OMA-Vol. 2 (2018), Issue 2, pp. 156–171 | Open Access Full-Text PDF

Abstract: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.

OMA-Vol. 2 (2018), Issue 2, pp. 142–155 | Open Access Full-Text PDF

Abstract: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.

OMA-Vol. 2 (2018), Issue 2, pp. 129–141 | Open Access Full-Text PDF

Abstract: In this paper, we find a solution of a new type of Langevin equation involving Hilfer fractional derivatives with impulsive effect. We formulate sufficient conditions for the existence and uniqueness of solutions. Moreover, we present Hyers-Ulam stability results.

OMA-Vol. 2 (2018), Issue 2, pp. 114–128 | Open Access Full-Text PDF

Abstract:Let \(\mathcal{A}\) be the class of analytic and univalent functions in the open unit disc \(\Delta\) normalized such that \(f(0)=0=f^{\prime }(0)-1.\) In this paper, for \(\psi \in \mathcal{A}\) of the form \(\frac{z}{f(z)}, f(z)=1+\sum\limits_{n=1}^{\infty }a_{_{n}}z^{n}\) and \(0\leq \alpha \leq 1,\) we introduce and investigate interesting subclasses \(\mathcal{H}_{\sigma }(\phi ), \;S_{\sigma }(\alpha ,\phi ), \; M_{\sigma }(\alpha ,\phi ),\) \( \Im _{\alpha} (\alpha ,\phi )\) and \(\beta _{\alpha}(\lambda ,\phi ) \left( \lambda \geq 0 \right)\) of analytic and bi-univalent Ma-Minda starlike and convex functions. Furthermore, we find estimates on the coefficients \(\left\vert a_{1}\right\vert\) and \(\left\vert a_{2}\right\vert\) for functions in these classess. Several related classes of functions are also considered.

OMA-Vol. 2 (2018), Issue 2, pp. 93–113 | Open Access Full-Text PDF

Abstract:In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish both the existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively.