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On a nonlocal Cauchy problem for differential inclusions. (English) Zbl 1080.34005

The paper concerns a system which consists of the differential inclusion \[ y'(t)\in F(t,y(t)) \] and the nonlocal condition \[ y(0)+\sum_{k=1}^{p}c_ky(t_k)=y_0. \] Here, \(F:[0,b]\times \mathbb{R}^n\to \mathbb{R}^n\) is a certain Carathéodory multi-valued map, whereas \(0\leq t_1<\cdots<t_p\leq b\) and \(c_1\neq0\), …, \(c_p\neq0\). In contrast with the paper by M. Benchohra and S. K. Ntouyas [Georgian Math. J. 7, 221–230 (2000; Zbl 0960.34049)], the \(F\)-values are not necessarily convex. Using either the Covitz-Nadler fixed-point theorem or the Bressan-Colombo selection theorem combined with Schaefer’s fixed-point theorem, there are derived two distinct results which state the existence of solutions to the system above.

MSC:

34A60 Ordinary differential inclusions

Citations:

Zbl 0960.34049
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