Some Results on Generalized Pata–Suzuki Type Contractive Mappings

Abstract

In this work, we establish new fixed point theorems for generalized Pata–Suzuki type contraction via -admissible mapping in metric spaces and to prove some fixed point results for such mappings. Moreover, we give an example to illustrate our main result. Consequently, the results presented in this paper generalize and improve the corresponding results of the literature.

1. Introduction and Preliminaries

The Banach contraction principle was introduced by Banach [1] which is one of the earlier and the most important result in fixed point theory. Because of its importance, over the years, many authors extended and generalized the Banach contraction principle in many directions.

The concept of almost contraction was given by Berinde and Pacurar [2]. Also this concept was compared with other contractions and proved some fixed point theorems by Berinde ([2–4]).

-condition was introduced by Suzuki ([5–7]). Suzuki [8] proved generalized versions of Edelstein theorem [9] for compact metric spaces.

Firstly, Samet et al. [10] introduced an interesting contraction which is called contractive and gave admissible mappings and investigated the existence and uniqueness of such mappings in the setting of complete metric spaces. Later, in addition to Samet et al. [10], some different fixed point theorems were introduced by Karapinar and Samet [11] and Babu et al. [12].

Recently, one of the famous generalizations of the Banach contraction principle for the existence and uniqueness of fixed points for self mappings on metric spaces is the theorem by Pata [13]. Since Pata’s fixed point theorem, some authors have studied this theorem in several ways (see [14–18]).

Using the ideas of Pata [13], we give a generalization of Pata-type contractions. The purpose of this paper is to introduce almost Pata–Suzuki type contractive mapping and to prove some fixed point results for such mappings with admissibility condition. These results are generalizations of various results in the literature.

Now, we give some definitions and fundamental results.

Let be a metric space; a point is said to be a fixed point of if . The set of all fixed points of is denoted by .

Berinde and Pacurar [2] introduced the concept of almost contraction as a generalization of Banach contraction principle.

Definition 1. (see [2]). Let be a metric space and be a mapping which is called an almost contraction if there exists a constant and such thatfor all .
In 2011, Pata [13] proved the following result.

Theorem 1 (see [13]). Let be a complete metric space and denote the class of all increasing function , which vanishes with continuity at zero. For an arbitrary , we denote . Let , and be fixed constants and and be a function. If for all , the following inequality is satisfied:for all , and then has a unique fixed point .

Suzuki ([5–7]) introduced condition as follows.

Definition 2. Let be a metric space and be a given self mapping on . h is said to satisfy C-condition if for all Samet et al. [10] gave admissibility condition and the definition of contractive mapping as follows.

Definition 3. (see [10]). Let be a metric space, be a map, and be a function.(i)If implies for all , then is said to be -admissible(ii)If h is α-admissible and and imply , then h is said to be triangular α-admissible. denotes the family of nondecreasing functions such that for each , where is the th iterate of .

Remark 1. Every function satisfies and for all .

Definition 4. (see [10]). Let be a metric space and be a mapping. We say that is an contractive mapping if there exist two functions and such that for all .

2. Almost Pata–Suzuki Type Contraction

In this section, we introduce concept of almost Pata–Suzuki type contractions in metric spaces. We establish some fixed point results for such contractions on metric spaces.

The following lemma is necessary in our Theorem’s proofs.

Lemma 1. Let be a metric space and be a sequence in such that as . If is not a Cauchy sequence, then there exist a and sequences of positive integers and with such that and and , , and .
From Lemma 1, we obtain and .
Let be a complete metric space. Along this paper, denotes the class of all increasing function , which vanishes with continuity at zero. For an arbitrary , we denote .
First, we begin with the following definition.

Definition 5. Let , and be fixed constants, and and be two functions. There exist two functions , and such that, if for all , and , satisfies the following inequality:which implieswhereThen, we say that is an almost Pata–Suzuki contractive mapping.
Now, we are ready to prove our first result.

Theorem 2. Let be a complete metric space. be an almost Pata–Suzuki contractive mapping. Assume that(i) is triangular admissible(ii)There exists such that (iii) is continuous(iv)For all , Then, has a unique fixed point, that is, , .

Proof. From the hypothesis (ii) of Theorem 2, there exists such that . Starting at the point the sequence is constructed by . If for any , then is a fixed point of . Consequently, assume that for all . First of all, we show that for all . Since is an admissible mapping, we haveBy induction, we obtainFrom the hypothesis (i) of Theorem 2, we haveHence, by induction, we obtainNow, we will show that is a decreasing sequence. Using Remark 1, since , we obtainfor some . If , then we get . In this way, we obtain , a contraction. Thus, we havethat is, is a decreasing sequence. Since is decreasing, it is convergent to a nonnegative real number. Let . Now, we will show that the sequence is bounded. From triangle inequality, we haveWe assert a claim thatOn the contrary, suppose thatFrom triangular inequality, we havewhich is a contradiction. Thus, (14) is satisfied. Since h is an almost -Pata-Suzuki contractive mapping and using (10) and Remark 1, we obtainUsing , we getfor some . By the same reason as in [13], the sequence is bounded. Let . Using Remark 1 and from (8), we havefor some . Taking limit as we obtain and thus .
Next, we demonstrate that is a Cauchy sequence. Assume that the sequence is not a Cauchy sequence. From Lemma 1, there exist subsequence and with such that , , , , and . We assert thatIf we assume thatthen we obtain a contradiction. If we take limit in inequality (21), we obtain contradiction. Since is an almost Pata–Suzuki contractive mapping, we haveTaking the limit as , we obtainand thenthat is, , which is a contradiction. Hence, the sequence is a Cauchy sequence in . By the completeness of , the sequence is convergent to some , that is, as . Since is continuous, as . By the uniqueness of the limit, we obtain ; that is, is a fixed point of .
Now, we observe that fixed point of is unique. Assume that and are fixed points of . Since satisfies the hypothesis (iv) of Theorem 2, is an almost Pata–Suzuki contractive mapping, Remark 1, and , we haveHence, we obtain that , and thus . Hence, has a unique fixed point in .
Now, we prove the following theorem without the assumption of continuity of .

Theorem 3. Let be a complete metric space. be an almost Pata–Suzuki contractive mapping. Assume that(i) is triangular admissible(ii)There exists such that (iii)If is a sequence in such that for all and as , then for all (iv)For all , Then, has a unique fixed point, that is, .

Proof. Following the proof of Theorem 2, we have already known that is a Cauchy sequence in . Since is complete, we have as . Now, we prove that . From (8) and hypothesis (iii) of Theorem 3, we have for all . For all , we assert thatWe assume thatSince is a decreasing sequence,which is a contradiction. Thus, (26) is held, and we havefor some . Taking the limit as , we obtainThus, we make an inference that and that is is a fixed point of . Similar to the proof of Theorem 2 the uniqueness of fixed point of can be obtained.
We give an example related to our main theorem.

Example 1. Let with the usual metric and define the mapping by . Let be defined as and , . It is easy to get is triangular admissible. Our aim is to prove that satisfies (5). For ,which impliesSince , we obtainand for , we can writeFor arbitrary , we can write inequality (34) as follows:Our aim is to prove that and such thatwhich satisfies for all , and every . We can find such thatwhich satisfies for each and some . If we choose such that , thenThus, we have thatwhich implieswhich satisfies for all and each . If , it can be seen that (5) is satisfied. Hence, the conditions of Theorem 2 are satisfied with . By an application of Theorem 2, has a unique fixed point in . It is seen that is the unique fixed point of in .
The following result is directly obtained from Theorem 2 by taking .

Corollary 1. Let be a function. There exist two functions and and such that if for all , satisfies the inequalitywhich implieswhereand assume that(i) is triangular admissible(ii)There exists such that (iii) is continuous(iv)For all , Then, has a unique fixed point, that is, .

Corollary 1 generalizes the results of Samet [10] and Karapınar [11].

Corollary 2. Let and be fixed constants and be a function. There exist two functions and and such that if for all , and , satisfies the inequalitywhich implieswhereand assume that(i) is triangular admissible(ii)There exists such that (iii) is continuous(iv)For all ,

Then, has a unique fixed point that is .

If we take in Theorem 2, then we say that is Pata–Suzuki contractive mapping and get the following corollary.

Corollary 3. Let , and be fixed constants; and be two functions. There exist two functions and such that if for all , and , satisfies the inequalitywhich implieswhereand assume that(i) is triangular admissible(ii)There exists such that (iii) is continuous(iv)For all , Then, has a unique fixed point, that is, .

If we take for all , and , in Theorem 2, then we get the following corollary.

Corollary 4. Let be a complete metric space, , and be fixed constants, and , and be a function. There exist a function and such that if for all , satisfies the inequalitywhich impliesis satisfied for all , and then has a unique fixed point .

If we take for all , and , , for all , in Theorem 2, then we get the following corollary.

Corollary 5. Let be a complete metric space, , and be fixed constants, and and be a function. There exists such that if for all , and , satisfies the inequalitywhich impliesis satisfied for all , and then has a unique fixed point .

Corollary 6. Let be a complete metric space, , and be fixed constants, and and be a function. There exists such that if for all , and , satisfies the inequalitywhich is satisfied for all , and then has a unique fixed point .

Corollary 6 generalizes the results of Pata [13] and Banach [1].

Data Availability

The data used to support the findings of this study are included in the references within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The first author would like to thank TUBITAK (the Scientific and Technological Research Council of Turkey) for their financial supports during her PhD studies.