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Research Article | Open Access

Volume 2012 |Article ID 756453 | https://doi.org/10.1155/2012/756453

Chi-Ming Chen, "Fixed Point Theorems for -Contractive Mappings in Ordered Metric Spaces", Journal of Applied Mathematics, vol. 2012, Article ID 756453, 10 pages, 2012. https://doi.org/10.1155/2012/756453

# Fixed Point Theorems for 𝜓 -Contractive Mappings in Ordered Metric Spaces

Academic Editor: Yansheng Liu
Received19 Sep 2011
Accepted25 Nov 2011
Published18 Jan 2012

#### Abstract

We obtain some new fixed point theorems for -contractive mappings in ordered metric spaces. Our results generalize or improve many recent fixed point theorems in the literature (e.g., Harjani et al., 2011 and 2010).

#### 1. Introduction and Preliminaries

Throughout this paper, by , we denote the set of all real nonnegative numbers, while is the set of all natural numbers. Let be a metric space, a subset of ; and a map. We say is contractive if there exists such that for all ,

The well-known Banach fixed point theorem asserts that if , is contractive and is complete, then has a unique fixed point in . It is well known that the Banach contraction principle  is a very useful and classical tool in nonlinear analysis. Also, this principle has many generalizations. For instance, a mapping is called a quasicontraction if there exists such that

for any . In 1974, Ćirić  introduced these maps and proved an existence and uniqueness fixed point theorem.

In 1972, Chatterjea  introduced the following definition.

Definition 1.1. Let be a metric space. A mapping is said to be a -contraction if there exists such that for all , the following inequality holds:

Choudhury  introduced a generalization of -contraction as follows.

Definition 1.2. Let be a metric space. A mapping is said to be a weakly -contraction if for all , where is a continuous function such that if and only if .

In [3, 4], the authors proved some fixed point results for the -contractions. In , Harjani et al. proved some fixed point results for weakly -contractive mappings in a complete metric space endowed with a partial order.

In the following, we assume that the function satisfies the following conditions:(C1) is a strictly increasing and continuous function in each coordinate, and(C2)for all , , , , and .

Example 1.3. Let denote Then, satisfies the above conditions (C1) and (C2).

Now, we define the following notion of a -contractive mapping in metric spaces.

Definition 1.4. Let be a partially ordered set and suppose that there exists a metric in such that is a metric space. The mapping is said to be a -contractive mapping, if for .

Using Example 1.3, it is easy to get the following examples of -contractive mappings.

Example 1.5. Let endowed with usual ordering and with the metric given by Let denote where , , , and , for all . Let denote Then, is a -contractive mapping.

Example 1.6. Let endowed with the coordinate ordering (i.e., and ) and with the metric given by Let denote where , , , , and , for all . Let denote Then, is a -contractive mapping.

In this paper, we obtain some new fixed point theorems for -contractive mappings in ordered metric spaces. Our results generalize or improve many recent fixed point theorems in the literature (e.g., [5, 6]).

#### 2. Main Results

We start with the following definition.

Definition 2.1. Let be a partially ordered set and . Then one says that is monotone nondecreasing if, for ,

We now state the main fixed point theorem for -contractive mappings in ordered metric spaces when the operator is nondecreasing, as follows.

Theorem 2.2. Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space, and let be a continuous and nondecreasing -contractive mapping. If there exists with , then has a fixed point in .

Proof. If , then the proof is finished. Suppose that . Since is nondecreasing mapping, by induction, we obtain that Put for . Then, for each , from (), and, as the elements and are comparable, we get and so we can deduce that, for each , Let we denote . Then, is a nonincreasing sequence and bounded below. Thus, it must converge to some . If , then by the above inequalities, we have Passing to the limit, as , we have which is a contradiction. So .
We next claim that that the following result holds.
For each , there is such that for all , We will prove () by contradiction. Suppose that () is false. Then, there exists some such that for all , there exist and with such that Using the triangular inequality: and letting , we have
Since is a -contractive mapping, we also have Letting . Then, we get a contradiction. It follows from () that the sequence must be a Cauchy sequence.
Similary, we also conclude that for each , and so we have that for each ,
Let us denote . Then, is a nonincreasing sequence and bounded below. Thus, it must converge to some . If , then by the above inequalities, we have Passing to the limit, as , we have which is a contradiction. So . By the above argument, we also conclude that is a Cauchy sequence.
Since is complete, there exists such that . Moreover, the continuity of implies that
So we complete the proof.

In what follows, we prove that Theorem 2.2 is still valid for not necessarily continuous, assuming the following hypothesis in (which appears in Theorem 1 of ).If is a nondecreasing sequence in , such that

Theorem 2.3. Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space. Assume that satisfies (), and let be a nondecreasing -contractive mapping. If there exists with , then has a fixed point in .

Proof. Following the proof of Theorem 2.2, we only have to check that . As is a nondecreasing sequence in and , then the condition () gives us that for every . Since is a nondecreasing -contractive mapping, we have Letting and using the continuity of , we have and this is a contraction unless , or equivalently, .

In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorems 2.2 and 2.3. This condition is the following and it appears in : In , it is proved that the above-mentioned condition is equivalent to the following:

Theorem 2.4. Adding condition () to the hypothesis of Theorem 2.2 (or Theorem 2.3) and the condition for all , (or, ) to the function , one obtains the uniqueness of the fixed point of .

Proof. Suppose that there exist which are fixed points of . We distinguish two cases.
Case 1. If and are comparable and , then is comparable to for all , and and this is a contradiction unless , that is, .Case 2. If and are not comparable, then there exists comparable to and . Monotonicity of implies that is comparable to and for all . We also distinguish two cases.Subcase 2.1. If there exists with , then we have and this is a contradiction unless , that is, .Subcase 2.2. For all with , since is a nondecreasing -contractive mapping, we have Using the above inequality, we claim that for each , If not, we assume that , then by the definition of and , we have which implies a contradiction. Therefore, our claim is proved.
This proves that the nonnegative decreasing sequence is convergent. Put , . We now claim that . If , then making , we getthis is a contradiction. So , that is, .
Analogously, it can be proved that .
Finally, the uniqueness of the limit gives us .
This finishes the proof.

In the following, we present a fixed point theorem for a -contractive mapping when the operator is nonincreasing. We start with the following definition.

Definition 2.5. Let be a partially ordered set and . Then one says that is monotone nonincreasing if, for ,
Using a similar argument to that in the proof of Theorem 3.1 of , we get the following point results.

Theorem 2.6. Let be a partially ordered set satisfying condition () and suppose that there exists a metric in such that is a complete metric space, and let be a nonincreasing -contractive mapping. If there exists with or , then . Moreover, if in addition, is compact and is continuous, then has a unique fixed point in .

Proof. If , then it is obvious that . Suppose that (the same argument serves for ). Since is nonincreasing the consecutive terms of the sequence are comparable, we have and so we conclude that for each ,
Thus, is a decreasing sequence and bounded below, and it must converge to . We claim that . If , then by the above inequalities and the continuity of , letting , we have which is a contradiction. So , that is, . Consequently, .
Further, since is continuous and is compact, we can find such that
and, therefore, is a fixed point of .
The uniqueness of the fixed point is proved as in Theorem 2.4.

#### Acknowledgment

The authors would like to thank the referees whose helpful comments and suggestions led to much improvement in this paper.

1. S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integerales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922. View at: Google Scholar
2. L. B. Ćirić, “A generalization of Banach's contraction principle,” Proceedings of the American Mathematical Society, vol. 45, no. 2, pp. 267–273, 1974. View at: Google Scholar | Zentralblatt MATH
3. S. K. Chatterjea, “Fixed-point theorems,” Comptes Rendus de l'Académie Bulgare des Sciences, vol. 25, pp. 727–730, 1972. View at: Google Scholar | Zentralblatt MATH
4. B. S. Choudhury, “Unique fixed point theorem for weakly C-contractive mappings,” Kathmandu University Journal of Scinece, Engineering and Technology, vol. 5, no. 1, pp. 6–13, 2009. View at: Google Scholar
5. J. Harjani, B. López, and K. Sadarangani, “Fixed point theorems for weakly C-contractive mappings in ordered metric spaces,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 790–796, 2011. View at: Publisher Site | Google Scholar
6. J. Harjani, B. López, and K. Sadarangani, “A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space,” Abstract and Applied Analysis, vol. 2010, Article ID 190701, 8 pages, 2010. View at: Google Scholar | Zentralblatt MATH
7. J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005.
8. A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004.