On non-local problems for third order equation with Caputo operator and non-linear loaded part

Abstract

This paper is devoted to proving the unique solvability of nonlocal problems with an integral conjugate condition for one class of third-order equations with a parabolic-hyperbolic operator including the Caputo fractional derivative and a nonlinear term containing the trace of the solution $u(x,0).$ Since the considered equation is of the third order, in which a first order differential operator with coefficients $a,$ $b$ and $c$ acts on a parabolic-hyperbolic second order operator, the coefficients $a,$ $b$ and $c$ influence essentially a well-defined formulation of boundary value problems. This is why, before providing complete formulation of the studied problems, we present the boundary conditions in their formulation for various cases of the behavior of the coefficients $a,$ $b$ and $c$. In the first part of the paper we formulate a nonlocal Problem I with an integral conjugate condition in the case $0