On a nonlocal Cauchy problem for differential inclusions
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 \(c_1\neq0\), \dots, \(c_p\neq0\). In contrast with the paper by \textit{M.~Benchohra} and \textit{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.
scientific article; zbMATH DE number 2163663