Fixed points of weakly K-nonexpansive mappings and a stability result for fixed point iterative process with an application

In this article, we introduce a new type of non-expansive mapping, namely weakly K-nonexpansive mapping, which is weaker than non-expansiveness and stronger than quasi-nonexpansiveness. We prove some weak and strong convergence results using weakly K-nonexpansive mappings. Also, we define weakly (α,K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\alpha ,K)$\end{document}-nonexpansive mapping and using it prove one stability result for JF-iterative process. Some prominent examples are presented illustrating the facts. A numerical example is given to compare the convergence behavior of some known iterative algorithms for weakly K-nonexpansive mappings. Moreover, we show by example that the class of α-nonexpansive mappings due to Aoyama and Kohsaka and the class of generalized α-nonexpansive mappings due to Pant and Shukla are independent. Finally, our fixed point theorem is applied to obtain a solution of a nonlinear fractional differential equation.


Introduction
Throughout this article, (B, · ) denotes a real Banach space, and D is a non-empty, closed and convex subset of B, unless otherwise stated. Let Υ : D → D be a self-mapping, and Fix(Υ ) denotes the set of all fixed points of Υ . Also, we use the notations u n u and u n → u for a sequence {u n } converging weakly and strongly to u, respectively.
The self-mapping Υ on D is said to be non-expansive (see [19]) if Υ x -Υ y ≤ xy for all x, y ∈ D and is said to be quasi-nonexpansive (see [19]) if Fix(Υ ) = ∅ and Υ xρ ≤ xρ for all x ∈ D and ρ ∈ Fix(Υ ). There are several extensions and generalizations of non-expansive mappings considered by many researchers.
In 1973, Hardy and Rogers [21] introduced the notion of generalized non-expansive mapping as below: © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Definition 1.1 ([21]) A mapping Υ : D → D is said to be a Generalized non-expansive mapping if for all x, y ∈ D, where α i ≥ 0 with 5 i=1 α i ≤ 1. Or equivalently, [11] with α i ≥ 0 and α 1 + 2α 2 + 2α 3 ≤ 1.
It is clear that if Fix(Υ ) = ∅, then Υ is a quasi-nonexpansive mapping.
In 2008, Suzuki [44] introduced a new generalization of non-expansive mappings, namely Condition (C) as below: It is also clear that a mapping with a fixed point satisfying Condition (C) is necessarily a quasi-nonexpansive mapping.
Furthermore, in 2017, Pant and Shukla [34] introduced a larger class of mappings, which contains both Suzuki-type mappings and α-nonexpansive mappings, and established some convergence theorem.
for all x, y ∈ D and for some 0 ≤ α < 1.
It can be easily prove that for both α-nonexpansive mappings and generalized αnonexpansive mappings, if Fix(Υ ) = ∅, then they are quasi-nonexpansive mappings. Thus, all the classes of mappings defined in (1.2), (1.3), (1.4), and (1.5) are wider than the class of non-expansive mappings and are narrower than the quasi-nonexpansive mappings.
Very recently, in 2020, Ali et al. [6] showed that the Suzuki Condition (C) and the generalized non-expansive mapping are independent.
On the other hand, the iterative processes have great importance in modern fixed point theory. To find a fixed point of a self-mapping defined on a metric type space, we often use the Picard iteration. On a distance space X for a mapping Υ , the Picard iterative process is defined by u n+1 = Υ u n with an initial guess u 1 ∈ X . Most of the researchers working on fixed point theory use this iterative process to obtain fixed points of a mapping [1,14,18,41].
In 1953, Mann [26] first initiated an iterative process to approximate the fixed point for non-expansive mappings with an initial guess u 1 ∈ D as: where {τ n } is a sequence in (0, 1). After that, Ishikawa [24] in 1974 introduced a two step iterative process with the help of two constant sequences {τ n } and {ξ n } in (0, 1) with an initial guess u 1 ∈ D as: which the convergence is faster than the Mann iterative process.
Very recently, in 2020, Ali et al. [6] have introduced a new iterative process called JFiterative process with an initial guess u 1 ∈ D, which is as follows: (1.8) where {ξ n } and {τ n } are two sequences in (0, 1).
Considering generalized non-expansive mappings, they proved in [6] that the iterative process given by (1.8) converges faster than the Mann iteration, Ishikawa iteration, Noor iteration, S-iteration, Picard-S iteration, and Thakur et al. iteration. Now, a natural question arises: How can we approximate the fixed point of such mappings using a certain iterative scheme if a mapping does not belong to any of nonexpansive, generalized non-expansive, Condition (C), α-nonexpansive and generalized αnonexpansive classes? In this paper, we answer this question only partially. Indeed, inspired by the papers [33] and [40], we introduce a new class of non-expansive mappings, namely weakly K -nonexpansive mappings, which is defined as follows: Definition 1.5 A mapping Υ : D → D is said to be a weakly K -nonexpansive mapping if there exists K ≥ 0 such that (1.9) It is to be noted that a weakly K -nonexpansive mapping does not guarantee the existence of fixed point. In particular, if K = 0, then (1.9) reduces to non-expansive mapping, and if Fix(Υ ) = ∅, then (1.9) reduces to quasi-nonexpansive mapping. Thus, the class of weakly K -nonexpansive mappings is larger than the that of non-expansive mappings and smaller than the class of quasi-nonexpansive mappings.
Here, using weakly K -nonexpansive mappings Υ with Fix(Υ ) = ∅, we establish a convergence theorem for the JF-iterative process to approximate fixed point for such mappings, and finally we compare its convergence rate by providing a numerical example with some other known iterative process.
In 1967, Ostrowski [32] was the first who studied the stability of iterative procedures in a metric space for the Picard iteration. Definition 1.6 ([10]) Let (X , d) be a metric space and Υ : X → X be a mapping. Let u 1 ∈ X and u n+1 = f (Υ , u n ) be a general iterative process involving the mapping Υ . Suppose that {u n } n converges to a fixed point ρ ∈ X of Υ . Let {x n } n ⊂ X be any sequence and let n := d(x n+1 , f (Υ , x n )) for all n ∈ N. Then the iterative process u n+1 = f (Υ , u n ) is Υ -stable (or stable with respect to the mapping Υ ) if and only if lim n→∞ n = 0 implies lim n→∞ x n = ρ.
In 1995, Osilike [31] proposed a new type of contractive mapping in a normed linear space X as: for all x, y ∈ X , there exists α ∈ [0, 1) and K ≥ 0 such that (1.10) Using this contractive condition, he proved that the Picard and Ishikawa iterating sequences are Υ -stable. Thereafter, in 2003, Imoru et al. [23] generalized the contractive mapping due to Osilike by replacing the constant K by a certain function as follows and proved some stability results for the Picard and Mann iterative processes (see, also [29] for the Ishikawa iterative process): For all x, y ∈ X , there exists α ∈ [0, 1) and a monotone increasing and continuous func- (1.11) Now, the question is: Does there exist a larger class of contractive mappings than that of (1.11) so that the stability results can be improved? We have also answered this question partially. Indeed, we employ another type of non-expansivity, namely the weakly (α, K)nonexpansive mappings defined as follows, and show by an example that there are such mappings, which do not satisfy (1.11), but they are weakly (α, K)-nonexpansive mappings. (1.12) Using weakly (α, K)-nonexpansive mapping (1.12), we prove stability results for the JFiterative process (1.8).

Preliminaries
In this section, we recall some basic definitions, preliminary facts, and Lemmas, which we have used in our main results.
A mapping Υ : D → B is said to be demiclosed at y ∈ B (see [19]) if for every sequence {u n } ⊂ D with u n x for some x ∈ D and Υ u n → y implies that Υ x = y.
A Banach space B is said to satisfy Opial's property(see [30]) if for any arbitrary sequence Let {u n } be a bounded sequence in B. Then, for every x ∈ D, we define (see [19]): • Asymptotic radius of {u n } relative to D by • Asymptotic centre of {u n } relative to D by Moreover, if B is uniformly convex, then it is well known that A(D, {u n }) is a singleton set.
A mapping Υ : D → D is said to satisfy Condition (I) (see [43]) if there exists a nondecreasing function ϕ :

Lemma 2.2 ([42])
Let B be a uniformly convex Banach space and 0 < r ≤ s n ≤ t < 1 for all n ∈ N. Suppose that {a n } and {b n } are two sequences in B satisfying lim sup n→+∞ a n ≤ s, lim sup n→+∞ b n ≤ s and lim sup n→+∞ s n a n + (1s n )b n = s for some s ≥ 0. Then lim n→+∞ a nb n = 0.

Lemma 2.3 ([15])
Let μ be a real number with 0 ≤ μ < 1 and { n } be a sequence of positive reals such that lim n→+∞ n = 0. Then, for any sequence of positive reals {x n } satisfying x n+1 ≤ n + μx n , we have lim n→+∞ x n = 0.

Some basic discussions
In this section, we discuss the nature of our weakly K -nonexpansive and weakly (α, K)nonexpansive mappings, compare them with the other previously defined mappings, and prove some basic properties of our newly defined non-expansive type mappings, which we used in our main results. Pant & Shukla [34] proved that every mapping satisfying Condition (C) (1.3) is a generalized α-nonexpansive mapping (1.5), but the reverse implication is not true.
Ali et al. [6] proved via some examples that the generalized non-expansive mapping (1.2) due to Hardy-Rogers and the Condition (C) (1.3) are independent.
First, we will prove that the class of α-nonexpansive mappings (1.4) and the class of generalized α-nonexpansive mapping (1.5) are independent. For this purpose, we consider the following example: Then, Υ is an α-nonexpansive mapping but not generalized α-nonexpansive mapping.
Then, Υ is a weakly K -nonexpansive mapping, whether it is neither a generalized αnonexpansive mapping nor satisfies the Suzuki Condition (C).

Proposition 3.4 For a weakly K -nonexpansive mapping
Proof Simply using the triangle inequality, we have for all x, y ∈ D.
Taking lim inf on both sides, we get lim inf n→+∞ u n -Υ x ≤ lim inf n→+∞ u nx and by Opial's property, we have Υ x = x, i.e., I -Υ is demiclosed at zero.

Stability results
In this section, first we present an example, which does not satisfy (1.11) but satisfies (1.12), and then we prove some stability results of JF-iterative process (1.8) for weakly (α, K)nonexpansive mappings (1.12).

Convergence results
In this section, we present some convergence results for weakly K -nonexpansive mappings using JF iterative algorithm (1.8). For this purpose, the following Lemmas are crucial. Proof Let ρ ∈ Fix(Υ ). Since Υ is a weakly K -nonexpansive mapping, so for every sequence {x n } ⊂ D, we can get Υ x nρ ≤ x nρ . Then using the iteration (1.8), we have Finally, using (5.3), we have Thus, we get { u nρ } n is a non-increasing sequence of reals, which is bounded below by zero. Hence, lim n→+∞ u nρ exists for all ρ ∈ Fix(Υ ). Proof First suppose that Fix(Υ ) = ∅ and let ρ ∈ Fix(Υ ). Then, from Lemma 5.1, we have lim n→+∞ u nρ exists and consequently {u n } becomes bounded.
Since Υ is weakly K -nonexpansive mapping, we have Υ u nρ = Υ u n -Υρ ≤ u nρ and therefore lim sup n→+∞ Υ u nρ ≤ θ . Now, taking lim inf on both sides of (5.4), we have Again by taking lim inf on both sides in (5.2), we have implying that lim n→+∞ w nρ = θ . Therefore, which implies that lim n→+∞ (1ξ n )(u nρ) + ξ n (Υ u nρ) = θ . Consequently, using Lemma 2.2, we can conclude that lim n→+∞ u n -Υ u n = 0. Conversely, suppose that {u n } be bounded and lim n→+∞ u n -Υ u n = 0. Since B is a uniformly convex Banach space, and D is a non-empty closed and convex subset of B, A(D, {u n }) is a singleton set, say {ρ}. Now, we claim that ρ is a fixed point of Υ . Using Therefore, Υρ ∈ A(D, {u n }) and consequently, Υρ = ρ, i.e., ρ is a fixed point of Υ , and we are done. Now, we are ready to prove a weak convergence result and a strong convergence result for a weakly K -nonexpansive mapping using the iterative scheme given by (1.8). Proof Suppose that Fix(Υ ) = ∅. Then, from Lemma 5.2, we have lim n→+∞ u n -Υ u n = 0. Since B is uniformly convex, it is reflexive, and hence there exists a sub-sequence {u n i } i of {u n } such that u n i ρ for some ρ ∈ D. Then, by Lemma 3.6, I -Υ is demiclosed at zero, where I is the identity mapping on D, i.e., ρ ∈ Fix(Υ ).
(ii) From Lemma 5.2, we have lim n→+∞ u n -Υ u n = 0. Again, from Condition (I), we have, which implies lim n→+∞ ϕ(d(u n , Fix(Υ ))) = 0 and hence lim n→+∞ d(u n , Fix(Υ )) = 0, which reduces to (i) and completes the proof. Now, we compare the behavior of convergence of some known iterative scheme for the weakly K -nonexpansive mappings by choosing the parameter sequences {τ n } and {ξ n } in (0, 1). Then, it can be easily checked that Υ is a weakly K -nonexpansive mapping for K = 2.
It is clear that x = 1 is the unique fixed point of Υ . Now, to approximate this fixed point, consider τ n = 5n 7n+4 and ξ n = 2n 3n+1 and let the initial guess be u 1 = 3. Using these sequences of scalars and the weakly K -nonexpansive mapping defined in (5.6), in Table 1, we compare the convergence behavior of the Mann-iteration, Ishikawa-iteration, Agarwal-iteration, Thakur-new iteration, M-iteration, M * -iteration, JF-iteration, and we stop the process when the result is correct up to 7-decimal places (i.e., we stop the process when the result comes 1.0000000).

Application to nonlinear fractional differential equation
During the last three decades, fractional differential calculus has became an interesting and fruitful area of research in science and engineering. It has several applications in the field of signal processing, fluid flow, diffusive transport, electrical networks, electronics, robotics, telecommunication, etc.; for more details, one can refer to ( [3-5, 7, 8, 17, 25], and [27]). Sometimes, it is observed that a particular nonlinear fractional differential equation may have no analytic solution. In this case, we need to find out an approximate solution. In this section, we will estimate an approximate solution of a nonlinear fractional differential equation using the iterative algorithm (1.8).
Type-I: Consider the fractional differential equation: with the boundary conditions y(0) = 0 and y(1) = 1, where f : [0, 1]×R → R is a continuous function, and D γ (= d γ dx γ ) denotes the fractional derivative of order γ . Let B = C[0, 1] be the Banach space of all continuous functions from [0, 1] to R equipped with the sup-norm. The Green function [12] corresponding to the equation (6.1) is defined by Now, we approximate the solution of the fractional differential equation (6.1) using the iterative scheme (1.8). Moreover, assume that f is a Lipschitz function with respect to the second variable, i.e., |f (x, y 1 )f (x, y 2 )| ≤ |y 1y 2 |, for all x ∈ [0, 1] and y 1 , y 2 ∈ B. Then JF-iterative sequence converges to a solution of the problem (6.1).
Proof We know that the solution of the fractional differential equation (6.1) in terms of Green's function is Then for all y 1 , y 2 ∈ B and x ∈ [0, 1], we have Thus, we get Υ y 1 -Υ y 2 ≤ y 1y 2 , ∀y 1 , y 2 ∈ B. Then, Υ is a non-expansive mapping and so is a weakly K -nonexpansive mapping, and hence the JF-iterative scheme converges to the solution of (6.1). Now, we present a numerical example, corresponding to the above theorem.
Then, the JF-iterative sequence converges to a solution of the fractional differential equation (6.4).

Conclusion
The main purpose of this paper is to introduce a new type of non-expansive mappings, which is different from any other previously defined non-expensive type mappings. We have used the latest JF-iterative algorithm to approximate fixed points for our new nonexpansive mappings, and we have established some weak and strong convergence theorems. Also, here, we have introduced the concept of (α, K)-nonexpansive mappings and have proved a stability result for the JF-iterative process, which is more general than other previous stability results. Furthermore, we have presented a numerical example for our mappings and have compared the convergence behavior of various iterative processes with respect to it. We have also shown that α-nonexpansive mappings and generalized α-nonexpansive mappings are independent of each other. Moreover, an application of our fixed point theorems is given to the nonlinear fractional differential equations.