1. Introduction
In 2007, Huang and Zhang ^{[4]} have introduced the concept of cone metric space it is generalization of the metric space. And proved fixed point theorems of contractive type mappings over cone metric spaces. After that, many authors generalized their fixed point theorems to different types of contraction mappings in cone metric spaces (see, ^{[1, 2, 3, 8]}). In 2012, Wasfi Shatanawi ^{[9]} proved some coincidence point results in cone metric spaces. In this paper, we prove a coincidence point theorem of two maps in ordered cone metric spaces. Our result extends and improves the results of ^{[9]}.
In this paper B is a real Banach space, θ denotes zero element of B.
Definition 1.1 (^{[9]}). Let B be a real Banach Space and P be a subset of B with int(P)≠∅ where P denotes the interior point of P. Then P is called a cone if the following conditions are satisfied:
(a). P is closed, non –empty and P≠{θ};
(b). a,b, x,y∈P implies ax+by∈P;
(c). xP∩ (P) implies x =θ.
Definition 1.2 (^{[4]}). Let P be a cone in a Banach Space B, define partial ordering ‘’ with respect to P by xy if and only if yx ∈P. We shall write xy to indicate xy but x≠y while x<<y will stand for yx∈Int P, where Int P denotes the interior of the set P. This Cone P is called an order cone. It can be easily shown that λ int(P) ⊆int (P) for all λ^{. }
Definition 1.3 (^{[4]}). Let B be a Banach Space and P⊂B be an order cone. The order cone P is called normal if there exists K>0 such that for all x,y∈B,
θxy implies ║x║≤K ║y║.
The least positive number K satisfying the above inequality is called the normal constant of P.
Definition 1.4 (^{[4]}). Let X be a nonempty set of B. Suppose that the map
d: X×X→B satisfies :
(d1).θd(x,y) for all x,y ∈X and
d(x,y) = θ if and only if x = y;
(d2).d(x,y) = d(y,x) for all x,y∈X;
(d3).d(x,y)d(x,z) +d(y,z) for all x,y,z∈X.
Then d is called a cone metric on X and (X, d) is called a cone metric space.
Definition 1.5 (^{[4]}). Let (X,d) be a cone metric space. We say that {x_{n}} is
(i). a Cauchy sequence if for every c in B with
c >>θ, there is N such that for all n, m>N,
d(x_{n,}x_{m}) <<c;
(ii). a convergent sequence if for any
c >>θ, there is an N such that for all n>N,
d(x_{n,} x) <<c, for some fixed x in X.
We denote this x_{n}→x
(as n→∞).
The space (X, d) is called a complete cone metric space if every Cauchy sequence is convergent.
The concept of weakly decreasing maps type A introduce by W. Shatanawi ^{[9]}.
Definition 1.6 (^{[9]}). Let (X, ⊆) be partially ordered set and let f, T: X→X be two maps. We say that f is weakly decreasing type A with respect to T if the following conditions hold:
(i). For all x∈X, we have that fx⊆fy for all y∈T^{1}(fx).
(ii). TX ⊆fX.
Definition 1.7 (^{[5]}). Let (X, d) be a cone metric space and f, g: X→X be two selfmaps. The pair {f, g} is said to be compatible if, for an arbitrary sequence {x_{n}}⊂X such that , and for arbitrary c∈int (P), there exists n_{0} such that d(fgx_{n}, gfx_{n})<< c whenever n > n_{0}. It is said to be weakly compatible if
fx = gx implies fgx = gfx.
Definition 1.8 (^{[1]}). For the mappings f, g: X→X.
If w=fz = gz for some z in X, then z is called a coincidence point of f and g and w is called a point of coincidence of f and g..
2. The Main Results
In this section, we prove a coincidence point theorem for two mappings in ordered cone metric spaces without using normal cone. Our result extends and improves the results of ^{[9]}.
The following Theorem is extends and improves Theorem 2.2 of ^{[9]}.
Theorem 2.1. Let (X, ⊆) be partially ordered set and (X, d) be a complete cone metric space over a solid cone P. Let f, T: X→X be two maps such that
 (1) 
for all x, yX for which fx and fy are comparable. Assume that f and T satisfy the following conditions:
(i). f is weakly decreasing type A with respect to T.
(ii). The pair {f, T} is compatible.
(iii). f and T are continuous.
If a_{i} (i= 1,2,3,4) are nonnegative real numbers with a_{1 }+ a_{2 }+ a_{3 }+ 2a_{4 }∈ [0,1), then f and T have a coincidence point in X, that is there exists a point uX such that fu = Tu.
Proof: Let x_{0 }X. Since TX⊆ fX, we can choose x_{1 }∈X such that Tx_{0 }= fx_{1}. Also since TX⊆ fX, we can choose
x_{2 }∈X such that Tx_{1 }= f x_{2}_{. }Continuing this process, we can construct a sequence {x_{n}} in X such that Tx_{n}= f x_{n+1}. Since x_{n}∈T^{1 }(f x_{n+1}), n∈N, then by using the assumption that f is weakly decreasing of type A with respect to T,
we have
By the condition (1) we have,
Putting, .
We obtain,
 (2) 
Thus, for n∈N, we have
Let n, m∈N with m > n. Then
Since, k∈ [0,1), we have
 (3) 
We shall show that {Tx_{n}} is a Cauchy sequence in (X, d). For this, let c >> θ be given.
Since, cint(P), then there exists a neighborhood of θ, N_{δ}(θ) = {yE:║y║< δ},δ>0, such that
Choose a natural number N_{1 }such that
Then for all n≥ N_{1} we have that
Hence, cd(Tx_{0}, Tx_{1}) c + N_{δ}(θ) ⊆int(P).
Thus, we have that for all n≥ N_{1}_{, }
 (4) 
By (3) and (4), it follows that
d(Tx_{n}, Tx_{m})<< c whenever n≥ N_{1}_{. }
Hence, {Tx_{n}} is a Cauchy sequence in X.
By the completeness of X, there is a
u∈X such that Tx_{n}→ u as n→+∞.
Since f and T are continuous, we have
T(Tx_{n}) →Tu as n→+ ∞, f (Tx_{n})→fuas n→+ ∞.
By the triangle inequality, we have
 (5) 
 (6) 
Let θ<<c be given. Then there exists k_{1 }= k_{1}(c) such that d(Tu,T(Tx_{n})) << for all n≥k_{1}.
Note that, fx_{n+1 }= Tx_{n} →u as n→+∞ and Tx_{n+1} →u as n→+∞.
Since {T, f} is compatible, we conclude that there is a
k_{2}_{ }= k_{2}(c) such that d(T(fx_{n+1}), f(Tx_{n+1})) << for all n ≥ k_{2}.
Finally, there is k_{3}_{ }= k_{3}(c) such that d(T(fx_{n+1}), fu) << for all n ≥ k_{3}.
Let k_{0} = max {k_{1, }k_{2,} k_{3}}. By (6) we obtain that
Since, c is arbitrary, we conclude that
d(Tu, fu) <<for each m∈N. Noting that → θ as m→∞, we have that
Since P is closed,  d (Tu, fu)∈P.
Thus, d(Tu, fu)∈P∩ (P).
Hence, d (Tu, fu) = θ.
Therefore, f and T have a coincidence point u∈X.
Remark 2.2. If we take a_{4} = 0 in the above Theorem 2.1, then we get the Theorem 2.2 of ^{[9]}.
Remark 2.3. If we take in the above Theorem 2.1
a_{1} =λ, and a_{2} = a_{3} = a_{4} = 0. Then we can get the following Corollary.
Corollary 2.4. Let (X, ⊆) be partially ordered set and (X, d) be a complete cone metric space over a solid cone P. Let f, T: X→X be two maps such that
 (7) 
for all x, yX for which fx and fy are comparable. Assume that f and T satisfy the following conditions:
(i). f is weakly decreasing type A with respect to T.
(ii). The pair {f, T} is compatible.
(iii). f and T are continuous.
If λ is a nonnegative real numbers with λ∈ [0,1), then f and T have a coincidence point in X.
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