Some applications of second-order differential subordination for a class of analytic function defined by the lambda operator
OMA-Vol. 4 (2020), Issue 2, pp. 170 – 177 Open Access Full-Text PDF
B. Venkateswarlu, P. Thirupathi Reddy, S. Sridevi, Sujatha
Abstract: In this paper, we introduce a new class of analytic functions by using the lambda operator and obtain some subordination results.
On properties of inner product type integral transformers
OMA-Vol. 4 (2020), Issue 2, pp. 160 – 169 Open Access Full-Text PDF
Benard Okelo
Abstract: In this paper, we give characterizations of certain properties of inner product type integral transformers. We first consider unitarily invariant norms and operator valued functions. We then give results on norm inequalities for inner product type integral transformers in terms of Landau inequality, Grüss inequality. Lastly, we explore some of the applications in quantum theory.
Boundary value problems for a class of stochastic nonlinear fractional order differential equations
OMA-Vol. 4 (2020), Issue 2, pp. 152 – 159 Open Access Full-Text PDF
McSylvester Ejighikeme Omaba, Louis O. Omenyi
Abstract: Consider a class of two-point Boundary Value Problems (BVP) for a stochastic nonlinear fractional order differential equation \(D^\alpha u(t)=\lambda\sqrt{I^\beta[\sigma^2(t,u(t))]}\dot{w}(t),\,\,00\) is a level of the noise term, \(\sigma:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous, \(\dot{w}(t)\) is a generalized derivative of Wiener process (Gaussian white noise), \(D^\alpha\) is the Riemann-Liouville fractional differential operator of order \(\alpha\in (3,4)\) and \(I^\beta,\,\,\beta>0\) is a fractional integral operator. We formulate the solution of the equation via a stochastic Volterra-type equation and investigate its existence and uniqueness under some precise linearity conditions using contraction fixed point theorem. A case of the above BVP for a stochastic nonlinear second order differential equation for \(\alpha=2\) and \(\beta=0\) with \(u(0)=u(1)=0\) is also studied.
Common fixed point results of \(s\)-\(\alpha\) contraction for a pair of maps in \(b\)-dislocated metric spaces
OMA-Vol. 4 (2020), Issue 2, pp. 142 – 151 Open Access Full-Text PDF
Abdissa Fekadu, Kidane Koyas, Solomon Gebregiorgis
Abstract: The purpose of this article is to construct fixed point theorems and prove the existence and uniqueness of common fixed point results of \(s-\alpha\) contraction for a pair of maps in the setting of \(b\) – dislocated metric spaces. Our results extend and generalize several well-known comparable results in the literature. The study procedure we used was that of Zoto and Kumari [1]. Furthermore, we provided an example in support of our main result.
On the existence of positive solutions of a state-dependent neutral functional differential equation with two state-delay functions
OMA-Vol. 4 (2020), Issue 2, pp. 132 – 141 Open Access Full-Text PDF
El-Sayed, A. M. A, Hamdallah, E. M. A, Ebead, H. R
Abstract: In this paper, we study the existence of positive solutions for an initial value problem of a state-dependent neutral functional differential equation with two state-delay functions. The continuous dependence of the unique solution will be proved. Some especial cases and examples will be given.
Exponential decay of solutions with \(L^{p}\) -norm for a class to semilinear wave equation with damping and source terms
OMA-Vol. 4 (2020), Issue 2, pp. 123 – 131 Open Access Full-Text PDF
Amar Ouaoua, Messaoud Maouni, Aya Khaldi
Abstract: In this paper, we consider an initial value problem related to a class of hyperbolic equation in a bounded domain is studied. We prove local existence and uniqueness of the solution by using the Faedo-Galerkin method and that the local solution is global in time. We also prove that the solutions with some conditions exponentially decay. The key tool in the proof is an idea of Haraux and Zuazua with is based on the construction of a suitable Lyapunov function.
Global solutions and general decay for the dispersive wave equation with memory and source terms
OMA-Vol. 4 (2020), Issue 2, pp. 116 – 122 Open Access Full-Text PDF
Mohamed Mellah
Abstract: This paper concerns with the global solutions and general decay to an initial-boundary value problem of the dispersive wave equation with memory and source terms.
Controllability for some nonlinear impulsive partial functional integrodifferential systems with infinite delay in Banach spaces
OMA-Vol. 4 (2020), Issue 2, pp. 104 – 115 Open Access Full-Text PDF
Patrice Ndambomve, Khalil Ezzinbi
Abstract: This work concerns the study of the controllability for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces. We give sufficient conditions that ensure the controllability of the system by supposing that its undelayed part admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of the work of K. Balachandran and R. Sakthivel (Journal of Mathematical Analysis and Applications, 255, 447-457, (2001)) and a host of important results in the literature, without assuming the compactness of the resolvent operator. An example is given for illustration.
On hyper-singular integrals
OMA-Vol. 4 (2020), Issue 2, pp. 100 – 103 Open Access Full-Text PDF
Alexander G. Ramm
Abstract: The integrals \(\int_{-\infty}^\infty t_+^{\lambda-1} \phi(t)dt\) and \(\int_0^t(t-s)^{\lambda -1}b(s)ds\) are considered, \(\lambda\neq 0,-1,-2…\), where \(\phi\in C^\infty_0(\mathbb{R})\) and \(0\le b(s)\in L^2_{loc}(\mathbb{R})\). These integrals are defined in this paper for \(\lambda\le 0\), \(\lambda\neq 0,-1,-2,…\), although they diverge classically for \(\lambda\le 0\). Integral equations and inequalities are considered with the kernel \((t-s)^{\lambda -1}_+\).
A differential inequality and meromorphic starlike and convex functions
OMA-Vol. 4 (2020), Issue 2, pp. 93 – 99 Open Access Full-Text PDF
Kuldeep Kaur Shergill, Sukhwinder Singh Billing
Abstract: In the present paper, we study a differential inequality involving certain differential operator. As a special case of our main result, we obtained certain differential inequalities implying sufficient conditions for meromorphic starlike and meromorphic convex functions of certain order.