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Existence of solutions for differential inclusions on closed moving constraints in Banach spaces. (English) Zbl 1182.34082

The author considers existence of solutions to the problem
\[ \begin{aligned} x'(t) &\in F(t,x(t)),\;t\in I\equiv [0,T],\\ x(t) &\in\Gamma(t),\;t\in I,\\ x(0) &= x_0\in\Gamma(0),\end{aligned} \]
where \(F: I\times E\to\) subsets of \(E\), \(E\) is a Banach space and \(\Gamma: I\to\) subsets of \(E\). Note that \(\Gamma\) represents a moving constraint. In Theorem 3.1, existence of solutions is proven under assumptions including that \(F\) is nonempty, convex and compact-valued, \(F(t,\cdot)\) is upper semicontinuous, \(F\) satisfies a boundedness condition, the graph of \(\Gamma\) is left closed, \(\Gamma\) is closed-valued and the following condition is satisfied: For each \(\varepsilon> 0\), there exists a closed subset \(I_\varepsilon\) of \(I\) with \(\lambda(I\setminus I_\varepsilon)<\varepsilon\) such that for any compact subset \(J\) of \(I_\varepsilon\) and any nonempty bounded subset \(Z\) of \(E\), we have \(\gamma(F(J\times Z))\leq\sup_{t\in J}\,w(t,\gamma(Z))\), where \(\lambda\) represents Lebesgue measure, \(w\) is a Kamke function and \(\gamma\) is a measure of strong noncompactness.
In Theorem 3.2, existence is proven under similar but different assumptions. Finally, in section 4, the author shows that a result in [H. Benabdellah, C. Castaing and M. A. Gamal Ibrahim [BV solutions of multivalued differential equations on closed moving sets in Banach spaces, Geometry in nonlinear control and differential inclusions (Warsaw, 1993), Warsaw: Polish Academy of Sciences, Banach Cent. Publ. 32, 53–81 (1995; Zbl 0836.34015)] is a special case of Theorem 3.2.

MSC:

34G25 Evolution inclusions
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations

Citations:

Zbl 0836.34015
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