A Solution Approach to Nonlinear Integral Equations in Generalized b-Metric Spaces

Jaradat, Mohammed M. M.; Ahmad, Abeeda; Rehman, Saif Ur; Muhammad Diaa, Nabaa; Jabeen, Shamoona; Haider, Muhammad Imran; Shamas, Iqra; Shlaka, Rawan A.

doi:https://doi.org/10.1155/2024/8847058

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 8847058 | https://doi.org/10.1155/2024/8847058

A Solution Approach to Nonlinear Integral Equations in Generalized b-Metric Spaces

Mohammed M. M. Jaradat,¹Abeeda Ahmad,²Saif Ur Rehman,²Nabaa Muhammad Diaa,³Shamoona Jabeen,⁴Muhammad Imran Haider,²Iqra Shamas,²and Rawan A. Shlaka⁵

Academic Editor: Francisco J. Garcia Pacheco

Received14 Mar 2023

Revised12 Dec 2023

Accepted30 Dec 2023

Published23 Jan 2024

Abstract

In this paper, we study some generalized contraction conditions for three self-mappings on generalized b-metric spaces to prove the existence of some unique common fixed-point results. To unify our results, we establish a supportive example for three self-mappings to show the uniqueness of a common fixed point for a generalized contraction in the said space. In addition, we present a supportive application of nonlinear integral equations for the validation of our work. The concept presented in this paper will play an important role in the theory of fixed points in the context of generalized metric spaces with applications.

1. Introduction

Fixed-point (FP) theory is one of the interesting areas of research in mathematics and other science fields. In this theory, Banach [1] introduced a valuable and important result for the existence and uniqueness of fixed point which is known as the “Banach Contraction Principle (BCP)” and stated as “a single-valued contractive type mapping on a complete metric space (M-space) has a unique FP.” BCP was later generalized in various directions, and many authors contributed to the theory of FP. Bhaskar and Lakshmikantham [2] established some FP results for a mixed monotone mapping in an ordered partial M-space using a weak contractivity type of mappings with an application. Jovanovic et al. [3] worked on common fixed point (CFP) results in M-spaces. Bojor [4] proved FP theorems for Reich-type contractions on M-spaces. Kutbi et al. [5] started to investigate CFP results for mappings with rational expressions. Batra et al. [6] presented a new extension of Kannan contractions and related FP results. Hussain [7] proved results for the solution of fractional differential equations using symmetric contraction. Debnath [8] studied Banach, Kannan, Chatterjia, and Reich-type contractive inequalities for multivalued mappings and proved CFP theorems. Rasham et al. [9] established some results for the family of multivalued mappings with the applications of functional and integral equations. Recently, Abbas et al. [10] studied the thermodynamic properties of the second-grade micropolar nanofluid flow past an exponential curved Riga stretching surface with Cattaneo–Christov double diffusion. Furthermore, in [11], Abbas et al. discussed the thermal analysis of MHD Casson-Sutterby fluid flow over exponential stretching curved sheet.

Bakhtin [12] gave the concept of b-metric space (b-M-space). After that, Czerwik [13] presented some FP results by using b-M-spaces. In 1998 Czerwik [14] studied some nonlinear set-valued contraction results in b-M-spaces. Boriceanu et al. [15] formulated the fractal operator theory by establishing it in b-M-spaces and verified some generalized CFP results. Aydi et al. [16] worked on an FP theorem for set-valued quasi-contractions in b-M-spaces. In [17], Roshan et al. proved CFP results for four self-mappings on b-M-spaces. Shatanawi et al. [18] extended contraction conditions using comparison functions on b-M-spaces. Alqahtani et al. [19] established CFP results on an extended b-M-space. Sintunavarat and Kumam [20] presented CFP theorems in complex-valued M-spaces with their applications. Recently, Bantan et al. [21] proposed integral equations in complex-valued b-M-spaces.

In 2006, Mustafa and Sims [22] introduced the idea of generalized metric space (GM-space). Mustafa et al. [23] proved an FP theorem for self-mappings on complete GM-spaces. Abbas and Rhoades [24] discussed CFP results for noncommuting mappings without continuity in a GM-space. In [25], Hussain et al. discussed the unification of -metric, partial metric, and GM-spaces. Gugnani et al. [26] formulated CFP results in GM-spaces and their applications. In 2012, Lakzain and Samet [27] and Mustafa et al. [28], respectively, established some FP and coincidence point results for -weakly contractive mappings in GM-spaces and ordered GM-spaces.

Aghajani et al. [29] introduced the idea of generalized b-metric space ( M-spaces). They proved some CFP results for four mappings satisfying a generalized weakly contractive condition in partially ordered complete b-M-spaces. Their results extended and improved several comparable results in the published literature. Roshan et al. [30], proved some CFP results for three mappings in discontinuous M-spaces. Cobzas and Czerwik [31] worked on the completion of M-spaces and proved some FP results. Aydi et al. [32] started to investigate a few coupled and tripled coincidence point results and also extended, complemented, and generalized several existing results in such spaces. In 2021, Gupta et al. [33] investigated various FP results on complete M-spaces and proved CFP results. Mustafa et al. [28] established some coupled coincidence point results for -weakly contractive mappings in the setup of partially ordered M-spaces. Makran et al. [34] provided generalized CFP results for multivalued mapping in M-spaces with an application. Mebawondu and Mewomo [35] gave the concept of Suzuki-type FP results in M-spaces. Recently, Mehmood et al. [36] established the notion of integral equations in complex-valued M-spaces and proved some CFP results.

The main purpose of this paper is to demonstrate some results for the existence and uniqueness of CFP using three self-maps satisfying the generalized contractive conditions in M-spaces with an illustrative example. Our results improve and modify many results presented in the literature. Further, we support our results by an application of the nonlinear integral equations to validate our work. This work is followed by Section 2, which consists of preliminary concepts. In Section 3, we establish some generalized CFP theorems on M-spaces with an illustrative example. In Section 4, we present an application of nonlinear integral equations to support our main work. Lastly, in Section 5, we discuss the conclusion of our work.

2. Preliminaries

Definition 1 (see [29]). Let be a nonempty set. A function is said to be a generalized b-metric space ( M-space) if the following axioms hold:(i)(ii)(iii)(iv), here, is a permutation of (symmetry)(v)For all . Then, the pair is said to be a M-space.

Example 1. Let and the mapping be defined as follows:for . Then, is a M-space with .

Definition 2 (see [29]). A M-space is said to be symmetric if .

Proposition 3 (see [29]). Let be a M-space. Then, for each it follows that(i)(ii)(iii)(iv)

Definition 4 (see [29]). Let be a M-space. A sequence in is said to be(i)-Cauchy sequence if for any such that (ii)Convergent to an element if for all given such that , whenever (iii)A pair is said to be complete if every -Cauchy sequence is -convergent in

Proposition 5 (see [29]). Let be a M-space. The following statements are equivalent:(i) is -convergent to (ii) as (iii) as

Proposition 6 (see [29]). Let be a M-space. The following statements are equivalent:(i) is -Cauchy sequence(ii) as

3. Main Results

In this section, we use the approaches of Aghajani et al. [29], Gupta et al. [33], and Mustafa et al. [28] to prove some modified rational contraction theorems with illustrative examples.

Theorem 7. Let be a M-space with coefficient and be three self-mappings which satisfyfor all with and . Then, the three self-mappings , and have a CFP in . Moreover, if , then and have a unique CFP in .

Proof. Fix . We now define an iterative sequence in as follows:By using (2), we haveAfter simplification, we obtainSimilarly, again by the view of (2),After simplification, we obtainBy a similar argument as in above, we can show thatNow, from (5), (7), and (8), we conclude thatHence, we have proved that the sequence is contractive under the M-space for three self-mappings. Therefore,Next, we will show that is a -Cauchy sequence in . For all and , using the rectangle inequality and (9), we haveSince we haveBy using Proposition 3 (ii), we have for with . If we take the limit as , we get . Hence, is a -Cauchy sequence. Since is complete, there is , such that as or . We now show that by contrary case, let . Then, by using the rectangular property of and by the view of (2), we have thatAfter simplification, we obtainNow, by taking limit and by using Proposition 3 (iii), we obtainwhich implies that is a contradiction, since Thus, which yields that
Next, we show that by contrary case. Let , then again by using the rectangular property of and by the view of (2), we have thatNow, by taking limit , we obtainHence, is a contradiction, since . Thus, which yields that
Now, we have to show that by contrary case. Let then by using the rectangular property of and by the view of (2), we have thatAfter simplification, we obtainNow, by taking limit , we obtain which is a contradiction. Thus, Hence, we have proved that is a CFP of , , and ; that is,
To this end, we prove the uniqueness of the CFP. Assume that is another CFP of the mappings and ; that is, Then, from (2), we have thatNow, by using Proposition 3 (iii), we get thatwhich implies that is a contradiction, since . Therefore, and so . The proof is complete.
By taking in Theorem 7, we get Corollary 8.

Corollary 8. Let be a M-space with coefficient and be three self-mappings which satisfyfor all and with . Then, the three self-mappings , and have a unique CFP in .

By specializing , in Theorem 7, we get Corollary 9.

Corollary 9. Let be a M-space with coefficient and be three self-mappings which satisfyfor all and with . Then, the three self-mappings , and have a unique CFP in .

Theorem 10. Let be a complete M-space with coefficient and be three self-mappings which satisfy, with , and Then, , and have a CFP in . Moreover, if , then have a unique CFP in .

Proof. Fix . We now define the iterative sequences in as follows:Now, by using (24), we haveHowever,And so,Thus, by combining (26) and (28), we obtainBy using Definition 1 (iii), we obtainNow, there are two possibilities: Possibility I. If , then (30) becomes Possibility II. If the , then (30) becomesFrom both cases, we obtain thatSimilarly, again by using (24),However,And so,By combining (34) and (36), we obtainBy applying a similar argument as in the above two cases, we obtainBy a similar argument as above, one can show thatNow, from (33), (38), and (39), we conclude thatHence, we have proved that the sequence is contractive under the M-space for three self-mappings. Therefore,Next, we will show that is a -Cauchy sequence in . For all and , using the rectangle inequality, we haveSince , so the above inequality yields thatBy using Proposition 3 (ii), we have for with . If we take the limit as , we get . Hence, is a -Cauchy sequence. Since is a complete -metric space, there is , such that as or . We now show that by contrary case, let . Then, by using the rectangular property of -metric space and by the view of (24), we have thatAfter simplification, we obtainBy taking limit , we obtainTo this end, we have two possibilities to consider: Possibility I. If be the maximum term, then And so, is a contradiction, since . Thus, Possibility II. If be the maximum term, thenAnd so, is a contradiction, since . Thus, Hence, from both possibilities, we get that
Next, we show that by contrary case and let . Then, by using the rectangular inequality of -metric space and by the view of (13), we have thatBy taking limit and after simplification, we obtainNow, as above, there are two possibilities: Possibility I. If , then (50) implies And so, is a contradiction, since . Thus, Possibility II. If , then (28) impliesAnd so, is a contradiction, since . Thus, Hence, from both possibilities, we get that .
Now, we have to show that by contrary case and let By using (24), we have thatAfter simplification, we obtainBy taking limit , we get , which is a contradiction. Thus, Hence, it is proved that is a CFP of , , and that is,
We now show the uniqueness. Assume that is another CFP of the mappings , and such thatThen, from (24), we have thatAfter simplification, we obtainBy using Proposition 3 (iii), we haveNote that is a maximum term in (58); therefore,which implies that is a contradiction, since ; therefore, Hence, the mappings , and have a unique CFP in . The proof is complete.
By taking in Theorem 10, we get Corollary 11.

Corollary 11. Let be a complete M-space with coefficient and be three self-mappings which satisfy, with and . Moreover, if then the three self-mappings , and have a unique CFP in .

Specializing in Theorem 10, we get Corollary 12.

Corollary 12. Let be a complete M-space with coefficient and be three self-mappings which satisfy, with . Moreover, if then the three self-mappings , and have a unique CFP in .

Example 2. Let be a M-space where and is defined as follows:for all . Now, we define the three self-mappings, that is, byNow, first, we calculate all the terms of (24) and we have thatAlso,Similarly, we can calculate the remaining terms of (13) as follows:and . Now, from (24), (64), and (65), we have thatHence, it is proved that all the conditions of Theorem 10 are satisfied with that is, and . The three self-mappings , and have a unique CFP in which is .

4. Application

In this section, we present an application to nonlinear integral equations (NLIEs) to support our results. The considered system of NLIEs is of the form as follows:where , for all where is the set of all real-valued continuous functions on and . A metric be defined as follows:

Then, easily one can prove that is a complete metric space. Now, we establish a theorem based on NLIEs to achieve the previous results on the existence of a common solution to support our work.

Theorem 13. Let be three self-mappings and let there exist satisfyingwhere and

Now, we define andwherefor all . Then, the system of NLIEs (68) has a unique common solution.

Proof. The integral operators be defined as follows:Now, we apply Theorem 7. Then, we may have the following three cases:(1)If be the maximum term in (71), then, from (69) and (70), we have that Thus, the mappings , and satisfy all the conditions of Theorem 7 with and in (2). Then, the given NLIEs, i.e., (35) have a unique common solution in .(2)If be the maximum term in (71), then, from (69) and (70), we have that Thus, the mappings , and satisfy all the conditions of Theorem 7 with and , in (2). Then, the given NLIEs (35) have a unique common solution in .(3)If be the maximum term in (71), thenThen further, we may have occurrence of the following four subcases: if be the minimum term in (77), then . Now, from (69) and (70), we have that if be the minimum term in (77) and , then . Now, from (69) and (70), we have that if be the minimum term in (77) and , then . Now, from (69) and (70), we have that if be the minimum term in (77), then . Now, from (69) and (70), we have thatThus, the subcases – satisfying all the conditions of Theorem 7 with and in (1) are satisfied. Then, the given system of NLIEs, i.e., (68), has a unique common solution in .

5. Conclusion

In this paper, we established some CFP theorems for three self-mappings on complete M-spaces. We proved the uniqueness of CFP by using some generalized rational-type contraction conditions in M-spaces without the continuity of self-mappings. We presented an illustrative example of a unique CFP for three self-mappings to justify our results. In addition, we presented an application of nonlinear integral equations to get the existing results for a unique common solution to support our work. By using this concept, one can define various rational-type contraction conditions for three or more single-valued and multivalued mappings in the context of generalized metric spaces such as generalized b-metric spaces, complex-valued generalized metric spaces, and complex-valued generalized b-metric spaces with applications of different types of differential equations and nonlinear integral equations.

Data Availability

No datasets were generated or analyzed during this current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors equally contributed to the finding and writing of this research work.

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Copyright © 2024 Mohammed M. M. Jaradat 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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