Open Journal of Mathematical Analysis (OMA)

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.

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.

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.

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.

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 \(M\dot{K}_{q,p(\cdot)}^{\alpha, \lambda}(\mathbb{R}^{n})\).

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 \(\frac{G}{{\nu}_2}\in \left[L^{s’}(\Omega ,{\nu}_2)\right]^N\).

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.

In this report we present new sixth order iterative methods for solving non-linear equations. The derivation of these methods is purely based on variational iteration technique. To check the validity and efficiency we compare of methods with Newton’s method, Ostrowski’s method, Traub’s method and modified Halleys’s method by solving some test examples. Numerical results shows that our developed methods are more effective. Finally, we compare polynomigraphs of our developed methods with Newton’s method, Ostrowski’s method, Traub’s method and modified Halleys’s method.

In this work we study the global solution, uniqueness and asymptotic behaviour of the nonlinear equation
\begin{eqnarray*}
u_{tt} – \Delta_{p} u = \Delta u – g*\Delta u
\end{eqnarray*}
where \(\Delta_{p} u\) is the nonlinear \(p\)-Laplacian operator, \(p \geq 2\) and \(g*\Delta u\) is a memory damping. The global solution is constructed by means of the Faedo-Galerkin approximations taking into account that the initial data is in appropriated set of stability created from the Nehari manifold and the asymptotic behavior is obtained by using a result of P. Martinez based on new inequality that generalizes the results of Haraux and Nakao.

This paper deals with the determination of a coefficient in the diffusion term of some degenerate /singular one-dimensional linear parabolic equation from final data observations. The mathematical model leads to a non convex minimization problem. To solve it, we propose a new approach based on a hybrid genetic algorithm (married genetic with descent method type gradient). Firstly, with the aim of showing that the minimization problem and the direct problem are well posed, we prove that the solution’s behavior changes continuously with respect to the initial conditions. Secondly, we chow that the minimization problem has at least one minimum. Finally, the gradient of the cost function is computed using the adjoint state method. Also we present some numerical experiments to show the performance of this approach.