Random evolution equations in Frechet spaces

Said Abbas,Fatima Mesri̇,Mouffak Benchohra,Amaria Arara

doi:10.31197/atnaa.451341

Abstract

This paper deals with the existence of random mild solutions for some classes of first and second order functional evolution equations with random effects in Frechet spaces. The technique used is a generalization of the classical Darbo fixed point theorem for Frechet spaces associated with the concept of measure of noncompactness

Highlights

This paper deals with the existence of random mild solutions for some classes of first and second order evolution equations with random effects in Fréchet spaces
The technique used is a generalization of the classical Darbo fixed point theorem for Fréchet spaces associated with the concept of measure of noncompactness
There has been a significant development in functional evolution equations in recent years; see the monographs [2, 3, 17, 23, 25] and the references therein

Summary

Introduction

There has been a significant development in functional evolution equations in recent years; see the monographs [2, 3, 17, 23, 25] and the references therein. In [1], an iterative method is used for the existence of mild solutions of evolution equations and inclusions. When our knowledge about the parameters of a dynamic system are of statistical nature, that is, the information is probabilistic, the common approach in mathematical modeling of such systems is the use of random differential equations or stochastic differential equations. We discuss the existence of random mild solutions for the evolution equation u (t, w) = A(t)u(t, w) + f (t, u(t, w), w); if t ∈ R+ := [0, ∞), w ∈ Ω,. We discuss the existence of random mild solutions for the following second order evolution problem. This paper initiates the existence of random mild solutions for evolution equations in Fréchet spaces with an application of a generalization of the classical Darbo fixed point theorem, and the concept of measure of noncompactness

Preliminaries

First Order Random Evolution Equations

Second Order Random Evolution Equations

An Example