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Nonlocal Cauchy problems governed by compact operator families. (English) Zbl 1083.34045

Let A be the infinitesimal generator of a compact semigroup of linear operators on a Banach space X. The authors establish the existence of mild solutions to the nonlocal Cauchy problem

u ' (t)=Au(t)+f(t,u(t)),t[t 0 ,t 0 +T],u(t 0 )+g(u)=u 0 ,

under some conditions on f and g, where f:[t 0 ,t 0 +T]×XX and g:C([t 0 ,t 0 +T];X)X are given functions. They assume a Lipschitz condition on f with respect to u, but they do not require any compactness assumption on g, opposed to S. Aizicovici and M. McKibben [Nonlinear Anal., Theory Methods Appl. 39, No. 5(A), 649–668 (2000; Zbl 0954.34055)] and L. Byszewski and H. Akca [Nonlinear Anal., Theory Methods Appl. 34, No. 1, 65–72 (1998; Zbl 0934.34068)], where the authors assume a compactness property for g, but do not require any Lipschitz condition on f.

34G20Nonlinear ODE in abstract spaces
47D06One-parameter semigroups and linear evolution equations