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Hongyan Guan, Jianju Li, "Common Fixed-Point Theorems of Generalized Weakly Contractive Mappings in Metric-Like Spaces and Application", Journal of Mathematics, vol. 2021, Article ID 6680381, 14 pages, 2021. https://doi.org/10.1155/2021/6680381
Common Fixed-Point Theorems of Generalized Weakly Contractive Mappings in Metric-Like Spaces and Application
In this paper, we prove some common fixed-point theorems of generalized weakly contractive mappings in metric-like spaces. We also give two examples to support our results. Meanwhile, we present an application to the existence of solutions for a system of integral equations by means of our results.
The Banach contraction mapping theorem  popularly known as Banach contraction mapping principle is a rewarding result in fixed-point theory. It has widespread applications in both pure and applied mathematics. This celebrated principle has been generalized by several authors. Recently, Saleem et al.  obtained fixed-point results of Suzuki-type generalized multivalued -almost contractions and coincidence and common fixed-point results of Suzuki-type generalized multivalued -almost contraction mapping in the setting of metric spaces. In 2019, Li et al.  defined a new contractive-type mapping called -contraction and proved some fixed-point and Suzuki-type fixed-point results in the context of complete metric spaces. Another extension idea is the promotion of spaces. In 1993, Czerwik  introduced the concept of the metric space, and he also obtained some fixed-point theorems of contractive mappings in metric space. Since then, some papers have generalized the fixed-point of different contractive conditions in metric space. For example, Aydi et al.  proved common fixed-point results for weak contractions on metric space, and Abbas et al.  also proved common fixed-point theorems of four mappings in metric space. In , Hussain et al. established some best proximity point and coupled best proximity point results in the context of complete -metric spaces based on the concepts of -proximal admissible mappings and simulation function. Lately, Lael et al.  replied to an open problem related to a -metric version of Banach’s fixed-point theorem and obtained some new fixed-point theorems for multivalued mappings in -metric spaces.
Inspired by Czerwik’s work, some researchers had studied the fixed-point theorems of various new types of contractive conditions in the generalized metric space and metric space. In 2011, Hussain and Shah in  introduced the notion of cone metric spaces, which means that it is a generalization of metric spaces and cone metric spaces. They also considered topological properties of cone metric spaces and results on KKM mappings in the setting of cone metric spaces. Fernandez et al.  introduced the concept of -cone metric spaces over a Banach algebra as a generalization of -cone metric spaces over a Banach algebra and -metric spaces and studied some coupled common fixed-point theorems for generalized Lipschitz mappings in this framework. In 2019, Kanwal et al.  generalized Nadler’s theorem in weak partial -metric space by using weak partial Hausdorff -metric spaces. In 2020, Abbas et al.  introduced the concepts of -contraction and monotone -contraction correspondence in fuzzy -metric spaces and obtained fixed-point results for these contractive mappings. Ansari et al.  introduced the concept of inverse -class function in -metric setting and established some fixed-point theorems. Recently, Saleem et al.  proved some new fixed-point theorems, coincidence point theorems, and common fixed-point theorems for multivalued -contractions involving a binary relation that is not necessarily a partial order, in the context of generalized metric spaces (in the sense of Jleli and Samet).
The concept of contractive mappings was introduced by Rhoades . Afterwards, some researchers introduced a few and weakly contractive conditions and discussed the existence of fixed and common fixed-point for these mappings [16–18]. In particular, Aghajani et al.  presented several common fixed-point results of generalized weak contractive mappings in partially ordered –metric spaces.
Our main concern is to study common fixed-point results involving generalized –weakly contractive conditions in metric-like space. Furthermore, we apply the given results to obtain existence of solutions of integral equations.
Throughout this paper, let denote the set of all positive integers, and . Put
In order to get our main results, we introduce some definitions and lemmas as follows.
Definition 1. (see ). Let be a nonempty set and be a given real number. A mapping is said to be a metric if and only if, for all , the following conditions are satisfied:(1) if and only if (2)(3)
The pair is called a metric space with parameter .
In general, the class of metric space is effectively larger than that of metric space, since a metric is a metric with . We can find several examples of metric spaces which are not metric spaces (see ).
Definition 2. (see ). Let be a nonempty set and be a given real number. A mapping is said to be a metric-like if and only if, for all , the following conditions are satisfied:(1) implies (2)(3)
The pair is called a metric-like space with parameter .
Remark 1. We should note that in a metric-like space , if and , then . But the converse need not be true, and may be positive for .
Example 1. Let , and let the mapping be defined by for all . Then, is a metric-like space with parameter .
Proof. We can infer from the convexity of the function () that holds. Then, for , we haveIn this case, is a metric-like space with parameter .
Definition 3. (see ). Let be a metric-like space with parameter and be a sequence in .(1)The sequence is said to be convergent to if .(2)The sequence is said to be a Cauchy sequence if and only if exists and is finite.(3) is said to be complete if for each Cauchy sequence in , there exists an such that .
Definition 4. (see ). Let and be two self-mappings on a nonempty set . If , for some , then is said to be the coincidence point of and , where is called the point of coincidence of and . Let denote the set of all coincidence points of and .
Definition 5. (see ). Let and be two self-mappings defined on a nonempty set . Then, and are said to be weakly compatible if they commute at every coincidence point, that is, for every .
Lemma 1. (see ). Let be a metric-like space with . We assume that and are convergent to and , respectively. Then, we haveIn particular, if , then we have . Moreover, for each , we haveIn particular, if , then
Lemma 2. (see ). Let be a metric-like space with . Then,(1)If , then (2)If is a sequence such that , then we have (3)If , then
3. Main Results
In this section, we will show the existence and uniqueness of common fixed-point for generalized weakly contractive mappings in complete metric-like space. Meanwhile, we give two examples to support our results.
Theorem 1. Let be a complete metric-like space with parameter and let be given self-mappings satisfying where is a closed subset of . If there are functions and such thatwherethen and have a unique coincidence point in . Moreover, and have a unique common fixed-point provided that and are weakly compatible.
Proof. Let . As , there exists such that . Now we define the sequences and in by for all . If for some , then we have and and have a coincidence point. Without loss of generality, we assume that (by Lemma 2, we know that ) for all . Applying (6) with and , we obtainwhereIf , for some , in view of (9) and (10), we haveIt follows from inequality (8) and the above inequalities thatwhich implies , that is, , a contradiction. Hence, and is a nonincreasing sequence, and so there exists such that
By virtue of (9) and (10), we haveIt follows thatNow suppose that . By taking the limit as in (12), we have , a contradiction. This yields thatNow we shall prove that . Suppose on the contrary that . It follows that there exists for which one can find sequences and of where is the smallest index for which , and
In view of the triangle inequality in metric-like space, we getUsing equality (15) and taking the upper limit as in the above inequality, we obtainAs the same arguments, we deduce the following results:In view of (18), we haveUsing (19) and (20), we obtainSimilarly, we deduce thatIt follows thatThrough the definition of , we havewhich yields thatAlso,It is easy to show thatApplying (6) with and , we getIn light of (26), one can obtainwhich implies thata contradiction to (28). It follows that is a Cauchy sequence in and . Since is complete metric-like space, there exists such thatFurthermore, we have since is closed. It follows that one can choose a such that , and one can write (32) asIf , taking and in contractive condition (6), we getwhereand then we obtainTaking the upper limit as in (34),which implies that
It follows that . That is, . Therefore, is a point of coincidence for and . We also conclude that the point of coincidence is unique. Assume on the contrary that there exist and ; applying (6) with and , we obtain thatHence, . That is, the point of coincidence is unique. Considering the weak compatibility of and , it can be shown that is a unique common fixed-point. This completes the proof.
Theorem 2. Let be a complete metric-like space with parameter and let be given self-mappings satisfying where is a closed subset of . If there are functions and such thatwherethen and have a unique coincidence point in . Moreover, and have a unique common fixed-point provided that and are weakly compatible.
Proof. It is the same as the proof of Theorem 1, and we also define the sequences and in by for all . We also suppose that for each , and it follows from (39) thatwhereIf we assume that for some ,then from inequalities (42) and (43), we get thatIn view of (41), we have the following inequality:which gives that , a contradiction to . It follows that . Hence, is a nonincreasing sequence. Consequently, the limit of the sequence is a nonnegative number, say . That is, .
According to (42) and (43), we haveSo,If , then letting in above inequality, we obtain that which implies that , i.e.,
Now we prove that . If not, as the proof of Theorem 1, there exists for which one can find sequences and of so that is the smallest index for which , and the following inequalities hold:We deduce the following equations according to the definitions of and :using (49), one can obtain thatTaking and in (39), we getTherefore, we haveand we conclude that which gives a contradiction to (52). Hence,
The completeness of ensures that there exists such thatIn view of the hypothesis is closed, we obtain that . It follows that one can choose such that , and we write the above equality asIf , putting and into contractive condition (39), we have