Nonlocal Integro-Differential Equations Without the Assumption of Equicontinuity on the Resolvent Operator in Banach Spaces

Abstract

In this work, we study the existence of mild solutions for the nonlocal integro-differential equation $$\begin{aligned} \left\{ \begin{array}{l} x'(t)=Ax(t)+\displaystyle \int _{0}^{t}B(t-s)x(s)\text {d}s+f(t,x_{t})\quad \text {for}\;\; t\in [0,b]\\ x_{0}=\phi +g(x)\in C([-r,0];X), \end{array} \right. \end{aligned}$$ without the assumption of equicontinuity on the resolvent operator and without the assumption of separability on the Banach space X. The nonlocal initial condition is assumed to be compact. Our main result is new and its proof is based on a measure of noncompactness developed in Kamenskii et al. (Condensing multivalued maps and semilinear differential inclusions in Banach spaces. Walter De Gruyter, Berlin, 2001) together with the well-known Monch fixed point Theorem. To illustrate our result, we provide an example in which the resolvent operator is not equicontinuous.