Mixed Problem for an Impulsive Parabolic Integro-Differential Equation with Involution and Nonlinear Conditions

Abstract

In this paper, we consider an impulsive homogeneous parabolic type partial integro-differential equation with degenerate kernel and involution. With respect to spatial variable \(x\) is used Dirichlet boundary value conditions and spectral problem is studied. The Fourier method of separation of variables is applied. The countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of unknown function. Theorem on a unique solvability of countable system of functional equations is proved. The method of successive approximations is used in combination with the method of contraction mapping. The unique solution of the impulsive mixed problem is obtained in the form of Fourier series. Absolutely and uniformly convergence of Fourier series is proved.