Common Fixed Point Theorems for Generalized Contractive Pair of Mappings in a Metric Space and Their Application to Fractional Calculus

Goel, Priya; Kumar, Manoj; Singh, Dimple; Kumar, Kamal

doi:https://doi.org/10.1155/2022/6760602

Journal of Applied Mathematics

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Research Article | Open Access

Volume 2022 | Article ID 6760602 | https://doi.org/10.1155/2022/6760602

Common Fixed Point Theorems for Generalized Contractive Pair of Mappings in a Metric Space and Their Application to Fractional Calculus

Priya Goel,¹Manoj Kumar,²Dimple Singh,¹and Kamal Kumar¹

Academic Editor: Yansheng Liu

Received26 Jul 2022

Revised21 Sept 2022

Accepted05 Oct 2022

Published16 Dec 2022

Abstract

In this manuscript, we have established relation-theoretic version of some common fixed point results in metric space for generalized -contractive pair of mappings furnished with an arbitrary binary relation . Recently, the concept of binary relation is well known leading trend in fixed point theory. Our results extend and unify several fixed point theorems present in the literature. An illustrative example is given to support our main theorem. Finally, we exploit our main result for proving existence and uniqueness results to established the solution of a fractional differential equation of Caputo type.

1. Introduction

Fixed point theory is a very extensive and active area of research which promises the existence and uniqueness of solution to many mathematical problems in various field of sciences. It begins with fixed point and proceeds through the Banach contraction principle [1]. Banach contraction principle is the most conspicuous tool in the field of fixed point theory which states that every contraction defined on a complete metric space possesses a unique fixed point that is a self- mapping defined on a complete metric space has a unique fixed point if there exists such that

Furthermore, for all in , where is the th iteration of . It ensures us about the existence and uniqueness of solutions to substantial problems in various directions of mathematics. Banach contraction principle is employed in various field of mathematics as well as other domain of sciences and proved many new fixed point results and their uniqueness related to contraction type of mappings. In , the first coincidence fixed point result is proved by Machuca [2], which was later improved by Goebel [3]. Further, in 1976, Jungck [4] has established the first common fixed point theorem. The principle of Banach contraction is generalized by many authors in numerous ways. Ran and Reurings [5] and Nieto and Rodriguez-Lopez [6] extended the Banach contraction principle in a new way by showing that if the metric space is endowed with an ordered binary relation, then it is sufficient to assume that the contraction condition holds only for those comparable elements. In , T. Bhaskar and Lakshmikantham [7] established a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order. In , Ben-El-Mechaiekh [8] generalized Ran–Reurings’ fixed point theorem using the concept of transitive binary relation in a complete metric space. Alam and Imdad [9] have presented a new variant of Banach contraction principle in a complete metric space under an universal binary relation and also utilized their result for transitive, strict order, preorder binary relation, etc. in .

In , Wardowski [10] introduced the concept of - contraction and stated a fixed point theorem for -contraction in complete metric spaces. In , Wardowski and Van Dung [11] introduced the notion of -weak contraction and stated a fixed point theorem for -weak contraction in metric space. Jleli and Samet [12] gave a generalization of Banach contraction principle in complete generalized metric space in . Later, in , Aggarwal et al. [13] discussed the existence and uniqueness of the common fixed point of expansive mappings in the complete G-metric space. In , Imdad et al. [14] demonstrated the idea of weak -contraction on metric space by generalizing the idea of -contraction introduced by JLeli and Samet and proved some relation-theoretical fixed point results for weak - contraction in a generalized metric space without completeness property with application in fractional calculus. In 2019, Alfaqih et al. [15] gave the notion of - contraction and proved some common and coincident fixed point results in metric space endowed with binary relations.

In , Khojasteh et al. [16] initiated the idea of -contraction by introducing a new function, known as simulation function. They generalized the Banach contraction principle by originating the idea of -contraction and proved some fixed point results in metric spaces. Moreover, many other researchers proved many coincidence and common fixed point results in various metric spaces using -contraction. Shahzad and Karapinar [17] demonstrated some coincidence fixed point results in metric spaces using the concept of -contractions. In -metric spaces, existence and the uniqueness of few operators are introduced by Rashid et al. [18]. They established a new type of contractive condition by combining the idea of simulation functions with admissible functions. Argoubi et al. [19] slightly modified the definition of simulation function by withdrawing a condition that is and proved some fixed point results for a pair of nonlinear operators satisfying nonlinear contraction results in partial ordered metric spaces based on -contraction.

In , Samet et al. [20] initiated the idea of -admissible functions and proposed a new category of contractive type mapping, known as -contractive type mapping. With the help of - admissibility, they established some fixed point results for - contractive type mappings. Motivated Samet et al. [20], Durmaz et al. [21] attained the existence and uniqueness of the solution of a fourth order two-point boundary value problem. Further, Wardowski and Van Dung [11] gave a generalized form of -contractive type mappings and obtained various fixed point results. Recently, Sarwar et al. [22] have proposed some fixed point results in metric interval and normed interval spaces using the concept of simulation functions and -admissibility in . Also, in , Kumar and Sharma [23] illustrated the idea of generalized -contractive type mapping and proved some fixed point results in metric spaces. Here, the results are proved in terms of -admissibility condition and a condition of existence a subsequence for some convergent sequence and for the validity of their results; they have also demonstrated some fixed point results in metric spaces endowed with a partial order.

Fixed point theory also plays a vital role in establishing the existence and the uniqueness of the solution of a fractional differential equation. In 1996, Delbosco and Rodino [24] established the existence and the uniqueness of the solution of a nonlinear fractional differential equation. In , Belmekki et al. [25], attained the existence of periodic solution of a nonlinear fractional differential equation. In [26], Zada et al. proved a coupled fixed point theorem and attained the solution of fractional variable order hybrid differential equations in . In addition, Abdo et al. [27], in , developed the theory of nonlinear system of pantograph-type fractional differential equations and proved fixed point result with the use of Banach contraction principle and Krasnoselskii’s fixed point theorem. In , Aydi et al. established some fixed point results in extended -metric space and existence and the uniqueness of a system of nonlinear fractional differential equation is obtained. For instance, the Caputo nonlinear fractional differential equation proposed by Kilbas, is given as where represents the Caputo fractional derivative of order , and is a continuous function. This nonlinear fractional differential equation can also be written as

In this paper, we have extended and generalized the results proved by Kumar and Sharma in [23] in . Our current study provides a new approach for proving fixed point results in terms of binary relation by relaxing the condition of existing a subsequence for some convergent sequence and modifying the condition of -admissibility. Our results show some fixed point results for generalized - contractive pair of mappings in a new manner using the conditions in terms of -continuity and relation in terms of closeness of functions. We also explore an example (Example 21) which shows that our results are better in comparison to the existing result [23]. In addition, we have attained the existence and the uniqueness of the solution of fractional differential equations in the context of our main result, and hence our study is also proved to be useful in finding the existence and the uniqueness of solutions of integral equations, boundary value problems, and fractional differential equations.

In this paper, we begin with introduction in Section 1. In Section 2, we have stated some fundamental definitions related to our work. In Section 3, we have demonstrated a relation-theoretical common fixed point result and its uniqueness for generalized -contractive pair of mappings with the help of binary relation. In Section 4, we have proved some more fixed point results with the help of our main result that is proved in Section 3. In the support of our main theorem, we have also provided an example in Section 5, for which our main result is applicable in finding the existence and the uniqueness of the solution for a given mathematical problem. In Section 6, we have applied our main result in finding the existence and the uniqueness of the solutions of fractional differential equation in Caputo sense. We have also given an example with graphical representation for the reliability of our attained result in fractional calculus.

2. Preliminaries

In this part, we have stated a few definitions associated with our work.

Definition 1 (see [28]). A subset of is termed as binary relation where is any nonempty set.

Definition 2 (see [29]). A sequence is called -preserving if for all , where is a binary relation defined on a nonempty set .

Definition 3 (see [29]). Let be any metric space and a binary relation defined on set . Thus, if each -preserving cauchy sequence in converges to a point in , then is called -complete.

Definition 4 (see [29]). Let us consider a metric space and a binary relation defined on set . Then, a self-mapping is known as -continuous at whenever for any -preserving sequence , then and if is continuous at every point of , then it is called -continuous.

Definition 5 (see [16]). A simulation function is a mapping requiring the following assumptions: (1);(2), for each ;(3)If we have two sequences and in such a manner that , then

Note: denotes the classes of all the simulation functions .

Definition 6 (see [20]). If we consider a self-mapping and a function , then is called -admissible if

Definition 7 (see [30]). Let be two self-mappings, and is a function. Then, is called -admissible with respect to if

Definition 8 (see [20]). , is the collection of functions where is required to hold: (1) is nondecreasing(2) for all , where is the th iterate of .

Definition 10 (see [20]). Consider a metric space and a self-mapping . Then, the self-mapping is known as -contractive mapping if there exist two functions and in such a manner that

Definition 11 (see [30]). Consider a metric space and two self-mappings . Then, the pair is said to be generalized -contractive pair of mappings if there exist two functions and such that where

Definition 12 (see [23]). Consider a metric space and two self-mappings . Then, the pair is said to be generalized -contractive pair of mappings with respect to if where and and

3. Main Results

In this part, we have proved few common fixed point results and their uniqueness for generalized -contractive pair of mappings by using the concept of binary relation and the simulation function.

Theorem 13. If we assume that be any -complete metric space equipped with a binary relation defined on and two self-mappings are such that , and the pair is a generalized -contractive pair of mappings. Then, and have a unique common fixed point if we have the following assumptions: (1) is nonempty(2) is -closed and -closed(3);(4) and are -continuous

Proof. In condition (1) we have given that is nonempty i.e., . So, let i.e., and construct a Picard sequence such that , and also it is given in condition (3) that is -closed, so we get Thus, is a -preserving sequence. Now, from the given condition , we have , therefore, we can choose a point such that . Hence, by continuing the same process and selecting the points , we get

Here, we have two cases.

Case I: If for some , then from (13) equation, we have for some . Thus, and have a common fixed point .

Case II: If and it is given that pair is a generalized - contractive pair of mappings, then

For and , the given inequality becomes

Now since are two self-mappings, so and from given condition (3), we have

Hence, by comparing Equations (18) and (19), we get

Using Condition (20) in Equation (14), we attain where

Now if then we get, which is a contradiction.

Thus, we have

By repeating the same process for and so on, we get

Thus, by substituting all these inequalities in Equation (25), we get

Using triangular inequality and above inequality, we get:

By letting, , we get which is a cauchy sequence in and since , so i.e., is a cauchy sequence, and sequence is also the -preserving. So, is the -preserving cauchy sequence in , and is the -complete. So, is the convergent sequence in , that is, , and . Since is the -continuous from the given condition (5), we have

Therefore, is a fixed point of . Now, since and is a cauchy sequence in . Hence, i.e. is also a cauchy sequence in and is the -continuous from the given condition (4), so

Thus, is also a fixed point of . Hence, is the common fixed point of and .

Uniqueness of common fixed point. Let be any other common fixed point point of and other than . Then, by definition of common fixed point, we have

So, we have to prove i.e., . By choosing and , we attain

Thus, if , then

By letting , we get , which is a contradiction. Hence,

By continuing the same process, we get

Similarly, we can prove that as .

[By triangular inequality].

as i.e., i.e. .

Thus, and have a unique common fixed point.

4. Consequences

Here, we have stated some other results with the help of Theorem (13).

Theorem 14. Consider a -complete metric space where is a binary relation defined on set and are two self-mappings with in such a manner that . Then, and have a unique common fixed point if we have the following assumptions: (1) is nonempty(2)(3) is -closed and -closed(4) and are -continuous

Proof. The result will hold directly from the given Theorem (13) by taking for all and and for all .

Theorem 15. Let be any -complete metric space where is a binary relation defined on set and is a self- mapping with such that . Then, has a unique fixed point if the following propositions hold: (1) is nonempty(2) is -closed(3) is -continuous

Proof. The result will hold directly from the given Theorem (13) by taking for all and , for all and .

Theorem 16. Suppose that be an -complete metric space where is a binary relation defined on set and are two mappings satisfying with . Now if we have the following conditions: (1) is nonempty(2)(3) is -closed and -closed(4) and are -continuousthen, and have a unique common fixed point.

Proof. The result can be hold directly from the given Theorem (13) by taking for each and .

Theorem 17. Assume that be any -complete metric space equipped with a binary relation defined on set and is a self-mapping satisfying the condition where . Thus, if we have the following assumptions: (1) is nonempty(2) is -closed(3) is -continuousthen, has a unique fixed point.

Proof. The result can be attained directly from the given Theorem (13) for for each and the self-map as .

Theorem 18. Consider a -complete metric space equipped with a binary relation defined on set and be a self-mappings in such a manner that , Then, has a unique fixed point if we have the following propositions: (1) is nonempty(2) is -closed(3) is -continuous

Proof. The result will hold directly from the given Theorem (13) by taking , and .

Theorem 19. Let us suppose that be any -complete metric space where is a binary relation defined on set and is a self-mapping with such that . Then, has a unique fixed point if we have the following conditions: (1) is nonempty(2) is -closed(3) is -continuous

Proof. The result can be attained directly from the given Theorem (13) by taking and for every .

Theorem 20. Let us suppose that be any -complete metric space equipped with a binary relation defined on set and is a self-mapping such that . Then, has a unique fixed point if we have the following conditions: (1) is nonempty(2) is -closed(3) is -continuous

Proof. The result will hold directly from the given Theorem (13) by taking for all and , and for all .

5. Example

Here, in this section, we have demonstrated an example in support of Theorem (13) which shows that our main Theorem (13) is applicable to determine the existence of a unique common fixed point for the given problem, but the existing result in [31] is not applicable.

Example 21. Consider the space with usual metric and a binary relation defined on . Then, a pair of mappings defined as have a unique common fixed point with .

Proof. For and , we have , , and . Hence, Thus, and hence, is not - admissible with respect to . Hence, solution of this mathematical problem can not be find out with the help of existing result Theorem (2.2) in [32]. (1)Clearly, is nonempty(2)Since and hence clearly, we have .(3)Since for all , we have . Thus, is and -closed(4)Also, and are -continuous, as for any sequence with , we have and .(5) and ..
.
Now by choosing an monotonically increasing function , we obtain
.
Thus, .
Hence, pair is generalized -contractive pair of mappings. Hence, all the conditions are satisfied of the above Theorem (13) and thus and have a unique common fixed point. Here, in our example, and have a unique common fixed point .

Graphical representation of solution: the graphical solution of Example 21 is given in Figure 1, which clearly shows that and have a unique common fixed point .

6. Application

6.1. Existence and Uniqueness of a Common Solution of Nonlinear Fractional Differential Equation of Caputo Type

The aim of this part is to introduce an application of Theorem (13) to obtain a common solution of nonlinear fractional differential equation of Caputo type for a pair of generalized -contractive pair of mapping in metric space. The Caputo nonlinear fractional differential equation proposed by Kilbas, is given as follows: where represents the Caputo fractional derivative of order and is a continuous function. This nonlinear fractional differential equation can also be represented in the form as follows:

A function is a solution of above defined nonlinear fractional differential equation whenever it is the solution of the fractional integral Equation (39) and vice-versa.

Theorem 22. Consider the space of all continuous functions constructed on closed interval , equipped with a binary relation . Let be the Banach space of all continuous functions from into with norm . This space defines the metric as follows: Now if we construct two self-maps as satisfying that . Then, and have a unique common fixed point if we have the following assumptions: (1),(2)there exist a continuous function satisfying that , where is some constant(3).

Proof. Obviously, is a complete metric space. (1)since is a self map, and , then for any , we have . Thus, and hence is nonempty(2)for any , as is a self-map. Therefore, and hence is -closed. In the same pattern, we can easily prove that is -closed.Thus, is -closed as well as -closed(3)from given condition (1), we have , i.e. , for each .(4)it is only required to prove that pair is generalized -contractive pair of mappingsBy using given condition , we get Now if we assume that , then by assuming , we get Then, by comparing Equations (45) and (6.6), we get This shows that pair is generalized -contractive pair of mapping. Therefore, all the hpothesis of Theorem (13) are satisfied, and hence, and have a unique fixed point.

Example 23. Consider the space of all continuous functions , defined on closed interval endowed with a metric defined as , and a binary relation . Then, the two self-maps are constructed as with and , have a unique common fixed point for and .

Proof. Since and . Then, . By putting , we get , at and at , we get By substituting , we get , and at , at and , we get , and at , at , we get Thus, all the hypotheses of Theorem (5.1) are fulfilled, and hence, and have a unique common fixed point. Here, clearly, and have a unique fixed point .
Graphical representation of solution: Figure 2 represents the solution of Example 23 in graphical form for the function , which shows that is the unique common fixed point of and .

Figure 3 represents the solution of Example 23 in graphical form for the function , which shows that is the unique common fixed point of and , and in this graph, is represented by .

7. Conclusion

The work presented here was carried out in the context of investigating some fixed point results for -contractive pair of mappings in a metric space under some binary relation assumed conditions. The notion of a binary relation is found to be more flexible in nature in the context of fixed point results. Due to the less restrictive nature of binary relations, the fixed point results attained in our paper have a much wider scope of applications. In the context of our main relation-theoretic common fixed point result, we also present an application in fractional calculus. In the future, we can try to prove the same results and many other results for some particular binary relations like transitive relations, symmetrical relations, reflexive relations, etc. The reader can also apply the concept of binary relation to other generalized metric spaces such as -metric space, nontriangular metric space, cone metric space, and so on.

Data Availability

The research data used to support the findings of this study are currently under embargo, while the research findings are commercialized. Requests for data, 6 months after the publication of this article, will be considered by the corresponding authors.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundamenta Mathematicae, vol. 3, no. 1, pp. 133–181, 1922.
View at: Publisher Site | Google Scholar
R. Machuca, “A coincidence theorem,” American Mathematical Monthly, vol. 74, no. 5, p. 569, 1967.
View at: Publisher Site | Google Scholar
K. Goebel, “A coincidence theorem,” Bulletin of the Polish Academy of Sciences Mathematics, vol. 16, pp. 733–735, 1968.
View at: Google Scholar
G. Jungck, “Commuting mappings and fixed points,” The American Mathematical Monthly, vol. 83, no. 4, pp. 261–263, 1976.
View at: Publisher Site | Google Scholar
A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceeding of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004.
View at: Publisher Site | Google Scholar
J. J. Nieto and R. R. Lopez, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, English Series, vol. 23, no. 12, pp. 2205–2212, 2007.
View at: Publisher Site | Google Scholar
T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis, Theory, Methods and Applications, vol. 65, no. 7, pp. 1379–1393, 2006.
View at: Publisher Site | Google Scholar
H. Ben-El-Mechaiekh, “A fixed point theorem based on Miranda,” Journal of Fixed Point Theory and Applications, vol. 2007, article 78706, 7 pages, 2007.
View at: Publisher Site | Google Scholar
A. Alam and M. Imdad, “Relation-theoretic contraction principle,” Journal of Fixed Point Theory and Applications, vol. 17, no. 4, pp. 693–702, 2015.
View at: Publisher Site | Google Scholar
D. Wardowski, “Fixed points of a new type of contractive mappings in complete metric spaces,” Fixed Point Theory and Applications, vol. 2012, no. 1, 2012.
View at: Publisher Site | Google Scholar
D. Wardowski and N. Van Dung, “Fixed points of F-weak contractions on complete metric spaces,” Demonstratio Mathematica, vol. 47, no. 1, pp. 146–155, 2014.
View at: Publisher Site | Google Scholar
M. Jleli and B. Samet, “A new generalization of the Banach contraction principle,” Journal of Inequalities and Applications, vol. 2014, no. 1, 2014.
View at: Publisher Site | Google Scholar
R. P. Aggarwal, E. Karapinar, D. Oregan, and R. L. De-Hierro, Fixed Point Theory in Metric Type Spaces, Springer, Cham, 2015.
View at: Publisher Site
M. Imdad, W. M. Alfaqih, and I. A. Khan, “Weak θ-contractions and some fixed point results with applications to fractal theory,” Advances in Differencial Equations, vol. 2018, no. 1, 2018.
View at: Publisher Site | Google Scholar
W. M. Alfaqih, M. Imdad, R. Gubran, and A. Khan, “Relation-theoretic coincidence and common fixed point results under (F,Rg)-contractions with an application,” Fixed Point Theory and Applications, vol. 2019, no. 1, 2019.
View at: Publisher Site | Google Scholar
F. Khojasteh, S. Shukla, and S. Radenovi, “A new approach to the study of fixed point theory for simulation functions,” Univerzitet u Nišu, vol. 29, no. 6, pp. 1189–1194, 2015.
View at: Publisher Site | Google Scholar
N. Shahzad and E. Karapinar, “On some fixed point theorems under (α , ψ , ϕ) -contractivity conditions in metric spaces endowed with transitive binary relations,” Fixed Point Theory and Applications, vol. 2015, no. 1, 2015.
View at: Publisher Site | Google Scholar
T. Rashid, N. Alharbi, N. Khan, Q. H. Khan, H. Aydi, and C. Ozel, “Order-theoretic metrical coincidence theorems involving (ϕ-ψ)-contractions,” Journal of Mathematical Analysis, vol. 10, no. 5, pp. 119–135, 2018.
View at: Google Scholar
H. Argoubi, B. Samet, and C. Vetro, “Nonlinear contractions involving simulation functions in a metric space with a partial order,” The Journal of Nonlinear Sciences and Applications, vol. 8, no. 6, pp. 1082–1094, 2015.
View at: Publisher Site | Google Scholar
B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for α-ϕ-contractive type mappings,” Nonlinear Analysis, vol. 75, no. 4, pp. 2154–2165, 2012.
View at: Publisher Site | Google Scholar
G. Durmaz, G. Minak, and I. Altun, “Fixed point results for α - ψ-contractive mappings including almost contractions and applications,” Abstract and Applied Analysis, vol. 2014, Article ID 869123, 10 pages, 2014.
View at: Publisher Site | Google Scholar
M. Sarwar, M. Ullah, H. Aydi, and M. De La Sen, “Near-fixed point results via Ƶ-contractions in metric interval and normed interval spaces,” Symmetry, vol. 13, no. 12, p. 2320, 2021.
View at: Publisher Site | Google Scholar
M. Kumar and R. Sharma, “Fixed point theorems for generalized β-ϕ-Z-contractive pair of mappings using simulation function,” Boletim da Sociedade Paranaense de Mathematica, vol. 39, no. 6, pp. 183–194, 2021.
View at: Publisher Site | Google Scholar
D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996.
View at: Publisher Site | Google Scholar
M. Belmekki, J. Nieto, and R. Rodríguez-López, “Existence of periodic solution for a nonlinear fractional differential equation,” Boundary Value Problems, vol. 2009, Article ID 324561, 18 pages, 2009.
View at: Publisher Site | Google Scholar
M. B. Zada, M. Sarwar, T. Abdeljawad, and A. Mukheimer, “Coupled fixed point results in Banach spaces with applications,” Nonlinear Analysis Methods in Applied Mathematics, vol. 9, no. 18, p. 2283, 2021.
View at: Publisher Site | Google Scholar
M. S. Abdo, T. Abdeljawad, K. D. Kucche, M. A. Alqudah, S. M. Ali, and M. B. Jeelani, “On nonlinear pantograph fractional differential equations with Atangana–-Baleanu-–Caputo derivative,” Advances in Differential Equations, vol. 2021, no. 1, 2021.
View at: Publisher Site | Google Scholar
S. Lipschutz, Schaum’s Outlines of Theory and Problems of Set Theory and Related Topic, MeGraw-Hill, New York, 1964.
A. Alam and M. Imdad, “Relation-theoretic metrical coincidence theorems,” Univerzitet u Nišu, vol. 31, no. 14, pp. 4421–4439, 2017.
View at: Google Scholar
P. Sahil, J. Kaur, and S. S. Bhatia, “Coincidence and common fixed point results for generalized α-ψ-contractive type mappings with applications,” 2013, http://arxiv.org/abs/1306.3498.
View at: Google Scholar
L. Ciric and V. Lakshmikantham, “Generalized 𝜶 - 𝝍 contractive type mappings and related fixed point theorems with applications,” Abstract and Applied Analysis, vol. 2012, Article ID 793486, 17 pages, 2012.
View at: Publisher Site | Google Scholar
M. Jleli, E. Karapinar, and B. Samet, “Further generalizations of the Banach contraction principle,” Journal of Inequalities and Applications, vol. 2014, no. 1, 2014.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Priya Goel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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