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 y-x ∈P. We shall write xy to indicate xy but x≠y while x<<y will stand for y-x∈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 self-maps. 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 n₀ such that d(fgx_n, gfx_n)<< c whenever n > n₀. 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 non-negative real numbers with a₁ + a₂ + a₃ + 2a₄ ∈ [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 x₀ X. Since TX⊆ fX, we can choose x₁ ∈X such that Tx₀ = fx₁. Also since TX⊆ fX, we can choose

x₂ ∈X such that Tx₁ = 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 N₁ such that

Then for all n≥ N₁ we have that

Hence, c-d(Tx₀, Tx₁) 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 k₁ = k₁(c) such that d(Tu,T(Tx_n)) << for all n≥k₁.

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

k₂ = k₂(c) such that d(T(fx_n+1), f(Tx_n+1)) << for all n ≥ k₂.

Finally, there is k₃ = k₃(c) such that d(T(fx_n+1), fu) << for all n ≥ k₃.

Let k₀ = max {k_1, k_2, k₃}. 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 a₄ = 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

a₁ =λ, and a₂ = a₃ = a₄ = 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 non-negative real numbers with λ∈ [0,1), then f and T have a coincidence point in X.

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