Nonlocal problem for a nonlinear fractional mixed type integro-differential equation with spectral parameters

Abstract

In this article, we consider a boundary value problem for a nonlinear partial integro-differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with positive spectral parameter in a negative rectangular domain. The partial integro-differential equation of mixed type depends on another real spectral parameter in integral part of the mixed equation. With respect to first variable this equation is a fractional integro-differential equation in the positive part of the considering segment, and is a second-order integro-differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method of separation variables and Fredholm method of degenerate kernels, the solutions of nonlinear boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of solution of the problem are proved for regular values of the spectral parameters.