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Dimension of the solution set for differential inclusions. (English) Zbl 0783.34008
Our paper is naturally divided into two short sections. In the first part we make some observations on results obtained by J. Saint-Raymond in [C. R. Acad. Sci., Paris, Sér. I 298, 71-74 (1984; Zbl 0561.54042); Fixed point theory and applications, Proc. Int. Conf., Marseille- Luminy/Fr. 1989, Pitman Res. Notes Math. Ser. 252, 359-375 (1991; Zbl 0757.47029)] concerning topological dimension of a fixed point set of convex-valued contraction. Slightly modifying the proofs, we generalize his theorems to the case of arbitrary closed convex subsets of a Banach space.
In the second section we apply the previous result to the solution set of a Cauchy problem with set-valued right-hand side. We prove that it has an infinite dimension if only values of the right-hand side function are at least one-dimensional.

34A60 Ordinary differential inclusions
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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