Solution of Nonlinear Fredholm Integral Equations on Almost <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.199599pt" id="M1" height="16.0985pt" version="1.1" viewBox="-0.0657574 -11.8989 25.27 16.0985" width="25.27pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g198-27" d="M407 519C374 464 344 443 287 443C240 443 200 472 200 528C200 573 239 648 335 648C442 648 532 615 613 583C557 535 504 459 456 342C449 343 444 343 439 343C392 343 348 315 313 278L330 262C369 291 396 294 425 294C430 294 434 294 437 293C400 197 365 125 317 78C266 94 218 109 175 109C107 109 59 82 59 51C59 16 88 -9 171 -9C229 -9 283 5 333 34C432 -1 479 -15 530 -15C655 -15 733 49 752 162L732 163C707 75 636 24 528 24C481 24 427 41 371 60C440 112 494 189 529 281C533 280 543 280 547 280C595 280 634 306 665 339L651 356C626 337 590 330 558 329C555 329 552 329 547 330C587 445 623 519 663 566C708 551 751 540 796 540C864 540 912 567 912 598S881 658 800 658C753 658 702 645 650 612C566 646 464 687 356 687C223 687 159 596 159 528C159 457 223 413 287 413C358 413 396 452 425 511L407 519ZM871 598C871 583 843 570 806 570C771 570 735 580 694 595C725 618 759 628 800 628C855 628 871 614 871 598ZM283 51C252 31 214 21 171 21C116 21 100 35 100 51C100 66 128 79 165 79C207 79 235 70 283 51Z"/></g><g transform="matrix(.012,0,0,-0.012,15.993,4.134)"><path id="g54-183" d="M645 0V53H381V555H319V53H55V0H645Z"/></g></svg>-</span>Contraction

Nallaselli, Gunasekaran; Gnanaprakasam, Arul Joseph; Mani, Gunaseelan; Javed, Khalil; Almalki, Yahya

doi:https://doi.org/10.1155/2023/7360236

Journal of Mathematics

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

Research Article | Open Access

Volume 2023 | Article ID 7360236 | https://doi.org/10.1155/2023/7360236

Solution of Nonlinear Fredholm Integral Equations on Almost -Contraction

Gunasekaran Nallaselli,¹Arul Joseph Gnanaprakasam,¹Gunaseelan Mani,²Khalil Javed,³and Yahya Almalki⁴

Academic Editor: A. Ghareeb

Received09 Sept 2022

Revised18 Nov 2022

Accepted16 Jan 2023

Published01 Feb 2023

Abstract

In this manuscript, we develop an orthogonal to basically -contraction and demonstrate various fixed point theorems of nonlinear Fredholm integral equation solutions in such a contraction. By using these ideas of discovering the fixed point theorems, we can also build the application of the Fredholm integral equation.

1. Introduction

The fixed point theorem is one of the fascinating subjects in nonlinear functional analysis because of how widely it may be used. Recently, Kojasteh et al. [1] proposed a simulation function that integrates certain known fixed point findings. Following that, Karapinar and Hima Bindu [2] explored fixed point findings on the almost -contraction. Roldán-López-de-Hierro et al. [3] somewhat adjusted their definition of simulation function and explored the presence and uniqueness of coincidence points of two nonlinear operators using this type of control function. Alharbi et al. [4] studied the existence and uniqueness of certain operators that establish a novel contractive condition by integrating the ideas of admissible function and simulation function in the setting of full b-metric spaces. Alqahtani et al. demonstrated that a fixed point exists and is unique for particular operators in [5]. These operators were thoroughly examined using simulation functions in -symmetric quasi-metric spaces. Aydi et al. [6] provided a number of fixed locations for -acceptable triangular contraction mappings. Vetro [7] identified a common fixed point and a point of a coincidence for two self-mappings and showed that such points are zeros of a given function defined in both metric space and partial metric space. Vetro [8] proposed the idea of ordered S-G-contraction by integrating the current notions of -contraction and Z-contraction. Radenovic et al. [9] directly established certain common fixed point solutions for two and three mappings under weak contractive circumstances, and some of these results are enhanced by utilizing various control function parameters. Samet et al. [10] developed a novel idea of -contractive type mappings, established a fixed point theorem for -contractive type mappings in full metric spaces, and improved several earlier findings in the literature (see, also [11]). Gordji et al. [12], in particular, brought the Banach contraction principle to the situation of an orthogonal set (briefly, O-set). Eshaghi Gordji and Habibi [13] proved the existence and uniqueness of the fixed point of the Cauchy problem for the first-order differential equation in the setting of orthogonal metric space, and this notion has been expanded in multiple ways by other writers (see, for example, [14, 15]). In the context of w-distances, Dhivya et al. [16] established a fixed point solution using nonlinear Fredholm integral equations. Sevinik-Adiguzel et al. [17] addressed the existence and uniqueness of solutions to a fixed point theorem of a specific form of nonlinear Volterra integral-dynamic equation on time scales. Kumari Panda et al. [18] achieved some fixed point findings and proposed a very easy solution for a Volterra integral problem utilizing the fixed point approach in the situation of dislocated extended b-metric space. Sahin [19] introduced a p-cyclic contraction mapping by combining the ideas of cyclic contraction mapping and p-contraction mapping and investigated the sufficient conditions for the existence of a solution to nonlinear Fredholm integral equations.

In this paper, we establish the fixed point theorems for a solution of nonlinear Fredholm integral equations on orthogonal almost -contraction. Various examples are presented to illustrate our obtained results.

2. Preliminaries

Throughout the paper, let , where represents the set of positive integers. Normally, denotes the set of all real numbers. In addition, we denote the set of nonnegative reals .

Khojasteh et al. [1] introduced a new control function namely a simulation function as follows:

Definition 1. (see [1]). A simulation function mapping from satisfies the following conditions: (1) (2) For each sequences in such that , thenWe observe that

Simulation function examples are given as follows.

Example 1. Let denote the collection of all simulation functions . A function , and is a simulation function.

Example 2. A continuous function on a self-map satisfies and a mapping defined byThen, is a simulation function.

-contraction introduced by Khojasteh et al. [1] as follows.

Definition 2. (see [1]). A self-map defined on a metric space is said to be -contraction with respect to (shortly, w.r.to) if

Here, recall the idea of an orthogonal set (or O-set), some examples, and its properties.

Definition 3. (see [12]). A binary relation defined on a nonvoid set satisfies the conditionsThen, the set is called an orthogonal set (shortly -set), and it is denoted by .

An example of O-set is given as follows.

Example 3. Let be the set of all ABO blood types were the most common factor considered in human paternity testing. A man who has type AB blood could not be a father of a child with type O blood, because he would pass on either the A or the B allele to all of his offspring. Define a binary relation on defined by if inherits types of blood to . If a child is a type of blood O, then we get . Hence, is an O-set, and is not have a unique blood type. In this case, a child may be blood type B, and we have .

Definition 4. (see [12]). A sequence defined on a nonvoid -set is said to be an orthogonal sequence (briefly, -sequence) if

Definition 5. (see [13]). A metric space is said to be an orthogonal metric space (shortly, OMS) if is an -set and is a metric space (shortly, MS).

Definition 6. (see [13]). In OMS , a self-map is orthogonal continuous (or -continuous) at if for each -sequence in with as , then we have as . Also, is said to be -continuous on if is -continuous in each .

Definition 7. (see [13]).(1)Let be an OMS, then the O-sequence in converges to in , iff as . We denote by .(2)Let be an OMS. We say that the O-sequence in is a Cauchy O-sequence if and only if as .(3)OMS is an orthogonal complete (briefly, -complete) if every Cauchy O-sequence is convergent.

Definition 8. (see [13]). A self-map on O-set is said to be -preserving if whenever .

The following are the main results of [1].

Theorem 1. Every -contraction on a complete MS has a unique fixed point.

In the next section, we use the following lemma in Radenovic et al. [9].

Lemma 1 (see [9]). Let be an O-sequence in a MS such that is nonincreasing and that

If is not a Cauchy O-sequence, then there exist a and two strictly increasing O-sequences and of positive integers such that the following O-sequences tend to when : .

Definition 9. A self-map is defined on a nonvoid O-set and a map . is said to be orthogonal extended -admissible (shortly, extended -admissible) if with ,If we put in (8), we say that is called orthogonal -admissible (shortly, -admissible).
Furthermore, if is extended -admissible, then we getSet , by (9), we get and .

3. Main Results

Definition 10. A self-map on an OMS is said to be orthogonal -contraction (shortly, contraction) with respect to (shortly, w.r.to) if, with , such that the following condition holds:

Definition 11. Let be a function and a self-map on OMS is called an orthogonal almost -contraction (shortly, almost contraction) w. r. to if , , and such that with wherewith

The following is our first main result:

Theorem 2. Let be a self-map on O-complete MS with an orthogonal element satisfying the following conditions:(i) is -preserving(ii) is an almost--contraction(iii) is an extended -admissible pair(iv) such that (v)either(a) is -continuous or(b) if there exists O-sequence in such that , then there is a subsequence of as such that Then, has a fixed point.

Proof. Consider is an O-set, there existsIt follows that or . LetSince (iv), there is a starting the initial point such that . We construct an O-sequence in by . Now, we arises two cases:(i)If such that , then we have . It is clearly is a fixed point of (shortly, fix ). Therefore, the proof is finished.(ii)If , for any , then we have , for each . On the contrary, where for , we desired that is the required fixed point, i.e., .Since is -preserving, we haveThis implies that is an -sequence.
Now, from (iii) and (iv), we desired thatContinuing in this way, we get for all . Furthermore, by regarding (9), we derive thatFirstly, we want to prove that is decreasing. On contrary, suppose that . Since (iv), we find thatwhich implies thatwhereHence, inequality (19) turns intoa contradiction from which we deduce that , for all since the O-sequence is decreasing. Next, we prove that . Since is decreasing and bounded below, from which it converges to in It is evident thatWe assume that . On contrary, suppose that . Then, from (22) and , putting limit as . Therefore,Consequently, and alsoAs follows, we prove that O-sequence is a Cauchy O-sequence. Contrary, we assume is not a Cauchy O-sequence, then there exists a positive number and two O-sequences such thatNow take and in (11), we haveimplieswhereDue to Lemma 1, we haveSinceUsing (26) and (31), we haveLet and , we have and letting in (29)Then by (31) and (34) and keeping in mind, we havea contradiction. Hence, proved our result and is a Cauchy O-sequence.
By the O-complete MS , the O-sequence approaches to some point as .Now, we will prove . Since is -continuous, suppose from (iiia), we haveSuppose we have (vb) and use the method of reductio ad absurdum. On contrary, assume that , i.e., . By (16), there exists a subsequence of such that .
It implies thatwhereBy putting in (38), with the previous, we havewhich is on contrary. Hence, has a fixed point of .

Theorem 3. In addition, to the hypothesis of Theorem 2, let us suppose we have .
Where the O-set is of all fix , then has a unique fixed point.

Proof. Now, to prove the uniqueness part, suppose that has two distinct fixed points, namely, with . On account of (vi), we have , which implieswherewithHence, expression (41) turns intoa contradiction and hence the proof.

Example 4. Let with a metric for all with . Let and considering and . Let be defined by and defined by

Since impliesTherefore, is almost -contraction w. r. to . Hence, all hypothesis of Theorem 3 holds, and hence, has a unique fixed point.

4. Immediately Consequence

The following will be a conclusion of our main results.

Theorem 4. A self-map on an O-complete MS and a mapping . Assume that and such that with where

Furthermore, we suppose, with , that(i)-preserving(ii) is an extended -admissible pair(iii) such that (iv)either(iva) is -continuous, or(ivb) if there exists O-sequence in such that , then there is a subsequence of as such that (v) we have , where the O-set is of all fix .

Then, has a fixed point.

Due to its similarity to the proof of Theorem 3, we omit this proof (and hence Theorem 2).

We omit the auxiliary function in Theorem 5, and we get the desirable result in the standard O-MSs.

Theorem 5. A self-map defined on an O-complete MS . Assume that and such that with where , and are in the above section Theorem 2. Then, has a unique fixed point.

Proof. Letting with , in Theorem 3.

Let be the family of all auxiliary functions and be -continuous with iff .

Theorem 6. A self-map defined on an O-complete MS and a mapping . Assume that with , and and such that with where , and are defined in the above section Theorem 2. Furthermore, suppose that with ,(i)-preserving(ii) is an extended -admissible pair(iii) such that (iv)either(iva) is -continuous, or(ivb) if there exists O-sequence in such that , then there exists a subsequence of as such that (v) we have , where the O-set is of all fix .Then, has a fixed point.

Proof. Assume , where . It shows that . As a conclusion, the required results are obtained as follows from Theorem 3.

The following theorem demonstrates the fixed point in integral function.

Theorem 7. A self-map on an O-complete MS and a mapping . Assume that with , , and such that with where , and are defined in Theorem 2. Furthermore, assume that with ,(i)-preserving(ii) is an extended- -admissible pair(iii) such that (iv)either(iva) is -continuous, or(ivb)if there exists O-sequence in such that and then there is a subsequence of as such that (v) we have , where the O-set is of all fix .

Then, has a fixed point.

Proof. Setting for all .
It is clear that . As a conclusion, the required results are obtained as follows from Theorem 3.

The following theorem demonstrates the fixed point in --contraction.

Theorem 8. A self-map on an O-complete MS and a function . Let and . Suppose there exist a upper semicontinuous function such that and . Suppose that with ,where , and are in the above section Theorem 2. Furthermore, suppose that with ,(i)-preserving(ii) is an extended - -admissible pair(iii) such that (iv)either(iva) is -continuous, or(ivb) if there exists O-sequence in such that , then there is a subsequence of as such that (v) we have , where the O-set is of all fix .

Then, has a fixed point.

Proof. Set .
It shows that (see e.g. [1–4]). Thus, the required results are obtained as follows from Theorem 3.

5. Application to Fredholm Integral Equations

In this section, we give an application of a solution for the nonlinear Fredholm integral equation via almost -contraction with OMS. Now, consider , be the O-complete MS of continuous real-valued functions defined on [c, d], and a functionwhere with such that and is a continuous function.

Theorem 9. Let with such that for all and . If

and for all , then (52) has a unique solution.

Proof. Set as with metric with . It is clear that the MS is O-complete. Now, the function defined byfor all . Clearly, is orthogonal MS on and a metric . Now, we have seen that satisfied almost -contraction in (11). Let in (11) and suppose that with and . Thus, we getFrom this, we havewhich implies that with . Therefore, we get with . For , it can be easy to verify that satisfies almost -contraction condition (11). Hence, satisfies all conditions of Theorem 2, and it has a unique fixed point. Hence, we get solution for the nonlinear Fredholm integral (52) that has a unique fixed point.

6. Conclusion

In this paper, we proved fixed point theorem of a solution to nonlinear Fredholm integral equations on an almost -contraction.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors equally contributed to this paper and read and approved the final manuscript.

Acknowledgments

The authors extend their appreciation to the Ministry of Education in KSA for funding this research work through the project number KKU-IFP2-H-6.

References

F. Khojasteh, S. Shukla, and S. Radenovic, “A new approach to the study of fixed point theorems via simulation functions,” Filomat, vol. 29, no. 6, pp. 1189–1194, 2015.
View at: Google Scholar
E. Karapinar and V. M. Hima Bindu, “Discussions on the almost Z-contraction,” Journal of Open Mathematics, vol. 18, pp. 448–457, 2020.
View at: Google Scholar
A. F. Roldán-López-de-Hierro, E. Karapinar, C. Roldán-López-de-Hierro, and J. Martínez-Moreno, “Coincidence point theorems on metric spaces via simulation functions,” Journal of Computational and Applied Mathematics, vol. 275, pp. 345–355, 2015.
View at: Google Scholar
A. S. S. Alharbi, H. H. Alsulami, and E. Karapinar, “On the power of simulation and admissible functions in metric fixed point theory,” Journal of Function Spaces, vol. 2017, Article ID 2068163, 7 pages, 2017.
View at: Publisher Site | Google Scholar
B. Alqahtani, A. Fulga, and E. Karapinar, “Fixed point results on -symmetric quasi-metric space via simulation function with an application to Ulam stability,” Mathematics, vol. 6, no. 10, 208 pages, 2018.
View at: Google Scholar
H. Aydi, A. Felhi, E. Karapinar, and F. A. Alojail, “Fixed points on quasi-metric spaces via simulation functions and consequences,” Journal of Mathematics Analysis, vol. 9, no. 2, pp. 10–24, 2018.
View at: Google Scholar
F. Vetro, “Points of coincidence that are zeros of a given function,” Results in Mathematics, vol. 74, no. 4, 159 pages, 2019.
View at: Publisher Site | Google Scholar
F. Vetro, “Fixed point results for ordered S-G-contractions in ordered metric spaces,” Nonlinear Analysis Modelling and Control, vol. 23, no. 2, pp. 269–283, 2018.
View at: Google Scholar
S. Radenovic, Z. Kadelburg, D. Jandrlic, and A. Jandrlic, “Some results on weakly contractive maps,” Bulletin of the Iranian Mathematical Society, vol. 38, no. 3, pp. 625–645, 2012.
View at: Google Scholar
B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for -contractive type mappings,” Nonlinear Analysis: Theory, Methods and Applications, vol. 75, no. 4, pp. 2154–2165, 2012.
View at: Google Scholar
E. Karapinar and B. Samet, “Generalized -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. E. Gordji, M. Ramezani, M. De La Sen, and Y. J. Cho, “On orthogonal sets and Banach fixed point theorem,” Fixed Point Theory (FPT), vol. 18, no. 2, pp. 569–578, 2017.
View at: Google Scholar
M. Eshaghi Gordji and H. Habibi, “Fixed point theory in generalized orthogonal metric space,” Journal of Linear and Topological Algebra (JLTA), vol. 6, no. 3, pp. 251–260, 2017.
View at: Google Scholar
M. Gunaseelan, G. Arul Joseph, M. Lakshmi Narayan, and M. Vishnu Narayan, “Fixed point theorems for orthogonal -Suzuki contraction mappings on -complete metric space with an applications,” Malaya Journal of Matematik, vol. 9, no. 1, pp. 369–377, 2021.
View at: Google Scholar
I. Beg, M. Gunseelan, and G. Arul Joseph, “Fixed point of orthogonal -Suzuki contraction mapping on -complete -metric spaces with applications,” Journal of Function spaces, vol. 2021, Article ID 6692112, 12 pages, 2021.
View at: Publisher Site | Google Scholar
P. Dhivya, M. Marudai, V. Rakocevic, and A. Fulga, “A solution to nonlinear Fredholm integral equations in the context of w-distances,” Advances in Difference Equations, vol. 2021, no. 1, pp. 1–23, 2021.
View at: Google Scholar
R. Sevinik-Adiguzel, E. Karapinar, and I. Erhan, “A solution to nonlinear Volterra integro-dynamic equations via fixed point theory,” Filomat, vol. 33, pp. 5331–2343, 2019.
View at: Google Scholar
S. Kumari Panda, E. Karapinar, and A. Atangana, “A numerical schemes and comparisons for fixed point results with applications to the solutions of Volterra integral equations in dislocated extended b-metric space,” Alexandria Engineering Journal, vol. 59, no. 2, pp. 815–827, 2020.
View at: Google Scholar
H. Sahin, “A new best proximity point result with an application to nonlinear Fredholm integral equations,” Mathematics, vol. 10, no. 4, p. 665, 2022.
View at: Publisher Site | Google Scholar

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Copyright © 2023 Gunasekaran Nallaselli 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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