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In this paper we prove the existence of solutions and Carathéodory’s type solutions of the dynamic Cauchy problemx    (  Δ  m  )    (  t  )=  f  (  t  ,  x  (  t  )  )  ,                 t  ∈  T  ,x  (  0  )=  0  ,x   Δ   (  0  )=    η   1   ,  …  ,    x    (  Δ  (  m  -  1  )  )    (  0  )  =    η    m  -  1    ,                η   1   ,  …    η    m  -  1    ∈  E  ,$$\matrix{\hfill {x^{(\Delta m)} (t)} &amp; { = f(t,x(t)),\;\;\;\;\;t \in T,} \hfill \cr \hfill {x(0)} &amp; { = 0,} \hfill \cr \hfill {x^\Delta (0)} &amp; { = \eta _1 , \ldots ,x^{(\Delta (m - 1))} (0) = \eta _{m - 1} ,\;\;\;\;\eta _1 , \ldots \eta _{m - 1} \in E,} \hfill \cr } $$    where  x  (Δ   m   )  denotes a  m th order Δ - derivative,  T  denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence ( a n  ) in  T  and  a n   → ∞),  E  — a Banach space and  f  is a continuous function or satisfies Carathéodory’s conditions and some conditions expressed in terms of measures of noncompactness.

Dynamic equations x (Δm) ( t ) = f ( t, x ( t )) on time scales