Existence Of Solutions Of The Dynamic Cauchy Problem In Banach Spaces

In this paper we obtain the existence of solutions and Caratheodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scalesx(triangle)(t) = f(t,x(t)), x(0)=x(o), t is an element of I-a,where f is continuous or f satisfies Caratheodory conditions and some conditions expressed in terms of measures of noncompactness. The Manch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.