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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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