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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
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