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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)f(t,u(t)),u(0)=g(u),0<t<T,(*)

in a real Banach space X. Here, A is a nonlinear m-accretive (possibly multivalued) operator on X, F:L 1 (0,T;X)L 1 (0,T;X) and g:L 1 (0,T;X)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 S. Aizicovici and Y. Gao [J. Appl. Math. Stochastic Anal. 10, No. 2, 145-156 (1997; Zbl 0883.34065)].


MSC:
34G25Evolution inclusions
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
47H06Accretive operators, dissipative operators, etc. (nonlinear)