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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

${u}^{\text{'}}\left(t\right)=Au\left(t\right)+f\left(t,u\left(t\right)\right),\phantom{\rule{1.em}{0ex}}t\in \left[{t}_{0},{t}_{0}+T\right],\phantom{\rule{1.em}{0ex}}u\left({t}_{0}\right)+g\left(u\right)={u}_{0},$

under some conditions on $f$ and $g$, where $f:\left[{t}_{0},{t}_{0}+T\right]×X\to X$ and $g:C\left(\left[{t}_{0},{t}_{0}+T\right];X\right)\to X$ 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 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 $f$.

##### MSC:
 34G20 Nonlinear ODE in abstract spaces 47D06 One-parameter semigroups and linear evolution equations