Nonlocal evolution inclusions under weak conditions

Shamas Bilal,Ovidiu Cârjă,Tzanko Donchev,Alina I Lazu,Nasir Javaid

doi:10.1186/s13662-018-1858-6

Abstract

We study evolution inclusions given by multivalued perturbations of m-dissipative operators with nonlocal initial conditions. We prove the existence of solutions. The commonly used Lipschitz hypothesis for the perturbations is weakened to one-sided Lipschitz ones. We prove an existence result for the multipoint problems that cover periodic and antiperiodic cases. We give examples to illustrate the applicability of our results.

Highlights

1 Introduction In this paper, we study the nonlinear evolution system with nonlocal initial condition
The commonly assumptions used to prove the existence of solutions are either that A is of compact type and F satisfies some upper semicontinuity, or F is Lipschitz continuous
In [22], the authors established an existence result for the nonlocal differential inclusion (1.1) assuming that X is separable with uniformly convex dual, F(·, x) is measurable, F(t, ·) is Lipschitz with the Lipschitz function p(·) ∈ L1(I, R+), and

Summary

Introduction

We study the nonlinear evolution system with nonlocal initial condition. See [2], where the authors assume that X∗ is uniformly convex (as in the present paper) and A generates a compact semigroup. The results of [1, 22] are not applicable in the important case of periodic or antiperiodic boundary conditions, that is, g(y) = y(T) or g(y) = –y(T) This problem is especially studied in the present paper. We first prove the existence of solutions to problem (1.1), assuming that F(s, ·) is one-sided Lipschitz, which is weaker than the commonly used Lipschitz condition. 1, which covers the remarkable periodic and antiperiodic cases For this specific case of (1.1), we obtain an existence result under a one-sided Lipschitz condition with negative constant on F. In the end of the paper, we give two examples to demonstrate the applicability of our results

Preliminaries

Findings

A multipoint problem