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

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 $0<b/a\le 1$. This problem is equivalently reduced to a Volterra type nonlinear integral equation and we prove its unique solvability by the successive approximations method.
The second part of the work is devoted to well-posed formulation and to studying other nonlocal problems, the formulations of which are related with other possible cases of $a$ and $b$. We provide a detailed study of Problem II. Then as remarks we described the way of studying other formulated problems.