Journal of Function Spaces

Volume 2019, Article ID 2789819, 8 pages

https://doi.org/10.1155/2019/2789819

## Convergence Analysis of an Accelerated Iteration for Monotone Generalized *α*-Nonexpansive Mappings with a Partial Order

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Correspondence should be addressed to Dao-Jun Wen; moc.361@newnujoad

Received 20 February 2019; Accepted 11 March 2019; Published 2 April 2019

Guest Editor: Tuncer Acar

Copyright © 2019 Yi-An Chen and Dao-Jun Wen. 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.

#### Abstract

In this paper, we introduce a new accelerated iteration for finding a fixed point of monotone generalized -nonexpansive mapping in an ordered Banach space. We establish some weak and strong convergence theorems of fixed point for monotone generalized -nonexpansive mapping in a uniformly convex Banach space with a partial order. Further, we provide a numerical example to illustrate the convergence behavior and effectiveness of the proposed iteration process.

#### 1. Introduction

Let be an ordered Banach space endowed with the partial order and be a nonempty closed convex subset of . A mapping is called monotone if whenever for all . Moreover, is said to be as follows:

Monotone nonexpansive if is monotone and such that Monotone quasinonexpansive if is monotone with such that where , the set of fixed points of , i.e., .

Monotone -nonexpansive if is monotone and there exists a constant such that Suzuki’s generalized nonexpansive if satisfy condition (C), i.e., implies which is an interesting generalization of nonexpansive mapping because it is weaker than nonexpansiveness and stronger than quasinonexpansiveness [1].

Monotone generalized -nonexpansive if is monotone and exists a constant such that impliesObviously, a monotone -nonexpansive mapping includes monotone nonexpansive (-nonexpansive) mapping as special case. Every monotone mapping satisfying condition (C) is a monotone generalized -nonexpansive mapping, but the converse is not true. Moreover, a monotone generalized -nonexpansive mapping includes nonexpansive, firmly nonexpansive, Suzuki’s generalized nonexpansive mapping as special cases and partially extends monotone -nonexpansive mapping [2].

In 1965, Browder [3] proved that every nonexpansive self-mapping of a closed convex and bounded subset has a fixed point in a uniformly convex Banach space. Since then, a number of iteration methods have been developed to approximate fixed point of nonexpansive mappings and some other relevant problems; see [4–16] and the references therein. In these algorithms, Mann iteration is a fundamental method approximating fixed points of nonexpansive mappings, which is defined bywhere and is a nonexpansive mapping. The other important iteration widely used to approximate fixed point of nonexpansive mapping is Ishikawa iteration, which is defined bywhere . Note that Ishikawa iteration (7) improves the rate of convergence of Mann iteration process for an increasing function due to Ishikawa [17] and Rhoades [18].

In 2007, Agrawal et al. [19] modified (7) and considered the following two-step iteration process: for an arbitrary , the sequence of is defined in the following manner:where and is a nearly asymptotically nonexpansive mapping. They claimed that this iteration process converges faster than the Mann iteration for some contractions.

Recently, Noor [20] modified (7) and further studied a three-step iteration process to solve the general variational inequalities: for an arbitrary defined a sequence bywhere and is a strong monotone mapping involved variational inequalities. Very recently, Abbas-Nazir [21] and Thakur et al. [22] modified Noor iteration (9) and introduced a new faster iteration process for solving the constrained minimization and feasibility problems and for finding the fixed point of Suzuki’s generalized nonexpansive mappings, respectively.

On the other hand, in 2004, Ran-Reurings [23] firstly introduced a fixed point theorem in a partially ordered metric space and some applications to matrix equations. They developed a new field only for comparable elements instead of the nonexpansive (or Lipschitz) condition in a partially ordered metric space, which has been successfully applied to solve not only the existence of fixed points but also a positive or negative solution of ordinary differential equations [24].

In 2015, Bin Dehaish-Khamsi [25] applied the Mann iteration (6) to the case of a monotone nonexpansive mapping in a Banach space endowed with a partial order. Moreover, they proved that generated by (6) weakly converges to and and are comparable.

In 2016, Song et al. [26] further extended the Mann iteration (6) to monotone -nonexpansive mappings and obtained some weak and strong convergence theorems in an ordered Banach space, which complemented the fixed point results of -nonexpansive mappings in Aoyama-Kohsaka [27]. However, in general, the monotone condition on comparable elements is a weaker assumption. In particular, the continuity property probably is not valid, which not only reduces the efficiency of numerical approach but also increases the difficulty of convergence analysis. This is also the main reason why Mann iteration has become popular in approximating the fixed point of monotone-type mappings [2, 25, 26]. Therefore, it is important and interesting to construct an iterative accelerator method for finding the fixed points problem of such class of monotone-type mappings.

Inspired and motivated by research going on in this area, we modify the iteration process (6), (8), and (9) to the case of monotone generalized -nonexpansive mappings and introduce a new accelerated iteration: for an arbitrary , sequence is defined byOur purpose is not only to extend Mann iteration of Bin Dehaish-Khamsi [25] and Song et al. [26] to an accelerated iteration for monotone generalized -nonexpansive mappings, but also to establish some weak and strong convergence theorems of fixed point for monotone generalized -nonexpansive mapping in a uniformly convex Banach space with a partial order. Furthermore, we provide a numerical example to illustrate the convergence behavior and effectiveness of the proposed iteration. The method and results presented in this paper extend and improve the corresponding results of [2, 17, 19, 20, 25, 26] and some others previously.

#### 2. Preliminaries

Recall that a Banach space with the norm is called uniformly convex if, for all , there exists a constant for which and implies A Banach space is said to satisfy the Opial property [5] if for each weakly convergent sequence in with weak limit , holds, for all with .

Let be a nonempty subset of a Banach space and be a bounded sequence in . For each , we define the following:(i)Asymptotic radius of at by .(ii)Asymptotic radius of relative to by .(iii)Asymptotic center of relative to by .

Note that is nonempty. Further, if is uniformly convex, then has exactly one point [28].

Recall also that an order interval is defined by where and . Throughout, we assume that the order intervals are closed and convex in an ordered Banach space .

Lemma 1 (see [1]). *Let be a nonempty closed convex subset of an ordered Banach space and be a monotone generalized -nonexpansive mapping. Then, for all with , the following inequalities hold.*(i)*,*(ii)* or* *.*

*Lemma 2 (see [2]). Let be a nonempty subset of an ordered Banach space and be a generalized -nonexpansive mapping. Then is closed.*

*Lemma 3 (see [2]). Let be a nonempty closed convex subset of a uniformly convex ordered Banach spaces . Let be a monotone generalized -nonexpansive mapping. Then if and only if is a bounded sequence for some with .*

*Lemma 4 (see [29]). A Banach space is uniformly convex if and only if there exists a continuous strictly increasing convex function with such thatwhere and .*

*Lemma 5 (see [30]). Let be a uniformly convex Banach space and be a sequence with . Suppose and are two sequences of such that and . Then *

*Lemma 6. Let be a nonempty closed convex subset of an ordered Banach space and be a monotone generalized -nonexpansive mapping. Then(i), with .(ii) is monotone quasinonexpansive if and with or .*

*Proof. *(i) From Lemma 1 (ii), in the first case, we havewhich implies thatIn the other case of Lemma 1 (ii), we further have which implies thatThe desired conclusion follows immediately from (17) and (19) for all and .

(ii) By the definition of monotone generalized -nonexpansive mapping, we havewhere , and so ; that is, is monotone quasinonexpansive.

*3. Main Results*

*3. Main Results*

*Lemma 7. Let be a nonempty closed convex subset of an ordered Banach space and be a monotone mapping. Assume that the sequence is defined by the iteration (10) and . Then*

*(i) ;*

*(ii) has at most one weak-cluster point . Moreover, for all provided weakly converges to a point .*

*Proof. *(i) Note that if are such that , then holds from the convex property defined on order intervals. This allows us to focus only on the proof of for any . By , we suppose that for . From (10), we have Since is monotone, we obtain . Using (10) again, we obtainwhich implies that . Similarly, we haveIt follows from (23) that . Consequently, , which further implies that .

(ii) The desired conclusion follows from (i) and Lemma 3.1 in Bin Dehaish-Khamsi [25].

*Theorem 8. Let be a nonempty closed convex subset of a uniformly convex ordered Banach space and be a monotone generalized -nonexpansive mapping. Suppose that the sequence defined by (10) is bounded and . Then .*

*Proof. *Since is a bounded sequence and , there exists a subsequence of such thatThe asymptotic center of with respect to is denoted by such that for all , such is unique. From the definition of asymptotic radius, we haveUsing Lemma 6 (i) and (24), we further obtainIt follows from the uniqueness of that , which shows that .

*Theorem 9. Let be a nonempty closed convex subset of a uniformly convex ordered Banach space with Opial property. Let be a monotone generalized -nonexpansive mapping with . Suppose that there exists a such that , then the sequences and satisfy the following conditions (i), ;(ii) and . Then the sequence generated by (10) weakly converges to a fixed point .*

*Proof. *Firstly, we show is bounded. Taking , without loss of generality, we assume . Associating with the monotone property of , we findFrom (10) and (27), we haveContinuing in this way, we can assume that , we get . Similarly, we have and . By Lemma 7 (i), we obtain which implies that . Therefore, the sequence is bounded, and so and are also bounded.

Secondly, we prove that . From (10) and Lemma 6 (ii), we haveSimilarly, from (10) and (30), we haveCombining (10), (30), and (31), we obtainwhich implies the limit of exists, i.e., . Also, it follows from (30) thatTogether (30), (31) with (32), we havewhich implies thatHence, . Noting that , we obtainMoreover, from (33) and (36), we can getOn the other hand, by the nonexpansive property defined on , we haveIt follows from (37), (38) and Lemma 5 thatFinally, we show that weakly converges to . By the boundness of , there exists a subsequence weakly converging and . From Lemma 6 (i) and (39), we can obtainArguing by contradiction, we suppose that . It follows from the Opial property of thatThis is a contradiction. Therefore, we conclude ; that is, . Moreover, if there exists another subsequence weakly converges . Similarly, we have . Note that exists andThis is a contradiction again. Consequently, and weakly converges to .

*Theorem 10. Let be a nonempty closed convex subset of a uniformly convex ordered Banach space and be a monotone generalized -nonexpansive mapping with . Suppose that there exists a such that , the sequences , and satisfy the following conditions(i), ;(ii) and . Then the sequence generated by (10) strong converges to a fixed point if and only if , where denotes the distance from to .*

*Proof. *Necessity is obvious. We only prove the sufficiency. Suppose that . From (31), exists. Thus By Theorem 9, we have that is bounded with . Without loss of generality, let be a subsequence of such that for all , where is a sequence in . Combining with (31), we haveIt follows from (44) thatThis shows that is a Cauchy sequence in . By Lemma 2, is closed, so converges to some . Moreover, by the triangle inequality, we have Taking implies that converges strongly to . From (31) again, exists, and the sequence converges strongly to .

*4. Numerical Example*

*4. Numerical Example*

*Example 1. *Define a mapping byNote that is not continuous. Setting , we obtain that is, is not a nonexpansive mapping. However, is a monotone mapping with and a monotone generalized -nonexpansive mapping.

*Numerical Results 4.2. *To illustrate the convergence of the proposed algorithm, we provide some numerical results of Example 1 and comparison with the other iterations previously.

*Firstly, we show the convergence behavior of scheme (10) with different initial points. To do this, we take , and set as stop criterion. From given , convergence behaviors of scheme (10) are displayed in Figure 1.*