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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.

MSC:
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
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