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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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