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Nonlinear nonlocal Cauchy problems in Banach spaces. (English) Zbl 1084.34002
The authors consider the abstract nonlinear problem with nonlocal initial conditions of the form: $u'(t)+ Au(t) \ni f(t,u(t)),\;t\in [0,T],\quad u(0) = g(u),$ in a real Banach space $$X$$, where $$u: [0,T]\to X$$, $$A: D(A)\subseteq X\to 2^X$$, $$f: [0, T]\times X\to X$$ and $$g: C([0, T]; X)\to \overline{D(A)}$$. Existence of an integral solution is proven under the assumptions that $$A$$ is a nonlinear, $$m$$-accretive operator such that the semigroup generated by $$-A$$ is compact, $$f$$ is continuous in $$t$$ and satisfies a Lipschitz condition in its second variable, and $$g$$ is continuous, among other conditions. The proof is accomplished through an application of Schauder’s fixed-point theorem. Two corollaries of this result are also given. The authors note that this extends to the fully nonlinear case a theorem in [J. Liang, J. H. Liu and T. Xiao, Nonlinear Anal., Theory Methods Appl. 57, No. 2(A), 183–189 (2004; Zbl 1051.65036)] and the results also cover examples which cannot be handled by a theorem in [S Aizicovici and M. McKibben, Nonlinear Anal., Theory Methods Appl. 39, No. 5(A), 649–668 (2000; Zbl 0954.34055)].

MSC:
 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 47H04 Set-valued operators 47H06 Nonlinear accretive operators, dissipative operators, etc. 47H20 Semigroups of nonlinear operators 47J35 Nonlinear evolution equations
Citations:
Zbl 0954.34055; Zbl 1051.65036
Full Text:
References:
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