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.


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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