Bressan, Alberto; Cellina, Arrigo; Fryszkowski, Andrzej A class of absolute retracts in spaces of integrable functions. (English) Zbl 0747.34014 Proc. Am. Math. Soc. 112, No. 2, 413-418 (1991). 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. Reviewer: S.-N.Patnaik (Delhi) Cited in 3 ReviewsCited in 30 Documents MSC: 34A60 Ordinary differential inclusions 49J45 Methods involving semicontinuity and convergence; relaxation 54C15 Retraction 54C20 Extension of maps 93B03 Attainable sets, reachability Keywords:multiple-valued functions; Banach space; Hausdorff metric; fixed points; absolute retract; differential inclusions; multiple-valued ordinary differential equations; Carathéodory solutions Citations:Zbl 0538.34007 × Cite Format Result Cite Review PDF Full Text: DOI 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 · doi:10.1216/rmjm/1181072798 [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 · doi:10.1016/0047-259X(77)90037-9 [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. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.