- Research Article
- Open Access
- Published:

# Fixed Point Theorems for Generalized Weakly Contractive Condition in Ordered Metric Spaces

*Fixed Point Theory and Applications*
**volume 2011**, Article number: 132367 (2011)

## Abstract

Fixed point results with the concept of generalized weakly contractive conditions in complete ordered metric spaces are derived. These results generalize the existing fixed point results in the literature.

## 1. Introduction and Preliminaries

There are a lot of generalizations of the Banach contraction mapping principle in the literature. One of the most interesting of them is the result of Khan et al. [1]. They addressed a new category of fixed point problems for a single self-map with the help of a control function which they called an altering distance function.

A function is called an altering distance function if is continuous, nondecreasing, and holds.

Khan et al. [1] given the following result.

Theorem 1.1.

Let be a complete metric space, let be an altering distance function, and let be a self-mapping which satisfies the following inequality:

for all and for some . Then has a unique fixed point.

In fact, Khan et al. [1] proved a more general theorem of which the above result is a corollary. Another generalization of the contraction principle was suggested by Alber and Guerre-Delabriere [2] in Hilbert Spaces by introducing the concept of weakly contractive mappings.

A self-mapping on a metric space is called weakly contractive if for each ,

where is positive on and .

Rhoades [3] showed that most results of [2] are still valid for any Banach space. Also, Rhoades [3] proved the following very interesting fixed point theorem which contains contractions as special case .

Theorem 1.2.

Let be a complete metric space. If is a weakly contractive mapping, and in addition, is continuous and nondecreasing function, then has a unique fixed point.

In fact, Alber and Guerre-Delabriere [2] assumed an additional condition on which is . But Rhoades [3] obtained the result noted in Theorem 1.2 without using this particular assumption. Also, the weak contractions are closely related to maps of Boyd and Wong [4] and Reich type [5]. Namely, if is a lower semicontinuous function from the right, then is an upper semicontinuous function from the right, and moreover, (1.2) turns into . Therefore, the weak contraction is of Boyd and Wong type. And if we define for and , then (1.2) is replaced by . Therefore, the weak contraction becomes a Reich-type one.

Recently, the following generalized result was given by Dutta and Choudhury [6] combining Theorem 1.1 and Theorem 1.2.

Theorem 1.3.

Let be a complete metric space, and let be a self-mapping satisfying the inequality

for all , where , are both continuous and nondecreasing functions with if and only if . Then, has a unique fixed point.

Also, Zhang and Song [7] given the following generalized version of Theorem 1.2.

Theorem 1.4.

Let be a complete metric space, and let , be two mappings such that for each ,

where is a lower semicontinuous function with for and ,

Then, there exists a unique point such that .

Very recently, Abbas and Doric [8] and Abbas and Ali Khan [9] have obtained common fixed points of four and two mappings, respectively, which satisfy generalized weak contractive condition.

In recent years, many results appeared related to fixed point theorem in complete metric spaces endowed with a partial ordering in the literature [10–25]. Most of them are a hybrid of two fundamental principle: Banach contraction theorem and the weakly contractive condition. Indeed, they deal with a monotone (either order-preserving or order-reversing) mapping satisfying, with some restriction, a classical contractive condition, and such that for some , either or , where is a self-map on metric space. The first result in this direction was given by Ran and Reurings [22, Theorem 2.1] who presented its applications to matrix equation. Subsequently, Nieto and Rodŕiguez-López [18] extended the result of Ran and Reurings [22] for nondecreasing mappings and applied to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions.

Further, Harjani and Sadarangani [26] proved the ordered version of Theorem 1.2, Amini-Harandi and Emami [12] proved the ordered version of Rich type fixed point theorem, and Harjani and Sadarangani [27] proved ordered version of Theorem 1.3.

The aim of this paper is to give a generalized ordered version of Theorem 1.4. We will do this using the concept of weakly increasing mapping mentioned by Altun and Simsek [11] (also see [28, 29]).

## 2. Main Results

We will begin with a single map. The following theorem is a generalized version of Theorems 2.1 and 2.2 of Harjani and Sadarangani [27].

Theorem 2.1.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping such that

where

, is continuous, nondecreasing, is lower semicontinuous functions, and if and only if . Also, suppose that there exists with . If

or

holds. Then, has a fixed point.

Proof.

If , then the proof is completed. Suppose that . Now, since , and is nondecreasing, we have

Put , and so . If there exists such that , then it is clear that , and so we are finished. Now, we can suppose that

for all .

First, we will prove that .

From (2.2), we have for

Now, we claim that

for all . Suppose that this is not true; that is, there exists such that . Now, since , we can use the (2.1) for these elements, then we have

This implies , by the property of , we have , which this contradict to (2.6). Therefore, (2.8) is true, and so the sequence is nonincreasing and bounded below. Thus there exists such that . Now suppose that . Therefore from (2.2)

This implies

and so there exist and a subsequence of such that .

By the lower semicontinuity of we have

From (2.1), we have

and taking upper limit as , we have

that is, . Thus, by the property of , we have , which is a contradiction. Therefore, we have .

Next, we show that is Cauchy.

Suppose that this is not true. Then, there is an such that for an integer , there exist integers such that

For every integer , let be the least positive integer exceeding satisfying (2.15) and such that

Now,

Then, by (2.15) and (2.16), it follows that

Also, by the triangle inequality, we have

By using (2.18), we get

Now, by (2.2), we get

and taking upper limit as and using (2.18) and (2.20), we have

This implies that there exist and a subsequence of such that

By the lower semicontinuity of , we have

Now, by (2.1), we get

which is a contradiction. Thus, is a Cauchy sequence. From the completeness of , there exists such that as . If is continuous, then it is clear that . If (2.4) holds, then we have for all . Therefore, for all , we can use (2.1) for and . Since

and so , we have

By the property of , we have . Thus, the proof is complete.

The following corollary is a generalized version of Theorems 1.2 and 1.3 of Harjani and Sadarangani [26].

Corollary 2.2.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping such that

where

, is a lower semicontinuous functions, and if and only if . Also, suppose that there exists with . If

or

holds. Then, has a fixed point.

Remark 2.3.

In Theorem 1.1 [22], it is proved that if

then for every ,

where is the fixed point of such that

and hence, has a unique fixed point. If condition (2.32) fails, it is possible to find examples of functions with more than one fixed point. There exist some examples to illustrate this fact in [18].

Example 2.4.

Let , and consider a relation on as follows:

It is easy to see that is a partial order on . Let be Euclidean metric on . Now, define a self map of as follows:

Now, we claim that the condition (2.1) of Theorem 2.1 is satisfied with . Indeed, if , then . Therefore, since , then the condition (2.1) is satisfied. Again, if and , then and are not comparative. Now, if , then and

Also, it is easy to see that the other conditions of Theorem 2.1 are satisfied, and so has a fixed point in . Also, note that the weak contractive condition of Theorem 1.3 of this paper and Corollary 2.2 of [7] is not satisfied.

Now, we will give a common fixed point theorem for two maps. For this, we need the following definition, which is given in [28].

Definition 2.5.

Let be a partially ordered set. Two mappings are said to be weakly increasing if and for all .

Note that two weakly increasing mappings need not be nondecreasing. There exist some examples to illustrate this fact in [11].

Theorem 2.6.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let , are two weakly increasing mappings such that

for all comparable , where

, is continuous, nondecreasing, is lower semicontinuous functions, and if and only if . If

or

or

holds. Then, and have a common fixed point.

Remark 2.7.

Note that in this theorem, we remove the condition "there exists an with " of Theorem 2.1. Again, we can consider the result of Remark 2.3 for this theorem.

Proof of Theorem 2.6.

First of all we show that if or has a fixed point, then it is a common fixed point of and . Indeed, let be a fixed point of . Now, assume . If we use (2.38) for , we have

which is a contradiction. Thus, , and so is a common fixed point of and . Similarly, if is a fixed point of , then it is also fixed point of . Now, let be an arbitrary point of . If , the proof is finished, so assume that . We can define a sequence in as follows:

Without lost of generality, we can suppose that the successive term of are different. Otherwise, we are again finished. Note that since and are weakly increasing, we have

and continuing this process, we have

Now, since and are comparable, then we can use (2.38) for these points, then we have

where

Now, if for some , then

and so, from (2.47) we have

which is a contradiction. So, we have for all . Similarly, we have for all . Therefore, we have for all

and so the sequence is nonincreasing and bounded below. Thus, there exists such that . This implies that . Suppose that . Therefore, from (2.39),

This implies , and so there exist and a subsequence of such that .

By the lower semicontinuity of , we have

Now, from (2.38), we have

and taking upper limit as , we have

which is a contradiction. Therefore, we have

Next, we show that is a Cauchy sequence. For this, it is sufficient to show that is a Cauchy sequence. Suppose it is not true. Then, we can find an such that for each even integer , there exist even integers such that

We may also assume that

by choosing to be smallest number exceeding for which (2.57) holds. Now, (2.56), (2.57), and (2.58) imply

and so

Also, by the triangular inequality,

Therefore, we get

On the other hand, since and are comparable, we can use the condition (2.38) for these points. Since

we have

This is a contradiction. Thus, is a Cauchy sequence in , so is a Cauchy sequence. Therefore, there exists a with .

If or is continuous hold, then clearly, . Now, suppose that (2.42) holds and . Since , then from (2.42), for all . Using (2.38), we have

or

so taking upper limit from the last inequality, we have

which is a contradiction. Thus, , and so .

Corollary 2.8.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let , be two weakly increasing mappings such that

for all comparable , where is a continuous, nondecreasing, is lower semicontinuous functions, and if and only if . If

or

or

holds. Then, and have a common fixed point.

Corollary 2.9.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let , be two weakly increasing mappings such that

for all comparable , where

, is a lower semicontinuous functions, and if and only if . If

or

or

holds. Then, and have a common fixed point.

## 3. Some Applications

In this section, we present some applications of previous sections, first we obtain some fixed point theorems for single mapping and pair of mappings satisfying a general contractive condition of integral type in complete partially ordered metric spaces. Second, we give an existence theorem for common solution of two integral equations.

Set is a Lebesgue integrable mapping which is summable and nonnegative and satisfies , for each .

Theorem 3.1.

Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping such that

where

, is continuous, nondecreasing, is lower semicontinuous functions, and if and only if . Also, suppose that there exists with . If

or

holds. Then, has a fixed point.

Proof.

Define by , then is continuous and nondecreasing with . Thus, (3.1) becomes

which further can be written as

where and . Hence by Theorem 2.1 has unique fixed fixed point.

Theorem 3.2.

Let be a partially ordered set, and suppose that there exists a metric in such that is a complete metric space. Let , be two weakly increasing mappings such that

for all comparable , where

, is continuous, nondecreasing, is lower semicontinuous functions, and if and only if . If

or

or

holds. Then, and have a common fixed point.

Proof.

Define by , then is continuous and nondecreasing with . Thus, (3.7) becomes

which further can be written as

where and . Hence, Theorem 2.6 has unique fixed fixed point.

Now, consider the integral equations

Let be a partial order relation on .

Theorem 3.3.

Consider the integral equations (3.14).

(i) and are continuous,

(ii)for each ,

(iii)there exist a continuous function such that

for each and comparable ,

(iv).

Then, the integral equations (3.14) have a unique common solution in .

Proof.

Let with the usual supremum norm; that is, , for . Consider on the partial order defined by

Then, is a partially ordered set. Also, is a complete metric space. Moreover, for any increasing sequence in converging to , we have for any . Also, for every , there exists which is comparable to and [21].

Define , by

Now, from (ii), we have for all ,

Thus, we have and for all . This shows that and are weakly increasing. Also, for each comparable , we have

Hence,

Put . Therefore,

for each comparable . Therefore, all conditions of Corollary 2.8 are satisfied. Thus, the conclusion follows.

## References

- 1.
Khan MS, Swaleh M, Sessa S:

**Fixed point theorems by altering distances between the points.***Bulletin of the Australian Mathematical Society*1984,**30**(1):1–9. 10.1017/S0004972700001659 - 2.
Alber YaI, Guerre-Delabriere S:

**Principle of weakly contractive maps in Hilbert spaces.**In*New Results in Operator Theory and Its Applications, Operator Theory: Advances and Applications*.*Volume 98*. Edited by: Gohberg I, Lyubich Yu. Birkhäuser, Basel, Switzerland; 1997:7–22. - 3.
Rhoades BE:

**Some theorems on weakly contractive maps.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(4):2683–2693. 10.1016/S0362-546X(01)00388-1 - 4.
Boyd DW, Wong JSW:

**On nonlinear contractions.***Proceedings of the American Mathematical Society*1969,**20:**458–464. 10.1090/S0002-9939-1969-0239559-9 - 5.
Reich S:

**Some fixed point problems.***Atti della Accademia Nazionale dei Lincei*1974,**57**(3–4):194–198. - 6.
Dutta PN, Choudhury BS:

**A generalisation of contraction principle in metric spaces.***Fixed Point Theory and Applications*2008,**2008:**-8. - 7.
Zhang Q, Song Y:

**Fixed point theory for generalized****-weak contractions.***Applied Mathematics Letters*2009,**22**(1):75–78. 10.1016/j.aml.2008.02.007 - 8.
Abbas M, Doric D:

**Common fixed point theorem for four mappings satisfying generalized weak contractive condition.***Filomat*2010,**24**(2):1–10. 10.2298/FIL1002001A - 9.
Abbas M, Khan MA:

**Common fixed point theorem of two mappings satisfying a generalized weak contractive condition.***International Journal of Mathematics and Mathematical Sciences*2009,**2009:**-9. - 10.
Agarwal RP, El-Gebeily MA, O'Regan D:

**Generalized contractions in partially ordered metric spaces.***Applicable Analysis*2008,**87**(1):109–116. 10.1080/00036810701556151 - 11.
Altun I, Simsek H:

**Some fixed point theorems on ordered metric spaces and application.***Fixed Point Theory and Applications*2010,**2010:**-17. - 12.
Amini-Harandi A, Emami H:

**A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(5):2238–2242. 10.1016/j.na.2009.10.023 - 13.
Beg I, Butt AR:

**Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(9):3699–3704. 10.1016/j.na.2009.02.027 - 14.
Beg I, Butt AR:

**Fixed points for weakly compatible mappings satisfying an implicit relation in partially ordered metric spaces.***Carpathian Journal of Mathematics*2009,**25**(1):1–12. - 15.
Gnana Bhaskar T, Lakshmikantham V:

**Fixed point theorems in partially ordered metric spaces and applications.***Nonlinear Analysis: Theory, Methods & Applications*2006,**65**(7):1379–1393. 10.1016/j.na.2005.10.017 - 16.
Ćirić L, Cakić N, Rajović M, Ume JS:

**Monotone generalized nonlinear contractions in partially ordered metric spaces.***Fixed Point Theory and Applications*2008,**2008:**-11. - 17.
Ćirić LB, Miheţ D, Saadati R:

**Monotone generalized contractions in partially ordered probabilistic metric spaces.***Topology and Its Applications*2009,**156**(17):2838–2844. 10.1016/j.topol.2009.08.029 - 18.
Nieto JJ, Rodríguez-López R:

**Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations.***Order*2005,**22**(3):223–239. 10.1007/s11083-005-9018-5 - 19.
Nieto JJ, Rodríguez-López R:

**Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations.***Acta Mathematica Sinica (English Series)*2007,**23**(12):2205–2212. 10.1007/s10114-005-0769-0 - 20.
Nieto JJ, Pouso RL, Rodríguez-López R:

**Fixed point theorems in ordered abstract spaces.***Proceedings of the American Mathematical Society*2007,**135**(8):2505–2517. 10.1090/S0002-9939-07-08729-1 - 21.
O'Regan D, Petruşel A:

**Fixed point theorems for generalized contractions in ordered metric spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(2):1241–1252. 10.1016/j.jmaa.2007.11.026 - 22.
Ran ACM, Reurings MCB:

**A fixed point theorem in partially ordered sets and some applications to matrix equations.***Proceedings of the American Mathematical Society*2004,**132**(5):1435–1443. 10.1090/S0002-9939-03-07220-4 - 23.
Saadati R, Vaezpour SM:

**Monotone generalized weak contractions in partially ordered metric spaces.***Fixed Point Theory*2010,**11**(2):375–382. - 24.
Saadati R, Vaezpour SM, Vetro P, Rhoades BE:

**Fixed point theorems in generalized partially ordered***G***-metric spaces.***Mathematical and Computer Modelling*2010,**52**(5–6):797–801. 10.1016/j.mcm.2010.05.009 - 25.
Shakeri S, Ćirić LJB, Saadati R:

**Common fixed point theorem in partially ordered***ℒ***-fuzzy metric spaces.***Fixed Point Theory and Applications*2010,**2010:**-13. - 26.
Harjani J, Sadarangani K:

**Fixed point theorems for weakly contractive mappings in partially ordered sets.***Nonlinear Analysis: Theory, Methods & Applications*2009,**71**(7–8):3403–3410. 10.1016/j.na.2009.01.240 - 27.
Harjani J, Sadarangani K:

**Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations.***Nonlinear Analysis: Theory, Methods & Applications*2010,**72**(3–4):1188–1197. 10.1016/j.na.2009.08.003 - 28.
Dhage BC:

**Condensing mappings and applications to existence theorems for common solution of differential equations.***Bulletin of the Korean Mathematical Society*1999,**36**(3):565–578. - 29.
Dhage BC, O'Regan D, Agarwal RP:

**Common fixed point theorems for a pair of countably condensing mappings in ordered Banach spaces.***Journal of Applied Mathematics and Stochastic Analysis*2003,**16**(3):243–248. 10.1155/S1048953303000182

## Acknowledgments

The authors thank the referees for their appreciation, valuable comments, and suggestions.

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Nashine, H., Altun, I. Fixed Point Theorems for Generalized Weakly Contractive Condition in Ordered Metric Spaces.
*Fixed Point Theory Appl* **2011, **132367 (2011). https://doi.org/10.1155/2011/132367

Received:

Accepted:

Published:

### Keywords

- Fixed Point Theorem
- Contractive Condition
- Lower Semicontinuity
- Cauchy Sequence
- Common Fixed Point