Iterative Construction of Fixed Points for Functional Equations and Fractional Differential Equations

Rahman, Latif Ur; Arshad, Muhammad; Thabet, Sabri T. M.; Kedim, Imed

doi:https://doi.org/10.1155/2023/6677650

Journal of Mathematics

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

Research Article | Open Access

Volume 2023 | Article ID 6677650 | https://doi.org/10.1155/2023/6677650

Iterative Construction of Fixed Points for Functional Equations and Fractional Differential Equations

Latif Ur Rahman,¹Muhammad Arshad,¹Sabri T. M. Thabet,²and Imed Kedim³

Academic Editor: Xiaolong Qin

Received14 Jan 2023

Revised24 Sept 2023

Accepted21 Oct 2023

Published13 Nov 2023

Abstract

This paper proposes some iterative constructions of fixed points for showing the existence and uniqueness of solutions for functional equations and fractional differential equations (FDEs) in the framework of CAT (0) spaces. Our new approach is based on the -iterative scheme and the class of mappings with the KSC condition. We first obtain some and strong convergence theorems using -iterative scheme. Using one of our main results, we solve a FDE from a broad class of fractional calculus. Eventually, we support our main results with a numerical example. A comparative numerical experiment shows that the -iterative scheme produces high accurate numerical results corresponding to the other schemes in the literature. Our results are new and generalize several comparable results in fixed point theory and applications.

1. Introduction

Fixed point theory in recent years has suggested very useful techniques for solving nonlinear problems (for details, see the survey article by Karapinar [1, 2]). Iterative solutions for functional equations and FDEs are a busy field of research on their own [3]. It is known that a sought solution of a functional equation or a FDE can be expressed as a fixed point of a certain linear or nonlinear operator acting on a subset of a suitable distance space (see, e.g., [4] and others). The existence as well as the iterative construction of the fixed point of this operator is always desirable. We know that, the existence of a fixed point is possible but to construct a suitable algorithm to approximate the value of the fixed point is not an easy work (see, e.g., [5, 6] and others). For example, the Banach Contraction Principle (BCP) [7] suggests a unique fixed point for contractions and suggests the Picard iteration [8], that is, , to approximate the values of this unique fixed point [9]. On the other hand, the Browder–Gohde–Kirk result (see Browder [10], Gohde [11], and Kirk [12]) proved that every nonexpansive mapping on a closed convex bounded subset of a uniformly convex Banach space (UCBS) has a fixed point. Notice that a self-map on a subset of a metric space is essentially called a contraction ifwhere .

A fixed point of in this case is any element, namely, with the property . The set of all fixed points of the operator will be denoted simply by throughout in this research paper. If (1) holds when we put then is known as nonexpansive. An example of a nonexpansive mapping for which Picard iteration does not converge is the following:

Example 1. Let and for all . It follows that is nonexpansive with and the Picard iteration is not convergent to 0.5 for all the starting value which is different from 0.5.

Example 1 suggests other iterative schemes instead of Picard iteration [8] which are convergent in the setting of nonexpansive mappings (or even generalized nonexpansive mappings). In 2008, Suzuki [13] introduced a condition on mappings, called condition.

Definition 1 (see [13]). The self-map of is said to satisfy the (C) condition of Suzuki if

The (C) condition is essentially weaker than the nonepensiveness property of any operator . For instance, see an example in [13].

Strongly motivated by Suzuki [13], the author Karapinar and Taş [14] suggested another condition for mappings.

Definition 2. (see [14]). The self-map of is said to satisfy KSC condition (or said to satisfy Kannan–Suzuki (C) condition) if

In fact, there are many iterative schemes in the literature, that are extensively used for approximating fixed points in different settings of mappings, (see e.g., Mann [15], Ishikawa [16], -iteration of Agarwal et al. [17], three-step iteration of Noor [18], Abbas [19], Thakur et al. [20], and others).

Ullah and Arshad [21] constructed a new iteration called -iteration and proved that this iteration is stable and suggests highly accurate results corresponding to other iterations of the literature. This iteration generates a sequence as follows:

In the scheme (4), the operator is a self-map of the set and the sequences and are in the interval (0,1). Although Ullah and Arshad [21] proved the convergence of the scheme (4) in the case of contractions. We extend here their main outcome to the more general setting of mappings satisfying the KSC condition. Using the same techniques, convergence of the above mentioned iterations can be proved on the same line of proof. Using a nontrivial example, we show that the iteration scheme suggests accurate results corresponding to the other iterations in this new setting of mappings.

2. Preliminaries

Now, we need some basic results of CAT (0) spaces. For more details on CAT (0) spaces, please see the books [22, 23].

We now state a result from [24].

Lemma 3. Suppose is any complete CAT (0) space and . Then,(a)If we have and there is a fixed element in the set [0,1], then one has a unique point , such that Sometimes, we may write as the unique point that satisfies (5).(b)If we have and is fix, then one has

Definition 4. Let be a bounded sequence in a metric space and closed convex. We denote and set the asymptotic radius of the sequence in the set as . We denote and set the asymptotic center of the sequence in the set as . If is a complete CAT (0) space then contains one and only one point.

The following is the definition of a convergence that can be considered as an analog of the weak convergence in a Banach space.

Definition 5. A bounded sequence, namely, in a complete CAT (0) space is said to be -convergent to a point, namely, (and denote it as ) if it is the case that the point is the unique asymptotic center for each choice of the subsequence of .

The CAT (0) version of the Opial’s [25] condition holds, that is, if is any -convergent sequence in a complete CAT (0) space with the limit , then for any , one has

Lemma 6 (see [26]). Suppose we have complete CAT (0) space . Then any bounded sequence admits a -convergent subsequence.

Lemma 7 (see [27]). Suppose we have complete CAT (0) space . If is convex and closed then the asymptotic center of any bounded sequence is contained in the space .

Definition 8. (see [28]). A self-map on a subset of a CAT (0) space is said to satisfy condition (I) if one has a function with , for each and for every point , where the notation is the distance of the point to the set .

Lemma 9 (see [14]). Suppose is any CAT (0) space and . Let be a self-map of satisfying KSC condition with . Then for any and , one has the following property:

Lemma 10 (see [14]). Suppose is any CAT (0) space and . Let be a self-map of satisfying KSC condition. Then for any , one has the following property:

Lemma 11 (see [14]). Suppose is any complete CAT (0) space and . Let be a self-map of satisfying KSC condition. Then, the following property holds:

Lemma 12 (see [29]). Consider , for every choice of . If and be two sequences in a complete CAT (0) space with and and for a real number , and some then .

3. Main Results

First, we define the CAT (0) space version of iterative scheme (4) as follows:

Now, using (11), we prove our man results. We first provide the following lemma that will play a key role.

Lemma 13. Suppose is any complete CAT (0) space and is closed and convex. Let be a self-map of satisfying the (KSC) condition with . Then the sequence generated by -iteration (11) satisfies exists for each .

Proof. Consider any point , then applying Lemma 9, one hasHence, we obtained for all , . This means that is essentially bounded as well as nonincreasing and hence it follows that exists for all .
Now, for the existence of a fixed point, we give the necessary and sufficient condition for mappings with condition defined on nonempty closed convex subsets of a UCBS as follows.

Theorem 14. Suppose is any complete CAT (0) space and is closed and convex. If is a self-map of satisfying KSC condition and is the sequence of -iteration (11). Then, , if and only if is bounded and satisfies .

Proof. First, we assume the case that the set and prove that is bounded with . For this, Lemma 13 suggests that is bounded and exists. Putfor some . We assume the nontrivial case, that is, when . Then in the view of the proof of Lemma 13, . It follows thatNow, from Lemma 9. So,Again, we see that, form the proof of Lemma 13. It follows that . So,Thus from (14) and (16), we haveFrom (17), we haveNow, applying Lemma 12 on (13), (15), and (18), we obtainFinally, we shall assume is bounded with the property and show that the set . For this, we may assume any point, namely, in the set . By Lemma 10, we haveThis implies that . Since is singleton, hence, we have and hence .
We first suggest a convergence result.

Theorem 15. Suppose is any complete CAT (0) space and is closed and convex. Let be a self-map of satisfying the KSC condition with . Then the sequence of the -iteration (11) -converges to some fixed point of provided that the space has Opial’s property.

Proof. Using Theorem 14, we have the sequence of iterates is bounded in the set and satisfies the condition . Set , where denotes any subsequences . We prove . To achieve the objective, we let , thus one can find a subsequence of such that . Applying Lemmas 6 and 7, one has a subsequence of such that converges to a point in . Now, using Theorem 14, we have . Also, is endowed with KSC condition, thereforeApplying limit on (21), it follows that . Hence, using Lemma 13, one has exists. The next aim is to obtain that . We prove this by contradiction, that is, we assume that . Keeping the uniqueness of asymptotic centers in mind, one hasSubsequently, we obtained . Since this is a contradiction, we conclude that and hence .
Eventually, we prove -converges to a fixed point of , that is, we need to show contains only one point. Suppose is a subsequence of and applying Lemmas 6 and 7, we have a -convergent subsequence of that -converges to a point in . Suppose and . Then since we proved already that and , we claim . Because if , then exists and keeping the uniqueness of asymptotic centers in mind, one haswhich is clearly a contradiction. Hence, we conclude that . It follows that . So -converges to a fixed point of .
The following theorem is based on compactness.

Theorem 16. Suppose is any complete CAT (0) space and is compact and convex. Let be a self-map of satisfying the KSC condition with . Then the sequence of the -iteration (11) converges strongly to some fixed point of .

Proof. As assumed, the set is convex and compact, so the sequence of iterates contained in the set and has a subsequence of that converges strongly to . Moreover, in the view of Theorem 14, we obtain . Hence, using these facts together with Lemma 10, we haveIt follows that . By Lemma 13, exists and hence is strongly convergent to .
Strong convergence without compactness of the domain is the following.

Theorem 17. Suppose is any complete CAT (0) space and is closed and convex. Let be a self-map of satisfying the KSC condition with . Then, the sequence of the -iteration (11) converges strongly to some fixed point of provided that .

Proof. The proof of this result is easy and hence we exclude the proof.

Theorem 18. Suppose is any complete CAT (0) space and is closed and convex. Let be a self-map of satisfying the KSC condition with . Then, the sequence of the -iteration (11) converges strongly to some fixed point of provided that the satisfies condition (I).

Proof. According to Theorem 14. Now, condition (I) of gives . Thus by Theorem 17, is strongly convergent in .

4. Numerical Example

In this section, first we give a numerical example of a mapping with condition which does not satisfy (C) condition and then we show that the sequence generated by -iteration process converges faster than some other well-known iteration schemes.

Example 2. Define a mapping on as follows:Now, we see that the above self-map is not enriched with condition. For example, if one chose and , then does not satisfy the condition. Eventually, we shall establish that this map is enriched with KSC condition. To achieve the objective, some elementary cases have been omitted, while nontrivial cases are considered as follows: C1: When , we have C2: When , we have C3: When and we have C4: When and , we have C5: When and , we have

Now, we draw a graph and table which show that the sequence of the scheme -iteration moving faster to the fixed point 0 of as compared to Thukar, Abbas, Noor, and Ishikawa iterative schemes. Assume that , , and . The iterative results are shown in Table 1 while the behavior of iterates are given in Figure 1. The effectiveness of iterative scheme is clear in both the table and graph.

We finish the paper with a nontrivial example.

Example 3. Let and . Put . Clearly, . Define on as follows:Here, the space is only a CAT (0) space but not a Banach space [22]. Also, is closed and convex. Now, let be the metric projection on , then by a well-known result (see, p178 in [22]) that is nonexpansive and hence it satisfies KSC condition. By our main results, the sequence (11) converges to a fixed point of .

5. Application to Differential Equations

In this section, we study the solution of a FDE in our new setting of mappings. This problem has been considered by some authors in the class of nonexpansive mappings [30] and other types of spaces [31, 32]. It is important to note here that our approach is alternative and based on the class of mappings with KSC. The main difference between our approach and classical approaches to the problems are that mappings with KSC are not necessarily continuous throughout on the their domains. Moreover, our iterative method is more effective and suggests very high accurate numerical results in less step of iterates. To achieve the objective, we follow the idea given by [33].

We consider the following general class of boundary value problems from fractional calculus:where , , and stands for the Caputo fractional derivative with order and .

Consider and Green’s function associated with (32) that reads as follows:

The main result is provided in the following way.

Theorem 19. Set a self-map by the following formula:

Ifthen, the iterates (11) associated with the (as defiend above) converge to the point of provided that , where denotes the set of all solutions of (32).

Proof. Since is a Green’s function to our problem, so by [34], the sought solution can be expressed as an integral form as follows:Now, for every choice of and , it follows thatConsequently, we obtainHence, satisfies the condition. In the view of Theorem 17, the sequence of the iterates converges to a fixed point of and hence to the solution of the given equation.

6. Conclusions

Existence as well as iterative constructional for the class of mappings satisfying the KSC condition is established under the iterative scheme in a CAT (0) space setting. We proved and strong convergence results for these mappings under certain mild conditions. It has been shown by providing an example that the class of mappings satisfying the KSC condition is different than the class of mappings satisfying condition. Eventually, we performed a comparative numerical experiment and proved that the iterative scheme in the class of KSC mappings is more effective than the many other iterative scheme. One application is also carried out. Our results refine and improve some main results due to Ullah and Arshad [21] from the case of the condition to the more general case of KSC condition. Similarly, our results extend the results of Abbas and Nazir [19], Agarwal et al. [17], Noor [18], Thakur et al. [20], and others.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Every author contributed equally to each part of the paper.

Acknowledgments

This study is supported via funding from the Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1445).

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Copyright © 2023 Latif Ur Rahman 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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