Abstract

A unique fixed point theorem for three self-maps under rational type contractive condition is established. In addition, a unique fixed point result for six continuous self-mappings through rational type expression is also discussed.

1. Introduction

Fixed point theory is one of the core subjects of nonlinear analysis. This theory is not constrained to mathematics; it is also applicable to other disciplines. It is closely linked with social and medical science, military applications, graph theory [1], game theory, economics [2], statistics, and medicine. This theory is divided into three categories: topological fixed point theory, metric fixed point theory, and discrete fixed point theory.

In metric fixed point theory, the first result proved by Banach [3] is known as Banach contraction principle. Many researchers extended this principle for the study of fixed points and common fixed points using different types of contraction such as weak contraction [4, 5], integral type contraction [6], rational type contraction [7], and T-Hardy Rogers type contraction [8]. For more details, see [911] and so forth.

Dass and Gupta [12] gave the extension of Banach’s contraction mapping principle by using a contractive condition of rational type. Jaggi [7] proved some unique fixed point results through contractive condition of rational type in metric spaces. Harjani et al. [13] studied the results of Jaggi in the setting of partially ordered metric spaces. Using generalized weak contractions Luong and Thuan [14] generalized the results of [13] through rational type expressions in the context of partially ordered metric spaces. Chandok and Karapinar [15] generalized the results of Harjani and established common fixed point results for weak contractive conditions satisfying rational type expressions in partially ordered metric spaces. Mustafa et al. [16] discussed fixed point results by almost generalized contraction via rational type expression which generalizes, extends, and unifies the results of Jaggi [7], Harjani et al. [13], and Luong and Thuan [14], respectively. Fixed point theorems for contractive type conditions satisfying rational inequalities in metric spaces have been developed in a number of works; see [1720] and so forth.

Mustafa and Sims [21] generalized the notion of metric space as an appropriate notion of generalized metric space called -metric space. They have investigated convergence in -metric spaces, introduced completeness of -metric spaces, and proved a Banach contraction mapping theorem and some other fixed point theorems involving contractive type mappings in -metric spaces using different contractive conditions. Later, various authors have proved some common fixed point theorems in these spaces (see [8, 2224]).

Sanodia et al. [25] used rational type contraction and investigated a unique fixed point theorem for single mapping in -metric spaces. Gandhi and Bajpai [26] generalized the result of Sanodia et al. and proved unique common fixed point results for three mappings in -metric space satisfying rational type contractive condition. Recently, Shrivastava et al. [27] established some unique fixed point theorem for some new rational type contraction.

The aim of this paper is to establish two common fixed point theorems satisfying rational type contraction. In the first result, we discuss the existence and uniqueness of common fixed point for three self-maps in the context of -metric space, while in the second one we studied the uniqueness of common fixed point for six continuous self-mappings in the setting of -metric through rational type expression.

2. Preliminaries

We recall some definitions that will be used in our discussion.

Definition 1 (see [21]). Let be a nonempty set and let be a function satisfying the following conditions:(1) implies that for all .(2) for all .(3) for all .(4) for all .Then, it is called -metric and the pair is a -metric space.

Proposition 2 (see [21]). Let be a -metric space. The following are equivalent:(1) is -convergent to .(2) as .(3) as .(4) as .

Definition 3 (see [22, 28]). A pair of self-mappings in a -metric space is said to be weakly commuting if Sanodia et al. [25] proved the following fixed point theorem in the setting of -metric space.

Theorem 4. Let be a -complete -metric space and let be a self-map satisfying the condition for all with . Then, has a unique common fixed point in .

Theorem 5. Let be a -complete -metric space and let be two self-maps such that satisfying the following condition: for all with . Then, and have a unique common fixed point in .

Gandhi and Bajpai [26] proved unique common fixed point results satisfying the following rational type contractive condition.

Theorem 6. Let be a -complete -metric space and let be three self-mappings satisfying the condition for all with . Then, , , and have a unique common fixed point in .

Currently, Shrivastava et al. [27] studied the following result.

Theorem 7. Let be a -complete -metric space and let be a self-map satisfying the condition for all with . Then, has a unique common fixed point in and is -continuous at .

3. Main Results

Our first new result is the following.

Theorem 8. Let be a -complete -metric space and let be three self-mappings satisfying the following condition:for all with , with , . Then, , , and have a common fixed point. Further, if implies , then , , and have a unique common fixed point in .

Proof. Let be arbitrary in ; we define a sequence by the following rules: Now, we have to show that is a -Cauchy sequence in . Consider ; from (6), we havewhich implies that where .
Similarly, Therefore, for all , we haveNow, for all , with , using rectangular inequality, the second axiom of the -metric, and (11), we have where as .
This shows that is a -Cauchy sequence. But is -complete -metric space so there exists in such that as tends to infinity.
Now, we assume that . Using condition (6), we haveAs is -Cauchy sequence and converges to , therefore, by taking limit , we get which is held only if implies that . Similarly, it can be shown that and . Hence, is a common fixed point of , and .
Uniqueness. Suppose that , , and have two common fixed points and such that . Since condition implies , we have that implies . Therefore, one can get the following: which is a contradiction. Therefore, the common fixed point is unique.

Corollary 9. Let be a -complete -metric space and let be three self-mappings satisfying the conditionfor all with with , . Then, , , and have a common fixed point. Further, if implies , then , , and have a unique common fixed point in .

Proof. The proof follows by taking in Theorem 8.

Corollary 10. Let be a -complete -metric space and let be three self-mappings satisfying the condition for all with with , . Then , , and have a common fixed point. Further, if implies , then , , and have a unique common fixed point in .

Proof. The proof follows by taking in Theorem 8.

Corollary 11. Let be a -complete -metric space and let be two self-mappings satisfying the conditionfor all with with , . Then, and have a common fixed point. Further, if implies , then and have a unique common fixed point in .

Proof. The proof follows by taking in Theorem 8.

By setting in Theorem 8, we have the following corollary.

Corollary 12. Let be a -complete -metric space and let be a self-mapping satisfying the conditionfor all with with , . Then, has a unique fixed point. Further, if implies , then has a unique common fixed point in .

The second main result in this section is the following.

Theorem 13. Let be a -complete -metric space. Let be six continuous self-maps and let , , and be weakly commuting pairs of self-mapping such that , , and , satisfying the conditionfor all with with , . Then have a common fixed point. Further, if implies , then have a unique common fixed point in .

Proof. Take as arbitrary point of . Since , we can find a point in such that . For , we can find a point in such that and for we can find a point in such that . Generally, for a point , choose such that ; for a point , choose such that ; and for a point , choose such that for .
Suppose and . Then, from condition (19), we haveHence, where . Continuing this procedure, in the end we getClearly, as . So, ; we get the following sequence: which is a Cauchy sequence in -complete -metric space and therefore converges to a limit point . But all subsequences of a convergent sequence converge; so, we have Since are weakly commuting mappings, thus we have Taking limit and noting that and are continuous mappings, we have which gives the notion that . Analogously, we can get and . We claim that and and then from condition (3)which is a contraction: Similarly, using similar arguments to those given above, we obtain a contradiction for and or for and . Hence, in all the cases, we conclude that . We prove that any fixed point of is a fixed point of , and . Assume that is such that . Now, we prove that . If it is not the case, then, for and , we getwhere which implies that ; in a similar argument, we can prove the other cases.
Uniqueness. Suppose that , and have two common fixed points and such that . Since condition implies , we have that implies , which can be written as or .
Therefore, one can get the following:

Theorem 13 produces the following corollaries.

Corollary 14. Let be a -complete -metric space and let be three self-maps and let , , and be weakly commuting pairs of self-mapping such that , , and , satisfying for all in with with . Then, , and have a unique common fixed point in .

Proof. It follows by taking in Theorem 13.

Corollary 15. Let be a -complete -metric space and let be three self-maps and let , , and be weakly commuting pairs of self-mapping such that , , and , satisfyingfor all in with with , . Then, , and have a common fixed point. Further, if implies , then , and have a unique common fixed point in .

Proof. It follows by taking in Theorem 13.

Corollary 16. Let be a -complete -metric space and let be three self-maps and let , , and be weakly commuting pairs of self-mapping such that , , and , satisfyingfor all with with , . Then, , and have a common fixed point. Further, if implies , then , and have a unique common fixed point in .

Proof. The proof follows by setting and in Theorem 13.

Competing Interests

The authors declare that they have no competing interests.