Gomaa, Adel Mahmoud Existence of solutions for differential inclusions on closed moving constraints in Banach spaces. (English) Zbl 1182.34082 Electron. J. Differ. Equ. 2009, Paper No. 22, 10 p. (2009). 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. Reviewer: Daniel C. Biles (Nashville) MSC: 34G25 Evolution inclusions 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations Keywords:differential inclusion; moving constraint; existence of solutions; measure of noncompactness Citations:Zbl 0836.34015 PDFBibTeX XMLCite \textit{A. M. Gomaa}, Electron. J. Differ. Equ. 2009, Paper No. 22, 10 p. (2009; Zbl 1182.34082) Full Text: EuDML EMIS