslant M_{b,\alpha,q}^{\frac{1}{2q}}(\frac{qb}{qb-(\alpha+1)})^{\frac{1}{2p}}(\frac{qb}{\alpha+1}-1)^{\frac{\alpha+1}{2qbp}}\|f\|^{1-\frac{\alpha+1}{qb}}_{2p,\nu_{\alpha}}\|r^bf\|^{\frac{\alpha+1}{qb}}_{2p,\nu_{\alpha}}.\] Moreover, \begin{alignat*}{2} \|\chi_\Sigma S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}&\leqslant(\mu_{\alpha}(\Sigma))^{\frac{1}{q}}\|S^{\alpha}_\psi(f)\|_{\infty,\mu_{\alpha}}\\ &\leqslant(\mu_{\alpha}(\Sigma))^{\frac{1}{q}}\|f\|_{2,\nu_{\alpha}}\|\psi\|_{2,\nu_{\alpha}}\\ &\leqslant M_{b,\alpha,q}^{\frac{1}{2q}}(\frac{qb}{qb-(\alpha+1)})^{\frac{1}{2p}}(\frac{qb}{\alpha+1}-1)^{\frac{\alpha+1}{2qbp}}(\mu_{\alpha}(\Sigma))^{\frac{1}{q}}\|f\|^{1-\frac{\alpha+1}{qb}}_{2p,\nu_{\alpha}} \|r^bf\|^{\frac{\alpha+1}{qb}}_{2p,\nu_{\alpha}}\|\psi\|_{2,\nu_{\alpha}}. \end{alignat*} This completes the proof of the second inequality.
• (iii) Let \(s>0\), from the inequality \[\displaystyle \big(\frac{r}{s}\big)^{\frac{\alpha+1}{2q}}\leqslant1+\big(\frac{r}{s}\big)^{\frac{\alpha+1}{q}},\] it follows that \[\|r^{\frac{\alpha+1}{2q}}f\|_{2p,\nu_{\alpha}}\leqslant s^{\frac{\alpha+1}{2q}}\|f\|_{2p,\nu_{\alpha}}+s^{\frac{-(\alpha+1)}{2q}}\|r^{\frac{\alpha+1}{q}}f\|_{2p,\nu_{\alpha}}.\] In particular, by choosing \(\displaystyle s=\|r^{\frac{\alpha+1}{q}}f\|^{\frac{q}{\alpha+1}}_{2p,\nu_{\alpha}}\|f\|^{\frac{-q}{\alpha+1}}_{2p,\nu_{\alpha}}\), we obtain \[\|r^{\frac{\alpha+1}{2q}}f\|_{2p,\nu_{\alpha}}\leqslant2\|f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\|r^{\frac{\alpha+1}{q}}f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}.\] Similarly, we prove that \[\|r^{\frac{\alpha+1}{2q}}f\|_{2,\nu_{\alpha}}\leqslant2\|f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}\|r^{\frac{\alpha+1}{q}}f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}.\] Thus, we deduce that \begin{alignat*}{2}\|\chi_\Sigma S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}&\leqslant C_1(\frac{\alpha+1}{2q},\psi)(\mu_{\alpha}(\Sigma))^{\frac{1}{2q}}\left(\|r^{\frac{\alpha+1}{2q}}f\|_{2p,\nu_{\alpha}}+\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{\frac{\alpha+1}{2q}}f\|_{2,\nu_{\alpha}}\right)\\ &\leqslant 2C_1(\frac{\alpha+1}{2q},\psi)(\mu_{\alpha}(\Sigma))^{\frac{1}{2q}}\Big(\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}\|r^{\frac{\alpha+1}{q}}f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}+\|f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\|r^{\frac{\alpha+1}{q}}f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\Big), \end{alignat*} which gives the result for \(b=\frac{\alpha+1}{q}\).

From the \(L^p\) local uncertainty principle, we can find the following \(L^p\) Heisenberg-Pauli-Weyl uncertainty principle for the Hankel-Stockwell transform.

Theorem 3. [\(L^p\) Heisenberg-Pauli-Weyl uncertainty principle for the Hankel-Stockwell transform] Let \(\psi\) be an admissible window function, \(p\in]1,2], q=\frac{p}{p-1}\), and \(d>0\). Then for every \(f\in L^p(d\nu_{\alpha})\), we have \[\|S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}\leqslant\left \{ \begin{array}{ll} C_1(b,d,\psi)\left(\|r^{b}f\|_{2p,\nu_{\alpha}}+\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{b}f\|_{2,\nu_{\alpha}}\right)^{\frac{d}{d+4b}} \||(a,r)|^dS^{\alpha}_\psi(f)\|^{\frac{4b}{d+4b}}_{q,\mu_{\alpha}},&\hbox{if \(0< b< \frac{\alpha+1}{q}\),}\\ C_2(b,d,\psi)\|f\|^{\frac{d}{4\alpha+4+dq}(q-\frac{\alpha+1}{b})}_{2p,\nu_{\alpha}} \|r^bf\|^{\frac{d(\alpha+1)}{b(4\alpha+4+dq)}}_{2p,\nu_{\alpha}}\||(a,r)|^dS^{\alpha}_\psi(f)\|^{\frac{4\alpha+4}{4\alpha+4+dq}}_{q,\mu_{\alpha}} ,&\hbox{if \(b>\frac{\alpha+1}{q}\)},\\ C_3(b,d,\psi)\Big(\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}+\|f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\Big)^{\frac{d}{2b+d}} \||(a,r)|^dS^{\alpha}_\psi(f)\|^{\frac{2b}{2b+d}}_{q,\mu_{\alpha}},&\hbox{if \(b=\frac{\alpha+1}{q}\)}, \end{array} \right. \] where \begin{align*} C_1(b,d,\psi)&=\frac{(C_1(b,\psi))^{\frac{d}{d+4b}}}{\big(2^{2\alpha+2}\Gamma(2\alpha+3)\big)^{\frac{db}{(\alpha+1)(d+4b)}}}\left(\left(\frac{d}{4b}\right)^{\frac{4b}{d+4b}}+\left(\frac{4b}{d}\right)^{\frac{d}{d+4b}}\right)^{\frac{1}{q}},\\ C_2(b,d,\psi)&=\frac{(C_2(b,\psi))^{\frac{dq}{4\alpha+4+dq}}}{(2^{2\alpha+2}\Gamma(2\alpha+3))^{\frac{d}{4\alpha+4+dq}}}\left(\left(\frac{dq}{4\alpha+4}\right)^{\frac{4\alpha+4}{4\alpha+4+dq}}+\left(\frac{4\alpha+4}{dq}\right)^{\frac{dq}{4\alpha+4+dq}}\right)^{\frac{1}{q}},\ \ \ \text{and}\\ C_3(b,d,\psi)&=\frac{(C_3(b,\psi))^{\frac{d}{d+2b}}}{(2^{2\alpha+2}\Gamma(2\alpha+3))^{\frac{d}{2q(d+2b)}}}\left(\left(\frac{d}{2b}\right)^{\frac{2b}{d+2b}}+\left(\frac{2b}{d}\right)^{\frac{d}{d+2b}}\right)^{\frac{1}{q}}.\end{align*}

Proof.

• (i)   Let \(0< b< \frac{\alpha+1}{q}, d>0\). For \(\rho>0\), let \(\displaystyle \widetilde{B}_\rho=\{(a,r)\in\mathbb{R}^*_+\times\mathbb{R}_+;\,a^2+r^2\leqslant\rho^2\}\). Then
  \begin{equation} \label{I1} \|S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}=\|\chi_{\widetilde{B}_\rho}S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}+\|\chi_{\widetilde{B}^c_\rho}S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}. \end{equation}

  (11)
  From Theorem 2, we get \begin{alignat*}{2}\|\chi_{\widetilde{B}_\rho}S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}&\leqslant C^q_1(b,\psi)(\mu_{\alpha}(\widetilde{B}_\rho))^{\frac{bq}{\alpha+1}}\left(\|r^{b}f\|_{2p,\nu_{\alpha}}+\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{b}f\|_{2,\nu_{\alpha}}\right)^q. \end{alignat*} On the other hand, \begin{alignat*}{2} \mu_{\alpha}(\widetilde{B}_\rho)=\frac{\rho^{4\alpha+4}}{2^{2\alpha+2}\Gamma(2\alpha+3)}. \end{alignat*} Using the previous result, we obtain
  \begin{alignat} {2}\label{I2}&\|\chi_{\widetilde{B}_\rho}S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}&\leqslant C^q_1(b,\psi)(\frac{\rho^{4\alpha+4}}{2^{2\alpha+2}\Gamma(2\alpha+3)})^{\frac{bq}{\alpha+1}}\left(\|r^{b}f\|_{2p,\nu_{\alpha}}+\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{b}f\|_{2,\nu_{\alpha}}\right)^q. \end{alignat}

  (12)
  Moreover,
  \begin{equation} \label{I3} \|\chi_{\widetilde{B}^c_\rho}S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}\leqslant\rho^{-dq}\||(a,r)|^dS^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}. \end{equation}

  (13)
  By Combining relations (11), (12) and (13), we get \begin{alignat*}{2}\|S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}} &\leqslant C^q_1(b,\psi)(\frac{\rho^{4\alpha+4}}{2^{2\alpha+2}\Gamma(2\alpha+3)})^{\frac{bq}{\alpha+1}}\left(\|r^{b}f\|_{2p,\nu_{\alpha}}+\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{b}f\|_{2,\nu_{\alpha}}\right)^q +\rho^{-dq}\||(a,r)|^dS^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}.\end{alignat*} We choose \begin{alignat*}{2}\rho&=\left(\frac{d(2^{2\alpha+2}\Gamma(2\alpha+3))^{\frac{bq}{\alpha+1}}}{4bC^q_1(b,\psi)}\right)^{\frac{1}{(d+4b)q}}\left(\frac{\||(a,r)|^dS^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}}{\|r^{b}f\|_{2p,\nu_{\alpha}}+\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{b}f\|_{2,\nu_{\alpha}}}\right)^{\frac{1}{d+4b}},\end{alignat*} hence, we obtain the first inequality.
• (ii)  Let \(b>\frac{\alpha+1}{q}, d>0\) and let \(\rho>0\). From Theorem 2, we obtain
  \begin{alignat} {2}\label{I4}\|\chi_{\widetilde{B}_\rho}S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}&\leqslant C^q_2(b,\psi)\mu_{\alpha}(\widetilde{B}_\rho)\|f\|^{q-\frac{\alpha+1}{b}}_{2p,\nu_{\alpha}} \|r^bf\|^{\frac{\alpha+1}{b}}_{2p,\nu_{\alpha}}\nonumber\\ &=C^q_2(b,\psi)\frac{\rho^{4\alpha+4}}{2^{2\alpha+2}\Gamma(2\alpha+3)}\|f\|^{q-\frac{\alpha+1}{b}}_{2p,\nu_{\alpha}} \|r^bf\|^{\frac{\alpha+1}{b}}_{2p,\nu_{\alpha}}. \end{alignat}

  (14)
  Combining the relations (11), (13) and (14), we get \begin{alignat*}{2}\|S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}&\leqslant C^q_2(b,\psi)\frac{\rho^{4\alpha+4}}{2^{2\alpha+2}\Gamma(2\alpha+3)}\|f\|^{q-\frac{\alpha+1}{b}}_{2p,\nu_{\alpha}} \|r^bf\|^{\frac{\alpha+1}{b}}_{2p,\nu_{\alpha}}+\rho^{-dq}\||(a,r)|^dS^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}.\end{alignat*} We choose \[\rho=\left(\frac{dq2^{2\alpha+2}\Gamma(2\alpha+3)\||(a,r)|^dS^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}}{(4\alpha+4)C^q_2(b,\psi)\|f\|^{q-\frac{\alpha+1}{b}}_{2p,\nu_{\alpha}} \|r^bf\|^{\frac{\alpha+1}{b}}_{2p,\nu_{\alpha}}}\right)^{\frac{1}{4\alpha+4+dq}},\] hence, we obtain the second inequality.
• (iii)   Let \(b=\frac{\alpha+1}{q}, d>0\) and let \(\rho>0\). From Theorem 2, we get
  \begin{alignat} {2}\label{I5} \|\chi_{\widetilde{B}_\rho}S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}&\leqslant C^q_3(b,\psi)(\mu_{\alpha}(\widetilde{B}_\rho))^{\frac{1}{2}}\Big(\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}+\|f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\Big)^q\nonumber\\ &=C^q_3(b,\psi)\frac{\rho^{2\alpha+2}}{\sqrt{2^{2\alpha+2}\Gamma(2\alpha+3)}}\Big(\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}+\|f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\Big)^q. \end{alignat}

  (15)
  Combining the relations (11), (13) and (15), we obtain \begin{alignat*}{2} \|S^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}\leqslant& C^q_3(b,\psi)\frac{\rho^{2\alpha+2}}{\sqrt{2^{2\alpha+2}\Gamma(2\alpha+3)}}\Big(\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}+\|f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\Big)^q\\ &+\rho^{-dq}\||(a,r)|^dS^{\alpha}_\psi(f)\|^q_{q,\mu_{\alpha}}.\end{alignat*} We choose \begin{alignat*}{2} \rho&=\left(\frac{dq\big(2^{2\alpha+2}\Gamma(2\alpha+3)\big)^{\frac{1}{2}}}{C^q_3(b,\psi)(2\alpha+2)}\right)^{\frac{1}{2\alpha+2+dq}}\times\left(\frac{\||(a,r)|^dS^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}}{\Big(\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}+\|f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\Big)}\right)^{\frac{q}{2\alpha+2+dq}}, \end{alignat*} hence, we obtain the result.

In the following, we shall use the \(L^p\) Heisenberg-Pauli-Weyl uncertainty principle to obtain a concentration uncertainty principle.

Definition 3. Let \(0 \leqslant \varepsilon < 1\) and let \(S\) be a measurable set of \(\mathbb{R}_+\). We say that \(f\in L^p(d\nu_{\alpha})\), \(p\in[1,2]\), is \(\varepsilon\)-concentrated on \(S\) in \(L^p(d\nu_{\alpha})\)-norm if there is a measurable function \(h\) vanishing outside \(S\) such that \[\|f-h\|_{p,\nu_{\alpha}}\leqslant\varepsilon\|f\|_{p,\nu_{\alpha}}.\] We introduce a projection operator \(P_S\) as \(P_S f(r)=f(r), \quad \mbox{if}\quad r\in S\) and \(P_S f(r)=0, \quad \mbox{if}\quad r\notin S\). Let \(0 \leqslant \varepsilon_S < 1\). Then \(f\) is \(\varepsilon_S\)-concentrated on \(S\) in \(L^p(d\nu_{\alpha})\)-norm if and only if \[\|f-P_Sf\|_{p,\nu_{\alpha}}\leqslant\varepsilon_S\|f\|_{p,\nu_{\alpha}}.\]

Definition 4. Let \(\psi\) be an admissible window function and \(\Sigma\) be a measurable set of \(\mathbb{R}^*_+\times\mathbb{R}_+\). We define a projection operator \(Q_\Sigma\) as \[Q_\Sigma f=(S^{\alpha}_\psi)^{-1}\Big(P_\Sigma(S^{\alpha}_\psi(f))\Big).\] Let \(0 \leqslant \varepsilon_{\Sigma} < 1\). Then \(S^{\alpha}_\psi\) is \(\varepsilon_{\Sigma}\) -concentrated on \(\Sigma\) in \(L^{q}(d\mu_{\alpha})\)-norm, \(1\leqslant q\leqslant2\) if and only if \begin{equation*}\|S^{\alpha}_\psi(f)-S^{\alpha}_\psi(Q_{\Sigma} f)\|_{q,\mu_{\alpha}}\leqslant\varepsilon_{\Sigma}\|S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}.\end{equation*}

Proposition 4. Let \(\psi\) be an admissible window function and \(\Sigma\) be a measurable set of \(\mathbb{R}^*_+\times\mathbb{R}_+\). Then, for every \(p > 2\) and \(\varepsilon> 0\), if \(S^{\alpha}_\psi\) is \(\varepsilon\)-concentrated in \(\Sigma\) with respect to the norm \(\|.\|_{2,\mu_{\alpha}}\), then \[(\mu_{\alpha}(\Sigma))^{\frac{p-2}{p}}\geqslant(1-\varepsilon^2)C^{1-\frac{2}{p}}_\psi\|\psi\|^{\frac{4}{p}-2}_{2,\nu_{\alpha}},\] where \(\displaystyle \mu_{\alpha}(\Sigma)=\int\int_{\Sigma}d\nu_{\alpha}(a)d\nu_{\alpha}(r).\)

Proof. Let \(f\in L^2(d\nu_{\alpha})\) and \(p>2\). As \(S^{\alpha}_\psi(f)\) is \(\varepsilon\)-concentrated in \(\Sigma\) with respect to the norm \(\|.\|_{2,\mu_{\alpha}}\), we have \[\|\chi_{\Sigma^c}S^{\alpha}_\psi(f)\|_{2,\mu_{\alpha}}\leqslant\varepsilon\sqrt{C_\psi}\|f\|_{2,\nu_{\alpha}}.\] Now, using relation (6), we get \[\|\chi_{\Sigma}S^{\alpha}_\psi(f)\|^2_{2,\mu_{\alpha}}\geqslant(1-\varepsilon^2)C_\psi\|f\|^2_{2,\nu_{\alpha}}.\] Applying Hölder's inequality, we obtain \[\|\chi_{\Sigma}S^{\alpha}_\psi(f)\|^2_{2,\mu_{\alpha}}\leqslant\|S^{\alpha}_\psi(f)\|^2_{p,\mu_{\alpha}}(\mu_{\alpha}(\Sigma))^{\frac{p-2}{p}}.\] By relation (7), we obtain \[\|\chi_{\Sigma}S^{\alpha}_\psi(f)\|^2_{2,\mu_{\alpha}}\leqslant C^{\frac{2}{p}}_\psi\|\psi\|^{2-\frac{4}{p}}_{2,\nu_{\alpha}}\|f\|^2_{2,\nu_{\alpha}}(\mu_{\alpha}(\Sigma))^{\frac{p-2}{p}}.\] Finally, \[(\mu_{\alpha}(\Sigma))^{\frac{p-2}{p}}\geqslant(1-\varepsilon^2)C^{1-\frac{2}{p}}_\psi\|\psi\|^{\frac{4}{p}-2}_{2,\nu_{\alpha}}.\]

Proposition 5. Let \(\psi\) be an admissible window function and \(f\in L^1(d\nu_{\alpha})\cap L^2(d\nu_{\alpha})\) such that \(\|S^{\alpha}_\psi(f)\|_{2,\mu_{\alpha}}=1\). If \(f\) is \(\varepsilon_S\)-concentrated on \(S\) in \( L^1(d\nu_{\alpha})\)-norm and \(S^{\alpha}_\psi(f)\) is \(\varepsilon_\Sigma\)-concentrated on \(\Sigma\) in \(L^2(d\mu_{\alpha})\)-norm, then \(\nu_{\alpha}(S)\geqslant C_\psi (1-\varepsilon_S)^2\|f\|^2_{1,\nu_{\alpha}},\) and \(\mu_{\alpha}(\Sigma)\|f\|^2_{2,\nu_{\alpha}}\|\psi\|^2_{2,\nu_{\alpha}}\geqslant 1-\varepsilon^2_\Sigma.\)

Proof. As \(S^{\alpha}_\psi(f)\) is \(\varepsilon_\Sigma\)-concentrated on \(\Sigma\) in \(L^2(d\mu_{\alpha})\)-norm and by the orthogonality of the projection operator \(P_\Sigma\), it follows that \begin{alignat*}{2}\|S^{\alpha}_\psi(f)\|^2_{2,\mu_{\alpha}}-\|S^{\alpha}_\psi(f)-P_\Sigma(S^{\alpha}_\psi(f))\|^2_{2,\mu_{\alpha}} &=\|P_\Sigma(S^{\alpha}_\psi(f))\|^2_{2,\mu_{\alpha}}&\geqslant1-\varepsilon^2_\Sigma,\end{alignat*} and thus \[1-\varepsilon^2_\Sigma\leqslant\|S^{\alpha}_\psi(f)\|^2_{\infty,\mu_{\alpha}}\mu_{\alpha}(\Sigma)\leqslant\mu_{\alpha}(\Sigma)\|f\|^2_{2,\nu_{\alpha}}\|\psi\|^2_{2,\nu_{\alpha}}.\] By the same way, \(f\) is \(\varepsilon_S\)-concentrated on \(S\) in \( L^1(d\nu_{\alpha})\)-norm, we obtain \[(1-\varepsilon_S)\|f\|_{1,\nu_{\alpha}}\leqslant\int_{S}|f(r)|d\nu_{\alpha}(r).\] Now, by the Cauchy-Schwarz inequality and the fact that \(\displaystyle \|f\|_{2,\nu_{\alpha}}=\frac{1}{\sqrt{C_\psi}}\), we get \[(1-\varepsilon_S)\|f\|_{1,\nu_{\alpha}}\leqslant\frac{\nu_{\alpha}^{\frac{1}{2}}(S)}{\sqrt{C_\psi}}.\]

Definition 5. Let \(\psi\) be an admissible window function and \(\Sigma\) be a measurable set of \(\mathbb{R}^*_+\times\mathbb{R}_+\). Let \(d>0\), \(f\in L^p(d\nu_{\alpha}),\,p\in[1,2]\) and \(0 \leqslant \varepsilon_\Sigma < 1\). We say that \(|(a,r)|^{d}S^{\alpha}_\psi\) is \(\varepsilon_\Sigma\) -concentrated on \(\Sigma\) in \(L^{q}(d\mu_{\alpha})\)-norm, if and only if \begin{alignat*}{2}\||(a,r)|^{d}S^{\alpha}_\psi(f)-|(a,r)|^{d}S^{\alpha}_\psi(Q_\Sigma f)\|_{q,\mu_{\alpha}}&\leqslant\varepsilon_\Sigma\||(a,r)|^{d}S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}. \end{alignat*}

Theorem 4. Let \(\psi\) be an admissible window function and \(\Sigma\) be a measurable set of \(\mathbb{R}^*_+\times\mathbb{R}_+\). Let \(f\in L^p(d\nu_{\alpha}),\,p\in]1,2]\), \(0 \leqslant \varepsilon_\Sigma < 1\) and \(d>0\). If \(|(a,r)|^{d}S^{\alpha}_\psi\) is \(\varepsilon_\Sigma\) -concentrated on \(\Sigma\) in \(L^{q}(d\mu_{\alpha})\)-norm, then \[\|S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}\leqslant\left \{ \begin{array}{ll} C_1(b,d,\psi)\left(\|r^{b}f\|_{2p,\nu_{\alpha}}+\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|r^{b}f\|_{2,\nu_{\alpha}}\right)^{\frac{d}{d+4b}}&\\ \times\Big(\frac{1}{1-\varepsilon_\Sigma}\||(a,r)|^{d}S^{\alpha}_\psi(Q_\Sigma f)\|_{q,\mu_{\alpha}}\Big)^{\frac{4b}{d+4b}},&\hbox{if \(0< b< \frac{\alpha+1}{q}\),}\\ [3mm] C_2(b,d,\psi)\|f\|^{\frac{d}{4\alpha+4+dq}(q-\frac{\alpha+1}{b})}_{2p,\nu_{\alpha}} \|r^bf\|^{\frac{d(\alpha+1)}{b(4\alpha+4+dq)}}_{2p,\nu_{\alpha}} \Big(\frac{1}{1-\varepsilon_\Sigma}\||(a,r)|^{d}S^{\alpha}_\psi(Q_\Sigma f)\|_{q,\mu_{\alpha}}\Big)^{\frac{4\alpha+4}{4\alpha+4+dq}} ,&\hbox{if \(b>\frac{\alpha+1}{q}\)},\\[3mm] C_3(b,d,\psi)\Big(\|\psi\|^{-\frac{2}{q}}_{2,\nu_{\alpha}}\|f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2,\nu_{\alpha}}+\|f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\|r^{b}f\|^{\frac{1}{2}}_{2p,\nu_{\alpha}}\Big)^{\frac{d}{2b+d}}&\\ \times\Big(\frac{1}{1-\varepsilon_\Sigma}\||(a,r)|^{d}S^{\alpha}_\psi(Q_\Sigma f)\|_{q,\mu_{\alpha}}\Big)^{\frac{2b}{2b+d}},&\hbox{if \(b=\frac{\alpha+1}{q}\)}. \end{array} \right. \]

Proof. Let \(f\in L^p(d\nu_{\alpha}),\,p\in]1,2]\). Since \(|(a,r)|^{d}S^{\alpha}_\psi\) is \(\varepsilon_\Sigma\) -concentrated on \(\Sigma\) in \(L^{q}(d\mu_{\alpha})\)-norm, then we have \begin{alignat*}{2}\||(a,r)|^{d}S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}&\leqslant\||(a,r)|^{d}S^{\alpha}_\psi(Q_\Sigma f)\|_{q,\mu_{\alpha}}+\varepsilon_\Sigma\||(a,r)|^{d}S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}. \end{alignat*} Thus, \[ \||(a,r)|^{d}S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}\leqslant\frac{1}{1-\varepsilon_\Sigma}\||(a,r)|^{d}S^{\alpha}_\psi(Q_\Sigma f)\|_{q,\mu_{\alpha}},\] and we obtain the result from Theorem 3.

Definition 6. Let \(\Sigma\) be a measurable set of \(\mathbb{R}^*_+\times\mathbb{R}_+\) and \(0\leqslant\eta< \sqrt{C_\psi}\). Then a nonzero function \(f\in L^p(d\nu_{\alpha}), 1\leqslant p\leqslant2\) is \(\eta\)-bandlimited on \(\Sigma\) in \(L^q(d\mu_{\alpha})\)-norm, if \begin{equation*} \|\chi_{\Sigma^c}S^{\alpha}_\psi(f)\|_{q,\mu_{\alpha}}\leqslant\eta\|f\|_{p,\nu_{\alpha}}. \end{equation*} where \(q=\frac{p}{p-1}\).

Corollary 1. Let \(\psi\) be an admissible window function.

• (i)     If \(0< b< \frac{\alpha+1}{2}\), then there exists a positive constant \(C\) such that for every function \(f\) which is \(\eta\)-bandlimited on \(\Sigma\) \[(\mu_{\alpha}(\Sigma))^{\frac{2b}{\alpha+1}}\Big(\|r^{b}f\|_{4,\nu_{\alpha}}+\|\psi\|^{-1}_{2,\nu_\alpha}\|r^{b}f\|_{2,\nu_{\alpha}}\Big)^2\geqslant C(C_\psi-\eta^2)\|f\|^2_{2,\nu_{\alpha}}.\]
• (ii)     If \(b>\frac{\alpha+1}{2}\), then there exists a positive constant \(C\) such that for every function \(f\) which is \(\eta\)-bandlimited on \(\Sigma\) \[\mu_{\alpha}(\Sigma)\|f\|^{2-\frac{\alpha+1}{b}}_{4,\nu_{\alpha}}\|r^bf\|^{\frac{\alpha+1}{b}}_{4,\nu_{\alpha}}\geqslant C(C_\psi-\eta^2)\|f\|^2_{2,\nu_{\alpha}}.\]

Proof. Since \(f\in L^2(d\nu_{\alpha})\) is \(\eta\)-bandlimited on \(\Sigma\), then \[\|\chi_{\Sigma}S^{\alpha}_\psi(f)\|^2_{2,\mu_{\alpha}}=C_\psi\|f\|^2_{2,\nu_{\alpha}}-\|\chi_{\Sigma^c}S^{\alpha}_\psi(f)\|^2_{2,\mu_{\alpha}}\geqslant(C_\psi-\eta^2)\|f\|^2_{2,\nu_{\alpha}}.\] For (i) and (ii), we use the local inequalities given respectively by Theorem 2.

According to the following Pitt's inequality for the Hankel transform [9], we obtain the Pitt's inequality for the Hankel-Stockwell transform.

Proposition 6. Let \( 0\leqslant\eta< \alpha+1\). For every \(f\in S_e(\mathbb{R}) \), we have

where \(\displaystyle C_{\eta,\alpha}=2^{-\eta}\left(\frac{\Gamma\left(\frac{2\alpha+2-\eta}{4}\right)}{\Gamma\left(\frac{2\alpha+2+\eta}{4}\right)}\right)^2\) and \(\Gamma(.)\) denotes the well known Eurler's gamma function.

Theorem 5. [Pitt's inequality the Hankel- Stockwell transform] Let \(\psi\) be an admissible window function and \( 0\leqslant\eta< \alpha+1\). For every \(f\in S_e(\mathbb{R}) \), the Pitt's inequality for the Hankel- Stockwell transform is given by \begin{alignat*}{2} C_\psi\int_0^{+\infty}|\lambda|^{-\eta}|\mathcal{H}_\alpha(f)(\lambda)|^2d\nu_\alpha(\lambda)&\leqslant C_{\eta,\alpha}\int_0^{+\infty}\int_0^{+\infty}|r|^{\eta}|S^{\alpha}_\psi(f)(a,r)|^2d\mu_\alpha(a,r). \end{alignat*}

Proof. For \(\eta=0\), the result follows from relation (6). Now suppose that \(0< \eta< \alpha+1\). For every \(f\in S_e(\mathbb{R})\) and by (16), we can write \begin{alignat*}{2} \int_0^{+\infty}|\lambda|^{-\eta}|\mathcal{H}_\alpha(S^\alpha_\psi(f)(a,.))(\lambda)|^2d\nu_\alpha(\lambda)&\leqslant C_{\eta,\alpha}\int_0^{+\infty}|r|^{\eta}|S^{\alpha}_\psi(f)(a,r)|^2d\nu_\alpha(r). \end{alignat*} Integrating with respect \(d\nu_\alpha(a)\), we get

By (1)-(4) and using Fubini's theorem, we obtain Relations (17) and (18) gives the Pitt's inequality for the Hankel-Stockwell transform.

Now, using the following logarithmic uncertainty principle for the Hankel transform [9], we obtain the logarithmic uncertainty principle for the Hankel-Stockwell transform.

Proposition 7. For every \(f\in S_e(\mathbb{R})\), the following inequality holds:

where \(\omega\) denotes the logarithmic derivative of the gamma function \(\Gamma\) [20,21].

Theorem 6. [Logarithmic uncertainty principle for the Hankel-Stockwell transform] Let \(\psi\) be an admissible window function. For every \(f\in S_e(\mathbb{R})\), we have \begin{alignat*}{2} C_\psi\int_0^{+\infty}\ln(t)|\mathcal{H}_\alpha(f)(t)|^2d\nu_\alpha(t)&+\int_0^{+\infty}\int_0^{+\infty}\ln(r)|S^{\alpha}_\psi(f)(a,r)|^2d\mu_\alpha(a,r)\geqslant C_\psi \Big(\ln2+\omega(\frac{\alpha+1}{2})\Big)\|f\|^2_{2,\nu_\alpha}. \end{alignat*}

Proof. Replacing \(f\) by \(S^{\alpha}_\psi(f) \) in the inequality (19), we obtain \begin{alignat*}{2} \int_0^{+\infty} \ln(t)|\mathcal{H}_\alpha(S^{\alpha}_\psi(f)(a,.))(t)|^2d&\nu_\alpha(t)+\int_0^{+\infty}\ln(r)|S^{\alpha}_\psi(f)(a,r)|^2d\nu_\alpha(r)\\ &\geqslant \left(\ln2+\omega\left(\frac{\alpha+1}{2}\right)\right)\int_0^{+\infty}|S^{\alpha}_\psi(f)(a,r)|^2d\nu_\alpha(r). \end{alignat*} Integrating both sides with respect to \(a\), we have

By (1), (4) and using Fubini's theorem, we obtain Hence, by (6), (20) and (21), we have \begin{alignat*}{2} C_\psi\int_0^{+\infty}\ln(t)|\mathcal{H}_\alpha(f)(t)|^2d\nu_\alpha(t)+\int_0^{+\infty}\int_0^{+\infty}\ln(r)|S^{\alpha}_\psi(f)(a,r)|^2d\mu_\alpha(a,r)\geqslant C_\psi \left(\ln2+\omega\left(\frac{\alpha+1}{2}\right)\right)\|f\|^2_{2,\nu_\alpha}. \end{alignat*}

Conflicts of Interest

The author declares no conflict of interest.