This paper is concerned with a version of Saint-Venant type decay estimates for solutions of the DIRICHLET problem for the NONLINEAR BIHARMONIC equation. Energy methods are used to establish a NONLINEAR INTEGRO-DIFFERENTIAL INEQUALITY for a quadratic functional. To this end, integral methods are used to establish exponential decay of solutions for the DIRICHLET problem being considered.

We study the existence, uniqueness, continuous dependence, and asymptotic expansion of solutions of the DIRICHLET problem for a NONLINEAR Kirchhoff wave equation. At first, we state and prove a theorem involving the local existence and uniqueness of a weak solution. Next, we establish a sufficient condition to get an estimate of the continuous dependence of the solution with respect to the NONLINEAR terms. Finally, an asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

In the present research paper we derive results about existence and uniqueness of solutions and Ulam{Hyers and Rassias stabilities of NONLINEAR Volterra{Fredholm delay integrodi erential equations. Pachpatte's INEQUALITY and Picard operator theory are the main tools that are used to obtain our main results. We concluded this work with applications of obtained results and few illustrative examples.

This paper deals with the problem of determining of an unknown coefficient in an inverse: elliptic boundary value problem. Using a nonconstant overspecified data, it has been shown that the: solution to this inverse problem exists and is unique.

In this paper, we establish some results on the existence of at least three weak solutions for a DIRICHLET problem involving p-Laplacian using a variational approach.

The aim of this research is to establish the analytic solution of partial differential equations with homogenous initial boundary conditions. Having hybrid fractional order derivative allows us to have classical boundary and initial conditions. The solution of the problem is obtained in terms of bivariate Mittag-Leffler function as a Fourier series by utilizing separation of variables method (SVM) and the inner product defined on L2 [0, l]. The presented examples illustrate the accuracy and effectiveness of the SVM for the fractional diffusion problems. The accuracy of the obtained solution can also be seen from the observation that as the fractional order α,tends to 1, the solution of the fractional diffusion problem tends to the solution of the diffusion problem.

In this paper we deal with the existence of weak solution for a $p(x)$-Kirchhoff type problem of the following form $$ \left\{\begin{array}{ll}-\left(a-b \int_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\, dx\right)\Delta(|\Delta u|^{p(x)-2}\Delta u) =\lambda |u|^{p(x)-2}u+g(x, u) & \text{ in } \Omega, \\ u=\Delta u=0 & \textrm{ on } \partial\Omega. \end{array}\right. $$ Using the Mountain Pass Theoem, we establish conditions ensuring the existence result.