Fixed Point Theorems for Generalized -Contractions in -Metric Spaces

Abstract

We introduce the notion of generalized contraction and establish some new fixed point theorems for this contraction in the setting of complete -metric spaces. The results presented in the paper improve, extend, and unify some known results. Finally, we give an example to illustrate them.

1. Introduction and Preliminaries

In 2006, Mustafa and Sims [1] introduced the notion of -metric space and studied the properties of it. Subsequently, many authors studied the fixed point theory in the setting of complete -metric spaces and obtained some fixed point theorems for different contractions (see [1–10]). In 2015, Agarwal et al. [11] presented a self-contained account of the fixed point theory (techniques and results) in -metric spaces. The book [11] contains almost all the research findings that relate to basic fixed point theorems, common fixed point theorems, and coupled fixed point theorems in -metric spaces and partially ordered -metric spaces (see [11] and the references therein).

In 2014, Jleli and Samet [12] introduced a new type of contraction called -contraction. Later, many authors have studied -contraction deeply (for example, see [13, 14]). Just recently, Zheng et al. [15] introduced the notion of contraction in metric spaces which generalized -contraction and other contractions (see [12, 15] and the references therein).

Inspired by [12, 15], we introduce the notion of generalized contraction and establish some new fixed point theorems for this contraction in the setting of complete -metric spaces. The results presented in the paper improve and extend the corresponding results of Agarwal et al. [11], Mustafa [4], Mustafa et al. [5], Mustafa and Sims [6], and Shatanawi [9]. Also, we give an example to illustrate them.

According to [12, 15], denote by the set of functions satisfying the following conditions:  is nondecreasing.For each sequence , if and only if .  is continuous on

And by the set of functions satisfies the following conditions: is nondecreasing.For each , . is continuous on .

Lemma 1 (see [15]). If , then and for each

Now we recall some basic definitions and give some lemmas that will be used in the paper.

Definition 2 (see [1, 11]). A -metric space is a pair , where is a nonempty set and is a function such that, for all , the following conditions are fulfilled: if . for all with . for all with . (symmetry in all 3). (rectangle inequality).In such a case, the function is called a -metric.

Example 3 (see [1, 11]). If is a nonempty subset of , then the function , given by for all , is a -metric on .

Example 4 (see [1, 11]). Let be the interval of nonnegative real numbers and let be defined by Then is a complete -metric on .

Definition 5 (see [1, 11]). Let be a -metric space; let and be a sequence. We say that(i)  -converges to , and we write if ; that is, for all there exists satisfying for all such that ;(ii) is -Cauchy if ; that is, for all there exists satisfying for all such that ;(iii) is complete if every -Cauchy sequence in is -convergent in .

Lemma 6 (see [1, 11]). Let be a -metric space, let and be a sequence. Then the following conditions are equivalent.(a)  -converges to .(b).(c).(d).(e).

Lemma 7 (see [1, 11]). Let be a G-metric space and be a sequence. Then the following conditions are equivalent.(a) is -Cauchy.(b).(c).(d).(e).

Lemma 8 (see [11]). Let be an asymptotically regular sequence in a G-metric space and suppose that is not Cauchy. Then there exist a positive real number and two subsequences and of such that, for all ,and also, for all given ,

Lemma 9 (see [11]). Let be a -metric space; then
for all .

2. Main Results

Based on the functions and , we give the following definition.

Definition 10. Let be a -metric space. A mapping is said to be a generalized contraction if there exist and such that, for any ,where

Theorem 11. Let be a complete -metric space and let be a generalized contraction. Then has a unique fixed point such that the sequence converges to for every .

Proof. Let be an arbitrary point. We define the sequence in by , for all . If for some , then is a fixed point for . Next, we assume that for all . Then for all . Applying inequality (4) with , , , we obtainwhereIf , then it follows from (4) thatwhich is a contradiction. Hence, for , Thus, (4) becomesRepeating this process, we getBy the definition of and , we haveBy , we obtainThus, is an asymptotically regular sequence.
In what follows, we shall prove that is a Cauchy sequence in .
Suppose, on the contrary, that, by Lemma 8, there exist a positive real number and two subsequences and of such that, for all ,and also, for all given ,Pick large enough, by (13), (15), and Lemma 9,Using the contractivity condition (4),Passing to limit as , then we get
By Lemma 1, , then , which is a contradiction. Thus, is a Cauchy sequence in .
Taking into account the fact that is complete, there exists such that converges to . In particular,Using the fact that is continuous on each variable,We claim that is a fixed point of . Suppose, on the contrary, if , then by (18), (19),Using the contractivity condition (4),Passing to limit as , then we haveBy Lemma 1, . Thenwhich is a contradiction. As a consequence, we conclude that .
Now, we will prove that has at most one fixed point. Suppose, on the contrary, that there exists another distinct fixed point of such that . Therefore, , and , and then by (4)and by Lemma 1, , which is a contradiction. Therefore, the fixed point of is unique.

Theorem 12. Let be a complete -metric space and let be a self-mapping. Assume that there exist and such that, for any ,whereThen has a unique fixed point such that the sequence converges to for every .

The following Theorem 13 is the main result of [5].

Theorem 13 (see [5]). Let be a complete -metric space and let be a self-mapping which satisfies the following condition, for all ,,where and . Then has a unique fixed point such that the sequence converges to for every .

Proof. Let ; then . And let , ; then and . Since Therefore,From Theorem 12, we can see that has a unique fixed point such that the sequence converges to for every .

The following Theorem 14 is the main result of [6].

Theorem 14 (see [6]). Let be a complete -metric space and let be a self-mapping which satisfies the following condition, for all ,where . Then has a unique fixed point such that the sequence converges to for every .

Proof. Let ; then . And let , ; then and . Since therefore,From Theorem 11, we can see that has a unique fixed point such that the sequence converges to for every .

Theorem 15. Let be a complete -metric space and let be a self-mapping. Assume that there exist and such that, for any ,Then has a unique fixed point such that the sequence converges to for every .

Theorem 16 (see [4]). Let be a complete -metric space and let be a self-mapping such that there exists satisfying, for any , Then has a unique fixed point such that the sequence converges to for every .

Proof. Let , ; then and .
is equivalent to ; that is, .
From Theorem 15, we can see that has a unique fixed point such that the sequence converges to for every .

Corollary 17. Let be a complete -metric space and let be a self-mapping. Assume that there exist and such that, for any , Then has a unique fixed point such that the sequence converges to for every .

Corollary 18. Let be a complete -metric space and let be a self-mapping such that there exists satisfying, for any , Then has a unique fixed point such that the sequence converges to for every .

Corollary 19. Let be a complete -metric space and let be a self-mapping. Assume that there exist and such that, for any , Then has a unique fixed point such that the sequence converges to for every .

Corollary 20. Let be a complete -metric space and let be a self-mapping such that there exists satisfying, for any , Then has a unique fixed point such that the sequence converges to for every .

Corollary 21. Let be a complete -metric space and let be a self-mapping. Assume that there exist and such that, for any , , we haveThen has a unique fixed point such that the sequence converges to for every .

Corollary 22. Let be a complete -metric space and let be a self-mapping which satisfies the following condition, for all ,where . Then has a unique fixed point such that the sequence converges to for every .

Corollary 23. Let be a complete -metric space and let be a self-mapping. Assume that there exist and such that, for any , Then has a unique fixed point such that the sequence converges to for every .

Corollary 24. Let be a complete -metric space and let be a self-mapping such that there exists satisfying, for any , Then has a unique fixed point such that the sequence converges to for every .

Theorem 25 (see [9]). Let be a complete -metric space and let be a mapping such that, for all , where is an increasing continuous function such that for .
Then has a unique fixed point and for every the sequence converges to

Proof. Let for all , and for all
Obviously, , .
By the definition of , we have Therefore, from Theorem 15, has a unique fixed point and for every the sequence converges to

Remark 26. In [9], the function is not required to be continuous. But due to Theorem of [16] and item 4.2.3 of [11], we can suppose that is continuous.
is continuous and nondecreasing, , and
is lower semicontinuous, .

Theorem 27 (see [11]). Let be a complete -metric space and let be a self-mapping. Assume that there exist and such that, for any ,Then has a unique fixed point such that the sequence converges to for every .

Proof. Due to Theorem of [16] and item 4.2.3 of [11], the condition where there exist and such that, for any , , is equivalent to the condition where there exist and such that, for any , , Let ; then and . is equivalent to ; that is, .
From Theorem 15, has a unique fixed point and for every the sequence converges to

Remark 28. According to fixed point theory of metric spaces, we divide contractions into different type in the setting of -metrics. Then Theorem 16 and Corollary 18 belong to Banach type, Corollaries 19–24 Kannan type [17], Theorem 25 Browder type [18], and Theorem 27 Choudhury type. To some extent, our results unify them.

3. Example

In this section, we give an example to illustrate our results.

Example 29. Let be endowed with the -metric for all Then is a complete -metric space. Define the mapping by At first, we observe that Theorems 16, 25, and 27 cannot be applied since for all ,
And Theorem 13 cannot be applied too. In fact, let , ; then , whileis equivalent to .
Since for all , we have , which yields a contradiction since .
By the same way, we can see that Theorem 14 cannot be applied.
Now, let the function be defined by And define by Obviously, , .
In what follows, we prove that is some Kannan-type contraction; that is, satisfies the condition of Corollary 23.
We consider three cases.

Case 1 (, or , ). In this case, we have

Case 2 ( or ). In this case, we have

Case 3 (, or ). In this case, we have Therefore, we have for all Thus, is a Kannan-type contraction.
So all the hypotheses of Corollary 23 are satisfied, thus has a fixed point. In this example is the fixed point.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by National Natural Science Foundation of China (nos. 11461002 and 11461003) and Guangxi Natural Science Foundation (2016GXNSFAA380003, 2016GXNSFAA380317, and 2017GXNSFAA198100).