Relaxation and weak solutions of nonlocal semilinear evolution systems

N Javaid,S Bilal,T Donchev,A I Lazu

doi:10.1186/s13662-019-2107-3

Abstract

We give a relatively short proof of the fact that the solution set of a nonlocal semilinear differential inclusion is dense in the weak solution set of the corresponding convexified system. Moreover, we prove a similar result for the solutions with continuous pseudoderivatives when the right-hand side is continuous with nonempty convex closed and bounded values.

Highlights

In this paper we study the following nonautonomous semilinear differential inclusion with nonlocal initial conditions:
The interest for this study comes from the fact that these mathematical models describe more accurately than the traditional Cauchy problems the evolution of various phenomena
In the present paper we prove a relaxation theorem for the nonlocal problem (1.1), assuming that F(·, ·) is almost continuous with closed bounded values and F(t, ·) is Lipschitz continuous, in general Banach spaces

Summary

Introduction

{A(t); t ∈ I} is a family of densely defined linear operators on E, F : I × E ⇒ E is a multifunction with nonempty closed bounded values, and g : C(I, E) → E is a given function. In the present paper we prove a relaxation theorem for the nonlocal problem (1.1), assuming that F(·, ·) is almost continuous with closed bounded values and F(t, ·) is Lipschitz continuous, in general Banach spaces. Recall that the continuous function x(·) is called the mild solution of (2.1) if it satisfies the integral equation t x(t) = T(t, 0)x0 + T(t, s)f (s) ds for any t ∈ I.

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Conclusion