A Fixed Point Theorem and an Application for the Cauchy Problem in the Scale of Banach Spaces

Abstract

The main aim of this paper is to prove the existence of the fixed point of the sum of two operators in setting of the cone-normed spaces with the values of cone-norm belonging to an ordered locally convex space. We apply this result to prove the existence of global solution of the Cauchy problem with perturbation of the form

{x'(t) = f[t, x(t)] + g[t, x(t)], t is an element of[0,infinity), x(0) = x(0)is an element of F-1,

in a scale of Banach spaces f(F-s; parallel to center dot parallel to(s)) : s is an element of(0, 1]}.