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Zbl 0954.34055
Aizicovici, Sergiu; McKibben, Mark
Existence results for a class of abstract nonlocal Cauchy problems.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 39, No.5, A, 649-668 (2000). ISSN 0362-546X

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)].
[J.Myjak (L'Aquila)]
MSC 2000:
*34G25 Evolution inclusions
34A12 Initial value problems for ODE
47H06 Accretive operators, etc. (nonlinear)

Keywords: nonlocal initial condition; $m$-accretive operator; compact semigroup; integral solution

Citations: Zbl 0883.34065

Cited in: Zbl 1084.34002 Zbl 1083.34045

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