In this section, we give the existence of coupled coincidence point theorems in ordered cone metric spaces lacking the mixed gmonotone property. Our first main result is the following theorem.
Theorem 3.1 Let (X,d,\u2aaf) be an ordered cone metric space over a solid cone P and let g:X\to X and F:X\times X\to X. Suppose that the following hold:

(i)
F(X\times X)\subseteq g(X) and g(X) is a complete subspace of X;

(ii)
g and F satisfy property (2.3);

(iii)
there exist {x}_{0},{y}_{0}\in X such that g{x}_{0}\asymp F({x}_{0},{y}_{0}) and g{y}_{0}\asymp F({y}_{0},{x}_{0});

(iv)
there exists {a}_{i}\ge 0 for i=1,2,\dots ,6 and {\sum}_{i=1}^{6}{a}_{i}<1 such that for all x,y,u,v\in X satisfying gx\asymp gu and gy\asymp gv,
holds;

(v)
if {x}_{n}\to x when n\to \mathrm{\infty} in X, then {x}_{n}\asymp x for n sufficiently large.
Then there exist
x,y\in X
such that
F(x,y)=gx\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}F(y,x)=gy,
that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Proof Starting from {x}_{0}, {y}_{0} (condition (iii)) and using the fact that F(X\times X)\subseteq g(X) (condition (i)), we can construct sequences \{g{x}_{n}\} and \{g{y}_{n}\} in X such that
g{x}_{n}=F({x}_{n1},{y}_{n1})\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}g{y}_{n}=F({y}_{n1},{x}_{n1})
(3.2)
for all n\in \mathbb{N}. By (iii), we get g{x}_{0}\asymp F({x}_{0},{y}_{0})=g{x}_{1}, and the condition (ii) implies that
g{x}_{1}=F({x}_{0},{y}_{0})\asymp F({x}_{1},{y}_{1})=g{x}_{2}.
Proceeding by induction, we get that g{x}_{n1}\asymp g{x}_{n} and, similarly, g{y}_{n1}\asymp g{y}_{n} for all n\in \mathbb{N}. Therefore, we can apply the condition (3.1) to obtain
\begin{array}{rcl}d(g{x}_{n},g{x}_{n+1})& =& d(F({x}_{n1},{y}_{n1}),F({x}_{n},{y}_{n}))\\ {\le}_{P}& {a}_{1}d(g{x}_{n1},g{x}_{n})+{a}_{2}d(F({x}_{n1},{y}_{n1}),g{x}_{n1})\\ +{a}_{3}d(g{y}_{n1},g{y}_{n})+{a}_{4}d(F({x}_{n},{y}_{n}),g{x}_{n})\\ +{a}_{5}d(F({x}_{n1},{y}_{n1}),g{x}_{n})+{a}_{6}d(F({x}_{n},{y}_{n}),g{x}_{n1})\\ =& {a}_{1}d(g{x}_{n1},g{x}_{n})+{a}_{2}d(g{x}_{n},g{x}_{n1})+{a}_{3}d(g{y}_{n1},g{y}_{n})\\ +{a}_{4}d(g{x}_{n+1},g{x}_{n})+{a}_{5}d(g{x}_{n},g{x}_{n})+{a}_{6}d(g{x}_{n+1},g{x}_{n1})\\ {\le}_{P}& {a}_{1}d(g{x}_{n1},g{x}_{n})+{a}_{2}d(g{x}_{n},g{x}_{n1})+{a}_{3}d(g{y}_{n1},g{y}_{n})\\ +{a}_{4}d(g{x}_{n+1},g{x}_{n})+{a}_{6}[d(g{x}_{n1},g{x}_{n})+d(g{x}_{n},g{x}_{n+1})]\\ {\le}_{P}& ({a}_{1}+{a}_{2}+{a}_{6})d(g{x}_{n1},g{x}_{n})+{a}_{3}d(g{y}_{n1},g{y}_{n})\\ +({a}_{4}+{a}_{6})d(g{x}_{n},g{x}_{n+1}),\end{array}
which implies that
(1{a}_{4}{a}_{6})d(g{x}_{n},g{x}_{n+1}){\le}_{P}({a}_{1}+{a}_{2}+{a}_{6})d(g{x}_{n1},g{x}_{n})+{a}_{3}d(g{y}_{n1},g{y}_{n}).
(3.3)
Similarly, starting with d(g{y}_{n},g{y}_{n+1})=d(F({y}_{n},{x}_{n}),F({y}_{n1},{x}_{n1})) and using g{x}_{n1}\asymp g{x}_{n} and g{y}_{n1}\asymp g{y}_{n} for all n\in \mathbb{N}, we get
(1{a}_{4}{a}_{6})d(g{y}_{n},g{y}_{n+1}){\le}_{P}({a}_{1}+{a}_{2}+{a}_{6})d(g{y}_{n1},g{y}_{n})+{a}_{3}d(g{x}_{n1},g{x}_{n}).
(3.4)
Combining (3.3) and (3.4), we obtain that
Now, starting from d(g{x}_{n+1},g{x}_{n})=d(F({x}_{n},{y}_{n}),F({x}_{n1},{y}_{n1})) and using g{x}_{n1}\asymp g{x}_{n} and g{y}_{n1}\asymp g{y}_{n} for all n\in \mathbb{N}, we get that
(1{a}_{2}{a}_{5})d(g{x}_{n},g{x}_{n+1}){\le}_{P}({a}_{1}+{a}_{4}+{a}_{5})d(g{x}_{n1},g{x}_{n})+{a}_{3}d(g{y}_{n1},g{y}_{n}).
Similarly, starting from d(g{y}_{n+1},g{y}_{n})=d(F({y}_{n},{x}_{n}),F({y}_{n1},{x}_{n1})) and using g{x}_{n1}\asymp g{x}_{n} and g{y}_{n1}\asymp g{y}_{n} for all n\in \mathbb{N}, we get that
(1{a}_{2}{a}_{5})d(g{y}_{n},g{y}_{n+1}){\le}_{P}({a}_{1}+{a}_{4}+{a}_{5})d(g{y}_{n1},g{y}_{n})+{a}_{3}d(g{x}_{n1},g{x}_{n}).
Again adding up, we obtain that
Finally, adding up (3.5) and (3.6), it follows that
d(g{x}_{n},g{x}_{n+1})+d(g{y}_{n},g{y}_{n+1}){\le}_{P}\lambda [d(g{x}_{n1},g{x}_{n})+d(g{y}_{n1},g{y}_{n})]
(3.7)
with
0\le \lambda =\frac{2{a}_{1}+{a}_{2}+2{a}_{3}+{a}_{4}+{a}_{5}+{a}_{6}}{2{a}_{2}{a}_{4}{a}_{5}{a}_{6}}<1,
(3.8)
since {\sum}_{i=1}^{6}{a}_{i}<1.
From the relation (3.7), we have
\begin{array}{rcl}d(g{x}_{n},g{x}_{n+1})+d(g{y}_{n},g{y}_{n+1})& {\le}_{P}& \lambda [d(g{x}_{n1},g{x}_{n})+d(g{y}_{n1},g{y}_{n})]\\ {\le}_{P}& {\lambda}^{2}[d(g{x}_{n2},g{x}_{n1})+d(g{y}_{n2},g{y}_{n1})]\\ \vdots \\ {\le}_{P}& {\lambda}^{n}[d(g{x}_{0},g{x}_{1})+d(g{y}_{0},g{y}_{1})].\end{array}
If d(g{x}_{0},g{x}_{1})+d(g{y}_{0},g{y}_{1})={0}_{E}, then ({x}_{0},{y}_{0}) is a coupled coincidence point of F and g. So, let {0}_{E}{<}_{P}d(g{x}_{0},g{x}_{1})+d(g{y}_{0},g{y}_{1}).
For any m>n\ge 1, repeated use of the triangle inequality gives
\begin{array}{c}d(g{x}_{n},g{x}_{m})+d(g{y}_{n},g{y}_{m})\hfill \\ \phantom{\rule{1em}{0ex}}{\le}_{P}d(g{x}_{n},g{x}_{n+1})+d(g{x}_{n+1},g{x}_{n+2})+\cdots +d(g{x}_{m1},g{x}_{m})\hfill \\ \phantom{\rule{2em}{0ex}}+d(g{y}_{n},g{y}_{n+1})+d(g{y}_{n+1},g{y}_{n+2})+\cdots +d(g{y}_{m1},g{y}_{m})\hfill \\ \phantom{\rule{1em}{0ex}}{\le}_{P}[{\lambda}^{n}+{\lambda}^{n+1}+\cdots +{\lambda}^{m1}][d(g{x}_{0},g{x}_{1})+d(g{y}_{0},g{y}_{1})]\hfill \\ \phantom{\rule{1em}{0ex}}{\le}_{P}\frac{{\lambda}^{n}}{1\lambda}[d(g{x}_{0},g{x}_{1})+d(g{y}_{0},g{y}_{1})].\hfill \end{array}
Since \frac{{\lambda}^{n}}{1\lambda}\to 0 as n\to \mathrm{\infty}, we get \frac{{\lambda}^{n}}{1\lambda}[d(g{x}_{0},g{x}_{1})+d(g{y}_{0},g{y}_{1})]\to {0}_{E} as n\to \mathrm{\infty}.
From ({p}_{4}), we have for {0}_{E}\ll c and large n,
\frac{{\lambda}^{n}}{1\lambda}[d(g{x}_{0},g{x}_{1})+d(g{y}_{0},g{y}_{1})]\ll c.
By ({p}_{3}), we get
d(g{x}_{n},g{x}_{m})+d(g{y}_{n},g{y}_{m})\ll c.
Since
d(g{x}_{n},g{x}_{m}){\le}_{P}d(g{x}_{n},g{x}_{m})+d(g{y}_{n},g{y}_{m})
and
d(g{y}_{n},g{y}_{m}){\le}_{P}d(g{x}_{n},g{x}_{m})+d(g{y}_{n},g{y}_{m}),
then by ({p}_{3}), we get d(g{x}_{n},g{x}_{m})\ll c and d(g{y}_{n},g{y}_{m})\ll c for n large enough. Therefore, we get \{g{x}_{n}\} and \{g{y}_{n}\} are Cauchy sequences in g(X). By completeness of g(X), there exist gx,gy\in g(X) such that g{x}_{n}\to gx and g{y}_{n}\to gy as n\to \mathrm{\infty}.
By (v), we have g{x}_{n}\asymp gx and gy\asymp g{y}_{n} for all n\ge 0. Now, we prove that F(x,y)=gx and F(y,x)=gy.
If g{x}_{n}=gx and g{y}_{n}=gy for some n\ge 0, from (3.1) we have
\begin{array}{rcl}d(F(x,y),gx)& {\le}_{P}& d(F(x,y),g{x}_{n+1})+d(g{x}_{n+1},gx)\\ =& d(F(x,y),F({x}_{n},{y}_{n}))+d(g{x}_{n+1},gx)\\ {\le}_{P}& {a}_{1}d(gx,g{x}_{n})+{a}_{2}d(F(x,y),gx)+{a}_{3}d(gy,g{y}_{n})\\ +{a}_{4}d(F({x}_{n},{y}_{n}),g{x}_{n})+{a}_{5}d(F(x,y),g{x}_{n})\\ +{a}_{6}d(F({x}_{n},{y}_{n}),gx)+d(g{x}_{n+1},gx)\\ {\le}_{P}& {a}_{1}d(gx,g{x}_{n})+{a}_{2}d(F(x,y),gx)+{a}_{3}d(gy,g{y}_{n})\\ +{a}_{4}d(g{x}_{n+1},gx)+{a}_{4}d(gx,g{x}_{n})+{a}_{5}d(F(x,y),gx)+{a}_{5}d(gx,g{x}_{n})\\ +{a}_{6}d(g{x}_{n+1},gx)+d(g{x}_{n+1},gx)\\ =& {a}_{2}d(F(x,y),gx)+{a}_{4}d(g{x}_{n+1},gx)+{a}_{5}d(F(x,y),gx)\\ +{a}_{6}d(g{x}_{n+1},gx)+d(g{x}_{n+1},gx),\end{array}
which further implies that
d(F(x,y),gx){\le}_{P}\frac{1+{a}_{4}+{a}_{6}}{1{a}_{2}{a}_{5}}d(g{x}_{n+1},gx).
Since g{x}_{n}\to gx, then for {0}_{E}\ll c, there exists N\in \mathbb{N} such that
d(g{x}_{n+1},gx)\ll \frac{(1{a}_{2}{a}_{5})c}{1+{a}_{4}+{a}_{6}}
for all n\ge N. Therefore,
Now, according to ({p}_{2}), it follows that d(F(x,y),gx)={0}_{E} and F(x,y)=gx. Similarly, we can prove that F(y,x)=gy. Hence, (x,y) is a coupled coincidence point of the mappings F and g.
So, we suppose that (g{x}_{n},g{y}_{n})\ne (gx,gy) for all n\ge 0. Using (3.1), we get
\begin{array}{rcl}d(F(x,y),gx)& {\le}_{P}& d(F(x,y),g{x}_{n+1})+d(g{x}_{n+1},gx)\\ =& d(F(x,y),F({x}_{n},{y}_{n}))+d(g{x}_{n+1},gx)\\ {\le}_{P}& {a}_{1}d(gx,g{x}_{n})+{a}_{2}d(F(x,y),gx)+{a}_{3}d(gy,g{y}_{n})\\ +{a}_{4}d(F({x}_{n},{y}_{n}),g{x}_{n})+{a}_{5}d(F(x,y),g{x}_{n})\\ +{a}_{6}d(F({x}_{n},{y}_{n}),gx)+d(g{x}_{n+1},gx)\\ {\le}_{P}& {a}_{1}d(gx,g{x}_{n})+{a}_{2}d(F(x,y),gx)+{a}_{3}d(gy,g{y}_{n})\\ +{a}_{4}d(g{x}_{n+1},gx)+{a}_{4}d(gx,g{x}_{n})+{a}_{5}d(F(x,y),gx)+{a}_{5}d(gx,g{x}_{n})\\ +{a}_{6}d(g{x}_{n+1},gx)+d(g{x}_{n+1},gx),\end{array}
which further implies that
\begin{array}{c}d(F(x,y),gx)\hfill \\ \phantom{\rule{1em}{0ex}}{\le}_{P}\frac{{a}_{1}+{a}_{4}+{a}_{5}}{1{a}_{2}{a}_{5}}d(gx,g{x}_{n})+\frac{1+{a}_{4}+{a}_{6}}{1{a}_{2}{a}_{5}}d(g{x}_{n+1},gx)+\frac{{a}_{3}}{1{a}_{2}{a}_{5}}d(gy,g{y}_{n}).\hfill \end{array}
Since g{x}_{n}\to gx and g{y}_{n}\to gy, then for {0}_{E}\ll c, there exists N\in \mathbb{N} such that d(g{x}_{n},gx)\ll \frac{(1{a}_{2}{a}_{5})c}{3({a}_{1}+{a}_{4}+{a}_{5})}, d(g{x}_{n+1},gx)\ll \frac{(1{a}_{2}{a}_{5})c}{3(1+{a}_{4}+{a}_{6})}, and d(g{y}_{n},gy)\ll \frac{(1{a}_{2}{a}_{5})c}{3{a}_{3}} for all n\ge N. Thus,
d(F(x,y),gx)\ll \frac{c}{3}+\frac{c}{3}+\frac{c}{3}=c.
Now, according to ({p}_{2}), it follows that d(F(x,y),gx)={0}_{E} and F(x,y)=gx. Similarly, F(y,x)=gy. Hence, (x,y) is a coupled coincidence point of the mappings F and g. □
Remark 3.2 In Theorem 3.1, the condition (ii) is a substitution for the mixed gmonotone property that has been used in most of the coupled coincidence point theorems so far. Therefore, Theorem 3.1 improves the results of Nashine et al. [33]. Moreover, it is an ordered version extension of the results of Abbas et al. [36].
Corollary 3.3 Let (X,d,\u2aaf) be an ordered cone metric space over a solid cone P and let g:X\to X and F:X\times X\to X. Suppose that the following hold:

(i)
F(X\times X)\subseteq g(X) and g(X) is a complete subspace of X;

(ii)
g and F satisfy property (2.3);

(iii)
there exist {x}_{0},{y}_{0}\in X such that g{x}_{0}\asymp F({x}_{0},{y}_{0}) and g{y}_{0}\asymp F({y}_{0},{x}_{0});

(iv)
there exist \alpha ,\beta ,\gamma \ge 0 and \alpha +\beta +\gamma <1 such that for all x,y,u,v\in X satisfying gx\asymp gu and gy\asymp gv,
d(F(x,y),F(u,v)){\le}_{P}\alpha d(gx,gu)+\beta d(gy,gv)+\gamma d(F(x,y),gu)
(3.9)
holds;

(v)
if {x}_{n}\to x when n\to \mathrm{\infty} in X, then {x}_{n}\asymp x for n sufficiently large.
Then there exist
x,y\in X
such that
F(x,y)=gx\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}F(y,x)=gy,
that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Putting g={I}_{X}, where {I}_{X} is the identity mapping from X into X in Theorem 3.1, we get the following corollary.
Corollary 3.4 Let (X,d,\u2aaf) be an ordered cone metric space over a solid cone P and let F:X\times X\to X. Suppose that the following hold:

(i)
X is complete;

(ii)
g and F satisfy property (2.4);

(iii)
there exist {x}_{0},{y}_{0}\in X such that {x}_{0}\asymp F({x}_{0},{y}_{0}) and {y}_{0}\asymp F({y}_{0},{x}_{0});

(iv)
there exists {a}_{i}\ge 0 for i=1,2,\dots ,6 and {\sum}_{i=1}^{6}{a}_{i}<1 such that for all x,y,u,v\in X satisfying x\asymp u and y\asymp v,
holds;

(v)
if {x}_{n}\to x when n\to \mathrm{\infty} in X, then {x}_{n}\asymp x for n sufficiently large.
Then there exist
x,y\in X
such that
F(x,y)=x\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}F(y,x)=y,
that is, F has a coupled fixed point (x,y)\in X\times X.
Our second main result is the following.
Theorem 3.5 Let (X,d,\u2aaf) be an ordered cone metric space over a solid cone P. Let F:X\times X\to X and g:X\to X be mappings. Suppose that the following hold:

(i)
F(X\times X)\subseteq g(X) and g(X) is a complete subspace of X;

(ii)
g and F satisfy property (2.3);

(iii)
there exist {x}_{0},{y}_{0}\in X such that g{x}_{0}\asymp F({x}_{0},{y}_{0}) and g{y}_{0}\asymp F({y}_{0},{x}_{0});

(iv)
there is some h\in [0,1/2) such that for all x,y,u,v\in X satisfying gx\asymp gu and gy\asymp gv, there exists
{\mathrm{\Theta}}_{x,y,u,v}\in \{d(gx,gu),d(gy,gv),d(F(x,y),gu)\}
such that
d(F(x,y),F(u,v)){\le}_{P}h{\mathrm{\Theta}}_{x,y,u,v};

(v)
if {x}_{n}\to x when n\to \mathrm{\infty} in X, then {x}_{n}\asymp x for n sufficiently large.
Then there exist
x,y\in X
such that
F(x,y)=gx\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}F(y,x)=gy,
that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Proof Since F(X\times X)\subseteq g(X) (condition (i)), we can start from {x}_{0}, {y}_{0} (condition (iii)) and construct sequences \{g{x}_{n}\} and \{g{y}_{n}\} in X such that
g{x}_{n}=F({x}_{n1},{y}_{n1})\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}g{y}_{n}=F({y}_{n1},{x}_{n1})
(3.11)
for all n\in \mathbb{N}. From (iii), we get g{x}_{0}\asymp F({x}_{0},{y}_{0})=g{x}_{1} and the condition (ii) implies that
g{x}_{1}=F({x}_{0},{y}_{0})\asymp F({x}_{1},{y}_{1})=g{x}_{2}.
By repeating this process, we have g{x}_{n1}\asymp g{x}_{n}. Similarly, we can prove that g{y}_{n1}\asymp g{y}_{n} for all n\in \mathbb{N}.
Since g{x}_{n1}\asymp g{x}_{n} and g{y}_{n1}\asymp g{y}_{n} for all n\in \mathbb{N}, from (iv), we have that there exist h\in [0,1/2) and
\begin{array}{rcl}{\mathrm{\Theta}}_{1}& \in & \{d(g{x}_{n1},g{x}_{n}),d(g{y}_{n1},g{y}_{n}),d(F({x}_{n1},{y}_{n1}),g{x}_{n})\}\\ =& \{d(g{x}_{n1},g{x}_{n}),d(g{y}_{n1},g{y}_{n}),{0}_{E}\}\end{array}
such that
d(g{x}_{n},g{x}_{n+1})=d(F({x}_{n1},{y}_{n1}),F({x}_{n},{y}_{n})){\le}_{P}h{\mathrm{\Theta}}_{1}.
Similarly, one can show that there exists
{\mathrm{\Theta}}_{2}\in \{d(g{x}_{n1},g{x}_{n}),d(g{y}_{n1},g{y}_{n}),{0}_{E})\}
such that
d(g{y}_{n},g{y}_{n+1})=d(F({y}_{n1},{x}_{n1}),F({y}_{n},{x}_{n})){\le}_{P}h{\mathrm{\Theta}}_{2}.
Now, denote {\delta}_{n}=d(g{x}_{n},g{x}_{n+1})+d(g{y}_{n},g{y}_{n+1}). Since the cases {\mathrm{\Theta}}_{1}={0}_{E} and {\mathrm{\Theta}}_{2}={0}_{E} are trivial, we have to consider the following four possibilities.
Case 1. d(g{x}_{n},g{x}_{n+1}){\le}_{P}hd(g{x}_{n1},g{x}_{n}) and d(g{y}_{n},g{y}_{n+1}){\le}_{P}hd(g{y}_{n1},g{y}_{n}). Adding up, we get that
{\delta}_{n}{\le}_{P}h{\delta}_{n1}{\le}_{P}2h{\delta}_{n1}.
Case 2. d(g{x}_{n},g{x}_{n+1}){\le}_{P}hd(g{x}_{n1},g{x}_{n}) and d(g{y}_{n},g{y}_{n+1}){\le}_{P}hd(g{x}_{n1},g{x}_{n}). Then
{\delta}_{n}{\le}_{P}2hd(g{x}_{n1},g{x}_{n}){\le}_{P}2hd(g{x}_{n1},g{x}_{n})+2hd(g{y}_{n1},g{y}_{n})=2h{\delta}_{n1}.
Case 3. d(g{x}_{n},g{x}_{n+1}){\le}_{P}hd(g{y}_{n1},g{y}_{n}) and d(g{y}_{n},g{y}_{n+1}){\le}_{P}hd(g{x}_{n1},g{x}_{n}). This case is treated analogously to Case 1.
Case 4. d(g{x}_{n},g{x}_{n+1}){\le}_{P}hd(g{y}_{n1},g{y}_{n}) and d(g{y}_{n},g{y}_{n+1}){\le}_{P}hd(g{y}_{n1},g{y}_{n}). This case is treated analogously to Case 2.
Thus, in all cases, we get {\delta}_{n}{\le}_{P}2h{\delta}_{n1} for all n\in \mathbb{N}, where 0\le 2h<1. Therefore,
{\delta}_{n}{\le}_{P}2h{\delta}_{n1}{\le}_{P}{(2h)}^{2}{\delta}_{n2}{\le}_{P}\cdots {\le}_{P}{(2h)}^{n}{\delta}_{0},
and by the same argument as in Theorem 3.1, it is proved that \{g{x}_{n}\} and \{g{y}_{n}\} are Cauchy sequences in g(X). By the completeness of g(X), there exist gx,gy\in g(X) such that g{x}_{n}\to gx and g{y}_{n}\to gy.
From (v), we get g{x}_{n}\asymp gx and gy\asymp g{y}_{n} for all n\ge 0. Now, we prove that F(x,y)=gx and F(y,x)=gy.
If g{x}_{n}=gx and g{y}_{n}=gy for some n\ge 0, from (iv) we have
\begin{array}{rcl}d(F(x,y),gx)& {\le}_{P}& d(F(x,y),g{x}_{n+1})+d(g{x}_{n+1},gx)\\ =& d(F(x,y),F({x}_{n},{y}_{n}))+d(g{x}_{n+1},gx)\\ {\le}_{P}& h{\mathrm{\Theta}}_{x,y,{x}_{n},{y}_{n}}+d(g{x}_{n+1},gx),\end{array}
where {\mathrm{\Theta}}_{x,y,{x}_{n},{y}_{n}}\in \{d(gx,g{x}_{n}),d(gy,g{y}_{n}),d(F(x,y),g{x}_{n})\}. Let c\in int(P) be fixed. If {\mathrm{\Theta}}_{x,y,{x}_{n},{y}_{n}}=d(gx,g{x}_{n})={0}_{E} or {\mathrm{\Theta}}_{x,y,{x}_{n},{y}_{n}}=d(gy,g{y}_{n})={0}_{E}, then for n sufficiently large, we have that
By property ({p}_{2}), it follows that F(x,y)=gx. If {\mathrm{\Theta}}_{x,y,{x}_{n},{y}_{n}}=d(F(x,y),g{x}_{n}), then we get that
\begin{array}{rcl}d(F(x,y),gx)& {\le}_{P}& hd(F(x,y),g{x}_{n})+d(g{x}_{n+1},gx)\\ {\le}_{P}& hd(F(x,y),gx)+hd(gx,g{x}_{n})+d(g{x}_{n+1},gx)\\ =& hd(F(x,y),gx)+d(g{x}_{n+1},gx).\end{array}
Now, it follows that for n sufficiently large,
\begin{array}{rcl}d(F(x,y),gx)& {\le}_{P}& \frac{1}{1h}d(g{x}_{n+1},gx)\\ {\le}_{P}& \frac{1}{1h}(1h)c\\ =& c.\end{array}
Therefore, again by property ({p}_{2}), we get that F(x,y)=gx. Similarly, we can prove that F(y,x)=gy. Hence, (x,y) is a coupled point of coincidence of F and g.
Then, we suppose that (g{x}_{n},g{y}_{n})\ne (gx,gy) for all n\ge 0. For this, consider
\begin{array}{rcl}d(F(x,y),gx)& {\le}_{P}& d(F(x,y),g{x}_{n+1})+d(g{x}_{n+1},gx)\\ =& d(F(x,y),F({x}_{n},{y}_{n}))+d(g{x}_{n+1},gx)\\ {\le}_{P}& h{\mathrm{\Theta}}_{x,y,{x}_{n},{y}_{n}}+d(g{x}_{n+1},gx),\end{array}
where {\mathrm{\Theta}}_{x,y,{x}_{n},{y}_{n}}\in \{d(gx,g{x}_{n}),d(gy,g{y}_{n}),d(F(x,y),g{x}_{n})\}. Let c\in int(P) be fixed. If {\mathrm{\Theta}}_{x,y,{x}_{n},{y}_{n}}=d(gx,g{x}_{n}) or {\mathrm{\Theta}}_{x,y,{x}_{n},{y}_{n}}=d(gy,g{y}_{n}), then for n sufficiently large, we have that
d(F(x,y),gx)\ll h\cdot \frac{c}{2h}+\frac{c}{2}=c.
By property ({p}_{2}), it follows that F(x,y)=gx. If {\mathrm{\Theta}}_{x,y,{x}_{n},{y}_{n}}=d(F(x,y),g{x}_{n}), then we get that
\begin{array}{rcl}d(F(x,y),gx)& {\le}_{P}& hd(F(x,y),g{x}_{n})+d(g{x}_{n+1},gx)\\ {\le}_{P}& hd(F(x,y),gx)+hd(gx,g{x}_{n})+d(g{x}_{n+1},gx).\end{array}
Now, it follows that for n sufficiently large,
\begin{array}{rcl}d(F(x,y),gx)& {\le}_{P}& \frac{h}{1h}d(gx,g{x}_{n})+\frac{1}{1h}d(g{x}_{n+1},gx)\\ \ll & \frac{h}{1h}\cdot \frac{1h}{h}\cdot \frac{c}{2}+\frac{1}{1h}(1h)\frac{c}{2}=c.\end{array}
Thus, again by property ({p}_{2}), we get that F(x,y)=gx.
Similarly, F(y,x)=gy is obtained. Hence, (x,y) is a coupled point of coincidence of the mappings F and g. □
Remark 3.6 It would be interesting to relate our Theorem 3.5 with Theorem 2.1 of Long et al. [39].
Putting g={I}_{X}, where {I}_{X} is the identity mapping from X into X in Theorem 3.5, we get the following corollary.
Corollary 3.7 Let (X,d,\u2aaf) be an ordered cone metric space over a solid cone P. Let F:X\times X\to X be mappings. Suppose that the following hold:

(i)
X is complete;

(ii)
F satisfies property (2.4);

(iii)
there exist {x}_{0},{y}_{0}\in X such that {x}_{0}\asymp F({x}_{0},{y}_{0}) and {y}_{0}\asymp F({y}_{0},{x}_{0});

(iv)
there is some h\in [0,1/2) such that for all x,y,u,v\in X satisfying x\asymp u and y\asymp v, there exists
{\mathrm{\Theta}}_{x,y,u,v}\in \{d(x,u),d(y,v),d(F(x,y),u)\}
such that
d(F(x,y),F(u,v)){\le}_{P}h{\mathrm{\Theta}}_{x,y,u,v}.

(v)
if {x}_{n}\to x when n\to \mathrm{\infty} in X, then {x}_{n}\asymp x for n sufficiently large.
Then there exist
x,y\in X
such that
F(x,y)=x\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}F(y,x)=y,
that is, F has a coupled fixed point (x,y)\in X\times X.