A fixed point theorem and an application for the Cauchy problem in the scale of Banach spaces

Abstract

The main aim of this manuscript is to prove the existence of the xed
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. To
illustrate our obtained result, we shall prove the existence of global solution
of the Cauchy problem with perturbation of the form.

x0 (t) = f [t; x (t)] + g [t; x (t)] ; t 2 [0;1);
x (0) = x0 2 F1;
in a scale of Banach spaces f(Fs; k:ks) : s 2 (0; 1]g.