Abstract
Theorems on the local extendability of selections for non-convex-valued maps of paracompact spaces into Banach spaces, i.e., infinite-dimensional analogs of the finite-dimensional Michael selection theorem are proved. We were able to obtain these results under an appropriate metric control of the local degree of nonconvexity on the valuesF(x), which naturally leads us to introduce the notion of equi-locally paraconvex families of sets. It is shown that all convex subsets of the integral curves of the differential equationy′=f(x,y) with a continuous right-hand sidef and the isometric images of such subsets form an equi-locally paraconvex family of subsets of a Euclidean space.
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Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 261–269, February, 1999.
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Semenov, P.V. Local paraconvexity and local selection theorems. Math Notes 65, 214–220 (1999). https://doi.org/10.1007/BF02679819
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DOI: https://doi.org/10.1007/BF02679819