## Nonlocal Cauchy problems for first-order multivalued differential equations.(English)Zbl 1011.34002

It is investigated the existence of solutions for the Cauchy problem $\dot{x} \in F(t,x(t)), \quad x(0)+ \sum_{k=1}^{m}a_{k}x(t_{k})=0,\quad t\in (0,T]. \tag{1}$ Here, $$F:J\times\mathbb{R}\to 2^{\mathbb{R}}$$ is a set-valued map, $$J=[0,T], 0 < t_{1}< t_{2}\ldots t_{m}<1.$$ The set of all bounded closed convex and nonempty subsets of $$\mathbb{R}$$ is denoted by $$bcc(\mathbb{R}).$$ The following conditions are assumed:
$$(H_{0})$$ $$a_{k}\neq 0$$ for each $$k=1,2,\ldots, m$$, and $$\sum_{k=1}^{m}a_{k}+1 \neq 0.$$
$$(H_{1})$$ $$F:J\times\mathbb{R}\to bcc(\mathbb{R}), (t,x)\to F(t,x)$$ is measurable in $$t$$ for each $$x\in \mathbb{R}$$ and upper semicontinuous with respect to $$x\in\mathbb{R}$$ for a.e. $$t\in J.$$ $$(H_{2})$$ $$|F(t,x)|\leq \psi(|x|)$$ for a.e. $$t\in J$$ and all $$x\in\mathbb{R}$$, where $$\psi:[0,\infty)\to (0,\infty)$$ is continuous nondecreasing and such that $$\limsup_{\rho\to\infty} \psi(\rho)/\rho=0.$$
Under these conditions the following theorem holds: If the assumptions $$(H_{0}), (H_{1}), (H_{2})$$ are satisfied, then the initial value problem (1) has least one solution.

### MSC:

 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34A60 Ordinary differential inclusions 34G20 Nonlinear differential equations in abstract spaces
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