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Mathematical Problems in Engineering
Volume 2011, Article ID 380784, 14 pages
http://dx.doi.org/10.1155/2011/380784
Research Article

Contractive Mapping in Generalized, Ordered Metric Spaces with Application in Integral Equations

1Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU), Tehran, Iran
2Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan
3Department of Mathematics, Amirkabir University of Technology, Tehran, Iran

Received 22 June 2011; Revised 1 October 2011; Accepted 3 October 2011

Academic Editor: Cristian Toma

Copyright © 2011 L. Gholizadeh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the concept of -distance on a complete, partially ordered -metric space and prove some fixed point theorems. Then, we present some applications in integral equations of our obtained results.

1. Introduction

The Banach fixed point theorem for contraction mapping has been generalized and extended in many directions [111]. Nieto and Rodríguez-López [10], Ran and Reurings [12], and Petrusel and Rus [13] presented some new results for contractions in partially ordered metric spaces. The main idea in [10, 12, 14] involves combining the ideas of an iterative technique in the contraction mapping principle with those in the monotone technique. Also, Mustafa and Sims [15] introduced the concept of -metric. Some authors [16, 17] have proved some fixed point theorems in these spaces. Recently, Saadati et al. [18], using the concept of -metric, defined an -distance on complete -metric space and generalized the concept of -distance due to Kada et al. [19].

In this paper, we extend some recent fixed point theorems by using this concept and prove various fixed point theorems in generalized partially ordered -metric spaces.

At first we recall some definitions and lemmas. For more information see [1518, 2023].

Definition 1 (see [15]). Let be a nonempty set. A function is called a -metric if the following conditions are satisfied:(i) if (coincidence), (ii) for all , where , (iii) for all , with , (iv), where is a permutation of (symmetry), (v) for all (rectangle inequality). A -metric is said to be symmetric if for all .

Definition 2. Let be a -metric space,(1)a sequence in is said to be -Cauchy sequence if, for each , there exists a positive integer such that for all , ; (2)a sequence in is said to be -convergent to a point if, for each , there exists a positive integer such that for all , .

Definition 3 (see [15]). Let be a -metric space. Then a function is called an -distance on if the following conditions are satisfied: (a) for all , (b)for any are lower semicontinuous, (c)for each , there exists a such that and imply .

Example 1 (see [18]). Let be a metric space and defined by for all . Then is an -distance on .

Example 2 (see [18]). In we consider the -metric defined by for all . Then defined by for all is an -distance on .

For more example see [18].

Lemma 1.1 (see [18]). Let be a metric space with metric and be an -distance on . Let be sequences in be sequences in converging to zero and let . Then one has the following.(1)If and for , then and hence . (2)If and for then and hence . (3)If for any with , then is a G-Cauchy sequence. (4)If for any then is a G-Cauchy sequence.

Definition 4 (see [18]). -metric space is said to be -bounded if there is a constant such that for all .

2. Fixed Point Theorems on Partially Ordered -Metric Spaces

Definition 5. Suppose is a partially ordered space and is a mapping of into itself. We say that is nondecreasing if for ,

Theorem 2.1. Let be a partially ordered space. Suppose that there exists a -metric on such that is a complete -metric space and is an -distance on such that is -bounded. Let and weakly compatible and be non-decreasing mapping such that (a); (b), , ,,for all and,(c)for every and with , ;(d)there exist that ; then and have a unique common fixed point in and .

Proof. Let that . By part (a), we can choose such that . Again from part (a), we can choose such that . Continuing this process we can construct sequences in such that, Now, since is non-decreasing mapping then, so, for all , Then, Now since, thus, So where Now, for any with and , we have, Since is -bounded and so, by Part (3) of Lemma 1.1, is a -Cauchy sequence. Since is -complete, converges to a point . Thus, for and by the lower semicontinuity of , we have Assume that . Since, so, , and, for every , that is a contraction. So, we have . Then, by (b), so, . Similarly, .
Now, Thus, By Part (c) of Definition 3, and consequently which implies that is a fixed point for . Now, So, it is enough to put , then is a common fixed point of and .
Uniqueness: Assume that there exist such that . Hence, we have, and so . Also, . On the other hand, which follows that, . Then by Part (c) of Definition 3, and .

The following corollary is a generalization of [24, Theorem 2.1].

Corollary 2.2. Let be a partially ordered space. Suppose that there exists a -metric on such that is a -metric space and is an -distance on such that be -bounded. Let and be weakly compatible and be a non-decreasing mapping such that (a) and either or is complete;(b)for all and , ;(c)for every and with ,  ;(d)there exist that ;then and have a unique common fixed point in and .

Definition 6 (see [25]). Let be the set of all functions such that is a continuous and nondecreasing function with for all and for each. The function is called a growth or control function of .
It is clear that

Theorem 2.3. Let be a partially ordered space. Suppose that there exists a -metric on such that is a complete -metric space and is an -distance on and is a non-decreasing mapping from into itself. Let be -bounded. Suppose that and Also, for every for every with . If there exists an with , then has a unique fixed point. Moreover, if , then .

Proof. If , then the proof is finished. Suppose that . since and is non-decreasing, we obtain For all and , We claim that for and with , We prove by, Since , so, By Part (c) of Lemma 1.1 is a -Cauchy sequence. Since is -complete, converges to a point . Let be fixed. By lower semicontinuity of , Assume that . Since , for every , which is a contraction. Therefore, we have .
Uniqueness: let be another fixed point of , then which is a contraction. Therefore, fixed point is unique. Now, if , we have, So .

Corollary 2.4. Let the assumptions of Theorem 2.3 hold and then has a unique fixed point.

Proof. From Theorem 2.3, has a unique fixed point . However, so is also a fixed point of . Since the fixed point of is unique, it must be the case that .

Corollary 2.5. Let the assumptions of Theorem 2.3 hold and satisfies, Then has a unique fixed point.

Proof. Take , and apply Theorem 2.3.

Theorem 2.6. Let be a partially ordered space. Suppose that there exists a -metric on such that is a complete -metric space, is an -distance on , and is a non-decreasing mapping from into itself. Let be -bounded. Suppose that where . Also for every , for every with . If there exists an with , then has a unique fixed point say and .

Proof. Let be an arbitrary point, and define the sequence by . By (2.35) and for all , But by Part (a) of Definition 3, Hence, which implies, Let , then and by repeated application of (2.40), we have Now, for any with and , we have, So, By Part (3) of Lemma 1.1, is a -Cauchy sequence. Since is -complete, converges to a point . Now, similar to proving Theorem 2.1, has a unique fixed point and .

Corollary 2.7. Let the assumptions of Theorem 2.6 hold and where , then has a unique fixed point.

Proof. The argument is similar to that used in the proof of Corollary 2.4.

3. Applications

In this section, we give an existence theorem for a solution of a class of integral equations. Denote by the set of all functions satisfying the following hypotheses: (i) is a Lebesgue-integrable mapping on each compact of , (ii)for every , we have ,(iii), where denotes to the norm of .

Now, we have the following results.

Theorem 3.1. Let be a partially ordered space. Suppose that there exists a -metric on such that is a complete -metric space and is an -distance on and is a non-decreasing mapping from into itself. Let be -bounded. Suppose that where . Also, suppose that for every for every with . If there exists an with , then has a unique fixed point.

Proof. Define by . It is clear that is nondecreasing and continuous. From (iii), we have Also, note that In general, we have . Thus, we have Therefore satisfies all the hypotheses of Definition 6. By inequality (3.1), we have . Therefore by Theorem 2.3, has a unique fixed point.

Now, our aim is to give an existence theorem for a solution of the following integral equation: Let be the set of all continuous functions defined on . Define by where . Then is a complete -metric space. Let . Then is an -distance on .

Define an ordered relation ≤ on by Then is a partially ordered set. Now, we prove the following result.

Theorem 3.2. Suppose the following hypotheses hold. (a) and are continuous. (b) is nondecreasing in its first coordinate and is nondecreasing. (c)There exist a continuous function such that for each comparable and each . (d) for some . Then the integral equation (3.6) has a solution .

Proof. Define by By hypothesis (b), we have that is nondecreasing.
Now, if for with , then for each there exists with such that So, we have Therefore, for each , we have By the continuity of , we have Thus, we have , a contradiction. Thus, Define by . For with , we have Moreover, take , then . Thus all the required hypotheses of Corollary 2.5 are satisfied. Thus there exists a solution of the integral equation (3.6).

Acknowledgment

The authors would like to thank the referee and area editor Professor Cristian Toma for providing useful suggestions and comments for the improvement of this paper.

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