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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 $$u'(t)=Au(t)+f(t,u(t)), \quad t\in[t_0,t_0+T], \quad u(t_0)+g(u)=u_0,$$ under some conditions on $f$ and $g$, where $f:[t_0,t_0+T]\times X\to X$ and $g:C([t_0,t_0+T];X)\to X$ 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 {\it S. Aizicovici} and {\it M. McKibben} [Nonlinear Anal., Theory Methods Appl. 39, No. 5(A), 649--668 (2000; Zbl 0954.34055)] and {\it L. Byszewski} and {\it 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
Full Text: DOI
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