The Over-Relaxed -Proximal Point Algorithm for General Nonlinear Mixed Set-Valued Inclusion Framework

Fixed Point Theory and Applications Volume 2011, Article ID 840978,12pages doi:10.1155/2011/840978

Research Article

The Over-Relaxed

A

-Proximal Point

Algorithm for General Nonlinear Mixed Set-Valued

Inclusion Framework

Xian Bing Pan,

_{Hong Gang Li,}

_{and An Jian Xu}

1_{Yitong College, Chongqing University of Posts and Telecommunications, Chongqing 400065, China} 2_{Institute of Applied Mathematics Research, Chongqing University of Posts and Telecommunications,}

Chongqing 400065, China

3_{College of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China}

Correspondence should be addressed to Xian Bing Pan,[email protected]

Received 16 November 2010; Accepted 10 January 2011

Academic Editor: T. Benavides

Copyrightq2011 Xian Bing Pan et al. 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.

The purpose of this paper is1a general nonlinear mixed set-valued inclusion framework for the over-relaxedA-proximal point algorithm based on theA,η-accretive mapping is introduced, and2 it is applied to the approximation solvability of a general class of inclusions problems using the generalized resolvent operator technique due to Lan-Cho-Verma, and the convergence of iterative sequences generated by the algorithm is discussed inq-uniformly smooth Banach spaces. The results presented in the paper improve and extend some known results in the literature.

1. Introduction

In recent years, various set-valued variational inclusion frameworks, which have wide applications to many fields including, for example, mechanics, physics, optimization and control, nonlinear programming, economics, and engineering sciences have been intensively

studied by Ding and Luo1, Verma2, Huang 3, Fang and Huang4, Fang et al. 5,

Lan et al. 6, Zhang et al. 7, respectively. Recently, Verma 8 has intended to develop

a general inclusion framework for the over-relaxedA-proximal point algorithm 9based

on the A-maximal monotonicity. In 2007-2008, Li 10,11 has studied the algorithm for a

new class of generalized nonlinear fuzzy set-valued variational inclusions involvingH, η

-monotone mappings and an existence theorem of solutions for the variational inclusions,

and a new iterative algorithm 12 for a new class of general nonlinear fuzzy mulitvalued

quasivariational inclusions involving G, η-monotone mappings in Hilbert spaces, and

quasivariational inclusions involvingA, η-accretive mappings, the stability 13 and the

convergence of the iterative sequences inq-uniformly smooth Banach spaces by using the

resolvent operator technique due to Lan et al.6.

Inspired and motivated by recent research work in this field, in this paper, a general

nonlinear mixed set-valued inclusion framework for the over-relaxed A-proximal point

algorithm based on the A, η-accretive mapping is introduced, which is applied to the

approximation solvability of a general class of inclusions problems by the generalized resolvent operator technique, and the convergence of iterative sequences generated by

the algorithm is discussed in q-uniformly smooth Banach spaces. For more literature, we

recommend to the reader1–17.

2. Preliminaries

LetX be a real Banach space with dual space X∗, and let ·,·be the dual pair betweenX

and X∗, let 2X _{denote the family of all the nonempty subsets of X, and let CBX denote}

the family of all nonempty closed bounded subsets ofX. The generalized duality mapping

Jq:X → 2X∗_{is single-valued ifX}∗_{is strictly convex14, orXis uniformly smooth space. In}

what follows we always denote the single-valued generalized duality mapping byJqin real

uniformly smooth Banach spaceXunless otherwise stated. We consider the followinggeneral

nonlinear mixed set-valued inclusion problem withA, η-accretive mappings (GNMSVIP).

Findingx∈Xsuch that

0∈FAx Mx, 2.1

whereA, F : X → X,η : X ×X → X be single-valued mappings;M : X → 2X _{be an}

A, η-accretive set-valued mapping. A special case of problem2.1is the following:

ifX X∗ is a Hilbert space,F 0 is the zero operator inX, andηx, y x−y,

then problem2.1becomes the inclusion problem 0 ∈ Mxwith a A-maximal

monotone mappingM, which was studied by Verma8.

Definition 2.1. LetXbe a real Banach space with dual spaceX∗, and let·,·be the dual pair

betweenX andX∗. Let A : X → X and η : X ×X → X be single-valued mappings. A

set-valued mappingM:X → 2X_{is said to be}

ir−stronglyη-accretive, if there exists a constantr >0 such that

y1−y2, Jq

ηx1, x2

≥rx1−x2q, ∀yi∈Mxi, i1,2; 2.2

iim-relaxedη-accretive, if there exists a constantm >0 such that

y1−y2, Jq

ηx1, x2

≥ −mx1−x2q, ∀x1, x2∈X, yi∈Mxi, i1,2; 2.3

iiic-cocoercive, if there exists a constantcsuch that

y1−y2, Jq

ηx1, x2

≥cy1−y2q, ∀x1, x2∈X, yi∈Mxi, i1,2; 2.4

iv A, η-accretive, ifMism-relaxedη-accretive andRAρMX X for every

Based on the literature6, we can define the resolvent operatorRA,η_ρ,Mas follows.

Definition 2.2. Let η : X×X → X be a single-valued mapping, A : X → X be a strictly

η-accretive single-valued mapping and M : X → 2X _{be an A, η-accretive set-valued}

mapping. The resolvent operatorRA,η_ρ,M:X → Xis defined by

RA,η_ρ,Mx AρM−1x ∀x∈X, 2.5

whereρ >0 is a constant.

Remark 2.3. TheA, η-accretive mappings are more general thanH, η-monotone mappings

and m-accretive mappings in Banach space or Hilbert space, and the resolvent operators

associated with A, η-accretive mappings include as special cases the corresponding

resolvent operators associated with H, η-monotone operators, m-accretive mappings, A

-monotone operators,η-subdifferential operators1–7,11–13.

Lemma 2.4see6. Letη :X×X → Xbeτ-Lipschtiz continuous mapping,A:X → Xbe an

r-stronglyη-accretive mapping, andM:X → 2X_{be anA, η-accretive set-valued mapping. Then}

the generalized resolvent operatorRA,η_ρ,M:X → Xisτq−1_{/r−mρ-Lipschitz continuous, that is,}

RA,η_ρ,Mx−RA,η_ρ,My≤ τ q−1

r−mρx−y ∀x, y∈X

, 2.6

whereρ∈0, r/m.

In the study of characteristic inequalities inq-uniformly smooth Banach spaces, Xu

14proved the following result.

Lemma 2.5. LetX be a real uniformly smooth Banach space. ThenX isq-uniformly smooth if and

only if there exists a constantcq >0such that for allx, y∈X,

_xyq

≤ xqqy, Jqxcqyq. 2.7

3. The Over-Relaxed

A

-Proximal Point Algorithm

This section deals with an introduction of a generalized version of the over-relaxed proximal point algorithm and its applications to approximation solvability of the inclusion problem of

the form2.1based on theA, η-accretive set-valued mapping.

LetM :X → 2X _{be a set-valued mapping, the set{x, y :y ∈ Mx}be the graph}

ofM, which is denoted byMfor simplicity, This is equivalent to stating that a mapping is

Mxto represent the unique y such thatx, y∈Mrather than the singleton set{y}. This

interpretation will depend greatly on the context. The inverseM−1ofMis{y, x:x, y∈

M}.

Definition 3.1. Let M : X → 2X _{be a set-valued mapping. The map M}−1_{, the inverse of}

M :X → 2X_{, is said to be generalu, t-Lipschitz continuous at 0 if, and only if there exist}

two constantsu, t ≥0 for anyw ∈Bt {w :w ≤t, w∈X}, a solutionx∗ of the inclusion

0∈Mxx∗∈M−10exist and thex∗such that

x−x∗ ≤uw ∀x∈M−1w, 3.1

holds.

Lemma 3.2. Let X be a q-uniformly smooth Banach space, η : X ×X → X be aτ-Lipschtiz

continuous mapping, A : X → X be an r-strongly η-accretive mapping,F : X → X be a ξ -Lipschtiz continuous mapping, and M : X → 2X _{be an A, η-accretive set-valued mapping. If} IkA−ARA,η_ρ,MA−ρFA, and for allx1, x2∈X,ρ >0andqγ >1

Ax1−Ax2, Jq

ARA,η_ρ,MAx1−ρFAx1

−ARA,η_ρ,MAx2−ρFAx2

≥γARA,η_ρ,MAx1−ρFAx1

−ARA,η_ρ,MAx2−ρFAx2 q

,

3.2

then

qγ−1AR_ρ,MA,ηAx1−ρFAx1

−ARA,η_ρ,MAx2−ρFAx2 q

Ikx1−Ikx2q≤cqAx1−Ax2q.

3.3

Proof. Let X be a q-uniformly smooth Banach space, η : X ×X → X be a τ-Lipschtiz

continuous mapping,A: X → Xbe anr-stronglyη-accretive mapping, andM: X → 2X

be an A, η-accretive set-valued mapping. Let us set Ik A − ARA,η_ρ,MA − ρFA and

si Axi−ρFAxixi ∈ X, i 1,2, then by usingDefinition 2.2, Lemmas2.4,2.5, and

3.2, we can have

Ikx1−Ikx2q

Ax1−A

RA,η_ρ,Ms1

−Ax2−A

≤cqAx1−Ax2q−q Ax1−Ax2, Jq

ARA,η_ρ,Ms1

−ARA,η_ρ,Ms2

ARA,η_ρ,MAx1−ρFAx1

−ARA,η_ρ,MAx2−ρFAx2 q

≤cqAx1−Ax2q

−qγARA,η_ρ,MAx1−ρFAx1

−ARA,η_ρ,MAx2−ρFAx2 q

ARA,η_ρ,MAx1−ρFAx1

−ARA,η_ρ,MAx2−ρFAx2 q

≤cqAx1−Ax2q

−qγ−1ARA,η_ρ,MAx1−ρFAx1

−ARA,η_ρ,MAx2−ρFAx2 q

.

3.4

Therefore,3.3holds.

Lemma 3.3. Let X be a q-uniformly smooth Banach space, η : X ×X → X be aτ-Lipschtiz

continuous mapping, A : X → X be an r-strongly η-accretive and nonexpansive mapping,

F : X → X be an ξ-Lipschtiz continuous mapping, and Ik A − ARA,η_ρ,MA −ρFA, and

M:X → 2X_{be anA, η-accretive set-valued mapping. Then the following statements are mutually}

equivalent.

iAn elementx∗∈Xis a solution of problem2.1.

iiFor ax∗∈X, such that

x∗RA,η_ρ,MAx∗−ρFAx∗. 3.5

iiiFor ax∗∈X, holds

Ikx∗ _Ax∗_−ARA,η ρ,M

Ax∗_−ρFAx∗_{0, 3.6}

whereρ >0is a constant.

Proof. This directly follows from definitions ofRA,η_ρ,MxandIk.

Lemma 3.4. Let X be a q-uniformly smooth Banach space, η : X ×X → X be aτ-Lipschtiz

continuous mapping,A:X → Xbe anr-stronglyη-accretive and nonexpansive mapping,F:X → Xbe anξ-Lipschtiz continuous andβ-stronglyη-accretive mapping, andIkA−ARA,η_ρ,MA−ρFA, andM:X → 2X_{be anA, η-accretive set-valued mapping. If the following conditions holds}

wherecq >0is the same as inLemma 2.5, andρ ∈0, r/m. Then the problem2.1has a solution

x∗∈X.

Proof. DefineN:X → Xas follows:

Nx RA,η_ρ,MAx−ρFAx, ∀x∈X. 3.8

For elementsx1, x2 ∈X, if letting

siAxi−ρFAxi i1,2, 3.9

then by3.1and3.3, we have

Nx1−Nx2RA,ηρ,Ms1−RA,ηρ,Ms2

≤ τq−1

r−mρAx1−Ax2−ρFAx1−FAx2.

3.10

By usingr-stronglyη-accretive ofA,β-stronglyη-accretive ofF, andLemma 2.5, we obtain

_{Ax1−Ax2−ρFAx1−FAx2}q

≤ Ax1−Ax2qcqρqFAx1−FAx2q

−qρFAx1−FAx2, JqAx1−Ax2

≤1cqρq_ξq_−qρβAx

1−Ax2q.

3.11

Combining3.10-3.11, by using nonexpansivity ofA, we have

Nx1−Nx2 ≤θ∗x1−x2, 3.12

where

θ∗ τ q−1

r−mρ

q

1cqρq_ξq_{−qρβ 1cqρ}q_ξq_{> qρβ.}

3.13

It follows from3.7–3.12thatN has a fixed point inX, that is, there exist a pointx∗ ∈ X

such thatx∗Nx∗, and

x∗Nx∗ RA,η_ρ,MAx∗−ρFAx∗. 3.14

Based on Lemma 3.3, we can develop a general over-relaxed A, η-proximal point

algorithm to approximating solution of problem2.1as follows.

Algorithm 3.5. LetXbe aq-uniformly smooth Banach space,η:X×X → Xbe aτ-Lipschtiz

continuous mapping,A:X → X be anr-stronglyη-accretive and nonexpansive mapping,

F : X → X be an β-strongly η-accretive mapping and ξ-Lipschitz continuous, and Ik

A−ARA,η_ρ,MA−ρFA, and M : X → 2X _{be an A, η-accretive set-valued mapping. Let}

{an}∞n0an≥1,{bn}∞n0and{ρn}∞n0be three nonnegative sequences such that

∞

n1

bn<∞, alim sup

n→ ∞ an ≥1, ρn↑ρ≤ ∞, 3.15

whereρn,ρ∈0, r/mn0,1,2,·,·,·and each satisfies condition3.7.

Step 1. For an arbitrarily chosen initial pointx0∈X, set

Ax1 1−a0Ax0 a0y0, 3.16

where they0satisfies

y0−A

RA,η_ρ0,MAx0−ρ0FAx0≤b0y0−Ax0. 3.17

Step 2. The sequence{xn}is generated by an iterative procedure

Axn1 1−anAxn anyn, 3.18

andynsatisfies

yn−ARA,η_ρn,MAxn−ρnFAxn≤bnyn−Axn, 3.19

wheren1,2,·,·,·.

Remark 3.6. For a suitable choice of the mappings A, η, F, M, Ik, and space X, then the

Algorithm 3.5can be degenerated to the hybrid proximal point algorithm 16,17and the

over-relaxedA-proximal point algorithm8.

Theorem 3.7. LetXbe aq-uniformly smooth Banach space. LetA, F:X → Xandη:X×X → X

be single-valued mappings, and letM:X×X → 2X _{be a set-valued mapping andFAM}−1_be

the inverse mapping of the mappingFAM:X → 2X_{satisfying the following conditions:} iη:X×X → Xisτ-Lipschtiz continuous;

iiA:X → Xbe anr-stronglyη-accretive mapping and nonexpansive;

iiiF:X → Xbe anξ-Lipschtiz continuous andβ-stronglyη-accretive mapping;

vtheFAM−1beu, t-Lipschitz continuous at 0u≥0;

vi{an}∞n0an≥1,{bn}∞n0and{ρn}∞n0be three nonnegative sequences such that

∞

n1

bn<∞, alim sup

n→ ∞ an≥1, ρn↑ρ≤ ∞, 3.20

whereρn,ρ∈0, r/mn0,1,2,·,·,·and each satisfies condition3.7,

viilet the sequence{xn} generated by the general over-relaxedA-proximal point algorithm

3.6be bounded andx∗be a solution of problem2.1, and the condition

Axn−Ax∗_{, JqAR}A,η ρ,M

Axn−ρFAxn−AR_ρ,MA,ηAx∗−ρFAx∗

≥γARA,η_ρ,MAxn−ρFAxn−ARA,η_ρ,MAx∗−ρFAx∗q,

3.21

0< cqa−1qaq−qa−1aγdq <1, 3.22

hold. Then the sequence {xn} converges linearly to a solution x∗ of problem2.1with convergence rateϑ, where

ϑ q

cqa−1qaq_q1−aaγdq_,

alim sup

n→ ∞ an, dlim supn→ ∞ dnlim supn→ ∞

q

cquq

qγ−1rq_uq_ρq n ,

∞

n1

bn<∞.

3.23

Proof. Let the x∗ be a solution of the Framework 2.1 for the conditions i–iv and

Lemma 3.4. Suppose that the sequence{xn}which generated by the hybrid proximal point

Algorithm 3.5is bounded, fromLemma 3.4, we have

Ax∗ _1−anAx∗ _anARA,η ρn,M

Ax∗_−ρnFAx∗_{. 3.24}

We infer fromLemma 3.3that any solution to2.1is a fixed point ofRA,η_ρn,MA−ρnFA. First,

in the light ofLemma 3.2, we show

RA,η_ρn,MAxn−ρnFAxn−x∗≤dnAxn−Ax∗, 3.25

wheredn q

cquq_/2γ−1rq_uq_ρq_{n<1 andR}A,η ρn,MAx

ForIk A−ARA,η_ρ,MA−ρnFA, and under the assumptionsincluding the condition

vii 3.21, thenIkxn → 0n → ∞since theFAM−1isu, t-Lipschitz continuous at 0.

Indeed, it follows thatRA,η_ρn,MAxn−ρnFAxn∈FAM−1ρn−1Ikxnfromρ−1n Ikxn∈

FAMRA,η

ρn,MAxn−ρnFAxn. Next, by using the conditionivand3.1, and setting

wρ−1n IkxnandzR A,η

ρn,MAxn−ρnFAxn, we have

RA,η_ρn,MAxn−ρnFAxn−x∗≤uρn−1Ikxn, ∀n > n. 3.26

Now applyingLemma 3.3, we get

RA,η_ρn,MAxn−ρnFAxn−x∗q

≤RA,η_ρn,MAxn−ρnFAxn−RA,η_ρn,MAx∗−ρnFAx∗q

≤uqρ−1n Ikxn−ρ−1n Ikx∗ q

≤

u ρn

q

Ikxn−Ikx∗q

≤

u ρn

q

−qγ−1rqRA,η_ρn,MAxn−ρnFAxn−RA,η_ρn,MAx∗−ρnFAx∗q

cqAxn−Ax∗q_.

3.27

Therefore,

RA,η_ρn,MAxn−ρnFAxn−x∗≤dnAxn−Ax∗, 3.28

wheredn q

cquq_/2γ−1rq_uq_ρq

n<1 andR A,η ρn,MAx

∗_−ρnFAx∗ _x∗_.

Next we start the main part of the proof by using the expression

Azn1 1−anAxn anA

RA,η_ρ

n,M

Let us setsn Axn−ρnFAxnands∗ Ax∗−ρnFAx∗for simple. We begin with

estimatingforan ≥ 1and later using3.2, the nonexpansivity ofA,3.21and 3.28as

follows:

Azn1−Ax∗q ≤1−anAxn anA

RA,η_ρn,Msn

−1−anAx∗ anARA,η_ρn,Ms∗q

≤1−anAxn−Ax∗ anARA,η_ρn,Msn−AR_ρA,η_n,Ms∗q

≤cq1−anAxn−Ax∗qanARA,η_ρn,Msn−ARA,η_ρn,Ms∗q

q1−anan Axn−Ax∗, JqARA,η_ρn,Msn−ARA,η_ρn,Ms∗

≤cqan−1qAxn−Ax∗qaqnRA,η_ρn,Msn−RA,η_ρn,Ms∗ q

q1−ananγRA,η_ρ

n,Msn−R

A,η ρn,Ms

∗q

≤cqan−1qAxn−Ax∗q

aqn−q1−ananγRA,ηρn,M

Axn−ρnFAxn−x∗q

≤cqan−1qaqnq1−ananγ

dqn

Axn−Ax∗q .

3.30

Thus, we have

Azn1−Ax∗q ≤θnAxn−Ax∗q, 3.31

where

θn q

cqan−1qaqnq1−ananγ

dqn<1, 3.32

andaqnq1−ananγ >0,an≥1,

_∞

n1bn<∞, anddn q

cquq_/2γ−1rq_uq_ρq_n<1.

SinceAxn1 1−anAxn anyn, we haveAxn1−Axn anyn−Axn. It

follows that

Axn1−Azn1

≤1−anAxn anyn−1−anAxn anRA,η_ρn,MAxn−ρnFAxn

≤anyn−R_ρA,η_n,MAxn−ρnFAxn

≤anbnyn−Axn.

Next, we can obtain

Axn1−Ax∗ ≤ Azn1−Ax∗Axn1−Azn1

≤ Azn1−Ax∗anbnyn−Axn

≤ Azn1−Ax∗bnAxn1−Axn

≤ Azn1−Ax∗bnAxn1−Ax∗bnAx∗−Axn. 3.34

This implies from3.38and3.39that

Axn1−Ax∗ ≤ θn₁₋bn

bn Axn−Ax

∗_{. 3.35}

SinceAis anr-stronglyη-accretive mappingand hence,Ax−Ay ≥ rx−y, this

implies from3.35that the sequence{xn}converges strongly tox∗for

θn q

cqan−1qaqnq1−ananγ

dqn<1, 3.36

wherean≥1,∞_n1bn <∞, anddn q

cquq_/2γ−1rq_uq_ρq_n<1.

Hence, we have

ϑlim sup

n→ ∞

θnbn

1−bn lim supn→ ∞ θn

q

cqa−1qaq_q1−aaγdq_.

3.37

By3.22, it follows that 0< ϑ <1 from the conditionvi, and the sequence{xn}generated

by the hybrid proximal pointAlgorithm 3.5converges linearly to a solutionx∗ of problem

2.1with convergence rateϑ. This completes the proof.

Corollary 3.8. LetXbe a Hilbert spaceq 2, cq 1,A : X → X be anr-strongly monotone

and nonexpansive mappingα 1,F 0is a zero operator,M :X → 2X _{be anA-maximal}

set-valued monotone.IkA−ARA,η_ρ,MA−ρnFA, and the condition3.21hold, theM−1beu-Lipschitz continuous at 0u≥0. Let{an}∞n0,{bn}∞n0and{ρn}∞n0be the same as inAlgorithm 3.5. If

0<1−a21−γd2−a1−2γ−1d2<1, 3.38

then the bounded sequence{xn}generated by the general over-relaxedA-proximal point algorithm converges linearly to a solutionx∗of problem2.1with convergence rateϑ, where

ϑ

anddlim sup_{n→ ∞}dnlim sup_{n→ ∞}

u2_/2γ−1r2_u2_ρ2_{n u}2_/2γ−1r2_u2_ρ2_,a

lim sup_{n→ ∞}an,∞_n1bn<∞.

This is Theorem 3.2 in 8, and if, in addition, γ 1, A I then we can have the

Proposition 2 in9.

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