Fixed point theorems and the Ulam–Hyers stability in non-Archimedean cone metric spaces

Let ( X , p ) be a metric space with a K-valued non-Archimedean metric p. In this paper, we prove the existence and approximation of a fixed point for operators F : X → X satisfying the contractive condition in the form p ( F ( x ) , F ( y ) ) ⩽ Q [ p ( x , y ) ] , where Q : K → K is an increasing operator. Then, we study the generalized Ulam–Hyers stability of fixed point equations. We next obtain an extension of the Krasnoselskii fixed point theorem for the sum of two operators. Finally, an application to functional equations is given.