In this manuscript, two new classes of generalized weakly contractions are introduced and common fixed point results concerning the new contractions are proved in the context of rectangular -metric spaces. Also, some examples are included to present the validity of our theorems. As an application, we provide the existence and uniqueness of solution of an integral equation.

1. Introduction

In the field of nonlinear analysis, Banach fixed point theorem, which is introduced by Banach [1], is a powerful and classical means to deal with problems on fixed points in metric spaces. It is widely used in many disciplines of mathematics and has been promoted in many aspects. One important extension is to extend the concept of metric spaces. -metric spaces and rectangular metric spaces are regarded as two well-known generalizations of metric spaces.

As a extension of a metric space, -metric space was firstly introduced by Czerwik [2], by modifying the third condition of metric function. In that paper, the author provided fixed point results for contraction conditions in this type space. Afterwards, some authors have obtained many excellent results concerning fixed point theory of a lot of new types of contractive mappings on -metric spaces. Generalizing the results of Berinde [3], Zada et al. [4] obtained fixed point results for mappings with rational type and Pacurar [5] got fixed point theorems of -contractions. In [6], common fixed point results for weak -contraction mappings were proved in this type spaces by Aydi et al. In 2019, problems about periodic common fixed point were studied by Hussain et al. [7]. Recently, in [8], Gopal et al. explored the latest researches and developments on theory of fixed point in the framework of -metric spaces. Younis et al. [9] introduced new fixed point results for the underlying mappings in the framework of dislocated -metric spaces. In [10], in -metric-like spaces, the authors extended the concept of Kannan mappings in view of -contraction. Lately, Younis et al. [11] presented the notion of graphical extended -metric spaces and discussed the framework of an open ball in this new type space.

In 2000, by changing triangular inequality to quadrilateral inequality, more general inequality, Branciari [12] introduced the concept of rectangular metric spaces. Also, the author extended the Banach contraction mapping principle for this new context. Subsequently, a lot of fixed point theorems of various contractive conditions in rectangular metric spaces were obtained. Lakzian et al. [13] established fixed point theorems dealing with -weakly contraction conditions in this type space, which was ulteriorly extended by Erhan et al. in [14]. Bari and Vetro [15] got common fixed point results on given functions with -weakly contractive conditions. In [16], George and Rajagopalan studied problems of common fixed points of -contractive mappings. Lately, in complete rectangular metric spaces, Wang and Pei-Sheng [17] gave generalised -contraction mappings which can be regarded as generalized Suzuki-Berinde type -contraction mappings and provided conditions which ensured this type mapping possesses a unique fixed point. By the help of -functions, in [18], some fixed point results were established by Budhia et al. In graphical rectangular -metric spaces, some errors from literature [19] were rectified by Younis et al. in [20].

Inspired by results of Czerwik [2] and Branciari [12], George et al. [21] extended -metric space and rectangular metric space by introduced rectangular -metric space. In that paper, the authors presented an analogue of Banach fixed point theorem and fixed point theorem of Kannan. After that, many researchers had solved problems of fixed point of new type of contractive mappings on this type space. Kadelburg and Radenovic [22] and Mitrovic [23] presented common fixed point theorems in this type space. In the setting of rectangular -metric spaces, a Boyd-Wong type theorem was studied by Ding et al. in [24]. Sukprasert et al. [25] presented the concept of weak altering distance function and discussed fixed point result of a new generalized contractive mapping. Roshan et al. [26] gave some fixed point theorems concerning almost generalized weakly contractive mappings and rational type contractions. In [27], Mitrovic obtained an analogue of Banach contractive mapping principle and solved an open problem arose in [21]. Recently, Sunarsini et al. [28] introduced a new extension of metric space named as complex valued rectangular metric space and gave an example of Banach contractive mapping principle at linear equation system. In [29], common coupled fixed point theorems concerning generalised contraction conditions were studied by George and Reshma. Lately, in ordered partial rectangular metric spaces, Asim et al. [30] established some ordered-theoretic fixed point results of Geraghty-weak contractive mappings.

In 1997, by using the notion of weak contractive mappings, Alber et al. [31] extended Banach contraction mapping principle in Hilbert spaces. In [32], weak contraction principle was generalized to metric spaces by Rhoades. After that, many authors had generalised the weak contraction principle. For example, in [33], the authors obtained the fixed point results involving - contraction conditions and applied them to solve quadratic integral equations. In [34], Jamal et al. used -weak contraction to extend coincidence point theorems obtained in partially ordered -metric spaces.


Hao and Guan [35] proved common fixed point result dealing with a new class of generalized weakly contraction conditions in complete -metric spaces as follows.

Theorem 1 (see [35]). Let be a complete -metric space with coefficient . Let be self-mappings such that is injective and where is closed. Assume that is a fixed number and is lower semicontinuous. If there exist and satisfying where then and possess a unique coincidence point in . Further, if and are weakly compatible, then and have a unique common fixed point.

Continuing in the same direction, our aim is to give two new classes of generalized weakly contractions and establish some common fixed point theorems dealing with the new contractions in the setting of rectangular -metric spaces. Moreover, we present some examples that elaborate the validity of our theorems. Also, as an application, we prove the existence of solution of an integral equation.

2. Preliminaries

First, we recall some definitions and lemmas as follows:

Definition 2 (see [2]). Let be a nonempty set and be a constant. A function is said to be a -metric iff (i) iff for (ii) for (iii)there exists a real number satisfying for

Usually, we call a -metric space with coefficient .

Definition 3 (see [12]). Let be a nonempty set. A function is said to be a rectangular metric iff (i) iff for (ii) for (iii) for and all different points

In general, we call a rectangular metric space.

Definition 4 (see [21]). Let be a nonempty set and be a constant. A function is said to be a rectangular -metric iff (i) iff for (ii) for (iii)there exists a real number satisfying for and all different points

As usual, we call a rectangular -metric space with coefficient .

Remark 5. It is obvious that a rectangular metric function becomes a metric function when and a rectangular -metric function becomes a rectangular metric function when , whereas the converse of this statement may not be true (see [21], Examples 1.4 and 1.5).

Example 1. Let , where , . Define with for and By calculation, we get is a rectangular -metric space as , whereas we obtain the following results: (1) is not a metric space, as(2) is not a rectangular metric space, as(3) is not a -metric space with , as

Example 2. Assume is a metric space. For , define . Then, is a rectangular -metric space with parameter .

Proof. One can verify easily the conditions (i) and (ii) hold by definition of In order to check (iii), we can infer from the following inequality: Then, for and all different points , we have That is, is a rectangular -metric space when .

Definition 6 (see [21]). Let be a rectangular -metric space with coefficient . A sequence in is called: (i)convergent sequence iff there is such that as (ii)a Cauchy sequence iff when

Furthermore, a rectangular -metric space is called completeness iff every Cauchy sequence is convergent.

Remark 9. In rectangular -metric spaces, one can show that the limit of a sequence may not unique and every convergent sequence in a rectangular -metric space may not be a Cauchy sequence(see [21], Example 1.7).

Definition 8 (see [36]). Let and be two self-maps defined on a nonempty set . If , for some then is called the point of coincidence of and , where is said to be the coincidence point of and . Let represent the collection of all coincidence points of and .

Definition 9 (see [36]). Let and be two self-maps defined on a nonempty set . Then, and are called weakly compatible mappings when they commute at each coincidence point, i.e., for each .

Lemma 10 (see [26]). Let be a rectangular -metric space with parameter . Assume that and are convergent to and , respectively. Then, one can get Moreover, if , then we have . Further, for , we deduce

3. Main Results

In this section, a few of new common fixed point results on generalized weakly contractive conditions in a complete rectangular -metric space will be presented. Moreover, two examples will be provided to prove the validity of our theorems.

Suppose is a rectangular -metric space. A mapping is named as a lower semicontinuous mapping if, for and is convergent to , one get

Let represent the set of all functions . We shall consider the contractive conditions defined by the family :

Lemma 11 (see [37]). Let be a nondecreasing and upper semicontinuous mapping. Then, for any iff as .

Definition 12. Let be a rectangular -metric space with coefficient . Let and be given functions and be a real number. A function is called --admissible function if, for all implies .

Definition 13. Let be a rectangular -metric space with coefficient . Let and be three given mappings. Suppose that is a real number and is a lower semicontinuous function. A mapping is called a generalized contractive mapping, if there exist and satisfying for all with and , where Let be a mapping. Set
If is a sequence in satisfying as , then there is a subsequence of with for
For one can get the condition of and

Theorem 14. Let be a complete rectangular -metric space with coefficient . Let be given self-mappings satisfying and is closed. Assume that is a lower semicontinuous mapping and . If (i) is --admissible(ii) is generalized contractive(iii)there is satisfying (iv)properties and are fulfilled(v) satisfies transitive property, i.e., for then and possess a unique point of coincidence. Furthermore, if and are weakly compatible, then and possess a unique common fixed point in .

Proof. It follows from condition (iii) that one can choose an with . Define sequences and in by for . If for some , then we deduce and and possess a point of coincidence. Next, we suppose that for . In light of contraction condition (i), we obtain Hence, for all , we deduce . Applying (14) with and , where If we assume that for some , according to (18), (19), and (20), we have which is a contradiction. Thus, It follows from (22) that is decreasing. It follows that there exists a real number satisfying In view of (18), (23), and (24), one can obtain If putting in (26), we obtain a contradiction. Hence, which implies that and In view of hypothesis (v), we have . Taking and in (14), we obtain where If for some , , according to (29), (30), and (31), we get which is a contradiction. It follows that Inequality (33) yields that is non-increasing and which yields that there exists satisfying In light of (32), (34), and (35), one can deduce Assume that Letting in (37), we derive which gives a contradiction. This yields that It follows that .
Now, we aim to show that is a Cauchy sequence. Assume on the contrary that, is not Cauchy. So, there exists for which we can choose sequences and of such that is the smallest index for which , In light of the rectangular inequality and (40) and (41), we have Taking the superior limit as , we have Similarly, It follows from (40), (41), and (42) that By (40), (41), (44), and (46), we get By the similar method, we have so is According to the definition of , we get Taking the superior limit as in (51), we get Also, we have It follows that The transitivity property of yields that . Taking and in (14), one can deduce