A New Approximation Method for Solving Variational Inequalities and Fixed Points of Nonexpansive Mappings

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Volume 2009, Article ID 520301,16pages doi:10.1155/2009/520301

Research Article

A New Approximation Method for Solving

Variational Inequalities and Fixed Points of

Nonexpansive Mappings

Chakkrid Klin-eam and Suthep Suantai

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

Correspondence should be addressed to Suthep Suantai,[email protected]

Received 3 June 2009; Revised 31 August 2009; Accepted 1 November 2009

Recommended by Vy Khoi Le

A new approximation method for solving variational inequalities and fixed points of nonexpansive mappings is introduced and studied. We prove strong convergence theorem of the new iterative scheme to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for the inverse-strongly monotone mapping which solves some variational inequalities. Moreover, we apply our main result to obtain strong convergence to a common fixed point of nonexpansive mapping and strictly pseudocontractive mapping in a Hilbert space.

Copyrightq2009 C. Klin-eam and S. Suantai. 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.

1. Introduction

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. 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:

θx 1

2Ax, x −

x, y ∀x∈FS, 1.1

whereAis a linear bounded operator,FSis the fixed point set of a nonexpansive mapping

S, andyis a given point inH.

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Recall that a mappingS:C → Cis callednonexpansiveifSx−Sy ≤ x−yfor allx, y∈C. The set of all fixed points ofS is denoted by FS, that is, FS {x ∈ C : x Sx}. A linear bounded operatorAisstrongly positiveif there is a constantγ > 0 with the property

Ax, x ≥ γx2 _{for all}_x_∈ _H_{. A self-mapping}_f _:_C _→ _C_{is a}_contraction_on_C_{if there is a}

constantα∈0,1such thatfx−fy ≤ αx−yfor allx, y∈C. We useΠC to denote the collection of all contractions onC. Note that eachf ∈Π_Chas a unique fixed point inC. A mappingB ofCinto His called monotoneif Bx−By, x−y ≥ 0 for allx, y ∈ C. The variational inequality problem is to findx∈Csuch that

Bx, y−x≥0 ∀y∈C. 1.2

The set of solutions of the variational inequality is denoted byV IC, B. A mappingB ofCtoHis calledinverse-strongly monotoneif there exists a positive real numberβsuch that

x−y, Bx−By≥βBx−By2 _∀_{x, y}_∈_C. _1.3

For such a case, B is β-inverse-strongly monotone. If B is a β-inverse-strongly monotone mapping ofCtoH, then it is obvious thatBis1/β-Lipschitz continuous.

In 2000, Moudafi1introduced the viscosity approximation method for nonexpansive mapping and proved that if H is a real Hilbert space, the sequence {xn} defined by the iterative method below, with the initial guessx0∈Cis chosen arbitrarily:

xn1αnfxn 1−αnSxn, n≥0, 1.4

where{α_n} ⊂ 0,1satisfies certain conditions, converges strongly to a fixed point ofSsay

x∈Cwhich is the unique solution of the following variational inequality:

I−fx, x−x≥0 ∀x∈FS. 1.5

In 2004, Xu2extended the results of Moudafi1to a Banach space. In 2006, Marino and Xu3introduced a general iterative method for nonexpansive mapping. They defined the sequence{x_n}by the following algorithm:

x0∈C, xn1αnγfxn I−αnASxn, n≥0, 1.6

where{αn} ⊂ 0,1andAis a strongly positive linear bounded operator, and they proved that ifCHand the sequence{αn}satisfies appropriate conditions, then the sequence{xn} generated by1.6converges strongly to a fixed point ofSsayx∈Hwhich is the unique solution of the following variational inequality:

A−γfx, x−x≥0 ∀x∈FS, 1.7

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For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of the variational inequalities, Iiduka and Takahashi4introduced following iterative process:

x0∈C, xn1αnu 1−αnSPCxn−λnBxn, n≥0, 1.8

where PC is the projection ofH onto C, u ∈ C,{αn} ⊂ 0,1 and {λn} ⊂ a, bfor some a, b with 0 < a < b < 2β. They proved that under certain appropriate conditions imposed on {α_n} and {λ_n}, the sequence {x_n} generated by 1.8 converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mappingsayx ∈Cwhich solves the variational inequality

x−u, x−x ≥0 ∀x∈FS∩V IC, B. 1.9

In 2007, Chen et al.5introduced the following iterative process:x0∈C,

xn1αnfxn 1−αnSPCxn−λnBxn, n≥0, 1.10

where {α_n} ⊂ 0,1 and {λ_n} ⊂ a, b for somea, b with 0 < a < b < 2β. They proved that under certain appropriate conditions imposed on {αn} and {λn}, the sequence {xn} generated by 1.10converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mappingsayx∈Cwhich solves the variational inequality

I−fx, x−x≥0 ∀x∈FS∩V IC, B. 1.11

In this paper, we modify the iterative methods 1.6 and 1.10 by purposing the following general iterative method:

x0 ∈C, xn1PCαnγfxn I−αnASPCxn−λnBxn, n≥0, 1.12

where PC is the projection of H ontoC,f is a contraction, A is a strongly positive linear bounded operator,B is a β-inverse strongly monotone mapping, {α_n} ⊂ 0,1and {λ_n} ⊂

a, bfor somea, bwith 0< a < b <2β.

We note that whenAIandγ 1, the iterative scheme1.12reduces to the iterative scheme1.10.

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2. Preliminaries

LetHbe real Hilbert space with inner product·,·,Ca nonempty closed convex subset of

H. Recall that the metricnearest pointprojectionP_Cfrom a real Hilbert spaceHto a closed convex subsetCofHis defined as follows: givenx∈H,PCxis the only point inCwith the propertyx−PCx inf{x−y:y ∈C}. In what followsLemma 2.1can be found in any standard functional analysis book.

Lemma 2.1. LetCbe a closed convex subset of a real Hilbert spaceH. Givenx∈Handy∈C, then

iyPCxif and only if the inequalityx−y, y−z ≥0for allz∈C,

iiPCis nonexpansive,

iiix−y, P_Cx−P_Cy ≥ P_Cx−P_Cy2 for allx, y∈H,

ivx−P_Cx, P_Cx−y ≥0for allx∈Handy∈C.

UsingLemma 2.1, one can show that the variational inequality1.2is equivalent to a fixed point problem.

Lemma 2.2. The pointu∈Cis a solution of the variational inequality1.2if and only ifusatisfies the relationuPCu−λBufor allλ >0.

We writexn xto indicate that the sequence{xn}converges weakly toxand write xn → xto indicate that{xn}converges strongly tox. It is well known thatHsatisfies the Opial’s condition6, that is, for any sequence{xn}withxn x, the inequality

lim inf

n→ ∞ xn−x<lim infn→ ∞ xn−y 2.1

holds for everyy∈Hwithx /y.

A set-valued mappingT :H → 2His calledmonotoneif for allx, y∈H,u∈Tx, and

v∈Tyimplyx−y, u−v ≥0. A monotone mappingT :H → 2Hismaximalif the graph

GTofTis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if forx, u∈H×H,x−y, u−v ≥0 for everyy, v∈GTimpliesu∈Tx. LetBbe an inverse-strongly monotone mapping ofCto

Hand letNCvbe normal cone toCatv∈C, that is,NCv{w∈H:v−u, w ≥0, ∀u∈C}, and define

Tv

⎧ ⎨ ⎩

BvNCv, if v∈C,

∅, if v /∈C. 2.2

ThenT is a maximal monotone and 0∈Tvif and only ifv∈V IC, B 7. In the sequel, the following lemmas are needed to prove our main results.

Lemma 2.3see8. Assume {an} is a sequence of nonnegative real numbers such thatan1 ≤

1−γ_na_nδ_n, n≥0, where{γ_n} ⊂0,1and{δ_n}is a sequence inRsuch that

i∞_n₁γ_n∞,

iilim sup_n_{→ ∞}δn/γn≤0or∞n1|δn|<∞.

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Lemma 2.4see9. LetCbe a closed convex subset of a real Hilbert spaceHand letT :C → C be a nonexpansive mapping such thatFT_/∅. If a sequence{x_n}in Cis such thatx_n zand

xn−Txn → 0, thenzTz.

Lemma 2.5see3. AssumeAis a strongly positive linear bounded operator on a Hilbert spaceH with coeﬃcientγ >0and0< ρ≤ A−1, thenI−ρA ≤1−ργ.

3. Main Results

In this section, we prove a strong convergence theorem for nonexpansive mapping and inverse strongly monotone mapping.

Theorem 3.1. LetHbe a real Hilbert space, letCbe a closed convex subset ofH, and letB:C → H be aβ-inverse strongly monotone mapping, also letAbe a strongly positive linear bounded operator ofHinto itself with coeﬃcientγ >0such thatA 1and letf : C → Cbe a contraction with coeﬃcient α0 < α < 1. Assume that0 < γ < γ/α. LetS be a nonexpansive mapping ofCinto itself such thatΩ FS∩V IC, B/∅. Suppose{xn}is the sequence generated by the following algorithm:x0∈C,

xn1PCαnγfxn I−αnASPCxn−λnBxn 3.1

for alln 0,1,2, . . ., where{αn} ⊂ 0,1and{λn} ⊂ 0,2β. If{αn}and{λn}are chosen so that λn∈a, bfor somea, bwith0< a < b <2β,

C1: lim

n→0αn0, C2:

∞

n1

αn ∞,

C3: ∞

n1

|αn1−αn|<∞, C4: ∞

n1

|λn1−λn|<∞,

3.2

then{xn}converges strongly toq ∈ Ω, whereq PΩγf I−Aqwhich solves the following variational inequality:

γf−Aq, p−q≤0 ∀p∈Ω. 3.3

Proof. First, we show the mappingI−λnBis nonexpansive. Indeed, sinceBis aβ-strongly monotone mapping and 0< λn<2β, we have that for allx, y∈C,

_I₋_λ_n_B_x₋_I₋_λ_n_B_y2_x₋_y₋_λ

nBx−By2

x−y2₋

2λnx−y, Bx−Byλ2nBx−By2

≤x−y2_λ

nλn−2βBx−By2

≤x−y2_,

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which implies that the mappingI−λnBis nonexpansive. Next, we show that the sequence

{xn}is bounded. PutynPCxn−λnxnfor alln≥0. Letu∈Ω, we have

_y_n₋_u_P_C_x_n₋_λ_n_Bx_n₋_P_C_u₋_λ_n_Bu

≤ xn−λnBxn−u−λnBu

≤ I−λnBxn−I−λnBu

≤ xn−u.

3.5

Then, we have

xn1−uPCαnγfxn I−αnASyn−PCu

≤αnγfxn−Au I−αnASyn−u

≤αnγfxn−Au1−αnγyn−u

≤αnγfxn−γfuαnγfu−Au1−αnγyn−u

≤αγαnxn−uαnγfu−Au1−αnγxn−u

1−γ−γααnxn−uαnγfu−Au

1−γ−γαα_nx_n−uγ−γαα_nγfu−Au

γ−γα

≤max

xn−u,

_γf_u₋_Au

γ−γα

.

3.6

It follows from induction that

xn−u ≤max

x0−u,

_γf_u₋_Au

γ−γα

, n≥0. 3.7

Therefore, {x_n} is bounded, so are {y_n},{Sy_n},{Bx_n}, and {fx_n}. Since I − λ_nB is nonexpansive andyn PCxn−λnBxn, we also have

_y_n₁₋_y_n_≤_x_n₁₋_λ_n₁_Bx_n₁₋_x_n₋_λ_n_Bx_n

≤ xn1−λn1Bxn1−xn−λn1Bxn|λn−λn1|Bxn

≤ I−λn1Bxn1−I−λn1Bxn|λn−λn1|Bxn

≤ xn1−xn|λn−λn1|Bxn.

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So we obtain

xn1−xnPCαnγfxn I−αnASyn−PCαn−1γfxn−1 I−αn−1ASyn−1

≤I−αnASyn−Syn−1

−αn−αn−1ASyn−1

γαnfxn−fxn−1

γαn−αn−1fxn−1

≤1−α_nγy_n−y_n₋1|αn−αn−1|ASyn−1

γααnxn−xn−1γ|αn−αn−1|fxn−1

≤1−αnγxn−xn−1|λn−1−λn|Bxn−1 |αn−αn−1|ASyn−1

≤1−αnγxn−xn−1|λn−1−λn|Bxn−1|αn−αn−1|ASyn−1

1−γ−γαα_nx_n−x_n₋1L|λn−1−λn|M|αn−αn−1|,

3.9

where L sup{Bxn−1 : n ∈ N}, M max{sup_n_∈_NASyn−1, sup_n_∈_Nγfxn−1}. Since

_∞

n1|αn−αn−1|<∞and

_∞

n1|λn−1−λn|<∞, byLemma 2.3, we havexn1−xn → 0. For u∈ΩanduPCu−λnBu, we have

xn1−u2PC

αnγfxn I−αnASyn−PCu2

≤αnγfxn−Au I−αnASyn−u2

≤αnγfxn−AuI−αnASyn−u2

≤αnγfxn−Au1−αnγyn−u2

≤αnγfxn−Au21−αnγyn−u2

2α_n1−α_nγγfx_n−Auy_n−u

≤αnγfxn−Au21−αnγI−λnBxn−I−λnBu2

2α_n1−α_nγγfx_n−Auy_n−u

≤αnγfxn−Au21−αnγxn−u2λnλn−2βBxn−Bu2

2αn1−αnγγfxn−Auyn−u

≤αnγfxn−Au2xn−u21−αnγab−2βBxn−Bu2

2αn1−αnγγfxn−Auyn−u.

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So, we obtain

−1−α_nγab−2βBx_n−Bu2

≤αnγfxn−Au2 xn−uxn1−uxn−u − xn1−u n

≤αnγfxn−Au2nxn−xn1xn−uxn1−u,

3.11

wheren2αn1−αnγγfxn−Auyn−u. Sinceαn → 0 andxn1−xn → 0, we obtain thatBx_n−Bu → 0 asn → ∞. Further, byLemma 2.1iii, we have

_y_n₋_u2_P

Cxn−λnBxn−PCu−λnBu2

≤xn−λnBxn−u−λnBu, yn−u

1

2

xn−λnBxn−u−λnBu2yn−u2

−xn−λnBxn−u−λnBu−yn−u2

≤ 1

2

xn−u2yn−u2−xn−yn−λnBxn−Bu2

1

2

xn−u2yn−u2−xn−yn2

1

2

2λ_nx_n−y_n, Bx_n−Bu−λ2_nBx_n−Bu2.

3.12

So, we obtain that

_y_n₋_u2

≤ xn−u2−xn−yn22λnxn−yn, Bxn−Bu−λ2nBxn−Bu2. 3.13

So, we have

xn1−u2 PC

αnγfxn I−αnASyn−PCu2

≤αnγfxn−Au I−αnASyn−u2

≤αnγfxn−AuI−αnASyn−u2

≤αnγfxn−Au21−αnγyn−u

2

≤αnγfxn−Au21−αnγyn−u22αn1−αnγγfxn−Auyn−u

≤αnγfxn−Au21−αnγxn−u2−1−αnγxn−yn2

21−αnγλnxn−yn, Bxn−Bu−1−αnγλ2nBxn−Bu2

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≤αnγfxn−Au2xn−u2−1−αnγxn−yn2

21−α_nγλ_nx_n−y_n, Bx_n−Bu −1−α_nγλ2_nBx_n−Bu2

2αn1−αnγγfxn−Auyn−u,

3.14

which implies

1−α_nγx_n−y_n2≤α_nγfx_n−Au2 x_n−ux_n1−uxn−xn1

21−α_nγλ_nx_n−y_n, Bx_n−Bu−1−α_nγλ2_nBx_n−Bu2

2αn1−αnγγfxn−Auyn−u.

3.15

Sinceα_n → 0,x_n1−xn → 0, andBxn−Bu → 0, we obtainxn−yn → 0 asn → ∞. Next, we have

_x_n₁₋_Sy_n_P_C

αnγfxn I−αnASyn−PCSyn

≤αnγfxn I−αnASyn−Syn

αnγfxn ASyn.

3.16

Sinceαn → 0 and{fxn},{ASyn}are bounded, we havexn1−Syn → 0 asn → ∞. Since

_x_n₋_Sy_n_≤_x_n₋_x_n₁_x_n₁₋_Sy_n_, _3.17

it implies thatxn−Syn → 0 asn → ∞. Since

xn−Sxn ≤xn−SynSyn−Sxn

≤xn−Synyn−xn,

3.18

we obtain thatxn−Sxn → 0 asn → ∞. Moreover, from

_y_n₋_Sy_n_≤_y_n₋_x_n_x_n₋_Sy_n_, _3.19

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Observe thatPΩγf I −Ais a contraction. Indeed, byLemma 2.5, we have that

I−A ≤1−γand since 0< γ < γ/α, we have

_P_Ω

γf I−Ax−PΩγf I−Ay≤γf I−Ax−γf I−Ay

≤γfx−fyI−Ax−y ≤γαx−y1−γx−y

1−γ−γαx−y.

3.20

Then Banach’s contraction mapping principle guarantees thatPΩγf I−Ahas a unique fixed point, say q ∈ H. That is,q P_Ωγf I−Aq. ByLemma 2.1i, we obtain that

γf−Aq, p−q ≤0 for allp∈Ω. Choose a subsequence{y_nk}of{y_n}such that

lim sup n→ ∞

γf−Aq, Syn−q lim k→ ∞

γf−Aq, Synk−q. 3.21

As{ynk}is bounded, there exists a subsequence{yn_kj}of{ynk}which converges weakly to p. We may assume without loss of generality thatynk p. Sinceyn−Syn → 0, we obtain Synk p. Sincexn−Sxn → 0,xn−yn → 0 and byLemma 2.4, we havep∈FS. Next, we show thatp∈V IC, B. Let

Tv

⎧ ⎨ ⎩

BvNCv, if v∈C,

∅, if v /∈C,

3.22

whereNCvis normal cone toCatv∈C, that is,NCv{w ∈H :v−u, w ≥0, ∀u∈C}. ThenTis a maximal monotone. Letv, w∈GT. Sincew−Bv∈NCvandyn∈C, we have

v−yn, w−Bv ≥0. On the other hand, byLemma 2.1ivand fromynPCxn−λnBxn, we have

v−yn, yn−xn−λnBxn≥0, 3.23

and hencev−yn,yn−xn/λnBxn ≥0. Therefore, we have

v−ynk, w≥v−ynk, Bv

≥v−ynk, Bv−

v−ynk,ynk_λ−xnk n Bxnk

v−ynk, Bv−Bxnk−ynk_λ−xnk n

v−ynk, Bv−Bynkv−ynk, Bynk−Bxnk−

v−ynk,ynk_λ−xnk n

≥v−ynk, Bynk−Bxnk−

v−ynk,ynk_λ−xnk n

.

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This impliesv−p, w ≥ 0 ask → ∞. SinceT is maximal monotone, we havep ∈ T−1₀

and hence p ∈ V IC, B. We obtain that p ∈ Ω. It follows from the variational inequality

γf−Aq, p−q ≤0 for allp∈Ωthat

lim sup n→ ∞

γf−Aq, Syn−q lim k→ ∞

γf−Aq, Synk−qγf−Aq, p−q≤0.

3.25

Finally, we provexn → q. By using3.5and together with Schwarz inequality, we have

_x_n₁₋_q2_P

Cαnγfxn I−αnASyn−PCq2

≤αnγfxn−Aq I−αnASyn−q2

≤I−αnASyn−q2α2nγfxn−Aq2

2αnI−αnASyn−q, γfxn−Aq

≤1−α_nγ2y_n−q2α2_nγfx_n−Aq2

2αnSyn−q, γfxn−Aq−2α2nASyn−q, γfxn−Aq

≤1−αnγ2xn−q2α2nγfxn−Aq2

2αnSyn−q, γfxn−γfq2αnSyn−q, γfq−Aq

−2α2_nASyn−q, γfxn−Aq

2αnSyn−qγfxn−γfq2αnSyn−q, γfq−Aq

2γααnyn−qxn−q2αnSyn−q, γfq−Aq

≤1−α_nγ2x_n−q2α2_nγfx_n−Aq2

2γααnxn−q22αnSyn−q, γfq−Aq

≤1−αnγ22γααnxn−q2

αn

2Syn−q, γfxn−Aqαnγfxn−Aq2

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1−2γ−γαα_nx_n−q2

αn

2Syn−q, γfq−Aqαnγfxn−Aq2

2αnASyn−qγfxn−Aqαnγ2xn−q2

.

3.26

Since{xn},{fxn}and{Syn}are bounded, we can take a constantη >0 such that

η≥γfxn−Aq22αnASyn−qγfxn−Aqαnγ2xn−q2 3.27

for alln≥0. It then follows that

_x_n₁₋_q2_≤

1−2γ−γααnxn−q2αnβn, 3.28

whereβ_n 2Sy_n−q, γfq−Aqηα_n. By lim sup_n_{→ ∞}γf −Aq, Sy_n−q ≤ 0, we get lim sup_n_{→ ∞}βn ≤ 0. By applying Lemma 2.3to3.28, we can conclude thatxn → q. This completes the proof

TakingAIandγ1 inTheorem 3.1, we get the results of Chen et al.5

Corollary 3.2see5, Proposition 3.1. LetH be a real Hilbert space, letCbe a closed convex subset ofH, and letB : C → Hbe a β-inverse strongly monotone mapping. Letf : C → Cbe a contraction with coeﬃcient α0 < α < 1and let Sbe a nonexpansive mapping ofCinto itself such thatΩ FS∩V IC, B_/∅. Suppose{x_n}is a sequence generated by the following algorithm:

x0∈C,

xn1αnfxn 1−αnSPCxn−λnBxn 3.29

C1: lim

n→0αn0, C2:

∞

n1

αn ∞,

C3: ∞

n1

|αn1−αn|<∞, C4: ∞

n1

|λn1−λn|<∞,

3.30

then {x_n} converges strongly to q ∈ Ω, which is the unique solution in the Ω to the following variational inequality:

f−Iq, p−q≤0 ∀p∈Ω. 3.31

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Corollary 3.3see5, Theorem 3.1. LetHbe a real Hilbert space, letCbe a closed convex subset ofH, and letB:C → Hbe aβ-inverse strongly monotone mapping. Letf:C → Cbe a contraction with coeﬃcient α0 < α < 1and letSbe a nonexpansive mapping ofCinto itself such thatΩ

FS∩V IC, B/∅. Suppose{xn}is a sequence generated by the following algorithm:x0, u∈C,

xn1αnu 1−αnSPCxn−λnBxn 3.32

C1: lim

n→0αn0, C2:

∞

n1

αn ∞,

C3: ∞

n1

|αn1−αn|<∞, C4: ∞

n1

|λn1−λn|<∞,

3.33

then {x_n} converges strongly to q ∈ Ω, which is the unique solution in the Ω to the following variational inequality:

u−q, p−q≤0 ∀p∈Ω. 3.34

4. Applications

In this section, we apply the iterative scheme 1.12 for finding a common fixed point of nonexpansive mapping and strictly pseudocontractive mapping and also applyTheorem 3.1

for finding a common fixed point of nonexpansive mapping and inverse strongly monotone mapping. Recall that a mappingT :C → Cis calledstrictly pseudocontractiveif there existsk with 0≤k <1 such that

_Tx₋_Ty2_≤_x₋_y2_k_I₋_T_x₋_I₋_T_y2 _∀_{x, y}_∈_C. _4.1

Ifk0, thenTis nonexpansive. PutBI−T, whereT :C → Cis a strictly pseudocontractive mapping withk. ThenBis1−k/2-inverse-strongly monotone. Actually, we have, for all

x, y∈C,

_I₋_B_x₋_I₋_B_y2_≤_x₋_y2_k_Bx₋_By2_. _4.2

On the other hand, sinceHis a real Hilbert space, we have

_I₋_B_x₋_I₋_B_y2_x₋_y2_Bx₋_By2₋

2x−y, Bx−By. 4.3

Hence, we have

x−y, Bx−By≥ 1−k

2 Bx−By

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Using Theorem 3.1, we firse prove a strongly convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping.

Theorem 4.1. LetHbe a real Hilbert space, letCbe a closed convex subset ofH, and letAbe a strongly positive linear bounded operator ofHinto itself with coeﬃcientγ > 0such thatA 1, so letf : C → Cbe a contraction with coeﬃcientα0 < α <1. Assume that0 < γ < γ/α. Let

Sbe a nonexpansive mapping ofCinto itself and letT be a strictly pseudocontractive mapping ofC into itself withβsuch thatFS∩FT/∅. Suppose{xn}is a sequence generated by the following algorithm:

x0∈C, xn1PC

αnγfxn I−αnAS1−λnxn−λnTxn 4.5

for alln0,1,2, . . ., where{αn} ⊂0,1and{λn} ⊂0,1−β. If{αn}and{λn}are chosen so that λn∈a, bfor somea, bwith0< a < b <1−β,

C1: lim

n→0αn0, C2:

∞

n1

αn ∞,

C3: ∞

n1

|αn1−αn|<∞, C4:

∞

n1

|λn1−λn|<∞,

4.6

then{x_n}converges strongly toq∈FS∩FT, such that

γf−Aq, p−q≤0 ∀p∈FS∩FT. 4.7

Proof. PutB I−T, thenBis1−k/2-inverse-strongly monotone andFT V IC, B andPCxn−λnBxn 1−λnxnλnTxn. So byTheorem 3.1, we obtain the desired result.

Corollary 4.2see5, Theorem 4.1. LetH be a real Hilbert space and letCbe a closed convex subset ofH. Letf :C → Cbe a contraction with coeﬃcientα0 < α <1, letSbe a nonexpansive mapping ofCinto itself, and letT be a strictly pseudocontractive mapping ofCinto itself withβsuch thatFS∩FT_/∅. Suppose{x_n}is a sequence generated by the following algorithm:

x0 ∈C, xn1αnfxn 1−αnS1−λnxn−λnTxn 4.8

for alln0,1,2, . . ., where{αn} ⊂0,1and{λn} ⊂0,1−β. If{αn}and{λn}are chosen so that λn∈a, bfor somea, bwith0< a < b <1−β,

C1: lim

n→0αn0, C2:

∞

n1

αn ∞,

C3: ∞

n1

|αn1−αn|<∞, C4: ∞

n1

|λn1−λn|<∞,

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then{xn}converges strongly toq∈FS∩FT, such that

f−Iq, p−q≤0 ∀p∈FS∩FT. 4.10

Theorem 4.3. LetHbe a real Hilbert space,Aa strongly positive linear bounded operator ofHinto itself with coeﬃcientγ >0such thatA 1and letf :H → Hbe a contraction with coeﬃcient

α0< α <1. Assume that0 < γ < γ/α. LetSbe a nonexpansive mapping ofHinto itself andB aβ-inverse strongly monotone mapping of H into itself such thatFS∩B−1₀_/_∅_{. Suppose}_{_x

n}is a sequence generated by the following algorithm:

x0∈H, xn1αnγfxn I−αnASxn−λnBxn 4.11

for alln 0,1,2, . . ., where{α_n} ⊂ 0,1and{λ_n} ⊂ 0,2β. If{α_n}and{λ_n}are chosen so that

λn∈a, bfor somea, bwith0< a < b <2β,

C1: lim

n→0αn0, C2:

∞

n1

αn ∞,

C3: ∞

n1

|αn1−αn|<∞, C4: ∞

n1

|λn1−λn|<∞,

4.12

then{xn}converges strongly toq∈FS∩B−10, such that

γf−Aq, p−q ≤0 ∀p∈FS∩B−10. 4.13

Proof. We haveB−1₀ _{V I}_{H, B}_{. So putting}_P

H I, byTheorem 3.1, we obtain the desired result.

Corollary 4.4see2, Theorem 4.2. LetHbe a real Hilbert space. Letfbe a contractive mapping ofHinto itself with coeﬃcientα0< α <1andSa nonexpansive mapping ofHinto itself andB aβ-inverse strongly monotone mapping of H into itself such thatFS∩B−1₀_/_∅_{. Suppose}_{_x

n}is a sequence generated by the following algorithm:

x0∈H, xn1αnfxn 1−αnSxn−λnBxn 4.14

C1: lim

n→0αn0, C2:

∞

n1

αn ∞,

C3: ∞

n1

|αn1−αn|<∞, C4: ∞

n1

|λn1−λn|<∞,

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then{xn}converges strongly toq∈FS∩B−10, such that

f−Iq, p−q≤0 ∀p∈FS∩B−10. 4.16

Remark 4.5. By takingA I,γ 1, andf ≡ u∈ Cin Theorems4.1and4.3, we can obtain Theorems 4.1 and 4.2 in4, respectively.

Acknowledgments

The authors would like to thank the referee for valuable suggestions to improve this manuscript and the Thailand Research Fund RGJ Project and Commission on Higher Education for their financial support during the preparation of this paper. C. Klin-eam was supported by the Royal Golden Jubilee Grant PHD/0018/2550 and the Graduate School, Chiang Mai University, Thailand.

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