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Perturbed Iterative Approximation of Solutions for Nonlinear General Open image in new window -Monotone Operator Equations in Banach Spaces

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Research Article

Abstract

We introduce and study a new class of nonlinear general Open image in new window -monotone operator equations with multivalued operator. By using Alber's inequalities, Nalder's results, and the new proximal mapping technique, we construct some new perturbed iterative algorithms with mixed errors for solving the nonlinear general Open image in new window -monotone operator equations and study the approximation-solvability of the nonlinear operator equations in Banach spaces. The results presented in this paper improve and generalize the corresponding results on strongly monotone quasivariational inclusions and nonlinear implicit quasivariational inclusions.

Keywords

Variational Inequality Complementarity Problem Monotone Operator Maximal Monotone Lipschitz Continuity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window be a real Banach space with the topological dual space of Open image in new window , let Open image in new window be the pairing between Open image in new window and Open image in new window , let Open image in new window denote the family of all subsets of Open image in new window and let Open image in new window denote the family of all nonempty closed bounded subsets of Open image in new window . We denote by the Open image in new window for all Open image in new window and Open image in new window . Let Open image in new window , Open image in new window and Open image in new window be nonlinear operators, and let Open image in new window be a general Open image in new window -monotone operator such that Open image in new window . We will consider the following nonlinear general Open image in new window -monotone operator equation with multivalued operator.

where Open image in new window is a constant and Open image in new window is the proximal mapping associated with the general Open image in new window -monotone operator Open image in new window due to Cui et al. [1].

It is easy to see that the problem (1.1) is equivalent to the problem of finding Open image in new window such that

Example 1.1.

If Open image in new window , then the problem (1.1) is equivalent to finding Open image in new window such that Open image in new window and
Based on the definition of the proximal mapping Open image in new window , (1.3) can be written as

Example 1.2.

If Open image in new window is a single-valued operator, then a special case of the problem (1.3) is to determine element Open image in new window such that

where Open image in new window is defined by Open image in new window for all Open image in new window . The problem (1.5) was studied by Xia and Huang [2] when M is a general H-monotone mapping. Further, the problem (1.5) was studied by Peng et al. [3] if Open image in new window , the identity operator, and M is a multivalued maximal monotone mapping.

Example 1.3.

If Open image in new window , Open image in new window , and Open image in new window are single-valued operators, and Open image in new window for all Open image in new window , then the problem (1.3) reduces to finding an element Open image in new window such that

which was considered by Verma [4, 5].

We note that for appropriate and suitable choices of Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window , it is easy to see that the problem (1.1) includes a number of quasivariational inclusions, generalized quasivariational inclusions, quasivariational inequalities, implicit quasivariational inequalities, complementarity problems, and equilibrium problems studied by many authors as special cases; see, for example, [4, 5, 6, 7] and the references therein.

The study of such types of problems is motivated by an increasing interest to study the behavior and approximation of the solution sets for many important nonlinear problems arising in mechanics, physics, optimization and control, nonlinear programming, economics, finance, regional structural, transportation, elasticity, engineering, and various applied sciences in a general and unified framework. It is well known that many authors have studied a number of nonlinear variational inclusions and many systems of variational inequalities, variational inclusions, complementarity problems, and equilibrium problems by using the resolvent operator technique, which is a very important method to find solutions of variational inequality and variational inclusion problems; see, for example, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and the references therein.

On the other hand, Verma [4, 5] introduced the concept of Open image in new window -monotone mappings, which generalizes the well-known general class of maximal monotone mappings and originates way back from an earlier work of the Verma [7]. Furthermore, motivated and inspired by the works of Xia and Huang [2], Cui et al. [1] introduced first a new class of general Open image in new window -monotone operators in Banach spaces, studied some properties of general Open image in new window -monotone operator, and defined a new proximal mapping associated with the general Open image in new window -monotone operator.

Inspired and motivated by the research works going on this field, the purpose of this paper is to introduce the new class of nonlinear general Open image in new window -monotone operator equation with multivalued operator. By using Alber's inequalities, Nalder's results, and the new proximal mapping technique, some new perturbed iterative algorithms with mixed errors for solving the nonlinear general Open image in new window -monotone operator equations will be constructed, and applications of general Open image in new window -monotone operators to the approximation-solvability of the nonlinear operator equations in Banach spaces will be studied. The results presented in this paper improve and extend some corresponding results in recent literature.

2. Preliminaries

In this paper, we will use the following definitions and lemmas.

Definition 2.1.

Let Open image in new window , Open image in new window , and Open image in new window be single-valued operators. Then

(i) Open image in new window is Open image in new window -strongly monotone, if there exists a positive constant Open image in new window such that
(ii) Open image in new window is Open image in new window -Lipschitz continuous, if there exists a constant Open image in new window such that
(iii) Open image in new window is Open image in new window -strongly accretive if for any Open image in new window , there exist Open image in new window and a positive constant Open image in new window such that
where the generalized duality mapping Open image in new window is defined by
(iv) Open image in new window is Open image in new window -relaxed cocoercive with respect to Open image in new window , if for all Open image in new window , there exist positive constants Open image in new window and Open image in new window such that
Example 2.2 (see [11, 12]).
  1. (1)

    Consider an Open image in new window -strongly monotone (and hence Open image in new window -expanding) operator Open image in new window . Then Open image in new window is Open image in new window -relaxed cocoercive with respect to Open image in new window .

     
  2. (2)

    Very Open image in new window -cocoercive operator is Open image in new window -relaxed cocoercive, while each Open image in new window -strongly monotone mapping is Open image in new window -relaxed cocoercive with respect to Open image in new window .

     

Remark 2.3.

The notion of the cocoercivity is applied in several directions, especially to solving variational inequality problems using the auxiliary problem principle and projection methods [5], while the notion of the relaxed cocoercivity is more general than the strong monotonicity as well as cocoercivity. Several classes of relaxed cocoercive variational inequalities have been studied in [4, 5].

Definition 2.4.

A multivalued operator Open image in new window is said to be

(ii) Open image in new window -relaxed monotone if, for any Open image in new window , Open image in new window , and Open image in new window , there exists a positive constant Open image in new window such that
(iii) Open image in new window - Open image in new window -Lipschitz continuous, if there exists a constant Open image in new window such that
where Open image in new window is the Hausdorff pseudometric, that is,

Note that if the domain of Open image in new window is restricted to closed bounded subsets Open image in new window , then Open image in new window is the Hausdorff metric.

Definition 2.5.

A single-valued operator Open image in new window is said to be

(i)coercive if

(ii)hemicontinuous if, for any fixed Open image in new window , the function Open image in new window is continuous at Open image in new window .

We remark that the uniform convexity of the space Open image in new window means that for any given Open image in new window , there exists Open image in new window such that for all Open image in new window , Open image in new window , Open image in new window and Open image in new window ensure the inequality Open image in new window . The function

is called the modulus of the convexity of the space Open image in new window .

The uniform smoothness of the space Open image in new window means that for any given Open image in new window , there exists Open image in new window such that Open image in new window holds. The function Open image in new window defined by

is called the modulus of the smoothness of the space Open image in new window .

We also remark that the space Open image in new window is uniformly convex if and only if Open image in new window for all Open image in new window , and it is uniformly smooth if and only if Open image in new window Moreover, Open image in new window is uniformly convex if and only if Open image in new window is uniformly smooth. In this case, Open image in new window is reflexive by the Milman theorem. A Hilbert space is uniformly convex and uniformly smooth. The proof of the following inequalities can be found, for example, in page 24 of Alber [16].

Lemma 2.6.

Let Open image in new window be a uniformly smooth Banach space, and let Open image in new window be the normalized duality mapping from Open image in new window into Open image in new window . Then, for all Open image in new window , we have

(i) Open image in new window ;

(ii) Open image in new window , where Open image in new window .

Definition 2.7.

Let Open image in new window be a Banach space with the dual space Open image in new window , Open image in new window be a nonlinear operator, and Open image in new window be a multivalued operator. The map Open image in new window is said to be general Open image in new window -monotone if Open image in new window is Open image in new window -relaxed monotone and Open image in new window holds for every Open image in new window .

This is equivalent to stating that Open image in new window is general Open image in new window -monotone if.

(i) Open image in new window is Open image in new window -relaxed monotone;

(ii) Open image in new window is maximal monotone for every Open image in new window .

Remark 2.8.
  1. (1)

    If Open image in new window , that is, Open image in new window is Open image in new window -relaxed monotone, then the general Open image in new window -monotone operators reduce to general Open image in new window -monotone operators (see, e.g., [1, 2]).

     
  2. (2)

    If Open image in new window is a Hilbert space, then the general Open image in new window -monotone operator reduces to the Open image in new window -monotone operator in Verma [7]. Therefore, the class of general Open image in new window -monotone operators provides a unifying frameworks for classes of maximal monotone operators, Open image in new window -monotone operators, Open image in new window -monotone operators, and general Open image in new window -monotone operators. For details about these operators, we refer the reader to [1, 2, 7] and the references therein.

     

Example 2.9.

Let Open image in new window be a reflexive Banach space with the dual space Open image in new window , Open image in new window a maximal monotone mapping, and Open image in new window a bounded, coercive, hemicontinuous, and relaxed monotone mapping. Then for any given Open image in new window , it follows from Theorem 3.1 in page 401 of Guo [10] that Open image in new window . This shows that Open image in new window is a general Open image in new window -monotone operator.

Example 2.10 (see [4]).

Let Open image in new window be a reflexive Banach space with Open image in new window its dual, and let Open image in new window be Open image in new window -strongly monotone. Let Open image in new window be locally Lipschitz such that Open image in new window is Open image in new window -relaxed monotone. Then Open image in new window is Open image in new window -monotone, which is equivalent to stating that Open image in new window is pseudomonotone (and in fact, maximal monotone).

Lemma 2.11 (see [1]).

Let Open image in new window be a reflexive Banach space with the dual space Open image in new window , let Open image in new window be a nonlinear operator, and let Open image in new window be a general Open image in new window -monotone operator. Then the proximal mapping Open image in new window is

(i) Open image in new window -Lipschitz continuous when Open image in new window is Open image in new window -strongly monotone with Open image in new window and Open image in new window ;

(ii) Open image in new window -Lipschitz continuous if Open image in new window is a strictly monotone operator and Open image in new window is an Open image in new window -strongly monotone operator.

3. Perturbed Algorithms and Convergence

Now we will consider some new perturbed algorithms for solving the nonlinear general Open image in new window -monotone operator equation problem (1.1) or (1.2) by using the proximal mapping technique associated with the general Open image in new window -monotone operators and the convergence of the sequences given by the algorithms.

Lemma 3.1.

Let Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window be the same as in (1.1). Then the following propositions are equivalent.

(1) Open image in new window is a solution of the problem (1.1), where Open image in new window and Open image in new window .

(2) Open image in new window is the fixed-point of the function Open image in new window defined by

where Open image in new window is a constant.

(3) Open image in new window is a solution of the following equation system:

where Open image in new window , Open image in new window and Open image in new window .

Lemma 3.2 (see [17]).

Let Open image in new window and Open image in new window be three nonnegative real sequences satisfying the following condition. There exists a natural number Open image in new window such that

where Open image in new window , Open image in new window , Open image in new window , Open image in new window . Then Open image in new window .

Algorithm 3.3.

Step 1.

Choose an arbitrary initial point Open image in new window .

Step 2.

Take any Open image in new window for Open image in new window

Step 3.

Choose sequences Open image in new window , Open image in new window , Open image in new window and Open image in new window such that for Open image in new window , Open image in new window are two sequences in Open image in new window and Open image in new window Open image in new window and Open image in new window are error sequences in Open image in new window to take into account a possible inexact computation of the operator point, which satisfies the following conditions:

(i) Open image in new window ;

(ii) Open image in new window ;

(iii) Open image in new window .

Step 4.

where Open image in new window is a constant.

Step 5.

If Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window ( Open image in new window ) satisfy (3.4) to sufficient accuracy, stop; otherwise, set Open image in new window and return to Step 2.

Algorithm 3.4.

where Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window are the same as in Algorithm 3.3.

Theorem 3.5.

Let Open image in new window be a uniformly smooth Banach space with Open image in new window for some Open image in new window , and let Open image in new window be the dual space of Open image in new window . Let Open image in new window be Open image in new window -strongly monotone and Open image in new window -Lipschitz continuous, and let Open image in new window be Open image in new window - Open image in new window -Lipschitz continuous. Suppose that Open image in new window is Open image in new window -relaxed cocoercive with respect to Open image in new window and Open image in new window -Lipschitz continuous, Open image in new window is Open image in new window -strongly monotone and Open image in new window -Lipschitz continuous, and Open image in new window is general Open image in new window -monotone, where Open image in new window is defined by Open image in new window for all Open image in new window . If, in addition, there exist constants Open image in new window and Open image in new window such that
then the following results hold:
  1. (1)

    the solution set of the problem (1.1) is nonempty;

     
  2. (2)

    the iterative sequence Open image in new window generated by Algorithm 3.3 converges strongly to the solution Open image in new window of the problem (1.1).

     

Proof.

Setting a multivalued function Open image in new window to be the same as (3.1), then we can prove that Open image in new window is a multivalued contractive operator.

Note that Open image in new window it follows from Nadler's result [18] that there exists Open image in new window such that
then we have Open image in new window . The Open image in new window -strongly monotonicity and Open image in new window -Lipschitz continuity of Open image in new window , the Open image in new window - Open image in new window -Lipschitz continuity of Open image in new window , the Open image in new window -relaxed cocoercivity with respect to Open image in new window and Open image in new window -Lipschitz continuity of Open image in new window , and the Open image in new window -Lipschitz continuity of Open image in new window , Lemma 2.6, and the inequality (3.8) imply that
Thus, it follows from (3.4) and Lemma 2.11 that
It follows from condition (3.6) that Open image in new window . Hence, from (3.11), we get
Since Open image in new window is arbitrary, we obtain Open image in new window . By using same argument, we can prove Open image in new window . It follows from the definition of the Hausdorff metric Open image in new window on Open image in new window that
and so Open image in new window is a multivalued contractive mapping. By a fixed-point theorem of Nadler [18], the definition of Open image in new window and (3.2), now we know that Open image in new window has a fixed-point Open image in new window , that is, Open image in new window and there exists Open image in new window such that

Hence, it follows from Lemma 3.1 that Open image in new window is a solution of the problem (1.1), that is, the solution set of the problem (1.1) is nonempty.

Next, we prove the conclusion (2). Let Open image in new window be a solution of problem (1.1). Then for all Open image in new window , we have
From Algorithm 3.3, the assumptions of the theorem 3.5 and Lemma 2.11, it follows that
Combining (3.17) and (3.18), we obtain
where Open image in new window is the same as in (3.11). Since Open image in new window , we know that Open image in new window and (3.19) implies
Since Open image in new window , it follows from Lemma 3.2 that the sequence Open image in new window strongly converges to Open image in new window . By Open image in new window , Open image in new window , and the Open image in new window -Lipschitz continuity of Open image in new window , we obtain

Thus, Open image in new window is also strongly converges to Open image in new window . Therefore, the iterative sequence Open image in new window generated by Algorithm 3.3 converges strongly to the solution Open image in new window of the problem (1.1) or (1.2). This completes the proof.

Based on Theorem 3.3 in [2], we have the following comment.

Remark 3.6.

If Open image in new window , Open image in new window is Open image in new window -strongly accretive and Open image in new window -Lipschitz continuous, Open image in new window is a strictly monotone and Open image in new window -Lipschitz continuous operator, Open image in new window is a general Open image in new window -monotone and Open image in new window -strongly monotone operator, Open image in new window is a single-valued operator and Open image in new window is Open image in new window -Lipschitz continuous, and Open image in new window is some constant such that

then (3.6) holds.

Theorem 3.7.

then there exists Open image in new window such that Open image in new window is a solution of the problem (1.3), and the iterative sequence Open image in new window generated by Algorithm 3.4 converges strongly to the solution Open image in new window of the problem (1.3).

Remark 3.8.

If Open image in new window for Open image in new window in Algorithm 3.4, Open image in new window and Open image in new window as the same in the problem (1.5), then the results of Theorem 3.4 obtained by Xia and Huang [2] also hold. For details, we can refer to [1, 2, 4, 5].

Remark 3.9.

If Open image in new window or Open image in new window or Open image in new window in Algorithms 3.3 and 3.4, then the conclusions of Theorems 3.5 and 3.7 also hold, respectively. The results of Theorems 3.5 and 3.7 improve and generalize the corresponding results of [2, 4, 5, 6, 7, 8, 9, 15, 17]. For other related works, we refer to [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and the references therein.

Notes

Acknowledgments

The authors are grateful to Professor Ram U. Verma and the referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (60804065), the Scientific Research Fund of Sichuan Provincial Education Department (2006A106, 07ZB151), and the Sichuan Youth Science and Technology Foundation (08ZQ026-008).

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© Xing Wei et al. 2009

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.College of Mathematics and Software ScienceSichuan Normal UniversityChengduChina
  2. 2.Department of MathematicsSichuan University of Science & EngineeringZigongChina

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