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Semilinear evolution equations with distributed measures. (English) Zbl 1338.37115

Summary: The aim of the paper is to provide, by an approach based on the Mönch fixed point theorem, existence results for the semilinear evolution problem with distributed measures \[ \begin{cases} dx=(-Ax+f(t,x))\,dt+dg, \quad t\in[0,1],\\ x(0)=x_{0}, \end{cases} \eqno{(1)} \] where \(-A\) is the infinitesimal generator of a (uniformly or strongly) continuous semigroup \(\{T(t),t\geq 0\}\) 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.

MSC:

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
47D06 One-parameter semigroups and linear evolution equations
47J35 Nonlinear evolution equations
58D25 Equations in function spaces; evolution equations
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
47H10 Fixed-point theorems
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