# Iterative Approaches to Find Zeros of Maximal Monotone Operators by Hybrid Approximate Proximal Point Methods

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## Abstract

The purpose of this paper is to introduce and investigate two kinds of iterative algorithms for the problem of finding zeros of maximal monotone operators. Weak and strong convergence theorems are established in a real Hilbert space. As applications, we consider a problem of finding a minimizer of a convex function.

### Keywords

Convex Subset Iterative Algorithm Nonexpansive Mapping Maximal Monotone Real Hilbert Space## 1. Introduction

where Open image in new window ( Open image in new window ) is a sequence of real numbers. However, as pointed out in [1], the ideal form of the method is often impractical since, in many cases, to solve the problem (1.2) exactly is either impossible or has the same difficulty as the original problem (1.1). Therefore, one of the most interesting and important problems in the theory of maximal monotone operators is to find an efficient iterative algorithm to compute approximate zeros of Open image in new window .

the sequence Open image in new window defined by (1.3) converges weakly to a zero of Open image in new window provided that Open image in new window . In [3], Güler obtained an example to show that Rockafellar's inexact proximal point method (1.3) does not converge strongly, in general.

Recently, many authors studied the problems of modifying Rockafellar's inexact proximal point method (1.3) in order to strong convergence to be guaranteed. In 2008, Ceng et al. [4] gave new accuracy criteria to modified approximate proximal point algorithms in Hilbert spaces; that is, they established strong and weak convergence theorems for modified approximate proximal point algorithms for finding zeros of maximal monotone operators in Hilbert spaces. In the meantime, Cho et al. [5] proved the following strong convergence result.

Theorem CKZ1.

Let Open image in new window be a real sequence in Open image in new window such that

(i) Open image in new window as Open image in new window ,

(ii) Open image in new window .

Then Open image in new window converges strongly to a zero Open image in new window of Open image in new window , where Open image in new window .

They also derived the following weak convergence theorem.

Theorem CKZ2.

where Open image in new window for all Open image in new window . Then the sequence Open image in new window converges weakly to a zero Open image in new window of Open image in new window .

respectively, where Open image in new window , Open image in new window , and Open image in new window with Open image in new window . Under appropriate conditions, they derived one strong convergence theorem for (1.11) and another weak convergence theorem for (1.12). In addition, for other recent research works on approximate proximal point methods and their variants for finding zeros of monotone maximal operators, see, for example, [7, 8, 9, 10] and the references therein.

Here, the first iteration step, Open image in new window + Open image in new window + Open image in new window , is to compute the prediction value of approximate zeros of Open image in new window ; the second iteration step, Open image in new window , is to compute the correction value of approximate zeros of Open image in new window . Therefore, there is no doubt that the iterative algorithms (1.13) and (1.14) are very interesting and quite reasonable.

In this paper, we consider the problem of finding zeros of maximal monotone operators by hybrid proximal point method. To be more precise, we introduce two kinds of iterative schemes, that is, (1.13) and (1.14). Weak and strong convergence theorems are established in a real Hilbert space. As applications, we also consider a problem of finding a minimizer of a convex function.

## 2. Preliminaries

Open image in new window is said to be maximal monotone if its graph is not properly contained in the one of any other monotone operator.

The class of monotone mappings is one of the most important classes of mappings among nonlinear mappings. Within the past several decades, many authors have been devoted to the study of the existence and iterative algorithms of zeros for maximal monotone mappings; see [1, 2, 3, 4, 5, 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. In order to prove our main results, we need the following lemmas. The first lemma can be obtained from Eckstein [1, Lemma 2] immediately.

Lemma 2.1.

Lemma 2.2 (see [30, Lemma 2.5, page 243]).

where Open image in new window , Open image in new window , and Open image in new window satisfy the conditions

(i) Open image in new window , Open image in new window , or equivalently Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window , Open image in new window .

Then Open image in new window .

Lemma 2.3 (see [28, Lemma 1, page 303]).

If Open image in new window , then Open image in new window exists.

Lemma 2.4 (see [11]).

Let Open image in new window be a uniformly convex Banach space, let Open image in new window be a nonempty closed convex subset of Open image in new window , and let Open image in new window be a nonexpansive mapping. Then Open image in new window is demiclosed at zero.

Lemma 2.5 (see [31]).

for all Open image in new window and Open image in new window with Open image in new window .

It is clear that the following lemma is valid.

Lemma 2.6.

## 3. Main Results

Assume that Open image in new window , where Open image in new window is the set of zeros of Open image in new window . Then Open image in new window for all Open image in new window , where Open image in new window is the set of fixed points of the resolvent Open image in new window .

Theorem 3.1.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty, closed, and convex subset of Open image in new window , and Open image in new window a maximal monotone operator with Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window . For any given Open image in new window , Open image in new window , and Open image in new window , find Open image in new window conforming to SVME (2.5), where Open image in new window with Open image in new window as Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window and Open image in new window ,

(iii) Open image in new window and Open image in new window .

where Open image in new window is a fixed point and Open image in new window is a bounded sequence in Open image in new window . Then the sequence Open image in new window generated by (3.2) converges strongly to a zero Open image in new window of Open image in new window , where Open image in new window , if and only if Open image in new window as Open image in new window .

Proof.

This shows that Open image in new window as Open image in new window .

Next, let us show the sufficiency. The proof is divided into several steps.

Step 1 ( Open image in new window is bounded).

This shows that the sequence Open image in new window is bounded.

Step 2 ( Open image in new window , where Open image in new window ).

The existence of Open image in new window is guaranteed by Lemma 1 of Bruck [12].

Step 3 ( Open image in new window as Open image in new window ).

This completes the proof.

Remark 3.2.

we can see that Theorem 3.1 still holds.

Corollary 3.3.

where Open image in new window with Open image in new window as Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window and Open image in new window ,

(iii) Open image in new window and Open image in new window .

where Open image in new window is a fixed point and Open image in new window is a bounded sequence in Open image in new window . If the sequence Open image in new window satisfies the condition Open image in new window as Open image in new window , then the sequence Open image in new window converges strongly to a fixed point Open image in new window of Open image in new window , where Open image in new window .

Proof.

Finally, from the proof of Theorem 3.1, we can derive the desired conclusion immediately.

From Theorem 3.1, we also have the following result immediately.

Corollary 3.4.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty, closed, and convex subset of Open image in new window , and Open image in new window a maximal monotone operator with Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window . For any Open image in new window , Open image in new window and Open image in new window , find Open image in new window conforming to SVME (2.5), where Open image in new window with Open image in new window as Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window ,

(ii) Open image in new window and Open image in new window ,

(iii) Open image in new window .

where Open image in new window is a fixed point. Then the sequence Open image in new window converges strongly to a zero Open image in new window of Open image in new window , where Open image in new window , if and only if Open image in new window as Open image in new window .

Proof.

In Theorem 3.1, put Open image in new window for all Open image in new window . Then, from Theorem 3.1, we obtain the desired result immediately.

Next, we give a hybrid Mann-type iterative algorithm and study the weak convergence of the algorithm.

Theorem 3.5.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty, closed, and convex subset of Open image in new window , and Open image in new window a maximal monotone operator with Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window . For any given Open image in new window , Open image in new window , and Open image in new window , find Open image in new window conforming to SVME (2.5), where Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window and Open image in new window .

where Open image in new window is a bounded sequence in Open image in new window . Then the sequence Open image in new window generated by (3.34) converges weakly to a zero Open image in new window of Open image in new window .

Proof.

Let Open image in new window be a weakly subsequential limit of Open image in new window such that Open image in new window converges weakly to Open image in new window as Open image in new window . From (3.40), we see that Open image in new window also converges weakly to Open image in new window . Since Open image in new window is nonexpansive, we can obtain that Open image in new window by Lemma 2.4. Opial's condition (see [23]) guarantees that the sequence Open image in new window converges weakly to Open image in new window . This completes the proof.

By the careful analysis of the proof of Corollary 3.3 and Theorem 3.5, it is not hard to derive the following result.

Corollary 3.6.

where Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window and Open image in new window .

where Open image in new window is a bounded sequence in Open image in new window . Then the sequence Open image in new window converges weakly to a fixed point Open image in new window of Open image in new window .

Utilizing Theorem 3.5, we also obtain the following result immediately.

Corollary 3.7.

Let Open image in new window be a real Hilbert space, Open image in new window a nonempty, closed, and convex subset of Open image in new window , and Open image in new window a maximal monotone operator with Open image in new window . Let Open image in new window be a metric projection from Open image in new window onto Open image in new window . For any Open image in new window , Open image in new window , and Open image in new window , find Open image in new window conforming to SVME (2.5), where Open image in new window and Open image in new window with Open image in new window . Let Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window .

Then the sequence Open image in new window converges weakly to a zero Open image in new window of Open image in new window .

## 4. Applications

In this section, as applications of the main Theorems 3.1 and 3.5, we consider the problem of finding a minimizer of a convex function Open image in new window .

Theorem 4.1.

Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window and Open image in new window ,

(iii) Open image in new window and Open image in new window .

where Open image in new window is a fixed point and Open image in new window is a bounded sequence in Open image in new window . If the sequence Open image in new window satisfies the condition Open image in new window as Open image in new window , then the sequence Open image in new window converges strongly to a minimizer of Open image in new window nearest to Open image in new window .

Proof.

By using Theorem 3.1, we can obtain the desired result immediately.

Theorem 4.2.

Let Open image in new window , Open image in new window , Open image in new window , and Open image in new window be real sequences in Open image in new window satisfying the following control conditions:

(i) Open image in new window and Open image in new window ,

(ii) Open image in new window ,

(iii) Open image in new window and Open image in new window .

where Open image in new window is a bounded sequence in Open image in new window . Then the sequence Open image in new window converges weakly to a minimizer of Open image in new window .

Proof.

We can obtain the desired result readily from the proof of Theorems 3.5 and 4.1.

## Notes

### Acknowledgment

This research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and the Dawn Program Foundation in Shanghai.

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