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 by‎ ‎$\sqsubseteq$ and $q$ be a c-distance on $X$‎. ‎Suppose $F‎ : ‎X \times‎ ‎X \to X$ and $g‎ : ‎X \to X$ be two continuous and commuting functions‎ ‎with $F(X \times X)\subseteq g(X)$.\ Let $F$ satisfy mixed‎ ‎g-monotone property and $q(F(x‎, ‎y)‎, ‎F(u‎, ‎v)) \preceq \frac{k}{2}‎ ‎(q(gx‎, ‎gu)+q(gy,gv))$ for some $k \in [0‎, ‎1)$ and all $x‎, ‎y‎, ‎u‎, ‎v‎ ‎\in X$ with $(gx \sqsubseteq gu)$ and $(gy \sqsupseteq gv)$ or $(gx‎ ‎\sqsupseteq gu)$ and $(gy \sqsubseteq gv)$.\ If there exist $x_0‎, ‎y_0 \in X$ satisfying $gx_0 \sqsubseteq F(x_0‎, ‎y_0)$ and $F(y_0‎, ‎x_0) \sqsubseteq gy_0$‎, ‎then there exist $x^*‎, ‎y^*\in X$ 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 $\{x_n\}$ in $X$ converging to‎ ‎$x$ we have $x_n \sqsubseteq x$ for all $n$.\ $(ii)$ for any‎ ‎nonincreasing sequence $\{y_n\}$ in $Y$ converging to $y$ we have $y‎ ‎\sqsubseteq y_n$ for all $n$‎, ‎then $F$ and $g$ have a coupled‎ ‎coincidence point $(x^*,y^*)$‎.