Coupled Coincidence Point Theorems for Nonlinear Contractions under C-distance in Cone Metric Spaces

In this paper, among others, we prove the following results: (1) Let (X, d) be a complete cone metric space partially ordered by v 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)) k2 (q(gx, gu) + q(gy, gv)) for some k ∈ [0, 1) and all x, y, u, v ∈ X with (gx v gu) and (gy w gv) or (gx w gu) and (gy v gv). If there exist x0, y0 ∈ X satisfying gx0 v F (x0, y0) and F (y0, x0) v gy0, then there exist x∗, y∗ ∈ X such that F (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 sequence {xn} in X converging to x we have xn v x for all n. (ii) for any nonincreasing sequence {yn} in Y converging to y we have y v yn for all n, then F and g have a coupled coincidence point (x∗, y∗).