Existence results for a class of abstract nonlocal Cauchy problems
The authors study the global existence of a solution to nonlinear evolution equations with nonlocal conditions of the form \[ u'(t) + Au(t) \ni f(t,u(t)), \quad u(0)=g(u), \quad 0<t<T, \tag{*} \] in a real Banach space \(X\). Here, \(A\) is a nonlinear \(m\)-accretive (possibly multivalued) operator on \(X\), \(F: L^1(0,T;X) \to L^1(0,T;X)\) and \(g:L^1(0,T;X) \to \overline {D(A)}\). Using the Schauder fixed point theorem, the Fryszkowski selection theorem and some properties of compact semigroups, the authors prove the existence of integral solutions. This work is a continuation of the paper by \textit{S. Aizicovici} and \textit{Y. Gao} [J. Appl. Math. Stochastic Anal. 10, No.~2, 145-156 (1997; Zbl 0883.34065)].
scientific article; zbMATH DE number 1389683