Journal of Function Spaces
Volume 2016 (2016), Article ID 6978439, 9 pages
http://dx.doi.org/10.1155/2016/6978439

Research Article

A New Approach to the Study of Fixed Point Theorems with -Distances via -Functions

Farzad Zarinfar,¹ Farshid Khojasteh,² and Seyyed Mansour Vaezpour³

¹Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
²Young Researcher and Elite Club, Arak-Branch, Islamic Azad University, Arak, Iran
³Department of Mathematics and Computer Sciences, Amirkabir University of Technology, Tehran, Iran

Received 5 June 2016; Revised 28 July 2016; Accepted 24 August 2016

Academic Editor: Francesca Vetro

Copyright © 2016 Farzad Zarinfar et al. 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.

Abstract.

We introduce some new generalization of fixed point theorems in complete metric spaces endowed with -distances via -functions. Our results extend many of known fixed point theorems such as Reich type contraction, Geraghty contraction, Meir-Keeler contraction, and -contraction. In addition, the result and corollaries show that our approach has a constructive attitude and many known and unknown results can be constructed in such way.

1. Introduction

Recently, extensive researches have been performed on various topics in different spaces equipped -distances, for example, fuzzy spaces [1], polish spaces [2], quasimetric spaces [3], metric spaces endowed with a graph [4], TVS-Cone metric spaces [5–8], partially ordered metric spaces [9–15], and other related papers such as [16, 17].

In 2014, Du and Khojasteh [18] introduced a very strong class of mappings called manageable function of which many of known contractions in the set of multivalued mappings have closely related to such class and obtain from them.

Moreover, the notion of simulation function was introduced by Khojasteh et al. [19]. They introduced -contraction with respect to simulation functions. The -contraction generalizes the Banach contraction and unifies several known types of contractions in complete metric space involving the combination of and and satisfies some particular conditions in such spaces.

Very recently, in order to extend the previous families of functions, Roldán López de Hierro and Shahzad [20] introduced a new generalized class of mappings called -functions and a new contraction called -contraction and show that every Geraghty contraction, Meir-Keeler contraction, -contraction, -contraction, and many of other known contractions are special case of -contraction.

In this work, we introduce some new generalization of fixed point theorems in complete metric spaces endowed with -distances via -functions and many of known fixed point theorems such as Reich type contraction, Geraghty contraction [9], Meir-Keeler contraction [21], and -contraction [19] which in such spaces are concluded as apparent corollaries.

2. Preliminaries

In the following, we assert some notions which are needed in continuance.

Definition 1 (see [22]). Let be a metric space. The mapping is said to be -distance on , if it satisfies the following axioms:

 (1) , for any ; (2) for any , is lower semicontinuous; (3) for any , there exists such that and implies .

Denote

Lemma 2 (see [22]). Let be a metric space and be -distance on . Assume that , are sequences in and , are sequences in which are convergent to and . Then the following holds:

(i)if and , for any , then ;(ii)if and , for any , then converges to ;(iii)if , for any with , then is a Cauchy sequence;(iv)if , for any , then is a Cauchy sequence.

Definition 3 (see [23]). A symmetric on a metric space is -distance on such that

 (SY): for all .

If is -distance satisfying , we then say that it is a symmetric -distance on .

Definition 4 (see [24]). A function is called L-function if

(a),(b) for all ,(c)for all , there exists such that , for all .

Moreover, for each we have .

Definition 5 (see [17, 21]). A mapping , from a metric space into itself, is called Meir-Keeler mapping, whenever, for all , there exists such that, for all , if , then .

Meir and Keeler [21] proved that every Meir-Keeler mapping on a complete metric space has a unique fixed point.

Definition 6 (see [19]). A simulation function is a mapping which satisfies the following conditions:

for all ,if are sequences in such that , then

Many examples of simulation functions can be found in [18, 19, 25]. Very recently, Roldán López de Hierro and Shahzad [20] introduced the following category of mappings which are connected to extend the simulation functions.

Definition 7. Let be a nonempty subset and let be a function. We say that is -function if it satisfies the following two conditions:

If is a sequence such that , for all , then .If are two sequences which converge to the same limit such that and , for all , then .

We denote by the family of all -functions whose domain is .

In some cases, for a given function , we will also consider the following property:

If are two sequences such that and , for all , then .

In the present research, we will introduce our main result and demonstrate that many of known results which have been invented in the set of metric spaces, with or without -distances, can be deduced easily from our main result. We draw the respondents’ attention to Alegre et al. [3, Theorem 2, Corollary 1] about Meir-Keeler contraction in metric spaces with -distances which can be acquired and generalized with Theorem 9 (for more detail see Corollary 13).

3. Main Result

The following definition plays the crucial role in our main result.

Definition 8. Let be a metric space, let be -distance on , and let be a mapping. We say that is -contraction, if there exists -function such that and

for all . In such case, we say that is -contraction with respect to . The family of all -contractions from into itself, with respect to , is denoted by or, when no confusion is possible, by .

In what follows, we assume that is a symmetric -distance, in the sense of Definition 3, unless the contrary is asserted. Now we are ready to give the main result.

Theorem 9. Let be a complete metric space, let be -distance on , and let be -contraction with respect to . Assume that, at least, one of the following conditions holds:

(a) is -continuous; that is, for any sequence , if then as .(b)The function satisfies .

Then has a unique fixed point.

Proof. Let be chosen arbitrary. If , then we have nothing to prove. So let . Define . Similarly one can define the Picard sequence , for all , such that . Let and divide the proof into three steps.
Step  1. is -asymptotically regular; that is,
is -contraction with respect to . Thus

By of Definition 7 we have and desired result is obtained.
Step  2. is a Cauchy sequence and so it converges to some .
On the contrary, assume that failed to be a Cauchy sequence. By (iii) of Lemma 2, failed to be -Cauchy sequence. Hence, there exists such that, for all , there exist subsequences and of and can be chosen as the smallest integer such that and for all

Obviously, , because

Also, applying the symmetry property we have

Therefore, . Now, taking , , and and by the fact that , we have

By of Definition 7, we have and this is a contradiction. Therefore, is -Cauchy sequence and by (iii) of Lemma 2 is Cauchy. Thus it converges to some .
Step  3. is a unique fixed point and .
If we consider a sequence , in a sense of Lemma 2, is -Cauchy sequence; that is, for all , , . Also as and is a lower semicontinuous, for all . Hence

It means that as .
Now suppose that holds. Since is -continuous mapping, and for all , specially and , we have

and so

Therefore, . By (i) of Lemma 2, we have and so is the fixed point of .
On the other hand, let hold. For all , suppose that and . By (8), we have . In addition, for all ,

Henceforth, if , then . So ; thus .
Now consider the following set:

One can consider two cases:

(a₁) is a finite set:  in this case, one can choose , such that , for all , and so , for all . Applying (11) concludes that . It means that and so applying (i) of Lemma 2 yields .(a₂) is infinite set:  in this case, one can find a subsequence of in . We have

 for all . Analogous the argument in the first case and applying (i) of Lemma 2 results again.

Moreover, if are two distinct fixed points for , then by considering we have

Applying deduces that and so . Applying the same argument on concludes that and (i) of Lemma 2 eventuates .

Example 10. The following items are some examples of -functions:

(a), where is a mapping such that , for all ,(b), where is a mapping such that, for all sequence ,

(c), where is -function in the sense of Definition 4,(d)every simulation function ,(e), where are two functions such that is nondecreasing and continuous from the right, is lower semicontinuous, and ,(f),(g),(h).

Proof. It is sufficient to show that and of Definition 7 and hold for all (a)–(h).

(a) 

Let be a sequence and . Then

 It means that

 and so taking limit on both sides of (17), we have

 By ratio test, and then concludes that .Let be two sequences which converge to the same limit , , and for all . Suppose that ; then and so

 By taking limit on both sides of (19), we have

 which is a contradiction. Thus, .Let be two sequences such that and . Therefore,

 so which implies that .

(b) 

Let be a sequence such that . Therefore,

 Since and , one can see easily

 Hence, is a strictly decreasing sequence of nonnegative real numbers and so converges to . On the contrary, assume that . Therefore,

 Taking limit on both sides of (24) deduces that . Since is a Geraghty function, so which contradicts . So .Let be two sequences which converge to the same limit , hold, and hold. On the contrary, assume that . Therefore,

 and so

 Taking limit on both sides of (26) deduces that . Since is a Geraghty function, it concludes that which contradicts . So .Let be two sequences such that and . Therefore,

 so which implies that .

(c) 

Let be a sequence and let . Then

 It means that

 Therefore, is a nonincreasing sequence; so is converges to . On the contrary, suppose that . By considering , there exists such that, for all , . Since , there exists such that, for all , we have . Therefore, and this is a contradiction to . So .Let be two sequences which converge to the same limit , hold, and hold. On the contrary, assume . Therefore,

 Also

 Since is -function, taking , there exists such that

 Since , there exists such that . Applying (32), one deduces that which is a contradiction, because . So .Let be two sequences such that and . Therefore,

 so which implies that .

(d) Suppose that is a simulation function.

Let be a sequence and let . Therefore . Hence is a strictly decreasing sequence so it converges to . On the contrary, suppose that . Let and . Then and are convergent to and . By condition we have

 and this is a contradiction. So .Let be two sequences which converge to the same limit , , and . In order to prove that , assume that . On the other hand, shows that so . Applying , one can yield that

 which is a contradiction. So .Let be two sequences such that and . Since is a simulation function . Hence, , which means that .

(e)First of all, we show that, for ,

 Let and ; we are going to show that . As is nondecreasing,

 which implies that . Hence and this is a contradiction. So we have .

Let be a sequence and let . By (36), . Hence is a strictly decreasing sequence so it converges to where ; hence , for all . On the contrary, suppose that . Therefore,

 and so . Since is continuous from the right and , we deduce that . Since is lower semicontinuous,

 Hence , so .Let be two sequences which converge to the same limit , verifying , and . By (36) one can see . In order to prove that , assume that . Therefore,

 and so . As is continuous from the right and , we deduce that . Since is lower semicontinuous,

 Hence, , so .Let be two sequences such that and . By (36), one can see and this implies that .

(f) 

Let be a sequence and let . It means that

 Hence,

 Therefore,

 Suppose that . If , then by letting we deduce that

 which is a contradiction. So .Let be two sequences which converge to the same limit , , and , for all . We have

 Therefore,

 Hence,

 Letting tend to infinity, we deduce , so and hence . It means that .Let be two sequences such that and . Therefore,

 so , which implies that .

(g) 

Let be a sequence and let . It means that

 Hence,

 Therefore,

 Suppose that . If , then taking limit on both sides of (52), we deduce that

 which is a contradiction. So .Let be two sequences which converge to the same limit , , and , for all . On the contrary, suppose that . We have

 Therefore,

 Hence,

 Letting , we deduce that , so and hence and this is a contradiction. So .Let be two sequences such that and . Therefore,

 so which implies that .

(h) 

Let be a sequence and let . It means that

 Thus,

 Therefore,

 Suppose that . If , then by letting , we deduce that

 which is a contradiction. So .Let be two sequences which converge to the same limit , , and , for all . We have

 Therefore,

 Hence,

 Letting , we deduce that and so .Let be two sequences such that and . Therefore,

 so which implies that .

The following corollary is the generalization of Reich’s theorem (see [21, 24]).

Corollary 11. Let be a complete metric space, let be -distance on , and let be a mapping such that

for all , where is a mapping such that , for all . Then, has a unique fixed point.

Proof. Defining , applying (a) of example, and using Theorem 9, one can conclude desired result.

The following corollary is the generalization of Geraghty fixed point theorem (see [9, Theorem 1.3]).

Corollary 12. Let be a complete metric space, let be -distance on , and let be a mapping such that

for all , where is a mapping such that, for all sequence ,

Then, has a unique fixed point.

Proof. Defining , applying (b) of example, and using Theorem 9, one can conclude desired result.

The following corollary is the generalization of Meir-Keeler fixed point theorem (also see [24, Theorem 1]).

Corollary 13. Let be a complete metric space, let be -distance on , and let be a mapping such that

for all , where is -function in a sense of Definition 4. Then has a unique fixed point.

Proof. Defining , applying (c) of example, and using Theorem 9, one can conclude desired result.

The following corollary is the generalization of Khojasteh’s result [19, Theorem 2.8] (see also [25]).

Corollary 14. Let be a complete metric space, let be -distance on , and let be a mapping such that

for all , where is a simulation function. Then has a unique fixed point.

Proof. Applying , (d) of example and considering Theorem 9, one can conclude desired result.

Corollary 15. Let be a complete metric space, let be -distance on , and let be a mapping such that

for all , where are two functions such that is nondecreasing and continuous from the right, is lower semicontinuous, and . Then, has a unique fixed point.

Proof. Considering , (e) of example, and applying Theorem 9, one can conclude desired result.

In the following, we are going to present some new corollaries to show that our results are constructive and new results can be built by -contractions.

Corollary 16. Let be a metric space, let be -distance on , and let be a self-mapping. Let, for every , with , the following hold:

Then, is -contraction.

Proof. Defining , applying (f) of example, and using Theorem 9, one can conclude desired result.

Corollary 17. Let be a metric space, let be -distance on , and let be a self-mapping. Let, for every , with , the following hold:

Then, is -contraction.

Proof. Considering , applying (g) of example, and using Theorem 9, one can conclude desired result.

Corollary 18. Let be a metric space, let be -distance on , and let be a self-mapping. Let for every , with ,

Then, is -contraction.

Proof. Defining , applying (h) of example, and using Theorem 9, one can conclude desired result.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

Farshid Khojasteh would like to thank to Science and Research Branch of Islamic Azad University to support this research.

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