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

- HemantKumar Nashine
^{1}and - Ishak Altun
^{2}Email author

**2011**:132367

https://doi.org/10.1155/2011/132367

© H. K. Nashine and I. Altun. 2011

**Received: **30 September 2010

**Accepted: **11 February 2011

**Published: **10 March 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.

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.

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.

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.

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.

holds. Then, has a fixed point.

Proof.

and so there exist and a subsequence of such that .

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

holds. Then, has a fixed point.

Remark 2.3.

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.

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.

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.

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

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

which is a contradiction. Thus, , and so .

Corollary 2.8.

holds. Then, and have a common fixed point.

Corollary 2.9.

## 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.

holds. Then, has a fixed point.

Proof.

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

Theorem 3.2.

holds. Then, and have a common fixed point.

Proof.

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

Let be a partial order relation on .

Theorem 3.3.

Consider the integral equations (3.14).

(iii)there exist a continuous function such that

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

Proof.

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].

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

## Declarations

### Acknowledgments

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

## Authors’ Affiliations

## References

- 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/S0004972700001659MATHMathSciNetView ArticleGoogle Scholar - 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.View ArticleGoogle Scholar - Rhoades BE:
**Some theorems on weakly contractive maps.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(4):2683–2693. 10.1016/S0362-546X(01)00388-1MATHMathSciNetView ArticleGoogle Scholar - Boyd DW, Wong JSW:
**On nonlinear contractions.***Proceedings of the American Mathematical Society*1969,**20:**458–464. 10.1090/S0002-9939-1969-0239559-9MATHMathSciNetView ArticleGoogle Scholar - Reich S:
**Some fixed point problems.***Atti della Accademia Nazionale dei Lincei*1974,**57**(3–4):194–198.MathSciNetMATHGoogle Scholar - Dutta PN, Choudhury BS:
**A generalisation of contraction principle in metric spaces.***Fixed Point Theory and Applications*2008,**2008:**-8.Google Scholar - Zhang Q, Song Y:
**Fixed point theory for generalized****-weak contractions.***Applied Mathematics Letters*2009,**22**(1):75–78. 10.1016/j.aml.2008.02.007MATHMathSciNetView ArticleGoogle Scholar - Abbas M, Doric D:
**Common fixed point theorem for four mappings satisfying generalized weak contractive condition.***Filomat*2010,**24**(2):1–10. 10.2298/FIL1002001AMATHMathSciNetView ArticleGoogle Scholar - 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.Google Scholar - Agarwal RP, El-Gebeily MA, O'Regan D:
**Generalized contractions in partially ordered metric spaces.***Applicable Analysis*2008,**87**(1):109–116. 10.1080/00036810701556151MATHMathSciNetView ArticleGoogle Scholar - Altun I, Simsek H:
**Some fixed point theorems on ordered metric spaces and application.***Fixed Point Theory and Applications*2010,**2010:**-17.Google Scholar - 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.2009.10.023MATHMathSciNetView ArticleGoogle Scholar - 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.2009.02.027MATHMathSciNetView ArticleGoogle Scholar - 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.MathSciNetMATHGoogle Scholar - 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.2005.10.017MATHMathSciNetView ArticleGoogle Scholar - 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.Google Scholar - 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.2009.08.029MATHMathSciNetView ArticleGoogle Scholar - 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/s11083-005-9018-5MATHMathSciNetView ArticleGoogle Scholar - 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/s10114-005-0769-0MATHMathSciNetView ArticleGoogle Scholar - 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/S0002-9939-07-08729-1MATHMathSciNetView ArticleGoogle Scholar - 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.2007.11.026MATHMathSciNetView ArticleGoogle Scholar - 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/S0002-9939-03-07220-4MATHMathSciNetView ArticleGoogle Scholar - Saadati R, Vaezpour SM:
**Monotone generalized weak contractions in partially ordered metric spaces.***Fixed Point Theory*2010,**11**(2):375–382.MATHMathSciNetGoogle Scholar - 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.2010.05.009MATHMathSciNetView ArticleGoogle Scholar - 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.Google Scholar - 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.2009.01.240MATHMathSciNetView ArticleGoogle Scholar - 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.2009.08.003MATHMathSciNetView ArticleGoogle Scholar - 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.MATHMathSciNetGoogle Scholar - 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/S1048953303000182MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.