Existence results for some integro-differential equations with state-dependent nonlocal conditions in Fréchet Spaces

Mamadou Abdoul Diop,Carlos Ogouyandjou,Kora Hafiz Bete,Reine Kakpo

doi:10.1515/msds-2020-0121

Abstract

Abstract In this work, we present existence of mild solutions for partial integro-differential equations with state-dependent nonlocal local conditions. We assume that the linear part has a resolvent operator in the sense given by Grimmer. The existence of mild solutions is proved by means of Kuratowski’s measure of non-compactness and a generalized Darbo fixed point theorem in Fréchet space. Finally, an example is given for demonstration.

Highlights

In this work, we present existence of mild solutions for partial integro-di erential equations with state-dependent nonlocal local conditions
We study existence of mild solutions for the following partial functional integrodi erential equation
We will work under the following assumptions : (A ) There exists a constant M > such that R(t) L(X) ≤ M for every t ∈ R+. (A ) (i) The function t → F(t, θ) is measurable on R+ for each θ ∈ C, and the function θ → F(t, θ) is continuous on C for a.e. t ∈ R+. (ii) There exists a function f ∈ Lloc(R+, R+) and a continuous nondecreasing function Ξ : R+ → R+

Summary

Introduction

Abstract: In this work, we present existence of mild solutions for partial integro-di erential equations with state-dependent nonlocal local conditions. The existence of mild solutions is proved by means of Kuratowski’s measure of noncompactness and a generalized Darbo xed point theorem in Fréchet space. We study existence of mild solutions for the following partial functional integrodi erential equation

Results

Conclusion