Forcing Strong Convergence of a Mann-Based Iteration for Nonexpansive and Monotone Operators in a Hilbert Space

Lv, Songtao

doi:https://doi.org/10.1155/2022/5450521

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Fixed Points and Computational Optimization

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Volume 2022 | Article ID 5450521 | https://doi.org/10.1155/2022/5450521

Forcing Strong Convergence of a Mann-Based Iteration for Nonexpansive and Monotone Operators in a Hilbert Space

Songtao Lv¹

Academic Editor: Sun Young Cho

Received24 Dec 2021

Accepted18 Jan 2022

Published10 Feb 2022

Abstract

Mann iteration is weakly convergent in infinite dimensional spaces. We, in this paper, use the nearest point projection to force the strong convergence of a Mann-based iteration for nonexpansive and monotone operators. A strong convergence theorem of common elements is obtained in an infinite dimensional Hilbert space. No compact conditions are needed.

1. Introduction: Preliminaries

In the real world, there are a lot of nonlinear phenomena, which can be modelled into variational inequalities and variational inclusions, such as signal processing, image recovery, and machine learning; see, e.g., [1–7] and the references therein. Fixed point methods are powerful and popular for dealing various nonlinear operator equations and inequalities in abstract spaces, in particular, for variational inequalities and variational inclusions. Recently, various efficient fixed point methods have been introduced and investigated; see, e.g., [8–13] and the references therein. Let be a nonlinear operator on a Hilbert space , which is endowed with inner product and induced norm . The fixed point set of is presented by . Recall that is said to be contractive iff there is a real number such that

Recall that is said to be nonexpansive iff

Recall that is said to be firmly nonexpansive iff

It is clear that the class of firmly nonexpansive mappings is a special class of nonexpansive mappings. One knows the projection operator (see below) is firmly nonexpansive. The class of nonexpansive operators is significant in various nonlinear equations and mathematical programming computation. It also has wide real applications in applied and industrial fields. For various iterative methods, Mann iteration is popular for dealing with fixed points of nonexpansive operators. It readswhere is a real number sequence in the interval . However, the Mann iteration is weakly convergent only in infinite dimensional spaces; see, e.g., [14] and the references therein. To force the strong convergence without possible compact assumptions, various regularized methods have been investigated in Hilbert spaces and Banach spaces recently; see, e.g., [15–19] and the references therein. One of the efficient regularized methods is the Halpern iteration, which readswhere is a real number sequence in the interval and is a fixed anchor. With some conditions on , it was proved that converges to , which is a special fixed point of , that is, the nearest point in to . Halpern [20] pointed out that conditions (c1) as and (c2) are necessary if the Halpern iteration scheme converges in norm. In view of (c2), the Halpern iteration may not be a fast iteration. Recently, a number of researchers investigated the problem of removing (c2) with the aid of projections; see, e.g., [21–24] and the references therein. In 2000, Moudafi [25] further proposed the viscosity approximation iteration, which reads as follows:where is a contraction. This approximation method, which improves the property of the class of nonexpansive mappings, is popular from the viewpoint of variational inequalities. Indeed, the fixed point also solves a monotone variational inequality with . Another popular regularized method is the hybrid projection method, which was considered by Nakajo and Takahashi [18] for fixed points of nonexpansive mappings first. Indeed, they studied the following algorithm:where is a closed, convex, and nonempty subset of and is the nearest point projection onto the intersection set. They obtained a strong convergence theorem for nonexpansive mappings in a real Hilbert spaces without compact assumption on . For more general nonlinear mappings though the projection-based method, we refer to [26–30] and the references therein.

Let be a convex and closed subset of a real Hilbert space . From now on, is borrowed to denote the nearest projection onto subset , i.e., . Let be a nonlinear mapping on . Recall that is said to be(1)Strongly monotone iff there exists a positive constant such that , (2)Monotone iff , (3)Cocoercive iff there exists a positive constant such that ,

Let be a multivalued nonlinear mapping. Next, we turn our attention to the class of multivalued mappings. is said to be a monotone mapping if and only if for all , , and . The symbol is used to stand for the set of zero points of . Mapping is said to be a maximally monotone mapping iff the graph of , , is not contained in the graph of any other monotone mapping properly. Let , where is the identity mapping and is a constant. This operator is called the resolvent of . Its domain is denoted by in this paper. It is clear .

Consider the following variational inclusion problem, which finds a point such that where is a multivalued maximally monotone mapping and is a -cocoercive mapping. For the inclusion problem, splitting methods (FB, PR, and DR) are popular for zero points of the sum of the monotone mappings. Splitting methods were considered by many authors for image recovery, signal processing, and machine learning. The FB-type splitting method means an iterative method for which each iteration involves only with the individual operators not the sum. In this paper, with the condition that the solution set is nonempty, we consider finding a such that where is a nonexpansive mapping with a nonempty fixed point set, is a multivalued maximally monotone mapping, and is a -cocoercive mapping. We establish a strong convergence with the aid of hybrid projection and FB splitting in a Hilbert space. Our strong convergence theorem requires less restriction on parameter sequences and the operators.

To show our main findings, we also need the following necessary tools.

The nearest point projection operator has the following property:

Lemma 1 (see [31]). Let be a Hilbert space, and let be a convex, closed, and nonempty subset of . Let be a nonexpansive mapping on . Then, is convex and closed.

Remark 1. Let be a Hilbert space, and let be a convex, closed, and nonempty subset of . Let be a -cocoercive mapping, and let be a multivalued maximally monotone operator. Then, , where is some constant and is the identity mapping. Besides, the resolvent is firmly nonexpansive. From Lemma 1, we have that is convex and closed.

Lemma 2 (see [31]). Let be a Hilbert space, and let be a convex, closed, and nonempty subset of . Let be a nonexpansive mapping on . Then, is demiclosed (let be a sequence weakly converging to , and let be . Then, is a fixed point of ).

2. Main Results

Theorem 1. Assume that is a Hilbert space and is a convex and closed subset in space . Assume that is a single-valued -cocoercive mapping from set to space and is a set-valued maximally monotone mapping from to . Assume that is a nonexpansive mapping from to , and is nonempty. Assume that and are positive real number sequences. Let be a sequence in set generated in the following iterative process:where is the resolvent mapping . Assume that and satisfy the conditions (i) with being a fixed real number and (ii) with and being two fixed real numbers. Then, the sequence converges strongly to .

Proof. From Lemma 1, we have that is convex and closed. From Remark 1, we have that is convex and closed. Hence, is convex and closed. This shows that the metric (nearest point) projection onto the set is well-defined.
Note that is equivalent to Let and be the points in . Then,where is a real number in . Adding the two inequalities above, we havethat is,It shows that . is convex. The closedness of is obvious. The definition of -cocoercive mappings send us to the situation is a nonexpansive mapping for each . Indeed, for any ,This indicates is a mapping of nonexpansive. Observe that . Indeed, from the nonexpansivity of the resolvent, we haveSo, we complete the proof .
On the contrary, it is obvious that is convex and closed. Next, one shows that . Borrowing , we have . Let be a given vector, and for some positive integer . There is a vector with . There holds for all . Borrowing , we get . Thus, . Hence, for all .
One next observes that is a bounded sequence. As we have showed that is convex and closed set in , a unique vector with is guaranteed. We have the construction of , that is, . So,for each . By , we obtainthat infers is a bounded sequence. Our next step shows as . Because and , one infers thatBorrowing the conclusion ( is a bounded sequence), one infers that the limit of exists. We may suppose that . Observethanks to ( and the property of the metric projection). By the limit of the limit of , one infers .
Note that is in . So,That indicates that as . Furthermore, as . Let . For any , -cocoercive and resolvent operators send us toSo,That is,By the fact that as , we have as . By the firm nonexpansivitity of the resolvent operator, we also havewhich holds thatSo, , where is some constant. By the requirement on the control parameter and the result that as and as . With a simple calculation, we have as . We have the fact that It holdsBy the assumption that is maximally monotone,for any . By the result that is a bounded sequence, there is a subsequence converges to weakly. The -cocoercive mappings yield . It holds It shows . Note that is demiclosed (Lemma 2). One asserts . One next shows that and converges to it strongly. Set . Since the functional is weakly lower semicontinuous, one hasOne gets . Since the framework is a Hilbert space, one gets as . This finishes this theorem.

Letwhere is a proper, convex, and lower semicontinuous function. Rockfellar [32] proved that is a multivalued maximally monotone operator. Let be a closed, convex, and nonempty subset of and be the indicator function of , that is,

Furthermore, we define the normal cone of at as follows:for any . Then, is proper, convex, and lower semicontinuous on . is a maximally monotone operator. Let . So, and ; we obtainwhere is the metric projection onto . This yields , where denotes the classical variational inequality, that is, find a point such that for all .

Corollary 1. Assume that is a Hilbert space and is a convex and closed subset in space . Assume that is a single-valued -cocoercive mapping from set to space . Assume that is a nonexpansive mapping from to and is nonempty. Assume that and are positive real number sequences. Let be a sequence in set generated in the following iterative process:

Assume that and satisfy the conditions (i) with being a fixed real number and (ii) with and being two fixed real numbers. Then, the sequence converges strongly to .

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that he has no conflicts of interest.

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Copyright

Copyright © 2022 Songtao Lv. 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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