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Existence results for a class of abstract nonlocal Cauchy problems. (English) Zbl 0954.34055
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 {\it S. Aizicovici} and {\it Y. Gao} [J. Appl. Math. Stochastic Anal. 10, No. 2, 145-156 (1997; Zbl 0883.34065)].

MSC:
 34G25 Evolution inclusions 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 47H06 Accretive operators, dissipative operators, etc. (nonlinear)
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