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a.e. on G admissible boundary value admissible couple Arzela-Ascoli theorem associate the following ball Bq Banach space boundary value problem bounded set BQ n Ker Caratheodory conditions center in zero closed subspace completely continuous mapping conditions 4.1 convex set convex-valued mapping given convex-valued mapping satisfying couple of problem d(Ny Da(u Darboux problem Definition Let denote Endom exists a ball exists at least fact is obvious following boundary value following condition following existence theorem following fact formula 1.5 given by formula K^-admissible couple least one solution Let f linear isomorphism linear mapping mapping and let mapping F Multi-valued boundary value multi-valued mapping open set ordinary differential equations partial differential equations point x e 3V problem 1.1 problem IV proof is completed Proposition real number right inverse satisfying conditions 5.1 satisfying the Caratheodory single-valued mapping solution of 1.1 solution of problem supess f u e G