Nonlinear nonlocal <inline-formula><tex-math id="M1">$ \psi $</tex-math></inline-formula>-Caputo fractional integro-differential equations of Sobolev type

In this paper, we investigate the nonlocal Cauchy problem for a class of nonlinear fractional $ \psi $-Caputo integro-differential equations of Sobolev type in Banach spaces. We first establish some new Darbo-type fixed point theorems, and then we present a solvability theorem for the nonlocal Cauchy problem for such nonlinear fractional equations of Sobolev type with the help of the new Darbo-type fixed point theorems. Moreover, we illustrate the applicability of our theoretical results with two examples.