Abstract and Applied Analysis
Volume 2017 (2017), Article ID 9718535, 10 pages
https://doi.org/10.1155/2017/9718535

Research Article

A New Class of Contraction in -Metric Spaces and Applications

Preeti Kaushik,¹ Sanjay Kumar,¹ and Kenan Tas²

¹Department of Mathematics, DCRUST, Murthal, Sonepat 131 039, India
²Department of Mathematics and Computer Science, Çankaya University, Ankara, Turkey

Correspondence should be addressed to Preeti Kaushik; preeti1785@gmail.com

Received 7 March 2017; Revised 27 April 2017; Accepted 8 May 2017; Published 11 June 2017

Academic Editor: Kunquan Lan

Copyright © 2017 Preeti Kaushik 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.

A novel class of --contraction for a pair of mappings is introduced in the setting of -metric spaces. Existence and uniqueness of coincidence and common fixed points for such kind of mappings are investigated. Results are supported with relevant examples. At the end, results are applied to find the solution of an integral equation.

1. Introduction and Prelims

Fixed point theorems in metric spaces and generalized metric spaces provide a tool to solve many problems and have applications in nonlinear analysis and in many other fields. Most of the problems of applied mathematics reduce to solve a given equality which in turn may be reduced to find the fixed points of a certain mapping or the common fixed points of pairs of mappings. In order to solve particular problems, researchers tried to generalize various contraction conditions, auxiliary mappings, and metric spaces. Here, in this paper, we will present common fixed point theorems for - -contractive mappings in the framework of b-metric space. Therefore, to have a clear understanding of the paper, we will discuss the b-metric space, Geraghty type mappings, and -admissible mappings step by step.

The b-metric space or metric type space was introduced by Czerwik [1] in 1993. In this interesting paper, Czerwik [1] generalized the Banach contraction principle in the context of complete b-metric spaces. After that many researchers reported the existence and uniqueness of fixed points of various operators in the setting of b-metric spaces (see, e.g., [2–13] and some references therein).

Let us have a look on definitions, examples, and properties of b-metric space.

Definition 1 (see [1, 4]). Let X be a nonempty space, and let be a given real number. A functional is said to be b-metric if the following conditions hold good:

(1) = 0 if and only if .(2).(3), for all , and .

If d satisfies all the above b-metric axioms, then the pair is called a b-metric or metric type space.

The class of b-metric spaces is larger than that of metric spaces, since a b-metric is a metric when . All the metric spaces are b-metric spaces but not vice versa. We can illustrate this with the help of the following example.

Example 2. Let and , = = = = , for and , .
Then

and if , the ordinary triangle inequality does not hold.

Definition 3 (see [1]). Let be a b-metric space.

(a)A sequence in is called b-convergent if and only if there exists such that as .(b)The sequence in is said to be b-Cauchy if and only if , as n, .

The b-metric space is called b-complete if every b-Cauchy sequence in is b-convergent.

The following lemmas are useful to prove our results.

Lemma 4 (see [9]). Let be a b-metric space and let be a sequence in such that

If is not a b-Cauchy sequence, then there exist and two sequences and of positive integers such that the four sequences

exist and the following hold:

Lemma 5 (see [9]). Let be a b-metric space and let and be b-convergent to and , respectively. Then one has

In particular, if , then one has . Moreover, for each , one has

Next, we address briefly the concept of Geraghty type mappings.

In 1973, Geraghty [14] generalized the Banach contraction principle in the setting of complete metric spaces by considering an auxiliary function. This function is known as Geraghty type function or mapping. Later on, many authors [11, 15–17] characterized the result of Geraghty in the context of various metric spaces and proved many interesting results. The definition of this new class of mapping is as follows.

Definition 6. Let denote the class of real functions satisfying the condition

For instance, consider the function given by for and ; here .

In 1973, Geraghty generalized the Banach contraction principle in the following form.

Theorem 7 (see [14]). Let be a complete metric space, and let be a self-map.
Suppose that there exists such that

holds for all . Then has a unique fixed point and for each the Picard sequence converges to when .

In 2011, Dukic et al. [16] introduced the Geraghty type functions in b-metric space as follows.

Definition 8 (see [16]). Let be a b-metric space with given . Consider the class of real functions satisfying the property

An example of a function in is given by for and .

Lastly, we discuss the -admissible mappings, examples, and alpha-admissibility for a pair of mappings.

The concept of -admissible mappings was introduced by Samet et al. [18] in 2012. They established some fixed point theorems for such mappings in complete metric spaces and showed some examples and applications to ordinary differential equations. Since then, many researchers extended the idea and generalized fixed point results for single-valued and multivalued -admissible mappings in various abstract spaces (see, e.g., [2, 3, 11, 19, 20] and more references in the literature). The following definitions and examples reveal the basics of -admissible mappings.

Definition 9 (see [18]). A self-mapping defined on a nonempty set is -admissible if, for all , one has

where is a given function under consideration.

Example 10. Let and assume that and by , for all , and

Then F is -admissible map.

In 2014, a new notion of h--admissible mapping was introduced by Rosa and Vetro [20]. The definition of this notion is as follows.

Definition 11 (see [20]). Let be two self-mappings defined on a nonempty set . And consider the map . Then mapping F is called h--admissible if, for all ,

Example 12. Let . We will define the mapping by

and consider the mappings by and for all . Then, the mapping F is h--admissible mapping.

Allahyari et al. [2] defined the -regular space with respect to some self-mapping defined on space as given below.

Definition 13 (see [2]). Let be a b-metric space. Suppose that and are two operators. is -regular with respect to if, for every sequence , for all and as ; then there exists a subsequence of such that, for all , .

Definition 14 (see [11]). Let X be a nonempty set. Then the map is called transitive if for one has

In 2014, Sintunavarat [11] proved the fixed point theorem for generalized --contraction mapping in metric space and established Ulam-Hyers stability and well-posedness via fixed point results. The purpose of this paper is to define --contraction mapping for a pair of mappings and to prove coincidence point and common fixed point theorems in the context of b-metric space. We will provide suitable examples to support our results. At the end, we will discuss some applications in the context of integral equations.

2. Coincidence and Common Fixed Point Theorems

First of all, we will present some definitions and results which will make our results easy to understand.

Definition 15. Let and be two self-mappings defined on a nonempty set . Then a point is called coincidence point of and if . Moreover, if , then is called common fixed point of F and .

Definition 16. Two self-mappings and defined on a nonempty set are weakly compatible if the maps commute at each coincidence point; that is, if , for some , then .

In 2014, Sintunavarat [11] defined the generalized --contraction mapping in metric spaces as follows.

Definition 17 (see [11]). Let F be a self-mapping on a nonempty set X and there exist two functions and . We say that F is --contraction mapping if the following condition holds:

for all , where .

Now, we will introduce our notions in the setting of b-metric space.

Definition 18. Let F and be two self-mappings defined on b-metric space with given and there exist two functions and . We say that F is - (b)-contraction with respect to if the following condition holds:

for all , where .

Next, we present the coincidence and common fixed point theorems for a new class of contraction in b-metric space.

Theorem 19. Let be a complete b-metric space and let be two self-mappings such that and one of these two subsets of X is complete. Suppose that F is - (b)-contraction with respect to mapping satisfying the following conditions:

(i)F is - -admissible and is transitive mapping.(ii)There exists such that (iii)X is -regular with respect to .

Then F and have a coincidence point.

Moreover, if F and are weakly compatible and hypothesis has one more additional assumption,

(A1)either or whenever and ,

then F and have a unique common fixed point.

Proof. Let such that . In order to prove that F and have a point of coincidence and using , define two sequences and in X such that

Now, if for any , then and F and have a point of coincidence.
Without loss of generality, one can suppose that for each .
Hence, is --admissible and . Similarly, . By induction, we can easily deduce

Step 1. We shall show that . Now, it follows from (19) and (21) for each that

This implies that

Further, from (23), it follows that

Step 2. Next, we show that is a Cauchy sequence in b-metric space . On the contrary, assume that is not a Cauchy sequence. Then there exists and subsequences of integers and with such that

Since and is transitive mapping, we can deduce easily, by using triangle inequality,

Now, using (19) and (26), we can consider

This gives

with for all .
Hence, by Lemma 4 and (29), we have

that is,

Also, implies that . However, this is not possible, as, using (4) of Lemma 4, we get that

as , which leads to contradicting (25).
Therefore, is a b-Cauchy sequence and by hypothesis we can suppose that is to be complete subspace of X (proof can be derived in a similar manner when is supposed to be complete). Then b-completeness of implies that b-converges to a point , where for some , or in other words

Step 3. Next, we will prove that . Since X is -regular with respect to and using (33), we have

To show , we consider

This in turns implies that

Thus, is a point of coincidence of F and .
Step 4. Next, we prove that F and have a common fixed point. Firstly, we claim that if and , then . By hypotheses, or . Suppose that and consider that

Then we get that

which is a contradiction.
Therefore, we conclude that . If we take , then we also get the same result.
Secondly, If F and are weakly compatible, then, for any , if (because F and have a point of coincidence, proven earlier in Step 3), then, using weak compatibility of F and , we get

Thus is a coincidence point of F and ; then, using the result in first part, , which leads to .
Therefore, is a common fixed point of F and .
We can prove the uniqueness of the common fixed point of F and by making use of condition (19) and assumption (A1) of hypothesis. The proof is very simple; therefore we do not go through details.

If we consider as identity mapping in the above theorem, we deduce the following corollary.

Corollary 20. Let be a complete b-metric space and let be continuous -admissible mapping satisfying the following condition:

for all and , where .
If is transitive mapping and there exists such that , then F has a fixed point.

Taking and in Corollary 20, we get the following variant of Geraghty theorem.

Corollary 21. Let be a complete b-metric space and let be self-mapping satisfying the following condition:

for all and . Then F has a unique fixed point and, for each , the Picard sequence converges to when .

The following lemma derived from [7] is very useful to prove our next theorem.

Lemma 22 (see [7]). Let be a sequence in a metric type space or b-metric space (X, d) such that

for some and each . Then is a Cauchy sequence in X.

Theorem 23. Let be a complete b-metric space and be -regular with respect to . And let be two self-mappings, where F is - -admissible and is transitive, and, for , one has . If and one of these two subsets of is complete, suppose that there exists such that

for all and .
Consider

Then F and have a coincidence point.

Moreover, if F and are weakly compatible and hypothesis has one more additional assumption,

(A1)either or whenever and ,

then F and have a unique common fixed point.

Proof. Let be arbitrary and, using condition , we can easily construct a Jungck sequence satisfying

Now, if for any , then and F and have a point of coincidence.
Without loss of generality, we assume that for each .
Hence, F is --admissible and . Similarly, . By induction, we can easily deduce

Step 1. We shall show that is a b-Cauchy sequence. Now, taking (43) and (46), we obtain, for each ,

This implies that

or

Now, if max = , then < .
Then (49) implies that

which is impossible as .
Therefore, we deduce that max .
This in turn implies that

Using Lemma 22, we obtain that is a b-Cauchy sequence and by hypothesis we can suppose that is to be complete subspace of (the proof when is similar). Then b-completeness of implies that b-converges to a point , where for some , or in other words

Step . Next, we will prove that . Since is -regular with respect to and using (52), we have

To show , we consider

This implies that

or

Also, and as , implying that

Then we have only two cases.
Case 1

Since > 0, it follows that .
Case 2

Since , it follows that
Uniqueness of the limit of a sequence implies that .
Using conditions (A1) and weak compatibility of F and of hypothesis, we can easily prove that F and have a unique common fixed point.

From the above theorem, one can deduce the following corollary easily.

Corollary 24. Let be a complete b-metric space and is continuous and -admissible. Suppose that there exists such that

for all and .
Consider

If is a transitive mapping and there exists such that , then has a fixed point.

Taking and , we get the following variant of Corollary 3.12 of [7].

Corollary 25. Let be a complete b-metric space and let be self-mapping. Suppose that there exists such that, for all ,

where

Then F has a unique fixed point

In what follows, we furnish illustrative examples wherein one demonstrates Theorems 19 and 23 on the existence and uniqueness of a common fixed point.

Example 26. Let be a b-metric space with metric d given by with . Consider the mappings defined by , and , and by , , and . Let us take , as for and .
And by

then one can examine easily that

Thus, in all cases, we get

for all , where .

Also we can check that F and satisfy all the other assumptions of Theorem 19 and thus the pair F and has a unique common fixed point .

Example 27. Let be a b-metric space with metric d given by with . Consider the mappings defined by , , and and by , , and . Let us take .
And by

Now, we verify Theorem 23 by considering the following cases.

Case 1 (take points 0 and 1).

Case 2 (take points 1 and 3).

Case 3 (take points 0 and 3).

Thus, in all cases, F and satisfy all the assumptions of Theorem 23 and thus the pair F and has a unique common fixed point .

3. An Application to an Integral Equation

This section deals with the applications of results proven in the previous section. Here, we will investigate the solution of integral equation through our results.

Consider the following integral equation:

where .

Let be the set of continuous real functions defined on . Define the b-metric by for all .

Consider . Then is a complete b-metric space with the constant .

for all and for all . Then the existence of a solution to (71) is equivalent to the existence of a fixed point of F. Now, we prove the following result.

Theorem 28. Let us suppose that the following hypotheses hold:

(i) is continuous.(ii)For all , there exists a continuous operator such that

 where .(iii)There exists a function such that, for all and for all with , we have

Then the integral equation (71) has a unique solution .

Proof. We define by

Then, for all , we have and , implying that for all , which proves that is a transitive mapping.
If for all , then . From (iii), we have , and so . Thus, F is -admissible.
From (iii), there exists such that .
Also we observe from (72) that

which in turn implies that .
Now, all the conditions of Corollary 24 hold and F has a unique fixed point , which means that is the unique solution for the integral equation (71).

4. Conclusion

To conclude, we can assert that our results are novel, interesting, and generalized while considering the alpha-admissible Geraghty type mappings. These results extend, improve, unify, and generalize many theorems based on alpha-admissible mappings in -metric spaces. Examples are presented in a simplest form and illustrate the theorems. Application to integral equation adds value to our research.

Conflicts of Interest

All the authors of this article declare that they have no conflicts of interest regarding the publication of this article.

Authors’ Contributions

All authors have equal contribution in writing this paper. All authors read and approved the final paper.

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