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First-order differential inclusions with nonlocal initial conditions. (English) Zbl 1025.34009
It is considered the following nonlocal initial value problem \[ x'(t)\in F(t,x(t)),\quad t\in (0,1],\qquad x(0)+\sum_{k=1}^{m}a_{k}x(t_{k})=x_{0}, \tag{1} \] where \(F:(0,1]\times \mathbb{R}\to 2^{\mathbb{R}}\) is a set-valued map, \(x_{0}\in \mathbb{R}\) is given, \(0<t_{1}<t_{2}<\cdots < t_{m}<1,\) and \(a_{k}\neq 0\) for all \(k=1,2,\ldots,m.\) If some assumptions are satisfied, then problem (1) has at least one solution.

34A60 Ordinary differential inclusions
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI
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