Abstract

We prove a fixed-point theorem for a class of maps that satisfy generalized -weak contractions depending on a given function. An example is given to illustrate our extensions.

1. Introduction

Because fixed-point theory has a wide array of applications in many areas such as economics, computer science, and engineering, it plays evidently a crucial role in nonlinear analysis. One of the cornerstones of this theory is the Banach fixed-point theorem, also known as the Banach contraction mapping theorem [1], which can be stated as follows.

Let be a contraction on a compete metric space ; that is, there is a nonnegative real number such that for all . Then the map admits one and only one point such that . Moreover, this fixed point is the limit of the iterative sequence for , where is an arbitrary starting point in . This theorem attracted a lot of attention because of its importance in the field. Many authors have started studying on fixed-point theory to explore some new contraction mappings to generalize the Banach contraction mapping theorem. In particular, Boyd and Wong [2] introduced the notion of -contractions. In 1997 Alber and Guerre-Delabriere [3] defined the -weak contraction which is a generalization of -contractions (see also [48]).

On the other hand, the notion of -contractions introduced and studied by the authors of the interesting papers in [911]. Following this trend, we explore in this paper another extension of -weak contractions in the context of -contractions.

2. Preliminaries

Let be a metric space. Boyd and Wong [2] introduced the notion of -contraction as follows. A map is called a -contraction if there exists an upper semicontinuous function such that for all . The concept of the -weak contraction was defined by Alber and Guerre-Delabriere [3] as a generalization of -contraction under the setting of Hilbert spaces and obtained fixed-point results. A map is a -weak contraction, if there exists a function such that for all provided that the function satisfies the following condition: Later Rhoades [7] proved analogs of the result in [3] in the context of metric spaces.

Theorem 2.1. Let be a complete metric space. Let be a continuous and nondecreasing function such that if and only if . If is a weak contraction, then has a unique fixed point.

In [12], Dutta and Choudhury proved an extension of Rhoades.

Theorem 2.2. Let be a complete metric space, and let be a self-mapping satisfying where are continuous and nondecreasing functions with if and only if . Then has a unique fixed point.

Zhang and Song [8] improved Theorem 2.1 and gave the following result which states the existence of common fixed points of certain maps in metric spaces.

Theorem 2.3. Let be a complete metric space, and let be self-mappings satisfying where and are lower semicontinuous functions with if and only if . Then have a unique common fixed point.

Combining the theorems above with the results of Dutta and Choudhury [12], Đoricorić [13] obtained the following theorem.

Theorem 2.4. Let be a complete metric space, and let be self-mappings satisfying where is a continuous and nondecreasing function with if and only if , and is a lower semicontinuous function with if and only if . Then have a unique common fixed point.

The notion of the -contraction is defined in ([10, 11]) as follows.

Definition 2.5. Let and be two self-mappings on a metric space . The mapping is said to be a -contraction if there exists such that

It can be easily seen that if is the identity map, then the -contraction coincides with the usual contraction.

Example 2.6. Let with the usual metric induced by . Consider the following self-mappings and on . It is clear that is not a contraction. On the contrary,

Definition 2.7 (see, e.g., [9, 11]). Let be a metric space. If is a convergent sequence whenever is convergent, then is called sequentially convergent.

The aim of this work is to give a proper extension of Đoricorić’s result of using the concept of -contraction, that is, the contraction depending on a given function. We will show the existence of a common fixed point for a class of certain maps.

3. Main Results

We start this section by recalling the following two classes of functions.

Let denote the set of all functions which satisfy (i)is continuous and nondecreasing, (ii) if and only if .

Similarly denotes the set of all functions which satisfy (i) is lower semi continuous, (ii) if and only if .

It is easy to see that belong to and ,   belong to .

We are ready to state our main theorem that is a proper extension of Theorem 2.4.

Theorem 3.1. Let be a complete metric space and an injective, continuous, and sequentially convergent mapping. Let be self-mappings. If there exist and such that for all , where then have a unique common fixed point.

Proof. We will follow the lines in the proof of the main result in [13]. By injection of , we easily check that if and only if is a common fixed point of and . Let . We define two iterative sequences and in the following way: We prove is a Cauchy sequence. For this purpose, we first claim that . It follows from property of that if is odd where Hence, we have If then , hence and which contradicts with and the property of . Thus, it follows from (3.5) that If is even then by the same argument above, we obtain Therefore, for all and is a nonincreasing sequence of nonnegative real numbers. Hence, there exists such that By the lower semicontinuity of , we have Taking the upper limits as on either side of we get that is, . By the property of , this implies that . It follows that and It is implied from (3.10) that Now, we claim that is a Cauchy sequence. Since , it is sufficient to prove that is a Cauchy sequence. Suppose on the contrary that is not a Cauchy sequence. Then, there exist and subsequences and of such that is the smallest index for which This means that From (3.18) and the triangle inequality, we get Letting and using (3.15), we get By the fact and using (3.15) and (3.20), we obtain Moreover, from and combining with (3.15) and (3.22), we conclude that Now, by the definition of and from (3.10), (3.15), and (3.20)–(3.24), we can deduce that Due to (3.1), we have Letting and using (3.22) and (3.25), we have It is a contradiction to for every . This proves that is a Cauchy sequence.
Since is a complete metric space, there exists such that . Since is sequentially convergent, we can deduce that converges to . By the continuity of , we infer that
We will show that . Indeed, suppose that , since is injective, we have . Hence, . Since we can seek such that for any Then, we have Therefore, for every . Since and letting , we arrive at We get a contradiction. Hence, . By the same argument, we get .
Let such that . Then, we have Thus This implies that , or . Since is injective, we have . The theorem is proved.

Remark 3.2. (1) In Theorem 3.1, if we choose for all , then we get Theorem 2.4.
(2) In Theorem 3.1, if we fix for all , then we obtain another extension of Theorem 2.3.
(3) In Theorem 3.1, if we choose , then we get the uniqueness and existence of fixed point of generalized -weak -contractions.
The following example shows that Theorem 3.1 is a proper extension of Theorem 2.4.

Example 3.3. Let and be the usual metric in . Consider the maps . It is easy to see that is the unique fixed point of and . We claim that and are not generalized -weak contraction. Indeed, if there exist lower semicontinuous functions with for and , such that then where . For and , we obtain It follows from (3.37) that Hence, . We arrive at a contradiction with for .
Consider the map , for all  . It is easy to see that is injective, continuous, and sequentially convergent. Let and , for all  . Now, we show that and are generalized -weak -contractions. It reduces to check the following inequality: We have It follows from (3.41) and (3.42) that for every . This proves that (3.40) is true.

By the same method used in the proof of Theorem 3.1, we get the following theorem.

Theorem 3.4. Let be a complete metric space and an injective, continuous, and sequentially convergent mapping. Let be self-mappings. If there exist and such that for all , then have a unique common fixed point.

Proof. It follows from the proof of Theorem 3.1 with necessary modifications.