#### Abstract

In this paper, the notion of sequential -metric spaces has been introduced as a generalization of usual -metric spaces, -metric spaces, metric spaces, and specially of -metric spaces. In view of this notion, we prove some fixed point theorems for some classes of -rational Geraghty JS-contractions over such spaces. A supporting example and an application have been given in order to examine the validity of the obtained results.

#### 1. Introduction and Preliminaries

There is a huge number of extensions of Banach contraction principle. Some of them focus on using different forms of contractive conditions and some of them focus on the various generalized metric spaces. There are many interesting generalization of metric type spaces such as -metric space , extended -metric space , -metric space , -metric spaces , -algebra valued metric space , -metric space, -metric space [6, 7], -metric space , and G-metric spaces . Also, for considering and analyzing more generalization of the concept of metric spaces, one can consider the following works dealing with controlled metric spaces and generalized -metric spaces [10, 11]. In the context of various metric type spaces which are the combination of the above mentioned spaces, several authors have proved different types of fixed point theorems . Now, we present some definitions of some generalized metric spaces which are pertinent to our research.

Let be the class of all strictly increasing continuous functions with and .

Definition 1 (see ). Let be a nonempty set. A function is a -metric if for some and for all , (1) if and only if (2)(3) for all In this case, the pair is called a -metric space, or an extended -metric space.
A -metric  is a -metric with , for some fixed . A -metric space reduces to an ordinary metric space provided that .
Let be a nonempty set and be a mapping. For any , let us define the set

Definition 2 (JS-metric space) (see ). Let be a mapping so that (1) implies (2)for every , we have (3)if and then , for some The pair is called a generalized metric space, usually known as -metric space.
In , Roy et al. introduced a new type of extended -metric spaces. First, let , be defined as above, where is substituted by .

Definition 3 (see ). Let be a nonempty set. A mapping is said to be a sequential -metric if for all : (a) implies (b)(c), where and so that for The triplet is called a sequential -metric space. We represent a sequential -metric space simply as .

Example 4 (see ). Let and the metric be defined by is a sequential -metric on for for all and for all .

Proposition 5 (see ). If is a -metric space, then is also a sequential -metric on .

Proposition 6 (see ). Let be a -metric space with coefficient . Let , where . Then, is a sequential -metric with for all .

Sedghi et al.  have recommended the notion of a -metric space as follows.

Definition 7 (see ). Let be a nonempty set. An -metric on is a function such that, (1) if and only if (2) for all for each .
The pair is nominated an -metric space.
Some examples of such -metric spaces are (1)Let be a normed space. Then, is an -metric on (2)Let be a normed space. Then, is an -metric on (3)Let be a nonempty set and be an ordinary metric on . Then, is an -metric on

Lemma 8 (see ). In an -metric space, we have .

Definition 9 (see ). Let be a nonempty set and . Suppose that a mapping satisfies: (1) iff (2) for all Then is called a modified -metric, and the pair is called a modified -metric space.
An -metric space is an -metric space where and every -metric space with parameter is an -metric space where .

Definition 10. A -metric is called symmetric if , for all

Example 11 (see ). Let be an -metric space. Then (1) is an -metric with (2) is an -metric with

In general, an -metric with nontrivial function need not to be jointly continuous in all its variables (see ). The following simple lemma is a modification of Lemma 1.11 of .

Lemma 12. Let be an -metric space. (1)Suppose that and are -convergent to and , respectively. Then, we haveIn particular, if , then, we have . (2)Suppose that -metric is symmetric and is -convergent to and is arbitrary. Then, we have

Proof. (1) Using the rectangle inequality in the -metric space, it is easy to see that The desired result is obtained via letting in the above inequalities and taking lower limit and the upper limit in the first and second inequality, respectively.
(2) Using the rectangle inequality, we see that The desired result is obtained via taking as in the first inequality and the as in the second inequality, respectively.☐

#### 2. Introduction to Sequential -Metric Spaces

In this section, we introduce a new type of extended -metric spaces. First, let us to define

where is a given mapping and is a nonempty set.

Definition 13. Let be a nonempty set. A mapping is said to be a sequential -metric if for all : (1) implies (2), where and so that for The triplet is called a sequential -metric space (SSPMS). It will be shown simply as

The above definition is an extension of Definition 13 of , that is, the definition of -metric spaces.

Example 14. Let the triplet be a sequential -metric space. If we define , then, will be a sequential -metric with the same control function provided the is subadditive.

According to the above example and Example 15, we construct the following example of -metrics.

Example 15. Let and the metric be defined by is a sequential -metric on for .

Proposition 16. Let be a -metric space with coefficient . Let , where . Then, is a sequential -metric for for all and .

Proof. Here, we show that satisfies all the conditions of Definition 25. (a) gives . Then which implies (b)For all , we have where .

Proposition 17. Let be a SSPMS with control function . Let , where . Then, is a sequential -metric with for all .

Proof. Here, we show that satisfies all the conditions of Definition 25. (a) gives . Then which implies (b)For all , we have where .

Definition 18. Let be a SSPMS. Also, let be a sequence in and . (i) is said to be convergent and converges to if (ii) is said to be Cauchy if (iii) is said to be complete if any Cauchy sequence in is convergent

Definition 19 (see ). Let and be two sequential -metric spaces. A mapping is called continuous at a point if for any there exists such that for any , whenever . is said to be continuous on if is continuous at each Point of .

Proposition 20. In a SSPMS if a sequence is convergent, then, it converges to unique element in .

Proposition 21. Let be a SSPMS and converges to some . Then,

Proof. Since converges to , so . Therefore, we have which implies

Proposition 21. Let be a Cauchy sequence in a SSPMS such that is continuous. If has a convergent subsequence which converges to an , then, also converges to .

Proof. From condition (3) of Definition 25, we have which implies that for all . Due to the Cauchyness of , it follows that and thus as which implies that as , since is continuous. Hence, converges to .

Proposition 22. In a SSPMS if a self mapping is continuous at , then for any sequence

Proof. Let be given. Since is continuous at , then for any there exists such that implies As converges to , so for , there exists such that for all . Therefore, for any , and thus as .

Definition 23. Let be a SSPMS with supporting function . Define for all and .

Remark 24. One can easily check that the collection forms a topology on
Another approach to define a sequential -metric is as follows:
Define where is a given mapping, and is a nonempty set.

Definition 25. Let be a nonempty set. A mapping is said to be a sequential -metric if for all : (1) implies (2), where and and for The triplet is called a sequential -metric space. It will be shown simply as

In this paper, we investigate the existence of unique fixed point for some rational contractions of Jleli-Samet and Geraghty type in ordered sequential -spaces. Our motivation is an interesting generalization of Banach contraction principle which is presented by Jleli et al. in .

#### 3. Main Results

Note that possesses the s.l.c.p. whenever is a nondecreasing sequence in such that , one has for all . From now on, by SSPMS, we mean a sequential -metric space and by SPRGJSC a -rational Geraghty JS-contraction.

Let denotes the class of all functions satisfying the following condition:

The following are two examples of Geraghty functions . (1) for all and for where (2) for all and for where

In the above examples, we assumed that In the case that, for instance, , then , and one can multiple this amount to the above functions .

Definition 26. Let be an ordered SSPMS. A mapping is called a SPRGJSC of type if for some and for all comparable elements , where

Theorem 27. Let be an ordered -complete SSPMS. Let be a -increasing mapping such that for some element . Suppose that be a SPRGJSC of type . If (I) is continuous, or(II) possesses the s.l.c.pthen has a fixed point. Moreover, the set of fixed points of is well ordered if and only if is a singleton.

Proof. Put Since and is -increasing, we obtain by induction that

We will do the proof in the following steps.

Step I. We will show that . Without any loss of generality, we may assume that for all Since for each , then by (16), we have because and if then from (19), we have which is a contradiction.
Hence, So, from (19), That is, is a decreasing sequence, so, there exists such that . We will prove that . Suppose on contrary that . Then, letting , from (22), we have which implies that . Now, as we conclude that which yields that , a contradiction. Hence,

Step II. Now, we show that the sequence is a -Cauchy sequence. Suppose that is not -Cauchy. Then for some we can find two subsequences and of such that and are the smallest index such that This means that From the definition of and the above limits, Now, from (16) and the above inequalities, we have a contradiction. Therefore, is a -Cauchy sequence. -Completeness of yields that -converges to a point .

Step III. is a fixed point of .
First, let is continuous, so, we have Now, let (II) holds. Using the assumption on , we have . Now, we show that . From definition of a sequential -metric, where Therefore, we deduce that , so, .
Finally, suppose that the set of fixed points of be well ordered. Assume that and are two fixed points of such that . Then by (16), we have Because

So, we get , a contradiction. Hence , and has a unique fixed point. Conversely, if has a unique fixed point, then the set of fixed points of is a singleton which is well ordered.

Corollary 28. Let be an ordered -complete SSPMS. Let be an -increasing mapping such that for some element . Suppose that where , , and If (I) is continuous, or(II) possesses the s.l.c.pthen has a fixed point. Moreover, the set of fixed points of is well ordered if and only if has one and only one fixed point.

Definition 29. Let is an ordered SSPMS. A mapping is called a SPRGJSC of type if there exists such that, for all comparable elements , where