Abstract
The aim of the paper is to provide, by an approach based on the Monch fixed point theorem, existence results for the semilinear evolution problem with distributed measures 1 $$ \textstyle\begin{cases} dx=(-Ax+f(t,x))\,dt+dg, \quad t\in[0,1], x(0)=x_{0}, \end{cases} $$ where −A is the infinitesimal generator of a (uniformly or strongly) continuous semigroup $\{T(t),t\geq0\}$ of bounded linear operators, f is not necessarily continuous and $g:[0,1]\to X$ is a regulated function. Working with Kurzweil-Stieltjes integrals and using a measure of non-compactness allows us to relax the assumptions on the semigroup, on f and g comparing to some already known results.
Highlights
1 Introduction The aim of the paper is to provide existence results for the semilinear evolution problem with distributed measures ( ), where –A is the infinitesimal generator of a continuous semigroup {T(t), t ≥ } of bounded linear operators, f : [, ] × X → X and g : [, ] → X
We are interested in discussing the matter of existence of mild solutions for the above problem under less restrictive assumptions: a regulated function g and a possibly discontinuous function f
When considering the Kurzweil-Stieltjes integral, even for regulated function g, this problem can be solved for non-reflexive spaces and non-uniformly continuous semigroups: as recalled in Remark, if the semigroup has bounded B-variation and g is regulated, the Kurzweil-Stieltjes integral t a
Summary
The aim of the paper is to provide existence results for the semilinear evolution problem with distributed measures ( ), where –A is the infinitesimal generator of a continuous semigroup {T(t), t ≥ } of bounded linear operators, f : [ , ] × X → X and g : [ , ] → X. We discuss the more general case when –A is the infinitesimal generator of a strongly continuous semigroup {T(t), t ≥ } of bounded linear operators In this setting, the problem becomes complicated due to the possibility for the Stieltjes integral to be not well defined in the Banach space. Our results extend and generalize some other recent theorems in the literature as well, see [ , ] or [ – ] (for the linear case)
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