Volume 3 (2019) – issue 1

OMA-Vol. 3 (2019), Issue 1, pp. 70 – 89 Open Access Full-Text PDF

Abstract: In this paper, we study the Cauchy problem of the fractional Navier-Stokes equations with Coriolis force in critical Fourier-Besov-Morrey spaces. By using the Fourier localization argument and the Littlewood-Paley theory, we get a local well-posedness results and global well-posedness results with small initial data belonging to the critical Fourier-Besov-Morrey spaces. Moreover; we prove that the corresponding global solution decays to zero as time goes to infinity, and we give the stability result for global solutions.

OMA-Vol. 3 (2019), Issue 1, pp. 61-69 Open Access Full-Text PDF

Abstract: We study the existence and uniqueness of positive solutions of the nonlinear fractional differential equation with integral boundary conditions \(\mathfrak{D}_{1}^{\alpha }x\left( t\right) =f\left( t,x\left( t\right) \right) ,\;\;\; 1<t\leq e, x\left( 1\right) =\lambda \int_{1}^{e}x\left( s\right) ds+d,\) where  \(\mathfrak{D}_{1}^{\alpha }\) is the Caputo-Hadamard fractional derivative of order \(0<\alpha \leq 1\). In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct an appropriate mapping and employ the Schauder fixed point theorem and the method of upper and lower solutions to show the existence of a positive solution of this equation. We also use the Banach fixed point theorem to show the existence of a unique positive solution. Finally, an example is given to illustrate our results.

OMA-Vol. 3 (2019), Issue 1, pp. 51-61 Open Access Full-Text PDF

Abstract: It is a well-known fact that the majority of rational difference equations cannot be solved theoretically. As a result, some scientific experts use manual iterations to obtain the exact solutions of some of these equations. In this paper, we obtain the fractional solutions of the following systems of difference equations:
$$
x_{n+1}=\frac{x_{n-1}y_{n-3}}{y_{n-1}\left( -1-x_{n-1}y_{n-3}\right) },\ \ \
y_{n+1}=\frac{y_{n-1}x_{n-3}}{x_{n-1}\left( \pm 1\pm y_{n-1}x_{n-3}\right) }
,\ \ \ n=0,1,…,
$$
where the initial data \(x_{-3},\ x_{-2},\ x_{-1},\ \)\ \(
x_{0},\ y_{-3},\ y_{-2},\ y_{-1}\) and \(\ \ y_{0}\;\) are arbitrary non-zero real numbers. All solutions will be depicted under specific initial conditions.

OMA-Vol. 3 (2019), Issue 1, pp. 42–51 Open Access Full-Text PDF

Abstract: We study the global existence and uniqueness of a solution to an initial boundary value problem for the Euler-Bernoulli viscoelastic equation \(u_{tt}+\Delta^{2}u-g_{1}\ast\Delta^{2} u+g_{2}\ast\Delta u+u_{t}=0.\) Further, the asymptotic behavior of solution is established.

OMA-Vol. 3 (2019), Issue 1, pp. 30–41 Open Access Full-Text PDF

Abstract: In this paper, we study the small oscillations of a visco-elastic fluid that is heated from below and fills completely a rigid container, restricting to the more simple Oldroyd model. We obtain the operatorial equations of the problem by using the Boussinesq hypothesis. We show the existence of the spectrum, prove the stability of the system if the kinematic coefficient of viscosity and the coefficient of temperature conductivity are sufficiently large and the existence of a set of positive real eigenvalues having a point of the real axis as point of accumulation. Then, we prove that the problem can be reduced to the study of a Krein-Langer pencil and obtain new results concerning the spectrum. Finally, we obtain an existence and unicity theorem of the solution of the associated evolution problem by means of the semigroups theory.

OMA-Vol. 3 (2019), Issue 1, pp. 19–29 Open Access Full-Text PDF

Abstract: By using the boundedness results for the commutators of the fractional integral with variable kernel on variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^{n})\), the boundedness results are established on variable exponent Herz-Morrey spaces \(M\dot{K}_{q,p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})\).

OMA-Vol. 3 (2019), Issue 1, pp. 07–18 Open Access Full-Text PDF

Abstract: In this article, we prove the existence and uniqueness of solutions for the Navier problem \( \Delta\big[\omega_1(x)\vert\Delta u\vert^{p-2}\Delta u+ \nu_1(x)\vert\Delta u\vert^{q-2}\Delta u\big] -{div}\big[\omega_2(x)\vert\nabla u\vert^{p-2}\nabla u +\nu_2(x)\vert\nabla u\vert^{s-2}\nabla u\big] = f(x) – { div}(G(x)),\) in \({\Omega},\) with
\(u(x) = {\Delta}u= 0,\) in \({\partial\Omega},\) where \(\Omega\) is a bounded open set of \(\mathbb{R}^N\) for \(N\geq 2\), \(\frac{f}{\omega_2}\in L^{p’}(\Omega , {\omega}_2)\) and \(\frac{G}{{\nu}_2}\in \left[L^{s’}(\Omega ,{\nu}_2)\right]^N\).

OMA-Vol. 3 (2019), Issue 1, pp. 01–06 Open Access Full-Text PDF

Abstract:In this paper, we present a new non-convex hybrid iteration algorithm for common fixed points of a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in the domains of Hilbert spaces.