The fixed point theorem by Arino-Gautier and Penot is used and we depend on
converting of the mentioned equations to the form of functional integral equations to
establish existence of pseudo-solutions to a Cauchy problem of differential equation of
arbitrary orders in Banach spaces.
The topic of fractional calculus ( derivatives and integrals of arbitrary orders) is enjoying
growing interest not only in Mathematics, but also in Physics, Engineering and Mathematical Biology.
The very rst approach via weak topology follows by Szep. Then more ideas are taken
from from papers by Kubiaczyk, Szufla or Kubiaczyk .
the existence of weak solutions for the initial value problem of the arbitrary
(fractional) orders differential equation in the reflexive Banach space E have been considered, for the first time, by Salem and El-Sayed. Let E be a Banach space with norm k : k and dual . Moreover, let Ew denote
the space E with its weak topology. By C = C[I;E] the Banach space of strongly
continuous functions x with jjxjjC = sup jjx(t)jjE; t in I