Convergence Analysis of an Iteration Process for a Class of Generalized Nonexpansive Mappings with Application to Fractional Differential Equations

Ullah, Kifayat; Thabet, Sabri T. M.; Kamal, Anwar; Ahmad, Junaid; Ahmad, Fayyaz

doi:https://doi.org/10.1155/2023/8432560

Discrete Dynamics in Nature and Society

On this page

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

Research Article | Open Access

Volume 2023 | Article ID 8432560 | https://doi.org/10.1155/2023/8432560

Convergence Analysis of an Iteration Process for a Class of Generalized Nonexpansive Mappings with Application to Fractional Differential Equations

Kifayat Ullah,¹Sabri T. M. Thabet,^2,3Anwar Kamal,¹Junaid Ahmad,⁴and Fayyaz Ahmad¹

Academic Editor: Rigoberto Medina

Received15 Dec 2022

Revised03 Mar 2023

Accepted04 Mar 2023

Published21 Mar 2023

Abstract

We consider the class of generalized α-nonexpansive mappings in a setting of Banach spaces. We prove existence of fixed point and convergence results for these mappings under the K^∗-iterative process. The weak convergence is obtained with the help of Opial’s property while strong convergence results are obtained under various assumptions. Finally, we construct two numerical examples and connect our K^∗-iterative process with them. An application to solve a fractional differential equation (FDE) is also provided. It has been eventually shown that the K^∗- iterative process of this example gives more accurate numerical results corresponding to some other iterative processes of the literature. The main outcome is new and improves some known results of the literature.

1. Introduction and Preliminaries

In recent years, theory of fixed points gained the attention of many authors [1, 2]. Whenever the ordinary analytical techniques cannot yield a solution to a differential or an integral equation, we are interested in finding the approximate value of the requested solution (see, e.g., the recent results in [3–5] and others). Before employing the appropriate iterative processes on such problems, one needs to convert it into a form of equation of fixed point. In this way a sequence is generated by the algorithm. The intended fixed point value of the equation of fixed point and the given equation’s solution is the limit of the series. In case of contraction mappings, Banach fixed point theorem [6] signals the fundamental Picard iteration . However, when the Picard iterative process for a given mapping does not converge, we employ alternative iterative procedures with different steps. One of the other iterative processes that have been studied by authors are the Mann [7], Noor [8], Ishikawa [9], and SP iteration (Phuengrattana and Suantai) [10]; S-iteration (Agarwal et al.) [11]; iteration (Karahan and Ozdemie) [12]; Picard–Mann hybrid [13]; Normal-S [14]; Krasnoselskii–Mann [15]; Abbas [16]; Picard-S [17]; and Thakur [18].

On the other hand, Ullah and Arshad [19] presented a new iterative process for generalized nonexpansive mappings and call it as a iterative process. The -iterative process reads as follows:where .

They demonstrated that among the many other iterative processes, the iteration (1) gives very high accurate results in very less steps of iterations in the setting of Suzuki mappings. We improve here their results to the larger class of generalized -nonexpansive mappings.

Definition 1. Let . Then, is referred to as(i)Nonexpansive if , for every two ;(ii)Satisfying Condition (C) (or Suzuki mapping) if implies , for every two ;(iii)Generalized -nonexpansive if implies , for every two and ;(iv)Satisfying condition [20] if there is nondecreasing with and at and for all .First time, a fixed point existence result for nonexpansive mappings established by Gohde [21] and Browder [22] in a setting of uniform convex Banach space (UCBS), and in the same year, Kirk [23] obtained the same result in a setting of reflexive Banach space. In [24], Suzuki suggested a very interesting generalized of nonexpansive mappings and any mapping of this class if named as a mapping satisfying condition (or Suzuki mapping). He established several convergence and existence results for these mappings in different Banach space settings. Since mappings with condition (C) are more general than the concept of nonexpansive mappings. Thus, Pant and Shukla [25] generalized the Suzuki mappings by introducing the class of generalized -nonexpansive mappings. They proved that every Suzuki mapping is generalized -nonexpansive but the converse is not valid in general; that is, they proved that the class of generalized -nonexpansive mappings properly includes the class of Suzuki mappings. Moreover, they used the Agarwal iteration [11] for establishing the main convergence results. The purpose of this work is to obtain the strong and weak convergence for the -iterative processes for generalized -nonexpansive mappings. In this way, we extend some main results of Pant and Shukla [25], Ullah and Arshad, and many others.

Definition 2 (see [26, 27]). Let be any nonvoid closed convex subset of a UCBS . is bounded. If is any fixed element then we set the following: (b1) For a bounded sequence at point , is termed as asymptotic radius; (b2) For a bounded sequence with the connection of , is termed as asymptotic radius; (b3) For a bounded sequence with the connection of , is termed as asymptotic center.

Definition 3 (see [28]). A Banach space space is said to satisfy the Opial’s condition in the case of any weakly convergent sequence whose weak limit is , one is able to obtain the following:In [25], the authors obtained some characterizations for the class of generalized -nonexpansive mappings. We write all these characterizations in the following proposition:

Proposition 1. Assume that is a self-map on any nonempty subset of a Banach space. Then, (p1) If satisfy condition , then is essentially generalized -nonexpansive. (p2) If is generalized -nonexpansive having a nonempty fixed point set, then for and in . (p3) If is generalized -nonexpansive, then is closed. Moreover, if the given space is strictly convex, is convex, then the set is also convex. (p4) If is generalized -nonexpansive, then for all pair of elements , one has (p5) If the space is endowed with the Opial condition, is generalized -nonexpansive, is any weakly convergent sequence to with the property , then .

Following lemma is a well-known property of any UCBS that is needed for our main results.

Lemma 1 (see [29]). Let be any UCBS. For , so that , and for some . Subsequently, we have .

2. Main Findings

To obtain our weak and strong convergence results, we need a key lemma as follows:

Lemma 2. If is generalized -nonexpansive self-map on a closed convex subset of a Banach space with and is a sequence of iterates obtained from the -iterative process (1). Subsequently, exists for each .

Proof. Let us take any . Using Proposition 1, we see thatThis implies thatHence, , which shows that is bounded and nonincreasing. This gives result exists for each .

Now we discuss about necessary and sufficient conditions that must be met in order for any generalized nonexpansive mapping in a Banach space to have fixed points.

Theorem 1. If is generalized -nonexpansive self-map on a closed convex subset of a UCBS and is a sequence of iterates obtained from the -iterative process (1). Subsequently, if and only if is bounded in and .

Proof. Let and . Applying Lemma 2, we get existence of and is bounded. Let be that limit; thus,As we have demonstrated in the proof of Lemma 2 thatThis together with (6) givesSince is in , so Proposition 1 (p₂) can be applied to obtain the following:Now by the proof of Lemma 2, we have the following:From (8) and (10), we haveBy (1) and (11), one hasIf and only ifLemma 1, can be applied to obtain,On the other hand we aim to demonstrate under the suppositions of a bounded in the sense that . We may select a point . If Proposition 1 is applied, then we must have the following:We observe that . As this set has only element in any UCBS setting, we deduce , hence the set is nonempty.

Among the convergence results, we first obtain our weak convergence result for the -iterative process in the setting of generalized -nonexpansive mappings as follows:

Theorem 2. If is generalized -nonexpansive self-map on a closed convex subset of a UCBS and is a sequence of iterates obtained from the -iterative process (1). Subsequently, is weakly convergent to a point of .

Proof. Using Theorem 1, the provided sequence is bounded. As is UCBS, a is reflexive Banach space. That is why, we can built a weakly convergent sequence of so that be the subsequence with as weak limit. Applying Theorem 1 on the subsequence, we may have . Hence, by Proposition 1, we have . It is enough to prove that is weakly convergent to . Thus, if does not weakly converge to . Then, a subsequence of and with converging weakly to and exists. Also by Proposition 1, , by Lemma 2 with Opial’s property, we getThis is a contradiction. Hence, we have . So, is weakly convergent to .

Now we discuss the strong convergence of the -iterative process for generalized -nonexpansive mappings on compact domains.

Theorem 3. If is generalized -nonexpansive self-map on a compact convex subset of a UCBS and is a sequence of iterates obtained from the-iterative process (1). Subsequently, is strongly convergent to a point of .

Proof. As the domain is a compact subset of and also due to the convexity of . Thus, a subsequence of exists with for some . In the view of Theorem 1, . Applying Proposition 1, one hasHence, if , then . In the view of Lemma 2, is the strong limit of . This finishes the proof.

A strong convergence of the -iterative process in the setting of generalized -nonexpansive mappings on a noncompact domain is established as follows. It should be noted that this result holds in general Banach spaces.

Theorem 4. If is generalized -nonexpansive self-map on a closed convex subset of a Banach space and is a sequence of iterates obtained from the -iterative process (1). Subsequently, is strongly convergent to a point of if.

Proof. Using Lemma 2, we have existence of , for each fixed point of . This provides us that exist. AccordinglyTwo subsequence and of and are, respectively, generated by the above limit. HenceFrom the proof of Lemma 2, we get nonincreasing , that is whyIt follows thatConsequently, is Cauchy sequence in as ; thus, converges to . Using Proposition 1, is closed; thus, . By Lemma 2, exists so, is the strong limit of .

Theorem 5. If is generalized -nonexpansive self-map on a closed convex subset of a UCBS and is a sequence of iterates obtained from the -iterative process (1). Subsequently, is strongly convergent to a point of if satisfies condition (I).

Proof. Using Theorem 1, we can haveDue to condition (I) of , we haveApplying (22) on (23), we getIt followsNow applying Theorem 4, converges strongly to .

3. Application to Fractional Differential Equations

Fractional calculus is important and an active field of research on its own [30–32]. It is well-known that fractional calculus has a crucial role in fluid, electromagnetic theory, and, especially, in electrical networks. In recent years, many papers appeared on the existence and approximation of solutions for certain FDEs (see e.g., Karapinar et al. [33] and others). However, all these authors used the concept of nonexpansive mappings to achieve the main objective that are continuous on their domain of definitions.

Our alternative in this paper is to solve a FDE in the setting of generalized -nonexpansive mappings that are in general discontinuous. Unlike, other iterative schemes, we suggest the iterative scheme (1) to find the solution for the following FDE.

Now we consider the following FDE and also assume that is a solution set of it:where , , and stands for the Caputo fractional derivative endowed with the order and .

Now we consider , where is the Banach space of continuous maps on to equiped with the maximum norm. The corresponding Green’s function with (26) is defined by

The main result is provided in the following way:

Theorem 6. If , then set an operator by the formula

Ifwhere is some real number in [0, 1). Subsequently, the iterates (1) associated with the (as defined above) essentially converges to some point of the solution set of (26) provided that .

Proof. Notice that the element of solves (26) if it solvesNow for every choice of and , it follows thatConsequently, we getHence, is generalized -nonexpansive mapping. In the view of Theorem 4, the sequence of iterates converges to a fixed point of and hence to the solution of the given equation.

4. Numerical Example

First, we construct a novel example of generalized -nonexpansive mappings on closed convex subset of a UCBS. Using this example, we perform a comparative numerical experiment using our and other iterative processes of the literature.

Example 1. Let and a self-map on by the following rule:In this case, we prove that is generalized -nonexpansive but does not satisfy the condition .

Proof. Let , then following are the all possible cases:

Case I. If , we have

Case II. If , we have

Case III. If and , we haveHence, for every two points . Now let and , then . Thus, does not satisfy condition (C).

To show the high accuracy of the proposed iteration, we compare it with the one-step Mann iteration [7], two-step Ishikawa [9], leading two-stepS-iteration of Agarwal [11] and a leading iterative scheme studied by Thakur [18]. We may take and . Table 1 shows some values for the initial value of . Additionally, Figure 1 offers detail on the behavior of the different schemes. Moreover, if , then further comparison is given in Table 2. For generalized -nonexpansive mapping it is evident that the iterative method performs better than the other methods.

We finish this section with an example. This example uses a subset of a two-dimensional Euclidian space.

Example 2. Let and set a self-map on by the following rule:Here is a generalized -nonexpansive mapping with fixed point (0, 0). The numerical results are shown in Table 3. In this case, it is also clear that our iterative scheme is moving fast to the fixed point (0, 0).

5. Conclusions

In this research, we obtained the following new finding:(i)We studied the iterative scheme of Ullah and Arshad for approximating fixed points of generalized -nonexpansive mappings.(ii)We successfully carried out some weak and strong convergence results under various mild conditions.(iii)We carried out an application of our main outcome for solution of a FBVP in a Banach space setting.(iv)A new example of generalized -nonexpansive mappings is constructed and proved that it exceeds properly the class of mappings with condition (C).(v)Using our new example, we showed that the iterative scheme is more effective and suggests very high accurate numerical results in the setting of generalized -nonexpansive mappings in the setting of generalized -nonexpansive mappings.(vi)Accordingly, our main outcome improved some recent results of Ullah and Arshad [19] form the case of mappings with condition (C) to the general case of mappings called generalized -nonexpansive mappings. In a similar way, our results are the improvement and refinements of the results due to Agarwal [11], Abbas [16], Thakur [18], and many others from the setting of nonexpansive and Suzuki mappings to the general setting of generalized -nonexpansive mappings.

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

Each author contributed equally and significantly to every part of this article. All authors read and approved the final version of the paper.

Acknowledgments

This research work was conducted while Junaid Ahmad was visiting the University of Lakki Marwat. Junaid Ahmad would like to thank Anwar Kamal and Kifayat Ullah for their kind hospitality.

References

N. Hussain, S. M. Alsulami, and H. Alamri, “Solving fractional differential equations via fixed points of Chatterjea maps,” Computer Modeling in Engineering and Sciences, vol. 135, no. 3, pp. 2617–2648, 2023.
View at: Publisher Site | Google Scholar
A. R. Khan, V. Kumar, and N. Hussain, “Analytical and numerical treatment of Jungck-Type iterative schemes,” Applied Mathematics and Computation, vol. 231, pp. 521–535, 2014.
View at: Publisher Site | Google Scholar
S. T. M. Thabet, M. B. Dhakne, M. A. Salman, and R. Gubran, “Generalized fractional Sturm- Liouville and Langevin equations involving Caputo derivative with nonlocal conditions,” Progress in Fractional Differentiation and Applications, vol. 6, no. 3, pp. 225–237, 2020.
View at: Google Scholar
S. T. M. Thabet and M. B. Dhakne, “On abstract fractional integro-differential equations via measure of noncompactness,” Advances in Fixed Point Theory, vol. 6, no. 2, pp. 175–193, 2016.
View at: Google Scholar
S. T. M. Thabet and M. B. Dhakne, “On positive solutions of higher order nonlinear fractional integro-differential equations with boundary conditions,” Malaya Journal of Matematik, vol. 7, no. 1, pp. 20–26, 2019.
View at: Publisher Site | Google Scholar
S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux equations integrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
View at: Publisher Site | Google Scholar
W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
View at: Google Scholar
M. A. Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 217–229, 2000.
View at: Google Scholar
S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974.
View at: Google Scholar
W. Phuengrattana and S. Suantai, “On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval,” Journal of Computational and Applied Mathematics, vol. 235, pp. 3006–3014, 2011.
View at: Google Scholar
R. P. Agarwal, D. O’Regan, and D. R. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings,” Journal of Nonlinear and Convex Analysis, vol. 8, no. 1, pp. 61–79, 2007.
View at: Google Scholar
I. Karahan and M. Ozdemir, “A general iterative method for approximation of fixed points and their applications,” Advances Fixed Point Theory, vol. 3, no. 3, pp. 510–526, 2013.
View at: Google Scholar
S. H. Khan, “A Picard-Mann hybrid iterative process,” Fixed Point Theory and Applications, vol. 2013, no. 1, 69 pages, 2013.
View at: Publisher Site | Google Scholar
D. R. Sahu and A. Petrusel, “Strong convergence of iterative methods by strictly pseudo-contractive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 17, pp. 6012–6023, 2011.
View at: Google Scholar
H. Afsharia and H. Aydi, “Some results about Krasnoselskii-Mann iteration process,” The Journal of Nonlinear Science and Applications, vol. 9, pp. 4852–4859, 2016.
View at: Google Scholar
M. Abbas and T. Nazir, “A new faster iteration process applied to constrained minimization and feasibility problems,” Matematicki Vesnik, vol. 66, pp. 223–234, 2014.
View at: Google Scholar
F. Gursoy and V. Karakaya, “A picard-s hybrid type iteration method for solving a differential equation with retarded argument,” 2014, https://arxiv.org/abs/1403.2546#:%7E:text=version%2C%20v2)%5D-,A%20Picard%2DS%20hybrid%20type%20iteration%20method%20for%20solving,differential%20equation%20with%20retarded%20argument%26text=We%20introduce%20a%20new%20iteration,fixed%20point%20of%20contractio.
View at: Google Scholar
B. S. Thakur, D. Thakur, and M. Postolache, “A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized nonexpansive mappings,” Applied Mathematics and Computation, vol. 275, pp. 147–155, 2016.
View at: Google Scholar
K. Ullah and M. Arshad, “New three step iteration process and fixed point approximation in Banach spaces,” Journal of Linear and Topological Algebra, vol. 8, p. 871, 2018.
View at: Google Scholar
H. F. Senter and W. G. Dotson, “Approximating fixed points of nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 44, pp. 375–380, 1974.
View at: Google Scholar
D. Gohde, “Zum prinzip der Kontraktiven abbildung,” Mathematische Nachrichten, vol. 30, pp. 251–258, 1965.
View at: Google Scholar
F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences (PNAS), vol. 54, pp. 1041–1044, 1965.
View at: Google Scholar
W. A. Kirk, “A fixed point theorem for mappings which do not increase distance,” The American Mathematical Monthly, vol. 72, pp. 1004–1006, 1965.
View at: Google Scholar
T. Suzuki, “Fixed point theorems and convergence theorems for some generalized nonexpansive mapping,” Journal of Mathematical Analysis and Applications, vol. 340, pp. 1088–1095, 2008.
View at: Google Scholar
R. Pant and R. Shukla, “Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces,” Numerical Functional Analysis and Optimization, vol. 38, no. 2, pp. 248–266, 2017.
View at: Google Scholar
W. Takahashi, Nonlinear Functional Analysis, Yokohoma Publishers, Yokohoma, Japan, 2000.
R. P. Agarwal, D. O’Regan, and D. R. Sahu, “Fixed point theory for lipschitzian-type mappings with applications series,” Topological Fixed Point Theory and Its Applications, Springer, New York, NY, USA, 2009.
View at: Google Scholar
Z. Opial, “Weak and strong convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.
View at: Google Scholar
J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, pp. 153–159, 1991.
View at: Google Scholar
K. Gopalan, S. T. Zubair, and T. Abdeljawad, “New fixed point theorems in operator valued extended hexagonal b-metric spaces,” Palestine Journal of Mathematics, vol. 11, pp. 48–56, 2022.
View at: Google Scholar
J. V. D. C. Sousa, T. Abdeljawad, and D. S. Oliveira, “Mild and classical solutions for fractional evolution differential equation,” Palestine Journal of Mathematics, vol. 11, no. 2, pp. 229–242, 2022.
View at: Google Scholar
N. Mlaiki, M. Hajji, T. Abdeljwad, and H. Aydi, “Generalized Meir-Keeler contraction mappings in controlledmetric type spaces,” Palestine Journal of Mathematics, vol. 10, no. 2, pp. 465–474, 2021.
View at: Google Scholar
E. Karapnar, T. Abdeljawad, and F. Jarad, “Applying new fixed point theorems on fractional and ordinary differential equations,” Advances in Difference Equations, vol. 421, pp. 1–25, 2019.
View at: Google Scholar
R. Chugh, V. Kumar, and S. Kumar, “Strong Convergence of a new three step iterative scheme in Banach spaces,” American Journal of Computational Mathematics, vol. 2, pp. 345–357, 2012.
View at: Google Scholar

Copyright

Copyright © 2023 Kifayat Ullah 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.

PDF Download Citation

Download other formats

Order printed copies