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A class of absolute retracts in spaces of integrable functions. (English) Zbl 0747.34014

The authors consider multiple-valued functions \(\phi:E\to 2^ E\), with closed bounded values, \(E\) a Banach space and \(\phi\) being contractive with respect to the usual Hausdorff metric (on the set \(2^ E\)). They consider the problem of obtaining some topological and set-theoretical information of the set of fixed points \(F\) (under the multiple-valued function), where \(F=\{u \in E \mid \phi \in \phi (u)\}\). The authors’ main results include a proof that \(F\) is an absolute retract when \(E=L^ 1(T)\), for some measure space \(T\) and if the values of \(\phi(u)\) are decomposable sets. This is related to an earlier result where this result is true when \(E\) is simply a Banach space. The authors then make some interesting applications to differential inclusions - a topic extensively treated in the book of J. P. Aubin and A. Cellina Differential inclusions. Set-valued maps and viability theory (1984; Zbl 0538.34007). Differential inclusions are directly related to multiple- valued ordinary differential equations with initial values. They obtain in this paper some set-theoretical information about the set of Carathéodory solutions of the differential inclusion or multi-valued ODE with initial value \(dx/dt \in K(t,x)\), \(x(0)=\xi\), for \(K:[0,T] \times \mathbb{R} \to 2^{\mathbb{R}^ n}\), where \(K\) is a Lipschitz continuous multi-function with compact values.

MSC:

34A60 Ordinary differential inclusions
49J45 Methods involving semicontinuity and convergence; relaxation
54C15 Retraction
54C20 Extension of maps
93B03 Attainable sets, reachability

Citations:

Zbl 0538.34007
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References:

[1] Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. · Zbl 0538.34007
[2] Alberto Bressan and Giovanni Colombo, Extensions and selections of maps with decomposable values, Studia Math. 90 (1988), no. 1, 69 – 86. · Zbl 0677.54013
[3] Arrigo Cellina, On the set of solutions to Lipschitzian differential inclusions, Differential Integral Equations 1 (1988), no. 4, 495 – 500. · Zbl 0723.34009
[4] Arrigo Cellina and António Ornelas, Representation of the attainable set for Lipschitzian differential inclusions, Rocky Mountain J. Math. 22 (1992), no. 1, 117 – 124. · Zbl 0752.34012
[5] R. M. Colombo, A. Fryszkowski, T. Rzezukowski, and V. Staicu, Continuous selections of solutions sets of Lipschitzean differential inclusions, Funk. Ekv. (to appear). · Zbl 0749.34008
[6] Andrzej Fryszkowski, Continuous selections for a class of nonconvex multivalued maps, Studia Math. 76 (1983), no. 2, 163 – 174. · Zbl 0534.28003
[7] Fumio Hiai and Hisaharu Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), no. 1, 149 – 182. · Zbl 0368.60006
[8] A. Ornelas, A continuous version of the Filippov-Gronwall inequality for differential inclusions, Rend. Sem. Mat. Univ. Padova (to appear). · Zbl 0719.34032
[9] B. Ricceri, Une proprieté topologique de l’ensemble des points fixes d’une contraction multivoque à valeurs convexes, preprint. · Zbl 0666.47030
[10] Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980.
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