On Pata Convex-Type Contractive Mappings

Aktay, Merve; Özdemir, Murat

doi:https://doi.org/10.1155/2022/6963446

Journal of Function Spaces

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Fixed Point Theory and its Applications in Ordinary and Fractional Differential and Integral Equations

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Research Article | Open Access

Volume 2022 | Article ID 6963446 | https://doi.org/10.1155/2022/6963446

On Pata Convex-Type Contractive Mappings

Merve Aktay¹and Murat Özdemir¹

Academic Editor: Marija Cvetkovic

Received30 Jun 2022

Accepted28 Jul 2022

Published08 Sept 2022

Abstract

In this work, we introduce weak Pata convex contractions and weak -Pata convex contractions via simulation functions in metric spaces to prove some fixed point results for such mappings. Also, we consider an example related to weak Pata convex contractions. Consequently, our results generalize and unify some results in the literature.

1. Introduction and Preliminaries

It is well known that Banach [1] pioneered in fixed point theory by introducing a novel notion, namely, Banach contraction principle in 1922. After this date, several authors generalized and extended this principle. A generalization was given by Pata [2] known as Pata contraction. Recently, Pata contraction has been studied by many authors. Some of the studies were for Pata contraction presented by [3–13].

Firstly, the concept of weak contraction was given by Alber et al. [14]. Zhang et al. and Rhoades’s results [15, 16] extend previous results given by Alber et al., and they obtained fixed point results for single-valued mappings in Banach spaces, and Rhoades [15] got a unique common fixed point of such contractions, respectively.

In 2012, Samet et al. [17] suggested a novel notion, the so-called -admissible. Later, Karapinar et al. [18] presented triangular -admissible mappings, and then, Arshad et al. [19] introduced -orbital admissible and triangular -orbital admissible mappings. Due to the importance, many authors studied such mappings. For more knowledge and different examples related to admissible mappings, one can see [20–25].

Istratescu [26–28] gave the concept of contractions known as the convex contraction of order and two-sided convex contraction mappings. Very recently on, Karapinar et al. [10] introduced the notion of -almost Istratescu contraction of type . Some notable generalizations related to Istratescu’s results were obtained by [29–35].

In a recent work, Khojasteh et al. [36] introduced the notion of -contraction using simulation functions. Later, Karapinar [37] and Argoubi et al. [38] studied such contractions. After that, some new studies were obtained related to simulation functions in [39–44].

The aim of this paper is to establish some fixed point results for weak Pata convex contractive mapping and weak -Pata convex contractive mapping via -admissible mappings by using simulation functions in metric spaces. Our results are generalization of recent fixed point results derived by Karapinar et al. ([10, 32, 45]), Alber et al. [14], Zhang et al. [16], Istratescu [26], Pata [2], and Banach [1] and some other related results in the literature.

Firstly, we start this section by recalling some definitions related to our work.

In the course of this manuscript, denote the set of real numbers and the set of natural numbers, respectively. Let

Alber et al. [14] gave the definition of weak contraction, stated below.

Definition 1. See [14]. Let be a metric space. A mapping is called weak contraction, if there exists a map with and for all such that for all .

The concept of -weak contraction was generalized by Zhang et al. [16] as generalized -weak contraction.

Definition 2. See [16]. Let be a metric space. A mapping is called generalized -weak contraction, if there exists a map with and for all such that for all , where

Samet et al. [17] and Karapinar et al. [18] introduced the following concepts, respectively.

Definition 3. Let be a metric space, be a map, and be a function. (i)[17] If implies for all , then is called admissible(ii)[18] If isadmissible and and imply , then is called triangular admissible

Example 4. Let , the mappings by and by Thus, is a triangular -admissible mapping.

Khojasteh et al. [36] gave the simulation function and -contraction as follows.

Definition 5. See [36]. A mapping is called a simulation function if it satisfies the following conditions:
()
()
() if and are sequences in such that , then .

Definition 6. See [36]. Let be a metric space and be a mapping. If there exists such that then, is called contraction with respect to

() condition was removed in the above definition of simulation function by Argoubi et al. [38] in 2015. Also, denotes the set of all simulation functions.

Example 7. See [36, 42, 44]. Let and be continuous functions with .
for all where for all For the above examples and other examples related to simulation functions, one can see [36, 37, 42, 44] and references therein.

The following two concepts were defined by Istratescu [26] as follows.

Definition 8. See [26]. Let be a metric space and be a self-mapping. For all , is called convex contraction of order if there exist such that and is called two-sided convex contraction mappings if there exist such that and In the course of this work, denotes the set of all increasing function , which vanishes with continuity at zero. For a random , we denote .

Introducing a novel generalization of the Banach contraction principle, Pata [2] proved Theorem 9.

Theorem 9. See [2]. Let be a metric space and and be fixed constants. and be functions. If for all the inequality is satisfied for all ; then has a unique fixed point, .
Pata-type contractions were studied by some authors. Karapinar et al. [11] introduced Pata-Ciric type contraction at a point. Alqahtani et al. [5] gave the -Pata–Suzuki contraction and fixed point results for such contractions. After that, Karapinar and Himabindu [11] proved some common fixed point results for Pata–Suzuki -contraction.

We recall here the following important Lemma 10 that we will use to proof of our main results.

Lemma 10. See [46]. Let be a metric space and be a sequence in such that as . If is not a Cauchy sequence, then there exist a and subsequences and of such that , and .

2. Main Results

The main objective of this work is to give some new fixed point theorems via a combination of convex contraction, weak contraction and Pata type contractive mappings by introducing the concept of weak -Pata convex contractions and weak Pata convex contractions in metric spaces. We will use simulation functions and admissible mappings when combining these concepts. Also, we will give an example that supports our conclusion.

In definitions and results in this paper, and will be considered as fixed constants, and also, we will consider the following equations:

At first, we begin our work by giving the following definitions.

Definition 11. Let be a metric space. We say that is weak Pata convex contractive mapping via simulation function if for all , and , there exist three functions and such that satisfies the inequality where is a continuous and nondecreasing function with and for all .

Definition 12. Let be a metric space. We say that is weak -Pata convex contractive mapping via simulation function if for all , and , there exist three functions , and such that satisfies the inequality where is a continuous and nondecreasing function with and for all .

Now, we are in a position to present our main theorems.

Theorem 13. Let be a complete metric space, and be a weak -Pata convex mapping via simulation function. Suppose that (i) is triangular -admissible(ii)there exists such that (iii) is continuous(iv)for all , .Then has a unique fixed point in .

Proof. From hypothesis (ii) of the Theorem 13, there exists such that . Firstly, we will show that for all . Since is an -admissible mapping, we have By induction, we obtain that Taking into account hypothesis (i) of the Theorem 13, we have Again by induction, we obtain that Now, we will show that is a nonincreasing sequence. Since is a weak -Pata convex contractive mapping via simulation function, we have From hypothesis (ii) of the Theorem 13, we get for some . If we assume that then we have . Hence, we have The inequality (20) is true for all For , we obtain which is a contradiction. Therefore, we obtain Analogously, as is a weak -Pata convex contractive mapping via simulation function, we have Now, we can write for some . In case that ; then we have . So, we have The inequality (24) is true for all For , we obtain which is again a contradiction. Therefore, we obtain By induction, since is a weak -Pata convex contractive mapping via simulation function, we have We have that for some . In case that ; then we have Again, the inequality (28) is true for all for ; we obtain is again a contradiction. Therefore, we obtain Consequently, we find that If the point is taken as the starting point, the sequence is constructed by , . If for any , then is a fixed point of . As a result, supposing that for all and let . So, we get that is a nonincreasing sequence. For this reason, there exists a such that We will demonstrate that . For this, we should demostrate that the sequence is bounded. Since is a nonincreasing sequence, we have By the triangle inequality, we have Since is a weak -Pata convex contractive mapping, we have Together with (35), we obtain where Now, we derive that Using , we get Conversely, we assume that is not bounded sequence. So, there exists a subsequence of such that . If we take in (39) inequality; then we have If we take limit in (40) inequality as , then we get which is a contradiction. Therefore, we demonstrate that the sequence is bounded. So, there exists such that for all . Following this line of work, we demonstrate that . Since is a weak -Pata convex contractive mapping, we have Since for all , we have Since the sequence is bounded, we have Now, we can write If we take the limit as in (45) inequality, then we obtain as that is . Now, we demonstrate that is a Cauchy sequence. On the contrary, assume that the sequence is not a Cauchy. From Lemma 10, there exist subsequence and with such that , , , , and . Since is a weak Pata convex contractive mapping, we have where Now, we have If we take the limit as , then we obtain and so, we have and thus, we get that , which is a contradiction. Therefore, we concluded that 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 .
Next, we will demonstrate the uniqueness of the fixed point. Suppose that and are two fixed points of Since satisfies the hypothesis (iv) of Theorem 13, is an weak -Pata convex contractive mapping; we have We obtain that for some , and so, we get . Hence, has a unique fixed point in , that is .
Following this line of work, Theorem 14 does not require the continuity of .

Theorem 14. Let be a complete metric space, and be a weak Pata-convex mapping. Suppose that (i) is triangular admissible(ii)there exists such that (iii) is continuous and for all, (iv)for all , Then, has a unique fixed point in .

Proof. Following the proof of Theorem 13, we have already proved that is a Cauchy sequence in . Since is complete, we have as . Taking into account hypothesis (iii) Theorem 14, we have . In the uniqueness of the limit, we obtain that . Next, we will prove that . On the contrary, we assume that is not fixed point of . So, we have for some . We obtain For in (54) which is a contradiction. Thus, we make an inference that , and so, is a fixed point of . Following the proof of Theorem 13, the uniqueness of fixed point of can be obtained.
Theorem 15 is other fundamental result of our work.

Theorem 15. Let be a complete metric space, and be a weak Pata convex contractive mapping via simulation function. On the assumption that all of the Theorem 13 hypotheses are satisfied, then has a unique fixed point.

Proof. In the proof of Theorem 13, we have got that Setting and now, we demonstrate that
is a nonincreasing sequence. Since is a weak Pata convex contractive mapping via simulation function, we have Using hypothesis (ii) of the Theorem 15, we get for some . Assuming that , then we have . Thus, we have and since , we have The inequality (59) is true for all For , we obtain which is a contradiction. Hence, we obtain Analogously, since is a weak Pata convex contractive mapping via simulation function, we have and we can write that for some . In case that then we have The inequality (64) is true for all For , we obtain is again a contradiction. Therefore, we obtain Again, by induction, since is a weak Pata convex contractive mapping via simulation function, we have and we have that for some . In case that then we have Again, the inequality (68) is true for all and for , we obtain is again a contradiction. Consequently, we can find that Starting at the point the sequence is constructed by , . If for any , then is a fixed point of . Hereby, assume that for all and let . Therefore, we get that is a nonincreasing sequence. Thereupon, there exists a such that We will demostrate that . For this, we should demostrate that the sequence is bounded. Since is a nonincreasing sequence, we have From the triangle inequality, we can write Since is a weak Pata convex contractive mapping via simulation function, we have Together with (71), we obtain that From (71) and , we have Now, we derive that Contrarily, supposing that is not bounded sequence. Thence, there exists a subsequence of such that . If we take in (76) inequality, then we have If we take limit in (77) inequality as , then we get that is a contradiction. Next, we show that the sequence is bounded. So, there exists such that for all . Following this line of work, we will demonstrate that . Since is a weak Pata convex contractive mapping via simulation function, we have where Since the sequence is bounded, we have Therefore, we have If the limit is taken as in (82) inequality, then we get as that is . Now, we demonstrate that is a Cauchy sequence. Contrarily, supposing that the sequence is not a Cauchy. From Lemma 10, we say that there exist subsequence and with such that , , , , and . Since is a weak Pata convex contractive mapping, we have where Now, we can write If we take the limit as , we get and so, we have that is, we get which is a contradiction. Therefore, we concluded that 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 will demonstrate that the fixed point is unique. Assuming that and are two fixed points of From hypothesis (iv) of Theorem 15 and since is an a weak Pata convex contractive mapping via simulation function, we have and so, we have We obtain that for some , and thus, we get . Hence, has a unique fixed point in .

Example 16. Let the usual metric space where . Let define the mappings by by and by It is easily seen that is a triangular -admissible mapping, and also, . Though the mapping, is discontinuous in and is continuous on . Now, we want to demonstrate that satisfies (11). For , we have Since and , for , we get that For arbitrary , as one can see, the above inequality turns into the following inequality, Now, our goal is to show that and such that holds for all , and every . We can find such that holds for each and some . If we choose such that , then Hence, we have that Now, we can write and for we have which satisfies for each and all . If , it can be seen that (11) is satisfied. Hence, all conditions of Theorem 15 are satisfied with and . By an application of Theorem 15, has a unique fixed point in .

Suppose that in Theorems 13 and 15; then we obtain the following corollaries.

Corollary 17. Let be a complete metric space and and be two functions. If for all , there exists a function, such that satisfies the inequality either where is a continuous and nondecreasing function with and for all , and assuming that all of the hypotheses of Theorem 13 are satisfied, then has a unique fixed point.

Karapinar’s contractive conditions [10, 32, 45] are a special case of ours, and also, Corollary 17 generalizes the results of Samet [17] and Istratescu [26–28].

Corollary 18. Let be a complete metric space and be a function. If for all , there exist two functions, such that satisfies the inequality either where is a continuous and nondecreasing function with and for all and assuming that all of the hypotheses of Theorem 13 are satisfied, then has a unique fixed point.

In comparison with recent results such as Alber et al. [14] and Zhang [16], our results are a generalization of them.

Corollary 19. Let be a complete metric space and be a function. If for all , there exists a function such that satisfies the inequality either and assuming that all of the hypotheses of Theorem 13 are satisfied, then has a unique fixed point.

Putting in Theorems 13 and 15, we can see the following results.

Corollary 20. Let be a complete metric space and and be two functions. If for all , and , there exists a function , such that satisfies the inequality either where is a continuous and nondecreasing function with and for all , and assuming that all of the hypotheses of Theorem 13 are satisfied, then has a unique fixed point.

Corollary 21. Let be a complete metric space and be a function. If for all , and , there exists a function , such that satisfies the inequality either where is a continuous and nondecreasing function with and for all , and assuming that all of the Theorem 13 hypotheses are satisfied, then has a unique fixed point.

Assume now that and in Theorem 13 and Theorem 15; then we get the following corollaries.

Corollary 22. Let be a complete metric space and and be two functions. If for all , satisfies the inequality either where is a continuous and nondecreasing function with and for all and assume that is continuous or is continuous. Then, has a unique fixed point that is .

Corollary 23. Let be a complete metric space and be a function. If for all , satisfies the inequality either where is a continuous and nondecreasing function with and for all and assume that is continuous or is continuous. Then, has a unique fixed point that is .

We derive that the main result of Pata [2] and Banach [1] can be expressed as a corollary of our main result.

3. Conclusion

We present the concept of weak -Pata convex contractions and weak Pata convex contractions in metric spaces in this paper. After that, we investigate the existence of a fixed point for our novel type contraction and we state some consequences. Our results generalize and merge the results derived by Istratescu [26] and Pata [2] and some other related results in the literature. Besides the corollaries in this paper, to underline the novelty of our given results, we give an example that shows that Theorem 15 is a genuine generalization of Istratescu’s results [26]. Our novel concept allows for further studies and applications.

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.

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