1. Introduction

Huang and Zhang ^[9] introduced the concept of a cone metric space, they replaced set of real numbers by an ordered Banach space and proved some fixed point theorems for contractive type conditions in cone metric spaces. Later on many authors have (for e.g., ^{[1, 2, 3, 5, 6, 8, 12]}) proved common fixed point theorems for different contractive type conditions in cone metric spaces. Some of the authors have studied fixed point theorems on partially ordered cone metric spaces (see, e.g., ^{[3, 4, 11]}). In 2011, Y. J. Cho, et. al. ^[7] introduced a concept of the c-distance in a cone metric spaces and proved some fixed point theorems in ordered cone metric spaces. In this paper, we obtained some fixed point theorems on c-distance in ordered cone metric spaces. Our results are improved and extended the results of Y. J. Cho, et. al. ^[7].

2. Preliminaries

2.1 ^[9] Definition

Let be a real Banach space and denotes the zero element in . be a subset of . The set is called a cone if and only if:

(a). is closed, non–empty and ;

(b). , , implies ;

(c). P ∩ (-P) = {}.

2.2 ^[9] Definition

Let P be a cone in a Banach space , define partial ordering ‘’ with respect by x y if and only if . We shall write xy to indicate x y but x ≠ y while will stand for , where int P denotes the interior of the set . This cone P is called an order cone.

2.3 ^[9] Definition

Let be a Banach space and be an order cone. The order cone is called normal if there exists such that for all ,

The least positive number satisfying the above inequality is called the normal constant of .

2.4 ^[9] Definition

Let X be a nonempty set of . Suppose that the map satisfies:

(d1). d(x,y) for all with and if and only if ;

(d2). for all ;

(d3). d(x, y) d(x, z) + d(z, y) for all .

Then d is called a cone metric on and is called a cone metric space.

It is clear that the concept of a cone metric space is more general than that of a metric space.

2.5 ^[9] Example

Let , = { such that : }, and such that , where is a constant. Then is a cone metric space.

2.6 ^[9] Definition

Let be a cone metric space. Then is said to be

(i) a convergent sequence if for any , there is a natural number such that for all , , for some fixed in X. We denote this (as

(ii) a Cauchy sequence if for every in with , there is a natural number such that for all , .

(iii) a cone metric space is said to be complete if every Cauchy sequence in is convergent.

2.7 ^[9] Lemma

Let be a cone metric space and P be a normal cone with normal constant . Let and be two sequences in with and . Then as .

2.8 ^[7] Remark

(1) If E is real Banach space with a cone P and a a, where and , then .

(2) If int P, a_n and a_n , then there exists a positive integer N such that for all .

The concept of c-distance introduced by Y. J. Cho, et. al. ^[7], which is a cone version of -distance of Kada et. al. ^[10].

2.9 ^[7] Definition

Let be a cone metric space. Then a function is called a c-distance on if the following are satisfied

(q1) q(x,y) for all ;

(q2) q(x, z) q(x, y) + q(y, z) for all ;

(q3) for each and , if q(x, y_n) u for some , then q(x, y) u whenever is a sequence in convergent to a point .

(q4) for all with , there exists with such that and imply .

2.10 ^[7] Lemma

Let be a cone metric space and be a cone distance on X. Let and be sequences in and . Suppose that is a sequence in converging to . Then the following are holds:

(1) If q(x_n , y) u_n and q(x_n , z) u_n , then .

(2) If q(x_n, y_n) u_n and q(x_n, z) u_n, then converges to z.

(3) If q(x_n , x_m) u_n for , then is a Cauchy sequence in X.

(4) If q(y , x_n) u_n, then is a Cauchy sequence in .

3. Fixed Point Theorems on c-distance

In this section we have extended the Theorem 3.1., and Theorem 3.3. of ^[7].

3.1. Theorem. Let be a partially ordered set and suppose that is a complete cone metric space. Let be a c-distance on and be a continuous and non-decreasing mapping with respect to . Suppose that the following assertions are hold:

(i) there exists ; with such that

for all with y x;

(ii) there exists such that x₀ f . Then f has a fixed point . If v = fv, then .

Proof: If , then the proof is finished. Suppose that then we construct a sequence in by . Since is non-decreasing with respect to , we obtain by the induction

We have,

And hence,

Put

q(x_n , x_n+1) q(x_n-1, x_n), for all .

Repeating this process, we get that

(1)

Let , then it follows from (1) that

Thus by Lemma (2.10 ) shows that is a Cauchy sequence in . Since is complete, there exists such that as . Finally, the continuity of and implies that . Thus we prove that is a fixed point of .

Suppose that . Then we have

Since, .

We have .

This completes the proof.

3.2. Theorem. Let be a partially ordered set and suppose that is a complete cone metric space and is a normal cone with normal constant L. Let q be a c-distance on and is a non-decreasing mapping with respect to . Suppose that the following assertions are hold:

(i) there exists with such that

for all with y x;

(ii) there exists x₀ ∈X such that x₀ f x_0.

(iii) inf{║q(x, y)║ + ║q(x, fx)║: x∈X} >0, for all with , then f has a fixed point . If , then .

Proof: If we take in the proof of Theorem 3.1, then we have

Moreover, converges to a point and for all , where

By , we have for all .

Since is normal cone with normal constant , we have

(2)

And

(3)

If , then by the hypothesis, (2) and (3) with , we have

This is a contradiction.

Therefore, we have . Suppose that holds, then from the above Theorem 3.1 we can easily prove .

This completes the proof.

3.3. Remark. If we choose in the above Theorem 3.1, then we get the Theorem 3.1 of ^[7].

3.4. Remark. If we choose in the above Theorem 3.2 , then we get the Theorem 3.2 of ^[7].

3.5. Conclusion. In this paper, we have extended the results of Y. J. Cho, et. al. ^[7]

Acknowledgements

The author is grateful to the reviewer to suggest improve this paper.

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