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Existence of solutions of the dynamic Cauchy problem in Banach spaces. (English) Zbl 1272.34122
Summary: We obtain the existence of solutions and Caratheodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales $$x^\Delta(t)=f(t,x(t)),\qquad x(0)=x_0,\quad t\in I_\alpha,$$ 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.

34N05Dynamic equations on time scales or measure chains
34G20Nonlinear ODE in abstract spaces
47N20Applications of operator theory to differential and integral equations
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