Coupled coincidence point theorems for nonlinear contractions under c-distance in cone metric‎ ‎spaces

Abstract

In this paper, among others, we prove the following results: (1) Let (X;d) be a complete cone metric space partially ordered byv and q be a c-distance on X. Suppose F : X X ! X and g : X ! X be two continuous and commuting functions with F (X X) g(X). Let F satisfy mixed g-monotone property and q(F (x;y);F (u;v)) k (q(gx;gu) +q(gy;gv)) for some k 2 (0; 1) and all x;y;u;v 2 X with (gx v gu) and (gy w gv) or (gxw gu) and (gyv gv). If there exist x0;y02 X satisfying gx0v F (x0;y0) andF (y0;x0)v gy0, then there existx ;y 2 X such thatF (x ;y ) = gx and F (y ;x ) = gy , that is, F and g have a coupled coincidence point (x ;y ). (2) If, in (1), we replace completeness of (X;d) by completeness of (g(X);d) and commutativity, continuity of mappings F and g by the condition: (i) for any nondecreasing sequencefxng in X converging to x we have xnv x for all n. (ii) for any nonincreasing sequencefyng in Y converging to y we have yv yn for all n, then F and g have a coupled coincidence point (x ;y ).