Urysohn integral equations approach by common fixed points in complex-valued metric spaces

Abstract

Recently, the complex-valued metric spaces which are more general than the metric spaces were first introduced by Azam et al. (Numer. Funct. Anal. Optim. 32:243-253, 2011). They also established the existence of fixed point theorems under the contraction condition in these spaces. The aim of this paper is to introduce the concepts of a C-Cauchy sequence and C-complete in complex-valued metric spaces and establish the existence of common fixed point theorems in C-complete complex-valued metric spaces. Furthermore, we apply our result to obtain the existence theorem for a common solution of the Urysohn integral equations x(t) = ∫ b a K1(t, s, x(s)) ds + g(t), x(t) = ∫ b a K2(t, s, x(s)) ds + h(t), where t ε [a, b] ⊆ ℝ, x, g, h ε C([a, b],ℝn) and K1, K2 : [a, b]×[a, b]×ℝn→ℝn. © 2013 Sintunavarat et al.