In this paper we obtain the existence of solutions and Carathéodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales (…), where f is continuous or f satisfies Carathéodory conditions and some conditions expressed in terms of measures of noncompactness. The Mönch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.
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In this paper we prove the existence of solutions and Carathéodory's type solutions of the dynamic Cauchy problem (…), where (…) denotes a mth order (…) - derivative,T denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence (…) E -a Banach space and f is a continuous function or satis .es Carathéodory's conditions and some conditions expressed in terms of measures of noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are an unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.
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