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On the solution sets of constrained differential inclusions with applications. (English) Zbl 0991.34011
Here, the authors study the existence and the structure of the solutions to the following initial value problem (IVP) \[ u'\in\varphi (t,u), \quad t\in J:=[0,T], \;u\in K,\qquad u(0)=x_{0}\in K, \] where \(\varphi: J\times K\to \mathbb{R}^{N}\) is a multivalued map with nonempty, compact and convex values, and \(K\) is a closed subset of the \(N\)-dimensional Euclidean space \(\mathbb{R}^{N}\), \(N\geq 1\). It is proved that if the multivalued map \(\varphi\) satisfies the following conditions
\((\varphi_{1})\) for almost all (a.a.) \(t\in J\), the map \(K\ni x\to \varphi(t,x)\) is upper semicontinuous and for all \(x\in K\), the map \(\varphi(\cdot,x)\) has a measurable selection, i.e. there is a measurable function \(w_{x}(t)\in\varphi(t,x)\) for a.a. \(t\in J\);
\((\varphi_{2})\) \(\varphi\) is (uniformly) bounded, i.e. there is \(c>0\) such that, for a.a. \(t\in J\) and all \(x\in K \;Sup_{y\in\varphi(t,x)} |y|\leq c\), then the set of all solutions to the above IVP is an \(\mathbb{R}_{\delta}\)-set in \(C(J,\mathbb{R}^{N})\). Some applications to the existence of periodic solutions as well as equilibria are given.

MSC:
34A60 Ordinary differential inclusions
34C25 Periodic solutions to ordinary differential equations
47H10 Fixed-point theorems
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