Iterative Algorithms for Finding Common Solutions to Variational Inclusion Equilibrium and Fixed Point Problems

Volume 2011, Article ID 915629,17pages doi:10.1155/2011/915629

Research Article

Iterative Algorithms for Finding Common

Solutions to Variational Inclusion Equilibrium and

Fixed Point Problems

J. F. Tan and S. S. Chang

Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China

Correspondence should be addressed to S. S. Chang,[email protected]

Received 30 October 2010; Accepted 9 November 2010

Academic Editor: Qamrul Hasan Ansari

Copyrightq2011 J. F. Tan and S. S. Chang. 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 main purpose of this paper is to introduce an explicit iterative algorithm to study the existence problem and the approximation problem of solution to the quadratic minimization problem. Under suitable conditions, some strong convergence theorems for a family of nonexpansive mappings are proved. The results presented in the paper improve and extend the corresponding results announced by some authors.

1. Introduction

Throughout this paper, we assume thatHis a real Hilbert space with inner product·,·and norm · ,Cis a nonempty closed convex subset ofH, andFT {x∈H :Tx x}is the set of fixed points of mappingT.

A mappingS:C → Cis called nonexpansive if

_{Sx−Sy≤x−y, ∀x, y∈C. 1.1}

Let A : H → H be a single-valued nonlenear mapping andM : H → 2H be a multivalued mapping. The so-called quasivariational inclusion problemsee1–3is to find

u∈Hsuch that

θ∈Au Mu. 1.2

Special Cases

IIfM∂φ:H → 2H, whereφ:H → Ê∪{∞}is a proper convex lower semi-continuous

function and∂φis the subdifferential ofφ, then the quasivariational inclusion problem1.2 is equivalent to findingu∈Hsuch that

Au, y−uφy−φu≥0, ∀y∈H, 1.3

which is called the mixed quasivariational inequalitysee4.

IIIfM ∂δ_C, whereCis a nonempty closed convex subset of H andδ_C : H →

0,∞is the indicator function ofC, that is,

δCx

⎧ ⎨ ⎩

0, x∈C,

∞, x /∈C, 1.4

then the quasivariational inclusion problem1.2is equivalent to findingu∈Csuch that

Au, v−u ≥0, ∀v∈C. 1.5

This problem is called the Hartman-Stampacchia variational inequalitysee5. The set of solutions to variational inequality1.5is denoted by VIA, C.

LetB : C → H be a nonlinear mapping andF : C×C → Ê be a bifunction. The

so-called generalized equilibrium problem is to find a pointu∈Csuch that

Fu, yBu, y−u ≥0, ∀y∈C. 1.6

The set of solutions to1.6is denoted by GEPsee5,6. IfB0, then1.6reduces to the following equilibrium problem: to findu∈Csuch that

Fu, y≥0, ∀y∈C. 1.7

The set of solutions to1.7is denoted by EP.

Iterative methods for nonexpansive mappings and equilibrium problems have been applied to solve convex minimization problemssee7–9. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:

min x∈F

1 2x

2_{, 1.8}

whereFis the fixed point set of a nonexpansive mappingTonH.

In 2010, Zhang et al.see10proposed the following iteration method for variational inclusion problem1.5and equilibrium problem1.6in a Hilbert spaceH:

Under suitable conditions, they proved the sequence {xn} generated by 1.9 converges strongly to the fixed pointx∗, which solves the quadratic minimization problem1.8.

Motivated and inspired by the researches going on in this direction, especially inspired by Zhang et al.10, the purpose of this paper is to introduce an explicit iterative algorithm to studying the existence problem and the approximation problem of the solution to the quadratic minimization problem 1.8 and prove some strong convergence theorems for a family of nonexpansive mappings in the setting of Hilbert spaces.

2. Preliminaries

LetHbe a real Hilbert space, andCbe a nonempty closed convex subset ofH. For anyx∈H, there exists a unique nearest point inC, denoted byPCx, such that

x−PCx ≤x−y, ∀y∈C. 2.1

Such a mappingPC fromHontoCis called the metric projection. It is well-known that the metric projectionP_C :H → Cis nonexpansive.

In the sequel, we usexn xandxn → xto denote the weak convergence and the strong convergence of the sequence{xn}, respectively.

Definition 2.1. A mappingA:H → His calledα-inverse strongly monotone if there exists anα >0 such that

Ax−Ay, x−y ≥αAx−Ay2_{, ∀x, y∈H. 2.2}

A multivalued mappingM:H → 2His called monotone if∀x, y∈H, u∈Mx, v∈My,

u−v, x−y ≥0. 2.3

A multivalued mappingM:H → 2His called maximal monotone if it is monotone and for anyx, u∈H×H, when

u−v, x−y ≥0 for everyy, v∈GraphM, 2.4

thenu∈Mx.

Proposition 2.2see11. LetA: H → Hbe anα-inverse strongly monotone mapping. Then, the following statements hold:

iAis an1/α-Lipschitz continuous and monotone mapping;

Lemma 2.3see12. LetXbe a strictly convex Banach space,Cbe a closed convex subset ofX, and{Tn :C → C}be a sequence of nonexpansive mappings. Suppose∞n1FTn/∅. Let{λn}be a

sequence of positive numbers withΣ∞_n1λn1. Then the mappingS:C → Cdefined by

Sx Σ∞

n1λnTnx, x∈C 2.5

is well defined. And it is nonexpansive and

FS ∞

n1

FTn. 2.6

Definition 2.4. Let H be a Hilbert space and M : H → 2H be a multivalued maximal monotone mapping. Then, the single-valued mappingJM,λ:H → Hdefined by

JM,λu IλM−1u, u∈H 2.7

is called theresolvent operator associated withM, whereλis any positive number andIis the identity mapping.

Proposition 2.5see11. iThe resolvent operatorJM,λassociated withMis single-valued and

nonexpansive for allλ >0, that is,

_{JM,λx−JM,λ}

y≤x−y, ∀x, y∈H, ∀λ >0. 2.8

iiThe resolvent operatorJM,λis 1-inverse strongly monotone, that is,

_{JM,λx−JM,λy}2_{≤x−y, J}

M,λx−JM,λy , ∀x, y∈H. 2.9

Definition 2.6. A single-valued mappingA:H → His said to be hemicontinuous if for any

x, y, z∈H, functiont→ Axty, zis continuous at 0.

It is well-known that every continuous mapping must be hemicontinuous.

Lemma 2.7see13. Let{xn}and{yn}be bounded sequences in a Banach spaceX. Let{βn}be a

sequence in0,1with

0<lim inf

n→ ∞ βn ≤lim sup_{n→ ∞} βn<1. 2.10

Suppose that

xn1

1−βnynβnxn, ∀n≥0,

lim sup n→ ∞

_{yn1−yn− xn1−xn}

Then,

lim

n→ ∞yn−xn0. 2.12

Lemma 2.8see14. LetXbe a real Banach space,X∗be the dual space ofX,T :X → 2X∗be a maximal monotone mapping, andP :X → X∗be a hemicontinuous bound monotone mapping with

DP X. Then, the mappingSTP :X → 2X∗is a maximal monotone mapping.

Lemma 2.9see15. LetXbe a uniformly convex Banach space, letCbe a nonempty closed convex subset ofX, andT :C → Cbe a nonexpansive mapping with a fixed point. Then,I−Tis demiclosed in the sense that if{xn}is a sequence inCsatisfying

xn x, I−Tn−→0, 2.13

then

I−Tx0. 2.14

Throughout this paper, we assume that the bifunction F : C×C → Ê satisfies the

following conditions:

H1Fx, x 0 for allx∈C;

H2Fis monotone, that is,

Fx, yFy, x≤0, ∀x, y∈C, 2.15

H3for eachx, y, z∈C,

lim t↓0 F

tz 1−tx, y≤Fx, y, _2.16

H4for eachx∈C,y→Fx, yis convex and lower semi-continuous.

Lemma 2.10see16. LetHbe a real Hilbert space,Cbe a nonempty closed convex subset ofH, andF:C×C → Êbe a bifunction satisfying the conditions (H1)–(H4). Letμ >0andx∈H. Then,

there exists a pointz∈Csuch that

Fz, y 1

μ

y−z, z−x ≥0, ∀y∈C. 2.17

Moreover, ifTμ:H → Cis a mapping defined by

Tμx

z∈C:Fz, y1

μ

y−z, z−x ≥0, ∀y∈C

, x∈H, 2.18

iTμis single-valued and firmly nonexpansive, that is, for anyx, y∈H,

_Tμx−Tμy2_≤T

μx−Tμy, x−y , 2.19

iiEPis closed and convex, andEPFTμ.

Lemma 2.11. i see11u∈His a solution of variational inclusion1.2if and only if

uJM,λu−λAu, ∀λ >0, 2.20

that is,

VIH, A, M FJM,λu−λAu, ∀λ >0. 2.21

ii see10u∈Cis a solution of generalized equilibrium problem1.6if and only if

uTμu−μBu, ∀μ >0, 2.22

that is,

GEPFTμu−μBu, ∀μ >0. 2.23

iii see10LetA:H → Hbe anα-inverse strongly monotone mapping andB:C → H

be aβ-inverse strongly monotone mapping. Ifλ ∈ 0,2αandμ ∈ 0,2β, thenVIH, A, Mis a closed convex subset inHand GEP is a closed convex subset inC.

Lemma 2.12see17. Assume that{an}is a sequence of nonnegative real numbers such that

an1≤1−γnanδn, ∀n≥1, 2.24

where{γn}is a sequence in0,1and{δn}is a sequence such that:

i∞_n1γn∞;

iilim sup_{n→ ∞}δ_n/γ_n≤0or∞_n1|δ_n|<∞.

Then,limn→ ∞an0.

3. Main Results

Theorem 3.1. LetHbe a real Hilbert space,Cbe a nonempty closed convex subset ofH,A:H → H

mapping. LetM : H → 2H be a maximal monotone mapping,{Tn : C → C}be a sequence of

nonexpansive mappings with∞_n1FTn/∅,S : C → Cbe the nonexpansive mapping defined by

2.5, andF:C×C → Êbe a bifunction satisfying conditions (H1)–(H4). Let{xn}be the sequence

defined by

xn1αnxn 1−αn

SPC1−tnJM,λI−λATμI−μBxn, 3.1

where the mappingTμ:H → Cis defined by2.18, andλ, μare two constants withλ∈0,2α, μ∈

0,2β, and

tn∈0,1, tn−→0n−→ ∞, ∞

n1

tn∞, 0< a < αn< b <1. 3.2

If

Ω:FS∩VIH, A, M∩GEP/∅, 3.3

whereVIH, A, Mand GEP is the set of solutions of variational inclusion1.2and generalized equilibrium problem1.6, respectively, then the sequence{xn}defined by3.1converges strongly to

x∗_{∈Ω, which is the unique solution of the following quadratic minimization problem:}

x∗2_min

x∈Ωx 2_.

3.4

Proof. We divide the proof ofTheorem 3.1into four steps.

Step 1The sequence{xn}is bounded. Set

unTμI−μBxn, ynJM,λI−λAun, znSPC1−tnyn. 3.5

Takingz∈Ω, then it follows fromLemma 2.11that

zTμz−μBzJM,λz−λAz SPCz. 3.6

un−z2 TμI−μBxn−Tμz−μBz2

≤I−μBxn−z−μBz2

≤ xn−z2μμ−2βBxn−Bz2,

3.7

_yn−z2 _J

M,λI−λAun−JM,λz−λAz2

≤ I−λAun−z−λAz2

≤ un−z2λλ−2αAun−Az2

≤ xn−z2λλ−2αAun−Az2μμ−2βBxn−Bz2.

3.8

This implies that

_{yn−z≤ un−z ≤ xn−z.} 3.9

It follows from3.1and3.9that

xn1−zαnxn 1−αnSPC1−tnyn−z

αnxn−z 1−αnSPC1−tnyn−SPCz

≤αnxn−z 1−αnSPC1−tnyn−SPCz

≤αnxn−z 1−αn1−tnyn−z

≤αnxn−z 1−αn1−tnyn−ztnz

≤αnxn−z 1−αn1−tnxn−ztnz

≤1−tn1−αnxn−ztn1−αnz

≤max{x_n−z,z}

≤max{xn−1−z,z}

≤ · · · ≤max{x1−z,z}M,

3.10

whereMmax{x1−z,z}. This shows that{xn}is bounded. Hence, it follows from3.9 that the sequence{un}and{yn}are also bounded.

It follows from3.5,3.6, and3.9that

zn−zSPC1−tnyn−SPCz≤1−tnyn−z

≤1−tnyn−ztnz ≤1−tnxn−ztnz ≤M.

3.11

Step 2. Now, we prove that

lim

n→ ∞xn−unnlim→ ∞un−ynnlim→ ∞xn−yn0, lim

n→ ∞xn−Sxn0.

3.12

SinceSPCis nonexpansive, from3.5and3.9, we have that

_{yn1−yn≤ un1−un ≤ xn1−xn,} 3.13

zn1−znSPC1−tn1yn1

−SPC1−tnyn

≤1−tn1yn1−1−tnyn

1−tn1

yn1−yn 1−tn1−1−tnyn

≤1−tn1yn1−yn|tn1−tn|yn

≤yn1−yn|tn1−tn|yn

≤ un1−un|tn1−tn|yn

≤ xn1−xn|tn1−tn|yn.

3.14

Letn → ∞in3.14, in view of conditiontn → 0n → ∞, we have

lim

n→ ∞zn1−zn − xn1−xn 0. 3.15

By virtue ofLemma 2.7, we have

lim

n→ ∞xn−zn0. 3.16

This implies that

lim

We derive from3.17that

lim n→ ∞

xn−z2− xn1−z2

lim n→ ∞

xn−xn122xn−xn1, xn1−z

≤ lim n→ ∞

xn−xn122xn−xn1 · xn1−z

0.

3.18

From3.1and3.8, we have

xn1−z2≤

αnxn−z 1−αn1−tnyn−z2

≤αnxn−z2 1−αn1−tnyn−z−tnz2

αnxn−z2 1−αn

1−tn2yn−z2−2tn1−tnz, yn−z t2nz2

≤αnxn−z2 1−αnyn−z2tnM1

≤αnxn−z2 1−αn

×xn−z2λλ−2αAun−Az2μμ−2βBxn−Bz2tnM1

xn−z2 1−αn

λλ−2αAun−Az2μμ−2βBxn−Bz2tnM1

,

3.19

where

M1sup n

z2_2λu

n−ynμxn−yn<∞, 3.20

that is,

1−αn

λ2α−λAun−Az2μ2β−μBxn−Bz2

≤ xn−z2− xn1−z2 1−αntnM1.

3.21

Letn → ∞, noting the assumptions thatλ ∈ 0,2α,μ ∈ 0,2β, from3.2and 3.18, we have

lim

By virtue ofLemma 2.10iand3.1, we have

un−z2Tμxn−μBxn−Tμz−μBz2

≤xn−μBxn−z−μBz, un−z

1

2

xn−μBxn−z−μBz2un−z2

−xn−z−μBxn−Bz−un−z2

≤ 1

2

xn−z2un−z2−xn−un−μBxn−Bz2

1

2

xn−z2un−z2− xn−un2

2μxn−un, Bxn−Bz −μ2Bxn−Bz2

.

3.23

Simplifying it, we have

un−z2≤ xn−z2− xn−un22μxn−un, Bxn−Bz −μ2Bxn−Bz2

≤ xn−z2− xn−un22μxn−un · Bxn−Bz

≤ xn−z2− xn−un2M1Bxn−Bz.

3.24

Similarly, in view ofProposition 2.5iiand3.1, we have

_yn−z2_J

M,λun−λAun−JM,λz−λAz2

≤un−λAun−z−λAz, yn−z

1

2

un−λAun−z−λAz2yn−z2

−un−λAun−z−λAz−yn−z2

≤ 1

2

un−z2yn−z2−un−yn−λAun−Az2

1

2

un−z2yn−z2−un−yn2

2λun−yn, Aun−Az −λ2Aun−Az2

.

Simplifying it, from3.24, we have

_yn−z2_{≤ u}

n−z2−un−yn22λun−yn, Aun−Az −λ2Aun−Az2

≤ un−z2−un−yn22λun−yn· Aun−Az

≤ xn−z2− xn−un2M1Bxn−Bz −un−yn2

M1Aun−Az.

3.26

From3.19and3.26, we have

xn1−z2≤αnxn−z2 1−αnyn−z2tnM1

≤αnxn−z2 1−αn

×xn−z2− xn−un2−un−yn2M1Bxn−BzAun−Aztn

xn−z2 1−αn

×M1Bxn−BzAun−Aztn− xn−un2−un−yn2

.

3.27

Letn → ∞nd in view of3.18and3.22, we have

lim n→ ∞

xn−un2un−yn2

0. 3.28

This shows that

lim

n→ ∞xn−unnlim→ ∞un−yn0, lim

n→ ∞xn−yn0.

3.29

Then, we have

xn1−Sxn1xn1−SxnSxn−Sxn1 ≤ xn1−SxnSxn−Sxn1

SPC1−tnyn−SPCxnSxn−Sxn1 ≤1−tnyn−xntnxnxn−xn1 −→0n−→ ∞.

Step 3sequence{xn}converges strongly tox∗ ∈Ω. Because{xn}is bounded, without loss of generality, we can assume thatxn x∗ ∈H. In view of3.12, it yields thatun x∗and

yn x∗. FromLemma 2.9and3.30, we know thatx∗ ∈FS. Next, we prove thatx∗∈GEP∩VIH, A, M.

SinceunTμxn−μBxn, we have

Fun, y 1_μy−un, un−xn−μBxn ≥0, ∀y∈C. 3.31

It follows from conditionH2that

1

μ

y−un, un−xn−μBxn ≥Fy, un, ∀y∈C. 3.32

Therefore,

y−un,un−_μxn Bxn

≥Fy, un, ∀y∈C. 3.33

For anyt∈0,1andy∈C, thenytty 1−tx∗∈C. From3.33, we have

yt−un, Byt ≥ yt−un, Byt −

yt−un, un−_μxn Bxn

Fyt, un

yt−um, Byt−Bum yt−un, Bun−Bxn

−

yt−un,un−_μxn

Fyt, un.

3.34

SinceBisβ-inverse strongly monotone, fromProposition 2.2iand3.12, we have

Bun−Bxn ≤ _β1un−xn −→0 n−→ ∞,

yt−un, Byt−Bun ≥βByt−Bun2≥0.

3.35

Letn → ∞in3.34, in view of conditionH4andun x∗, we have

yt−x∗, Byt ≥Fyt, x∗. 3.36

It follows from conditionsH1,H4and3.36that

0Fyt, yt≤tFyt, y 1−tFyt, x∗

≤tFyt, y 1−tyt−x∗, Byt

tFyt, y 1−tty−x∗, Byt ,

that is,

0≤Fyt, y 1−ty−x∗, Byt . 3.38

Lettto 0 in3.38, we have

Fx∗_{, yy−x}∗_{, Bx}∗ _{≥0, ∀y∈C. 3.39}

This shows thatx∗∈GEP.

Step 4now, we prove thatx∗ ∈VIH, A, M. SinceAisα-inverse strongly monotone, from Proposition 2.2i, we know thatAis an 1/α-Lipschitz continuous and monotone mapping andDA H, whereDAis the domain ofA. It follows fromLemma 2.8thatMAis maximal monotone. Letv, f∈GraphMA, that is,f−Av∈Mv. SinceynJM,λun−

λAun, we haveun−λAun∈IλMyn, that is, 1/λun−yn−λAun∈Myn. By virtue of the maximal monotonicity ofM, we have

v−yn, f−Av−1_λun−yn−λAun≥0. 3.40

Therefore we have

v−yn, f ≥

v−yn, Av_λ1un−yn−λAun

v−yn, Av−AynAyn−Aun 1_λun−yn.

3.41

SinceAis monotone, this implies that

v−yn, f ≥0v−yn, Ayn−Aun

v−yn,_λ1un−yn.

3.42

Since

_{un−yn−→0, Aun−Ayn−→0, yn x}∗_{, n−→ ∞, 3.43}

from3.42, we have

lim

n→ ∞v−yn, f

v−x∗_{, f ≥0. 3.44}

Summing up the above arguments, we have proved that

x∗_{∈Ω:FS∩VIH, M, A∩GEP. 3.45}

On the other hand, for anyz∈Ω, we have

zn−z2SPC1−tnyn−SPCz2

≤1−tnyn−z2yn−z−tnyn2

yn−z2−2tnyn, yn−zt2nyn2

yn−z2−2tnyn−z, yn−z −2tnz, yn−zt2nyn2

1−2tnyn−z22tnz, z−ynt2nyn2

≤1−2tnxn−z22tnz, z−ynt2nyn2,

3.46

and so we have

xn1−z2αnxn−z 1−αnzn−z2

≤αnxn−z2 1−αn

1−2tnxn−z22tnz, z−yn t2nyn2

1−2tn1−αnxn−z221−αntnz, z−yn t2nyn2

≤1−2tn1−bxn−z221−αntnz, z−yn t2nyn2.

3.47

Putzx∗in3.47, we have

xn1−x∗2≤

1−γnxn−x∗2δn, 3.48

whereγn 2tn1−bandδn 21−αntnx∗, x∗−yntn2yn2. Sinceyn x∗, it is easy to see that_n∞₁γn∞and limn→ ∞δn/γn 0. ByLemma 2.12, we conclude thatxn → x∗as

n → ∞, wherex∗is the unique solution of the following quadratic minimization problem:

x∗2_min

x∈Ωx 2_.

3.49

This completes the proof ofTheorem 3.1.

In Theorem 3.1, if T Tn ∀n ≥ 1, then the following corollary can be obtained immediately.

Corollary 3.2. LetHbe a real Hilbert space,Cbe a nonempty closed convex subset ofH,A:H →

mappings withFT/∅. LetF : C×C → Ê be a bifunction satisfying conditions (H1)–(H4). Let

{xn}be the sequence defined by

xn1 αnxn 1−αn

TPC1−tnJM,λI−λATμI−μBxn 3.50

where the mappingTμ:H → Cis defined by2.18, andλ, μare two constants withλ∈0,2α, μ∈

0,2β, and

tn∈0,1, tn−→0n−→ ∞, ∞

n1

tn∞, 0< a < αn< b <1. 3.51

If

Ω1:FT∩VIH, A, M∩GEP/∅, 3.52

whereVIH, A, Mand GEP are the sets of solutions of variational inclusion1.2and generalized equilibrium problem1.6, then the sequence{xn}defined by3.50converges strongly tox∗ ∈Ω1,

which is the unique solution of the following quadratic minimization problem:

x∗2_min

x∈Ω1 x2_.

3.53

InTheorem 3.1, ifM ∂δC : H → 2H, whereδC : H → 0,∞is the indicator function of C, then the variational inclusion problem 1.2 is equivalent to variational inequality 1.5, that is, to find u ∈ C such that Au, v−u ≥ 0, for all v ∈ C. Since

M∂δC, JM,λPC. Consequently, we have the following corollary.

Corollary 3.3. LetHbe a real Hilbert space,Cbe a nonempty closed convex subset ofH,A:H →

Hbe anα-inverse strongly monotone mapping andB: C → Hbe aβ-inverse strongly monotone mapping. LetM∂δC :H → 2Hand{T :C → C}be a nonexpansive mappings withFT/∅.

LetF:C×C → Êbe a bifunction satisfying conditions (H1)–(H4). Let{xn}be the sequence defined

by

xn1 αnxn 1−αn

T1−tnPCI−λATμI−μBxn, 3.54

where the mappingTμ:H → Cis defined by2.18, andλ, μare two constants withλ∈0,2α, μ∈

0,2β, and

tn∈0,1, tn−→0n−→ ∞, ∞

n1

tn∞, 0< a < αn< b <1. 3.55

If

whereVIA, C and GEP are the sets of solutions of variational inclusion 1.5 and generalized equilibrium problem1.6, then the sequence{xn}defined by3.54converges strongly tox∗ ∈Ω2,

which is the unique solution of the following quadratic minimization problem:

x∗2_min

x∈Ω2 x2_.

3.57

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