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Fixed point theorems for a generalized almost -contraction with respect to S in ordered metric spaces
Journal of Inequalities and Applications volume 2012, Article number: 263 (2012)
Abstract
In this paper, the existence theorems of fixed points and common fixed points for two weakly increasing mappings satisfying a new condition in ordered metric spaces are proved. Our results extend, generalize and unify most of the fundamental metrical fixed point theorems in the literature.
MSC:47H09, 47H10.
1 Introduction and preliminaries
The classical Banach contraction principle is one of the most useful results in nonlinear analysis. In a metric space, the full statement of the Banach contraction principle is given by the following theorem.
Theorem 1.1 Letbe a complete metric space and. If T satisfies
for all, where, then T has a unique fixed point.
Due to its simplicity and usefulness, it has become a very popular tool in solving existence problems in many branches of mathematical analysis and its has many applications in solving nonlinear equations. Then, several authors studied and extended it in many direction; for example, see [1–19] and the references therein.
Despite these important features, Theorem 1.1 suffers from one drawback: the contractive condition (1.1) forces T to be continuous on X. It was then natural to ask if there exist weaker contractive conditions which do not imply the continuity of T. In 1968, this question was answered in confirmation by Kannan [20], who extended Theorem 1.1 to mappings that need not be continuous on X (but are continuous at their fixed point, see [21]).
On the other hand, Sessa [22] introduced the notion of weakly commuting mappings, which are a generalization of commuting mappings, while Jungck [23] generalized the notion of weak commutativity by introducing compatible mappings and then weakly compatible mappings [24].
In 2004, Berinde [25] defined the notion of a weak contraction mapping which is more general than a contraction mapping. However, in [26] Berinde renamed it as an almost contraction mapping, which is more appropriate. Berinde [25] proved some fixed point theorems for almost contractions in complete metric spaces. Afterward, many authors have studied this problem and obtained significant results (see [27–36]). Moreover, in [25] Berinde proved that any strict contraction, the Kannan [26] and Zamfirescu [37] mappings as well as a large class of quasi-contractions are all almost contractions.
Let T and S be two self mappings in a metric space . The mapping T is said to be a S-contraction if there exists such that for all .
In 2006, Al-Thagafi and Shahzad [38] proved the following theorem which is a generalization of many known results.
Theorem 1.2 ([[38], Theorem 2.1])
Let E be a subset of a metric spaceand S, T be two selfmaps of E such that. Suppose that S and T are weakly compatible, T is an S-contraction andis complete. Then S and T have a unique common fixed point in E.
Recently Babu et al.[39] defined the class of mappings satisfying condition (B) as follows.
Definition 1.3 Let be a metric space. A mapping is said to satisfy condition (B) if there exist a constant and some such that
for all .
They proved a fixed point theorem for such mappings in complete metric spaces. They also discussed quasi-contraction, almost contraction and the class of mappings that satisfy condition (B) in detail.
In recent year, Ćirić et al.[40] defined the following class of mappings satisfying an almost generalized contractive condition.
Definition 1.4 Let be a metric space, and let . A mapping T is called an almost generalized contraction if there exist and such that
for all , where .
Definition 1.5 Let be a partial ordered set. We say that are comparable if or holds.
Definition 1.6 Let be a partial ordered set. A mapping is said to be nondecreasing if , whenever and .
Definition 1.7 Let be a partial ordered set. Two mappings are said to be strictly increasing if and for all .
In 2004, Ran and Reurings [41] proved the following result.
Theorem 1.8 Letbe a partially ordered set such that every pairhas a lower and an upper bound. Suppose that d is a complete metric on X. Letbe a continuous and monotone mapping. Suppose that there existssuch thatfor all comparable. If there existssuch that, then T has a unique fixed point.
Ćirić et al. in [40] established fixed point and common fixed point theorems which are more general than Theorem 1.8 and several comparable results in the existing literature regarding the existence of a fixed point in ordered spaces.
In this paper, we introduce a new class which extends and unifies mappings satisfying the almost generalized contractive condition and establish the result on the existence of fixed points and common fixed points in a complete ordered space. This result substantially generalizes, extends and unifies the main results of Ćirić et al. [[40], Theorems 2.1, 2.2, 2.3, 2.6], Theorem 1.8, and several comparable results in the existing literature regarding the existence of a fixed and a common fixed point in ordered spaces.
2 Fixed point theorems for a generalized almost -contraction
First we introduce the notion of generalized almost -contraction mappings.
Definition 2.1 Let be a partially ordered set, and let a metric d exist on X. A mapping is called a generalized almost -contraction if there exist two mappings which and such that
for all comparable , where .
Theorem 2.2 Letbe a partially ordered set, and let a complete metric d exist on X. Letbe a strictly increasing continuous mapping with respect to ≤ and a generalized almost-contraction mapping. If there existssuch that, then T has a unique fixed point in X.
Proof If , then is fixed of T and we finish the proof. Now, we may assume that , that is, . We construct the sequence in X by
for all . Since T is strictly increasing, we have that . Thus, , which implies that the sequence is strictly increasing. Note that
Since and are comparable, we get
If for some , , then we get
which is a contradiction. Thus, we have
for all . Therefore, from (2.4), we have
for all . Hence,
for all . Now, for positive integers m and n with , we have
Since , if we take the limit as , then , which implies that is a Cauchy sequence. Since X is complete, there exists a such that as . It follows from the continuity of T that implies that . Therefore, z is a fixed point of T. □
Example 2.3 Let , the partial order ≤ be defined by if and only if and d be the usual metric on X. Let be defined by for all . It can be easily checked that T is a generalized almost -contraction mapping with and . Moreover, T is strictly increasing on X, and there exists such that . Therefore, T satisfies the conditions of Theorem 2.2 and thus T has a fixed point 0.
The following corollaries follow immediately from the Theorem 2.2 with and .
Corollary 2.4 ([[40], Theorem 2.1])
Letbe a partially ordered set, and let a complete metric d exist on X. Letbe a strictly increasing continuous mapping with respect to ≤. Suppose that there existandsuch that
for all comparable, where. If there existssuch that, then T has a fixed point in X.
Corollary 2.5 Letbe a partially ordered set, and let a complete metric d exist on X. Letbe a strictly increasing continuous mapping with respect to ≤. Assume that there exist two mappingswithandsuch that
for all comparable, where. If there existssuch that, then T has a fixed point in X.
Proof Since (2.10) is a special case of the generalized almost -contraction, the result follows from Theorem 2.2. □
Corollary 2.6 Letbe a partially ordered set, and let a complete metric d exist on X. Letbe a strictly increasing continuous mapping with respect to ≤. Assume that there exist two mappingswithandsuch that
for all comparable. If there existssuch that, then T has a fixed point in X.
Proof Since (2.11) is a special case of the generalized almost -contraction, the result follows from Theorem 2.2. □
Theorem 2.7 Letbe a partially ordered set, and let a complete metric d exist on X. Letbe a strictly increasing mapping with respect to ≤ and a generalized almost-contraction mapping with a continuous mapping ϕ. If there existssuch thatand for an increasing sequencein X converging towe haveandfor all, then T has a fixed point in X.
Proof Suppose that for all . Following similar arguments to those given in Theorem 2.2, we obtain an increasing sequence such that as for some . By the given hypothesis, we have for all . Therefore,
Taking the limit as , we get that . Since , we have , that is, , which is a contradiction. Therefore, there exists such that which implies that is a fixed point of T. □
3 Common fixed point theorems for a generalized almost -contraction with respect to S
Definition 3.1 Let be a partially ordered set, and let a metric d exist on X, and . A mapping T is called a generalized almost -contraction with respect to S if there exist two mappings with and and such that
for all comparable , where .
Remark 3.2 If we take where and where , then the generalized almost -contraction with respect to S reduces to an almost generalized contraction of Ćirić et al. in [40].
The following theorem deals with the existence of a common fixed point of two weakly increasing mappings which is more general and covers more than the result of Ćirić et al. in [40].
Theorem 3.3 Letbe a partially ordered set, and let a complete metric d exist on X. Letbe two strictly weakly increasing mappings with respect to ≤ and T be a generalized almost-contraction with respect to S. If either S or T is continuous, then there exists a common fixed point of S and T in X.
Proof Let be an arbitrary point in X. We construct the sequence in X such that
for all . Since S and T are strictly weakly increasing, we get
and
for all . Therefore, , that is, the sequence is strictly increasing. Note that
for all . Since and are comparable, we get
If for some , , then we get
which is a contradiction. Thus,
for all . Therefore, from (3.6), we have
for all . Similarly, it can be proved that
for all . It follows from the proof of Theorem 2.2 that
for all . Similarly to the proof of Theorem 2.2, we get is a Cauchy sequence. It follows from the completeness of X that there exists a such that as . Suppose that T is continuous, then . This implies that . Therefore, z is a fixed point of T. Suppose that . Since . Therefore,
which is a contradiction. Therefore, and then z is a common fixed point of S and T. Similarly, it can be proved that S and T have a common fixed point if S is continuous. □
The following corollaries are a generalization and extension of Corollaries 2.4 and 2.5 in Ćirić et al. in [40].
Corollary 3.4 Letbe a partially ordered set, and let a complete metric d exist on X. Letbe two strictly weakly increasing mappings with respect to ≤. Assume that there exist two mappingswithandandsuch that
for all comparable, where. If either S or T is continuous, then there exists a common fixed point of S and T in X.
Proof Since (3.11) is a special case of the generalized almost -contraction with respect to S, the result follows from Theorem 3.3. □
Corollary 3.5 Letbe a partially ordered set, and let a complete metric d exist on X. Letbe two strictly increasing mappings with respect to ≤. Assume that there exist two mappingswithandandsuch that
for all comparable. If either S or T is continuous, then there exists a common fixed point of S and T in X.
Proof Since (3.12) is a special case of the generalized almost -contraction with respect to S, the result follows from Theorem 3.3. □
Now, we have the following result of the continuity on the set of common fixed points. Let denote the set of all common fixed points of S and T.
Theorem 3.6 Letbe a partially ordered set, and let a complete metric d exist on X. Letand T be a generalized almost-contraction mapping with respect to S. Ifand for any sequencein X withasfor some, we havefor all, then S and T are continuous at.
Proof Fix . Let be any sequence in X converging to z and then for all . From the notion of a generalized almost -contraction with respect to S, we get
for all . Letting , we have . Therefore, S is continuous at . Similarly, it can be shown that T is continuous at . □
Remark 3.7 In fixed point theory, after the remarkable paper of Huang and Zhang [42], cone metric spaces have been considered by several authors. The results and theorems in this paper can be also generalized to cone metric spaces.
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Acknowledgements
This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission. The first author would like to thank the Research Professional Development Project Under the Science Achievement Scholarship of Thailand (SAST) for financial support during the preparation of this manuscript. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138). This project was partially completed during the first and third authors’ last visit to Kyungnam University.
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Sintunavarat, W., Kim, J.K. & Kumam, P. Fixed point theorems for a generalized almost -contraction with respect to S in ordered metric spaces. J Inequal Appl 2012, 263 (2012). https://doi.org/10.1186/1029-242X-2012-263
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DOI: https://doi.org/10.1186/1029-242X-2012-263