A General Iterative Approach to Variational

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Volume 2011, Article ID 284363,20pages doi:10.1155/2011/284363

Research Article

A General Iterative Approach to Variational

Inequality Problems and Optimization Problems

Jong Soo Jung

Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

Correspondence should be addressed to Jong Soo Jung,jungjs@mail.donga.ac.kr Received 4 October 2010; Accepted 14 November 2010

Academic Editor: Jen Chih Yao

Copyrightq2011 Jong Soo Jung. 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.

We introduce a new general iterative scheme for finding a common element of the set of solutions of variational inequality problem for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a Hilbert space and then establish strong convergence of the sequence generated by the proposed iterative scheme to a common element of the above two sets under suitable control conditions, which is a solution of a certain optimization problem.

Applications of the main result are also given.

1. Introduction

LetH be a real Hilbert space with inner product ·,·and induced norm · . LetCbe a nonempty closed convex subset ofHandS :C → Cbe self-mapping onC. We denote by FSthe set of fixed points ofSand byP_C the metric projection ofHontoC.

LetAbe a nonlinear mapping ofCintoH. The variational inequality problem is to find au∈Csuch that

v−u, Au ≥0, ∀v∈C. 1.1

We denote the set of solutions of the variational inequality problem1.1by VIC, A. The variational inequality problem has been extensively studied in the literature; see1–5and the references therein.

Recently, in order to study the problem1.1coupled with the fixed point problem, many authors have introduced some iterative schemes for finding a common element of the set of the solutions of the problem1.1and the set of fixed points of nonexpansive mappings;

see 6–9 and the references therein. In particular, in 2005, Iiduka and Takahashi 8

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introduced an iterative scheme for finding a common point of the set of fixed points of a nonexapansive mappingSand the set of solutions of the problem1.1for an inverse-strong monotone mappingA:x₁∈Cand

x_n1α_nx 1−α_nSPCxn−λ_nAx_n, n≥1, 1.2 where{αn} ⊂ 0,1 and{λn} ⊂ 0,2α. They proved that the sequence generated by1.2 strongly converges strongly toP_FS∩VIC,Ax. In 2010, Jung10provided the following new composite iterative scheme for the fixed point problem and the problem1.1:x1x∈Cand

ynαnfxn 1−αnSPCxn−λnAxn, x_n1

1−β_n

y_nβ_nSP_C

y_n−λ_nAy_n

, n≥1, 1.3

wheref is a contraction with constantk ∈ 0,1,{αn},{βn} ∈ 0,1, and{λn} ⊂ 0,2α. He proved that the sequence{xn}generated by1.3strongly converges strongly to a point in FS∩VIC, A, which is the unique solution of a certain variational inequality.

On the other hand, the following optimization problem has been studied extensively by many authors:

minx∈Ω

μ

2Bx, x1

2x−u²−hx, 1.4

where Ω _∞

n1C_n, C₁, C₂, . . . are infinitely many closed convex subsets of H such that _∞

n1C_n/∅,u∈H,μ≥0 is a real number,Bis a strongly positive bounded linear operator on Hi.e., there is a constantγ >0 such thatBx, x ≥γx², for allx∈H, andhis a potential function forγf i.e.,hx γfxfor all x ∈ H. For this kind of optimization problems, see, for example, Deutsch and Yamada11, Jung10, and Xu12,13whenΩ _N

i1C_iand hx x, bfor a given pointbinH.

In 2007, related to a certain optimization problem, Marino and Xu14introduced the following general iterative scheme for the fixed point problem of a nonexpansive mapping:

x_n1α_nγfxn I−α_nBSxn, n≥0, 1.5 where {αn} ∈ 0,1 and γ > 0. They proved that the sequence {xn} generated by 1.5 converges strongly to the unique solution of the variational inequality

B−γf

x^∗, x−x^∗

≥0, x∈FS, 1.6

which is the optimality condition for the minimization problem

x∈FSmin 1

2Bx, x −hx, 1.7

where h is a potential function for γf. The result improved the corresponding results of Moudafi15and Xu16.

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In this paper, motivated by the above-mentioned results, we introduce a new general composite iterative scheme for finding a common point of the set of solutions of the variational inequality problem1.1for an inverse-strongly monotone mapping and the set of fixed points of a nonexapansive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a common point of the above two sets, which is a solution of a certain optimization problem. Applications of the main result are also discussed. Our results improve and complement the corresponding results of Chen et al.6, Iiduka and Takahashi8, Jung10, and others.

2. Preliminaries and Lemmas

LetHbe a real Hilbert space and letCbe a nonempty closed convex subset ofH. We write x_n xto indicate that the sequence{xn}converges weakly tox.x_n → ximplies that{xn} converges strongly tox.

First we recall that a mappingf :C → Cis a contraction onCif there exists a constant k ∈ 0,1such thatfx−fy ≤ kx−y,x, y ∈ C. A mapping T : C → Cis called nonexpansive ifTx−Ty ≤ x−y, x, y∈C. We denote byFTthe set of fixed points ofT. For every pointx∈H, there exists a unique nearest point inC, denoted byP_Cx, such that

x−PCx ≤x−y 2.1

for all y ∈ C. P_C is called the metric projection of H onto C. It is well known that P_C is nonexpansive andPC satisfies

x−y, PCx−PC

y

≥PCx−PC

y² 2.2

for everyx, y∈H. Moreover,PCxis characterized by the properties:

x−y²≥ x−PCx²y−PCx², uP_Cx⇐⇒

x−u, u−y

≥0, ∀x∈H, y∈C.

2.3

In the context of the variational inequality problem for a nonlinear mappingA, this implies that

u∈VIC, A⇐⇒uP_Cu−λAu, for anyλ >0. 2.4

It is also well known thatHsatisfies the Opial condition, that is, for any sequence{xn}with xn x, the inequality

lim inf

n→ ∞ xn−x<lim inf

n→ ∞ xn−y 2.5

holds for everyy∈Hwithy /x.

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A mappingAofCintoH is called inverse-strongly monotone if there exists a positive real numberαsuch that

x−y, Ax−Ay

≥αAx−Ay² 2.6 for allx, y ∈C; see4,7,17. For such a case,Ais calledα-inverse-strongly monotone. We know that ifAI−T, whereTis a nonexpansive mapping ofCinto itself andIis the identity mapping ofH, thenAis 1/2-inverse-strongly monotone and VIC, A FT. A mappingA ofCintoHis called strongly monotone if there exists a positive real numberηsuch that

x−y, Ax−Ay

≥ηx−y² 2.7

for allx, y∈C. In such a case, we sayAisη-strongly monotone. IfAisη-strongly monotone andκ-Lipschitz continuous, that is,Ax−Ay ≤ κx−yfor allx, y ∈ C, thenAisη/κ²- inverse-strongly monotone. If Ais an α-inverse-strongly monotone mapping ofC intoH, then it is obvious thatAis 1/α-Lipschitz continuous. We also have that for allx, y ∈Cand λ >0,

I−λAx−I−λAy²x−y

−λ

Ax−Ay²

x−y²−2λx−y, Ax−Ayλ²Ax−Ay²

≤x−y²λλ−2αAx−Ay².

2.8

So, ifλ≤2α, thenI−λAis a nonexpansive mapping ofCintoH. The following result for the existence of solutions of the variational inequality problem for inverse strongly-monotone mappings was given in Takahashi and Toyoda9.

Proposition 2.1. LetCbe a bounded closed convex subset of a real Hilbert space and let A be an α-inverse-strongly monotone mapping ofCintoH. Then, VIC, Ais nonempty.

A set-valued mappingT :H → 2^His called monotone if for allx, y∈H,f ∈Tx, and g∈Tyimplyx−y, f−g ≥0. A monotone mappingT :H → 2^His maximal if the graphGT ofTis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if forx, f∈H×H,x−y, f−g ≥0 for every y, g∈GTimpliesf ∈Tx. LetAbe an inverse-strongly monotone mapping ofCintoH and letNCvbe the normal cone toCatv, that is,NCv{w∈H:v−u, w ≥0, for allu∈C}, and define

Tv

⎧⎨

⎩

AvN_Cv, v∈C,

∅, v /∈C. 2.9

ThenT is maximal monotone and 0∈Tvif and only ifv∈VIC, A; see18,19.

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We need the following lemmas for the proof of our main results.

Lemma 2.2. In a real Hilbert spaceH, there holds the following inequality:

xy²≤ x²2

y, xy

, 2.10

for allx, y∈H.

Lemma 2.3Xu12. Let{sn}be a sequence of nonnegative real numbers satisfying

s_n1≤1−λ_nsnβ_nγ_n, n≥1, 2.11

where{λn}and{βn}satisfy the following conditions:

i{λn} ⊂0,1and_∞

n1λ_n∞or, equivalently,_∞

n11−λ_n 0;

iilim sup_n_{→ ∞}βn/λ_n≤0 or_∞

n1|βn|<∞;

iiiγ_n≥0 n≥1, _∞

n1γ_n<∞.

Then lim_n_{→ ∞}s_n0.

Lemma 2.4Marino and Xu14. Assume that A is a strongly positive linear bounded operator on a Hilbert spaceHwith constantγ >0 and 0< ρ≤ B⁻¹. ThenI−ρB ≤1−ργ.

The following lemma can be found in20,21 see alsoLemma 2.2in22.

Lemma 2.5. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, and letg : C → R∪ {∞}be a proper lower semicontinunous diﬀerentiable convex function. Ifx^∗is a solution to the minimization problem

gx^∗ inf

x∈Cgx, 2.12

then

gx, x−x^∗

≥0, x∈C. 2.13

In particular, ifx^∗solves the optimization problem

minx∈C

μ

2Bx, x1

2x−u²−hx, 2.14

then

u γf−

IμB

x^∗, x−x^∗

≤0, x∈C, 2.15

wherehis a potential function forγf.

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3. Main Results

In this section, we present a new general composite iterative scheme for inverse-strongly monotone mappings and a nonexpansive mapping.

Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceHsuch thatC ± C⊂ C. Let A be anα-inverse-strongly monotone mapping ofCintoHandSa nonexpansive mapping of Cinto itself such thatFS∩VIC, A/∅. Letu∈Cand letBbe a strongly positive bounded linear operator onCwith constantγ ∈0,1andfa contraction ofCinto itself with constantk ∈0,1.

Assume thatμ >0 and 0< γ <1μγ/k. Let{xn}be a sequence generated by

x₁x∈C, ynαn

uγfxn

I−αn

IμB

SPCxn−λnAxn, x_n1

1−β_n

y_nβ_nSP_C

y_n−λ_nAy_n

, n≥1,

IS

where{λn} ⊂0,2α,{αn} ⊂0,1, and{βn} ⊂0,1. Let{αn},{λn}, and{βn}satisfy the following conditions:

iα_n → 0 n → ∞;_∞

n1α_n∞;

iiβ_n⊂0, afor alln≥0 and for somea∈0,1;

iiiλ_n∈c, dfor somec,dwith 0< c < d <2α;

iv_∞

n1|αn1−α_n|<∞,_∞

n1|βn1−β_n|<∞,_∞

n1|λn1−λ_n|<∞.

Then{xn}converges strongly toq∈FS∩VIC, A, which is a solution of the optimization problem

x∈FS∩VIC,Amin μ

2Bx, x1

2x−u²−hx, OP1

Proof. We note that from the control conditioni, we may assume, without loss of generality, thatαn≤1μB⁻¹. Recall that ifBis bounded linear self-adjoint operator onH, then

Bsup{|Bu, u|:u∈H, u1}. 3.1

Observe that

I−αn

IμB u, u

1−αn−αnμBu, u

≥1−α_n−α_nμB

≥0,

3.2

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which is to say thatI−αnIμBis positive. It follows that I−α_n

IμBsup I−α_n

IμB u, u

:u∈H, u1 sup

1−α_n−α_nμBu, u:u∈H, u1

≤1−αn

1μγ

<1−α_n 1μ

γ.

3.3

Now we divide the proof into several steps.

Step 1. We show that{xn}is bounded. To this end, letz_n P_Cxn−λ_nAx_nandw_nP_Cyn− λ_nAy_nfor every n ≥ 1. Letp ∈ FS∩VIC, A. SinceI −λ_nAis nonexpansive andp PCp−λnApfrom2.4, we have

z_n−p≤xn−λ_nAx_n−

p−λ_nAp

≤x_n−p. 3.4

Similarly, we have

wn−p≤yn−p. 3.5

Now, setB IμB. Letp∈FS∩VIC, A. Then, fromISand3.4, we obtain

y_n−pαnuα_n

γfxn−Bp

I−α_nB

Sz_n−p

≤ 1−

1μ

γαnzn−pαnu α_nγfxn−f

pα_nγf p

−Bp

≤ 1−

1μ

γαnzn−pαnu α_nγkx_n−pα_nγf

p

−Bp

1− 1μ

γ−γk

α_nx_n1−p

1μ γ−γk

α_n γf

p

−Bpu 1μ

γ−γk .

3.6

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From3.5and3.6, it follows that x_n1−p1−βn

yn−p βn

Swn−p

≤

1−β_ny_n−pβ_nw_n−p

≤

1−βnyn−pβnyn−p y_n−p

≤max

⎧⎨

⎩xn−p, γf

p

−Bpu 1μ

γ−γk

⎫⎬

⎭.

3.7

By induction, it follows from3.7that xn−p≤max

⎧⎨

⎩x1−p, γf

p

−Bpu 1μ

γ−γk

⎫⎬

⎭ n≥1. 3.8

Therefore, {xn} is bounded. So {yn}, {zn}, {wn},{fxn}, {Axn}, {Ayn}, and {BSzn} are bounded. Moreover, sinceSzn−p ≤ xn−pandSwn−p ≤ yn−p,{Szn}and{Swn} are also bounded. And by the conditioni, we have

y_n−Sz_nα_nuγfxn

− IμB

Sz_n αn

uγfxn

−BSzn−→0 asn−→ ∞. 3.9 Step 2. We show that limn→ ∞xn1−xn0 and limn→ ∞yn1−yn0. Indeed, sinceI−λnA andP_Care nonexpansive andz_nP_Cxn−λ_nAx_n, we have

zn−z_n−1 ≤ xn−λ_nAx_n−x_n−1−λ_n−1Ax_n−1

≤ xn−x_n−1|λn−λ_n−1|Axn−1. 3.10

Similarly, we get

wn−w_n−1 ≤yn−y_n−1|λn−λ_n−1|Ay_n−1. 3.11 Simple calculations show that

y_n−y_n−1α_n

uγfxn

I−α_nB

Sz_n−α_n−1

uγfx_n−1

−

I−α_n−1B Sz_n−1 α_n−α_n−1

uγfx_n−1−BSz_n−1 α_nγ

fxn−fx_n−1

I−α_nB

Szn−Sz_n−1.

3.12

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So, we obtain

y_n−y_n−1≤ |α_n−α_n−1|

uγfx_n−1BSz_n−1 α_nγkxn−x_n−1

1− 1μ

γα_n

zn−z_n−1

≤ |αn−α_n−1|

uγfxn−1BSzn−1 α_nγkxn−x_n−1

1− 1μ

γα_n

xn−x_n−1 |λn−λ_n−1|Axn−1.

3.13

Also observe that

x_n1−xn 1−βn

yn−y_n−1

βn−β_n−1

Sw_n−1−y_n−1

βnSwn−Sw_n−1. 3.14

By3.11,3.13, and3.14, we have x_n1−x_n ≤

1−β_ny_n−y_n−1β_n−β_n−1Sx_n−1y_n−1 β_nwn−w_n−1

≤

1−βnyn−y_n−1βnyn−y_n−1βn|λn−λ_n−1|Ay_n−1 β_n−β_n−1Sw_n−1y_n−1

≤y_n−y_n−1|λn−λ_n−1|Ay_n−1β_n−β_n−1Swn−1y_n−1

≤ 1−

1μ γ−γk

αn

xn−x_n−1

|αn−α_n−1|

uγfx_n−1BSz_n−1 |λn−λ_n−1|Ay_n−1Axn−1

βn−β_n−1Swn−1y_n−1

≤ 1−

1μ γ−γk

αn

xn−x_n−1

M₁|αn−α_n−1|M₂|λn−λ_n−1|M₃β_n−β_n−1,

3.15

whereM1sup{uγfxnBTnzn:n≥1},M2sup{AynAxn:n≥1}, and M₃sup{Swnyn:n≥1}. From the conditionsiandiv, it is easy to see that

nlim→ ∞

1μ γ−γk

α_n 0,

∞ n1

1μ γ−γk

α_n∞, ∞

n2

M₁|αn−α_n−1|M₂|λn−λ_n−1|M₃β_n−β_n−1<∞.

3.16

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ApplyingLemma 2.3to3.15, we obtain

nlim→ ∞x_n1−x_n0. 3.17

Moreover, by3.10and3.13, we also have

n→ ∞limzn1−z_n0, lim

n→ ∞y_n1−y_n0. 3.18

Step 3. We show that lim_n_{→ ∞}xn−yn0 and limn→ ∞xn−Szn0. Indeed,

x_n1−y_nβ_nSw_n−y_n

≤βn

Swn−SznSzn−yn

≤a

wn−z_nSz_n−y_n

≤ay_n−x_nSz_n−y_n

≤ayn−x_n1xn1−xnSzn−yn

3.19

which implies that

x_n1−yn≤ a 1−a

x_n1−x_nSz_n−y_n. 3.20

Obviously, by3.9andStep 2, we havexn1−yn → 0 asn → ∞. This implies that

x_n−y_n≤ x_n−x_n1x_n1−y_n−→0 asn−→ ∞. 3.21 By3.9and3.21, we also have

xn−Sz_n ≤x_n−y_ny_n−Sz_n−→0 asn−→ ∞. 3.22

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Step 4. We show that limn→ ∞xn−zn0 and limn→ ∞yn−zn0. To this end, letpFS∩ VIC, A. Sincez_nP_Cxn−λ_nAx_nandpP_Cp−λ_np, we have

y_n−p² α_n

uγfxn−Bp

I−α_nB

Sz_n−p²

≤

α_nuγfxn−BpI−α_nBSz_n−p²

≤αnuγfxn−Bp² 1−αn

1μ

γzn−p² 2αn

1−αn

1μ

γuγfxn−Bpzn−p

≤αnuγfxn−Bp²

1−αn

1μ

γxn−p²λnλn−2αAxn−Ap² 2αn

1−αn

1μ

γγufxn−Bpzn−p

≤αnuγfxn−Bp²xn−p²

1−α_n 1μ

γ

cd−2αAx_n−Ap² 2α_nuγfxn−Bpz_n−p.

3.23

So we obtain

− 1−αn

1μ γ

cd−2αAxn−Ap²

≤αnγufxn−Bp²xn−pyn−pxn−p−yn−p 2α_nγufxn−Bpz_n−p

≤αnγufxn−Bp²xn−py_n−pxn−yn 2α_nγufxn−Bpz_n−p.

3.24

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Sinceαn → 0 from the conditioniandxn−yn → 0 fromStep 3, we haveAxn−Ap → 0n → ∞. Moreover, from2.4we obtain

z_n−p²P_Cxn−λ_nAx_n−P_C

p−λ_nAp²

≤

x_n−λ_nAx_n−

p−λ_nAp

, z_n−p

1 2

xn−λnAxn−

p−λnAp²zn−p²

−xn−λnAxn−

p−λnAp

−

zn−p²

≤ 1 2

xn−p²zn−p²− xn−zn² 2λn

x_n−z_n, Ax_n−Ap

−λ²_nAx_n−Ap² ,

3.25

and so

zn−p²≤xn−p²− xn−zn²2λn

xn−zn, Axn−Ap

−λ²_nAxn−Ap². 3.26

Thus

y_n−p²≤α_nuγfxn−Bp² 1−α_n

1μ

γz_n−p² 2α_n

1−α_n 1μ

γγufxn−Bpz_n−p

≤α_nuγfxn−Bp²x_n−p²− 1−α_n

1μ γ

xn−z_n²

2 1−αn

1μ γ

λn

xn−zn, Axn−Ap

− 1−αn

1μ γ

λ²_nAxn−Ap² 2α_nuγfxn−Bz_n−p.

3.27

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Then, we have

1−αn

1μ γ

xn−zn²

≤α_nuγfxn−Bp²x_n−py_n−px_n−p−y_n−p 2

1−α_n 1μ

γ λ_n

x_n−z_n, Ax_n−Ap

− 1−α_n

1μ γ

λ²_nAx_n−Ap² 2αnuγfxn−Bpzn−p

≤αnuγfxn−Bp²xn−pyn−pxn−yn 2

1−αn

1μ γ

λn

xn−zn, Axn−Ap

− 1−αn

1μ γ

λ²_nAxn−Ap² 2α_nuγfxn−Bpz_n−p.

3.28

Sinceα_n → 0,xn−y_n → 0 andAxn−Au → 0, we getxn−z_n → 0. Also by3.21 y_n−z_n≤y_n−x_nxn−z_n −→0 n−→ ∞. 3.29

Step 5. We show that lim_n_{→ ∞}Szn−z_n0. In fact, since

Szn−z_n ≤Sz_n−y_ny_n−z_n

αnuγfxn−BSznyn−zn,

3.30

from3.9and3.29, we have limn→ ∞Szn−zn0.

Step 6. We show that

lim sup

n→ ∞

u γf−

IμB

q, y_n−q

lim sup

n→ ∞

u

γf−B

q, y_n−q

≤0, 3.31

whereqis a solution of the optimization problemOP1. First we prove that

lim sup

n→ ∞

u

γf−B

q, Szn−q

≤0. 3.32

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Since{zn}is bounded, we can choose a subsequence{zni}of{zn}such that

lim sup

n→ ∞

u

γf−B

q, Sz_n−q lim

i→ ∞

u

γf−B

q, Sz_n_i−q

. 3.33

Without loss of generality, we may assume that{zni}converges weakly toz∈C.

Now we will show thatz∈FS∩VIC, A. First we show thatz∈FS. Assume that z /∈FS. Sincezni zandSz /z, by the Opial condition andStep 5, we obtain

lim inf

i→ ∞ zni −z<lim inf

i→ ∞ zni−Sz

≤lim inf

i→ ∞ zni−SzniSzni−Sz lim inf

i→ ∞ Szni−Sz

≤lim inf

i→ ∞ zni−z,

3.34

which is a contradiction. Thus we havez∈FS.

Next, let us show thatz∈VIC, A. Let

Tv

⎧⎨

⎩

AvNCv, v∈C,

∅, v /∈C. 3.35

ThenT is maximal monotone. Letv, w∈GT. Sincew−Av∈N_Cvandz_n∈C, we have

v−z_n, w−Av ≥0. 3.36

On the other hand, fromzn PCxn−λnAxn, we havev−zn, zn−xn−λnAzn ≥ 0 and hence

v−zn,zn−xn

λ_n Axn ≥0. 3.37

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Therefore, we have

v−z_n_i, w ≥ v−z_n_i, Av

≥ v−zni, Av −

v−zni,z_n_i−x_n_i λ_n_i Axni

v−z_n_i, Av−Ax_n_i−zni −xni

λni

v−zni, Av−Azniv−zni, Azni −Axni −

v−zni,z_n_i−x_n_i λni

≥ v−z_n_i, Az_n_i−Ax_n_i −

v−z_n_i,z_n_i−x_n_i λ_n_i

.

3.38

Sincezn−xn → 0 inStep 4andAisα-inverse-strongly monotone, we havev−z, w ≥0 asi → ∞. SinceT is maximal monotone, we havez∈T⁻¹0 and hencez∈VIC, A.

Therefore,z∈FS∩VIC, A. Now fromLemma 2.5andStep 5, we obtain

lim sup

n→ ∞

u

γf−B

q, Szn−q lim

i→ ∞

u

γf−B

q, Szni−q

lim

i→ ∞

u

γf−B

q, z_n_i−q

u

γf−B

q, z−q

≤0.

3.39

By3.9and3.39, we conclude that

lim sup

n→ ∞

u

γf−B

q, y_n−q

≤lim sup

n→ ∞

u

γf−B

q, yn−Szn

lim sup

n→ ∞

u

γf−B

q, Szn−q

≤lim sup

n→ ∞

u

γf−B

qyn−Sznlim sup

n→ ∞

u

γf−B

q, Szn−q

≤0.

3.40

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Step 7. We show that limn→ ∞xn−q0 and limn→ ∞un−q 0, whereqis a solution of the optimization problemOP1. Indeed fromISandLemma 2.2, we have

x_n1−q²≤yn−q² α_n

uγfxn−Bq

I−α_nB

Sz_n−q

≤

I−α_nB

Sz_n−q²2α_n

uγfxn−Bq, y_n−q

≤ 1−

1μ

γα_n2z_n−q²2α_nγ

fxn−f q

, y_n−q 2α_n

uγf q

−Bq, y_n−q

≤ 1−

1μ γαn

₂xn−q²2αnγkxn−qyn−q 2αn

u

γf−B

q, yn−q

≤ 1−

1μ

γα_n₂x_n−q²2α_nγkx_n−qy_n−x_nx_n−q 2α_n

u

γf−B

q, y_n−q

1−2 1μ

γ−γk

α_nx_n−q² α²_n

1μ

γ₂x_n−q²2α_nγkx_n−qy_n−x_n 2α_n

u

γf−B

q, y_n−q ,

3.41

that is,

x_n1−q² ≤ 1−2

1μ γ−γk

α_nx_n−q² α²_n

1μ γ₂

M₄²2αnγkyn−xnM4

2α_n u

γf−B

q, y_n−q 1−α_nx_n−q²β_n,

3.42

whereM₄sup{xn−q:n≥1},α_n21μγ−γkαn, and

βnαn

αn

1μγ₂

M²₄2γkyn−xnM42 u

γf−B

q, yn−q

. 3.43

From i, yn −x_n → 0 in Steps 3, and6, it is easily seen that α_n → 0, _∞

n1α_n ∞, and lim sup_n_{→ ∞}βn/αn ≤ 0. Hence, byLemma 2.3, we concludexn → qasn → ∞. This completes the proof.

As a direct consequence ofTheorem 3.1, we have the following results.

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Corollary 3.2. Let H, C, S, B, f, u, γ, γ, k, and μ be as in Theorem 3.1. Let {xn} be a sequence generated by

x1x∈C, ynαn

uγfxn

I−αn

IμB Sxn, x_n1

1−β_n

y_nβ_nSy_n, n≥1,

3.44

where {αn} and {βn} ⊂ 0,1. Let {αn} and {βn} satisfy the conditions (i), (ii), and (iv) in Theorem 3.1. Then {xn} converges strongly to q ∈ FS, which is a solution of the optimization problem

x∈FSmin μ

2Bx, x1

Corollary 3.3. Let H, C, A, B, f, u, γ, γ, k, and μ be as in Theorem 3.1. Let {xn} be a sequence generated by

x1x∈C, ynαn

uγfxn

I−αn

IμB

PCxn−λnAxn, x_n1

1−β_n

y_nβ_nP_C

y_n−λ_nAy_n

, n≥1,

3.45

where{λn} ⊂0,2α,{αn} ⊂0,1, and{βn} ⊂0,1. Let{αn},{λn}and{βn}satisfy the conditions (i), (ii), (iii), and (iv) in Theorem 3.1. Then {xn}converges strongly to q ∈ VIC, A, which is a solution of the optimization problem

x∈V IC,Amin μ

2Bx, x1

Remark 3.4. 1 Theorem 3.1 and Corollary 3.3 improve and develop the corresponding results in Chen et al.6, Iiduka and Takahashi8, and Jung10.

2Even thoughβ_n 0 forn≥1, the iterative scheme3.44inCorollary 3.2is a new one for fixed point problem of a nonexpansive mapping.

4. Applications

In this section, as in6,8,10, we prove two theorems by usingTheorem 3.1. First of all, we recall the following definition.

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A mappingT :C → Cis called strictly pseudocontractive if there existsαwith 0≤α <1 such that

Tx−Ty²≤x−y²αI−Tx−I−Ty² 4.1 for everyx, y ∈ C. Ifk 0, thenT is nonexpansive. PutA I−T, whereT : C → Cis a strictly pseudo-contractive mapping with constantα. ThenAis1−α/2-inverse-strongly monotone; see2. Actually, we have, for allx, y∈C,

I−Ax−I−Ay²≤x−y²αAx−Ay². 4.2

On the other hand, sinceHis a real Hilbert space, we have I−Ax−I−Ay²x−y²Ax−Ay²−2

x−y, Ax−Ay

. 4.3

Hence we have

x−y, Ax−Ay

≥ 1−α

2 Ax−Ay². 4.4

UsingTheorem 3.1, we found a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudo-contractive mapping.

Theorem 4.1. LetH,C,S,B,f,u,γ,γ,k, andμbe as inTheorem 3.1. LetTbe anα-strictly pseudo- contractive mapping ofCinto itself such thatFS∩FT/∅. Let{xn}be a sequence generated by

x1x∈C, y_n α_n

uγfxn

I−α_n

IμB

S1−λ_nxnλ_nTx_n, x_n1

1−β_n

y_nβ_nS

1−λ_nynλ_nTy_n

, n≥1,

4.5

where {λn} ⊂ 0,1−α,{αn} ⊂ 0,1, and{βn} ⊂ 0,1. Let {αn},{λn}, and{βn} satisfy the conditions (i), (ii), (iii), and (iv) inTheorem 3.1. Then{xn}converges strongly toq∈FS∩FT, which is a solution of the optimization problem

x∈FS∩FTmin μ

2Bx, x1

Proof. PutAI−T. ThenAis1−α/2-inverse-strongly monotone. We haveFT VIC, A andPCxn−λnAxn 1−λnxnλnTxn. Thus, the desired result follows fromTheorem 3.1.

UsingTheorem 3.1, we also obtain the following result.

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Theorem 4.2. LetHbe a real Hilbert space. Let Abe anα-inverse-strongly monotone mapping of HintoHand S a nonexpansive mapping ofHinto itself such thatFS∩A⁻¹0/∅. Letu∈H, and letBbe a strongly positive bounded linear operator on Hwith constant γ > 0 andf : H → H a contraction with constantk ∈0,1. Assume thatμ >0 and 0 < γ < 1μγ/k. Let{xn}be a sequence generated by

x1x∈H, y_nα_n

uγfxn

I−α_n

IμB

Sxn−λ_nAx_n, x_n1

1−β_n

y_nβ_nS

y_n−λ_nAy_n

, n≥1,

4.6

where {λn} ⊂ 0,2α, {αn} ⊂ 0,1, and {βn} ⊂ 0,1. Let {αn}, {λn}, and {βn} satisfy the conditions (i), (ii), (iii), and (iv) inTheorem 3.1. Then{xn}converges strongly toq∈FS∩A⁻¹0, which is a solution of the optimization problem

x∈FS∩Amin⁻¹0

μ

2Bx, x1

Proof. We haveA⁻¹0 VIH, A. So, puttingP_H I, byTheorem 3.1, we obtain the desired result.

Remark 4.3. 1Theorems4.1and4.2complement and develop the corresponding results in Chen et al.6and Jung10.

2In all our results, we can replace the condition_∞

n1|αn1−αn|<∞on the control parameter{αn}by the conditionα_n ∈0,1forn≥1, lim_n_{→ ∞}α_n/α_n1 112,13or by the perturbed control condition|αn1−α_n|< oαn1 σ_n,_∞

n1σ_n<∞23.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and Technology2010-0017007.

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