System of General Variational Inequalities Involving Different Nonlinear Operators Related to Fixed Point Problems and Its Applications

Fixed Point Theory and Applications Volume 2011, Article ID 689478,17pages doi:10.1155/2011/689478

Research Article

System of General Variational Inequalities

Involving Different Nonlinear Operators Related to

Fixed Point Problems and Its Applications

Issara Inchan

_{and Narin Petrot}

1_{Department of Mathematics and computer, Faculty of Science and Technology,}

Uttaradit Rajabhat University, Uttaradit 53000, Thailand

2_{Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand}

3_{Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand}

Correspondence should be addressed to Narin Petrot,[email protected]

Received 5 October 2010; Revised 11 November 2010; Accepted 9 December 2010

Academic Editor: Qamrul Hasan Ansari

Copyrightq2011 I. Inchan and N. Petrot. 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.

By using the projection methods, we suggest and analyze the iterative schemes for finding the approximation solvability of a system of general variational inequalities involving different nonlinear operators in the framework of Hilbert spaces. Moreover, such solutions are also fixed points of a Lipschitz mapping. Some interesting cases and examples of applying the main results are discussed and showed. The results presented in this paper are more general and include many previously known results as special cases.

1. Introduction

methods. It is also worth noting that the projection methods have been applied widely to problems arising especially from complementarity, convex quadratic programming, and variational problems.

On the other hand, in 1985, Pang 4 studied the variational inequality problem on the product sets, by decomposing the original variational inequality into a system of variational inequalities, and discussed the convergence of method of decomposition for system of variational inequalities. Moreover, he showed that a variety of equilibrium models, for example, the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium problem, and the general equilibrium programming problem, can be uniformly modelled as a variational inequality defined on the product sets. Later, it was noticed that variational inequality over product sets and the system of variational inequalities both are equivalent; see 4–7 for applications. Since then many authors, see, for example, 8–

11, studied the existence theory of various classes of system of variational inequalities by exploiting fixed point theorems and minimax theorems. Recently, Verma 12 introduced a new system of nonlinear strongly monotone variational inequalities and studied the approximate solvability of this system based on a system of projection methods. Additional research on the approximate solvability of a system of nonlinear variational inequalities is according to Chang et al.13, Cho et al.14, Nie et al.15, Noor16, Petrot17, Suantai and Petrot18, Verma19,20, and others.

Motivated by the research works going on this field, in this paper, the methods for finding the common solutions of a system of general variational inequalities involving different nonlinear operators and fixed point problem are considered, via the projection method, in the framework of Hilbert spaces. Since the problems of a system of general variational inequalities and fixed point are both important, the results presented in this paper are useful and can be viewed as an improvement and extension of the previously known results appearing in the literature, which mainly improves the results of Chang et al.13and also extends the results of Huang and Noor21, Verma20to some extent.

2. Preliminaries

LetCbe a closed convex subset of real HilbertH, whose inner product and norm are denoted by·,·and · , respectively.

We begin with some basic definitions and well-known results.

Definition 2.1. A nonlinear mappingS:H → His said to be aκ-Lipschitzian mappingif there

exists a positive constantκsuch that

Sx−Sy ≤κx−y, ∀x, y∈H. 2.1

In the caseκ1, the mappingSis known as a nonexpansive mapping. IfSis a mapping, we will denote byFSthe set of fixed points ofS, that is,FS {x∈H:Sxx}.

LetCbe a nonempty closed convex subset ofH. It is well known that, for eachz∈H, there exists a unique nearest point inC, denoted byPCz, such that

Such a mappingPC is called themetric projection ofH ontoC. We know thatPC is nonexpansive. Furthermore, for allz∈Handu∈C,

uPCz⇐⇒ u−z, w−u ≥0, ∀w∈C. 2.3

For the nonlinear operators T, g : H → H, thegeneral variational inequality problem

write GVIT, g, Cis to findu∈Hsuch thatgu∈Cand

Tu, gv−gu ≥0, ∀gv∈C. 2.4

The inequality of the type2.4was introduced by Noor22. It has been shown that a large class of unrelated odd-order and nonsymmetric obstacle, unilateral, contact, free, moving, and equilibrium problems arising in regional, ecology, physical, mathematical, engineering, and physical sciences can be studied in the unified framework of the problem2.4; see22–

24 and the references therein. We remark that, if the operator g is the identity operator, the problem 2.4 is nothing but the originally variational inequality problem, which was originally introduced and studied by Stampacchia1.

Applying2.3, one can obtain the following result.

Lemma 2.2. LetCbe a closed convex set inHsuch thatC⊂gH. Thenu∈His a solution of the

problem2.4if and only ifgu PCgu−ρTu, whereρ >0is a constant.

It is clear, in view ofLemma 2.2, that the variational inequalities and the fixed point problems are equivalent. This alternative equivalent formulation is suggest in the study of the variational inequalities and related optimization problems.

LetTi, gi :H → Hbe nonlinear operator, and letri be a fixed positive real number,

for eachi 1,2,3. SetΞ {T1, T2, T3} andΛ {g1, g2, g3}. Thesystem of general variational

inequalities involving three different nonlinear operators generated byr1,r2, andr3 is defined as

follows.

Findx∗, y∗, z∗∈H×H×Hsuch that

r1T1y∗g1x∗−g1

y∗_{, g}

1x−g1x∗ ≥0, ∀g1x∈C, r2T2z∗g2

y∗_−g

2z∗, g2x−g2

y∗ _{≥0, ∀g}

2x∈C,

r3T3x∗g3z∗−g3x∗, g3x−g3z∗

≥0, ∀g3x∈C.

2.5

We denote by SGVIDΞ,Λ, Cthe set of all solutionsx∗, y∗, z∗of the problem2.5.

By using2.3, we see that the problem2.5is equivalent to the following projection problem:

g1x∗ PC

g1

y∗_−r

1T1y∗

,

g2

y∗_P

Cg2z∗−r2T2z∗

,

g3z∗ PC

g3x∗−r3T3x∗

,

2.6

We now discuss several special cases of the problem2.5.

iIf g1 g2 g3 g, then the system 2.5 reduces to the problem of finding

x∗_{, y}∗_{, z}∗_{∈H×H×Hsuch that}

r1T1y∗gx∗−g

y∗_{, gx−gx}∗ _{≥0, ∀gx∈C,}

r2T2z∗g

y∗_−gz∗_{, gx−gy}∗ _{≥0, ∀gx∈C,}

r3T3x∗gz∗−gx∗, gx−gz∗ ≥0, ∀gx∈C.

2.7

We denote by SGVIDΞ, g, Cthe set of all solutionsx∗, y∗, z∗of the problem2.7.

iiIfT1 T2 T3T, then the system2.7reduces to the following system of general

variational inequalities,write SGVIT, g, C, for shot: findx∗, y∗, z∗∈Hsuch that

r1Ty∗gx∗−g

y∗_{, gx−gx}∗ _{≥0, ∀gx∈C,}

r2Tz∗g

y∗_−gz∗_{, gx−gy}∗ _{≥0, ∀gx∈C,}

r3Tx∗gz∗−gx∗, gx−gz∗ ≥0, ∀gx∈C.

2.8

iiiIf g I : the identity operator, then, from the problem 2.7, we have the following system of variational inequalities involving three different nonlinear operators

write SVIDΞ, C, for shot: findx∗, y∗, z∗∈H×H×Hsuch that

r1T1y∗x∗−y∗, x−x∗ ≥0, ∀x∈C, r2T2z∗y∗−z∗, x−y∗ ≥0, ∀x∈C, r3T3x∗z∗−x∗, x−z∗ ≥0, ∀x∈C.

2.9

ivIfT1 T2 T3 T, then, from the problem2.9, we have the following system of

variational inequalitieswrite SVIT, C, for shot: findx∗, y∗, z∗∈H×H×Hsuch

that

r1Ty∗x∗−y∗, x−x∗ ≥0, ∀x∈C, r2Tz∗y∗−z∗, x−y∗ ≥0, ∀x∈C, r3Tx∗z∗−x∗, x−z∗ ≥0, ∀x∈C.

2.10

vIfr3 0, then the problem2.10reduces to the following problem: findx∗, y∗∈

H×Hsuch that

r1Ty∗x∗−y∗, x−x∗ ≥0, ∀x∈C, r2Tx∗y∗−x∗, x−y∗ ≥0, ∀x∈C.

2.11

viIfr2 0, then the problem 2.11reduces to the following problem: findx∗ ∈ H

such that

Tx∗_{, x−x}∗ _{≥0, ∀x∈C, 2.12}

which is, in fact, the originally variational inequality problem, introduced by Stampacchia1.

This shows that, roughly speaking, for suitable and appropriate choice of the operators and spaces, one can obtain several classes of variational inequalities and related optimization problems. Consequently, the class of system of general variational inequalities involving three different nonlinear operators problems is more general and has had a great impact and influence in the development of several branches of pure, applied, and engineering sciences. For the recent applications, numerical methods, and formulations of variational inequalities, see1–27and the references therein.

Now we recall the definition of a class of mappings.

Definition 2.3. The mapping T : H → His said to be ν-strongly monotoneif there exists a

constantν >0 such that

Tx−Ty, x−y≥νx−y2_{, ∀x, y∈H. 2.13}

In order to prove our main result, the next lemma is very useful.

Lemma 2.4see28. Assume that{an}is a sequence of nonnegative real numbers such that

an1≤1−λnanbncn, ∀n≥n0, 2.14

wheren0 is a nonnegative integer,{λn}is a sequence in0,1withΣ∞_n1λn ∞,bn ◦λn, and

Σ∞

n1cn <∞, thenlimn→ ∞an0.

Denotation. LetΩ ⊂ H×H×H. In what follows, we will put the symbolΩ1 : {x ∈ H :

x, y, z∈Ω}.

3. Main Results

We begin with some observations which are related to the problem2.5.

Remark 3.1. Ifx∗, y∗, z∗∈SGVIDΞ,Λ, C, by2.6, we see that

x∗_x∗_−g

1x∗ PCg1

y∗_−r

1T1y∗

, 3.1

providedC ⊂g1H. Consequently, ifSis a Lipschitz mapping such thatx∗ ∈FS, then it

follows that

x∗_Sx∗ _Sx∗_−g

1x∗ PC

g1

y∗_−r

1T1y∗

The formulation3.2is used to suggest the following iterative method for finding common elements of two different sets, which are the solutions set of the problem2.5and the set of fixed points of a Lipschitz mapping. Of course, since we hope to use the formulation

3.2as an initial idea for constructing the iterative algorithm, hence, from now on, we will assume thatgi :H → Hsatisfies a conditionC⊂giHfor eachi1,2,3. Now, in view of

the formulations2.6and3.2, we suggest the following algorithm.

Algorithm 1. Letr1,r2, andr3be fixed positive real numbers. For arbitrary chosen initialx0∈

H, compute the sequences{xn},{yn}, and{zn}such that

g3zn PCg3xn−r3T3xn,

g2

ynPCg2zn−r2T2zn,

xn1 1−αnxnαnS

xn−g1xn PC

g1

yn−r1T1yn

,

3.3

where{αn}is a sequence in0,1andS:H → His a mapping.

In what follows, ifT :H → His aν-strongly monotone andμ-Lipschitz continuous mapping, then we define a function ΦT : 0,∞ → −∞,∞, associated with such a

mappingT, by

ΦTr

1−2rνr2_μ2_{, ∀r∈0,∞. 3.4}

We now state and prove the main results of this paper.

Theorem 3.2. Let C be a closed convex subset of a real Hilbert space H. LetTi : H → H be

νi-strongly monotone and μi-Lipschitz mapping, and let gi : H → H beλi-strongly monotone

and δi-Lipschitz mapping for i 1,2,3. Let S : H → H be a τ-Lipschitz mapping such that

SGVIDΞ,Λ, C₁∩FS/∅. Put

pi1δ2

i −2λi 3.5

for eachi1,2,3. If

ipi∈0,μi−

μ2

i −ν2i/2μi∪μi

μ2

i −νi2/2μi,1, for eachi1,2,3,

ii|ri−νi/μ2i|<

ν2

i −μ2i4pi1−pi/μ2i, for eachi1,2,3,

iiiτ 3_i1ΦTiri pi/1−pi<1,

iv∞_n0αn∞,

then the sequences{xn},{yn}, and{zn}generated byAlgorithm 1converge strongly tox∗,y∗, and

Proof. Letx∗, y∗, z∗∈SGVIDΞ,Λ, Cbe such thatx∗∈FS. By2.6and3.2, we have

g3z∗ PC

g3x∗−r3T3x∗

,

g2

y∗_P

Cg2z∗−r2T2z∗

,

x∗ _1−αnx∗_αnSx∗_−g

1x∗ PC

g1

y∗_−r

1T1y∗

.

3.6

Consequently, by3.3, we obtain

xn1−x∗1−αnxnαnS

xn−g1xn PCg1

yn−r1T1yn

−x∗

≤1−αnxn−x∗αnSxn−g1xn PCg1

yn−r1T1yn

−Sx∗_−g

1x∗ PC

g1

y∗_−r

1T1y∗ ≤1−αnxn−x∗

αnτxn−x∗−g1xn−g1x∗

yn−y∗−g1

yn−g1

y∗

yn−y∗_−r

1

T1yn−T1y∗.

3.7

By the assumption thatT1isν1-strongly monotone andμ1-Lipschitz mapping, we obtain

yn−y∗_−r

1T1yn−T1y∗2yn−y∗2−2r1yn−y∗, T1yn−T1y∗r12T1yn−T1y∗2 ≤ yn−y∗2−2r1ν1yn−y∗2r12μ12yn−y∗2

1−2r1ν1r₁2μ2₁

yn−y∗2

ΦT1r1

2_y

n−y∗2.

3.8

Notice that

yn−y∗yn−y∗−g2

yn−g2

y∗_g

2

yn−g2

y∗

≤yn−y∗_−g

2

yn−g2

y∗_g

2

yn−g2

y∗_. 3.9

Now we consider,

yn−y∗_−g

2yn−g2y∗2yn−y∗2−2yn−y∗, g2yn−g2y∗g2yn−g2y∗2 ≤ yn−y∗2_−2λ

2yn−y∗2

δ2

2yn−y∗2

1−2λ2δ22

yn−y∗2

p2

2_yn−y∗2_,

sinceg2isλ2-strongly monotone andδ2-Lipschitz mapping. And

g2

yn−g2

y∗_PCg

2zn−r2T2zn

−PCg2z∗−r2T2z∗ ≤g2zn−g2z∗−r2T2zn−T2z∗

≤zn−z∗−g2zn−g2z∗zn−z∗−r2T2zn−T2z∗.

3.11

By the assumptions of T2 and g2, using the same lines as obtained in 3.8and 3.10, we

know that

zn−z∗−r2T2zn−T2z∗2≤ΦT2r2

2_z

n−z∗2, 3.12 zn−z∗−g2zn−g2z∗2 ≤

p2

2_z

n−z∗2, 3.13

respectively.

Substituting3.12and3.13into3.11, we have

g2

yn−g2

y∗ _≤ΦT

2r2 p2

zn−z∗. 3.14

Combining3.9,3.10, and3.14yields that

yn−y∗ _≤p

2yn−y∗

ΦT2r2 p2

zn−z∗_{. 3.15}

Observe that,

zn−z∗_zn−z∗_−g

3zn−g3z∗

g3zn−g3z∗ ≤zn−z∗−g3zn−g3z∗g3zn−g3z∗,

3.16

g3zn−g3z∗≤xn−x∗−g3xn−g3x∗xn−x∗−r3T3xn−T3x∗. 3.17

Using the assumptions ofT3andg3, we know that

xn−x∗−r3T3xn−T3x∗2≤ΦT3r3

2_x

n−x∗2, 3.18 _xn−x∗_{−g3xn−g3x}∗2_≤p

3

2_{xn−x∗2, 3.19}

zn−z∗_−g

3zn−g3z∗≤p3zn−z∗, 3.20

respectively. Substituting3.18and3.19into3.17, we have

g3zn−g3z∗ ≤

ΦT3r3 p3

xn−x∗. 3.21

Combining3.16,3.20, and3.21yields that

zn−z∗ _≤p

3zn−z∗

ΦT3r3 p3

This implies that

zn−z∗ ≤

ΦT3r3 p3

1−p3

xn−x∗. 3.23

Substituting3.23into3.15, we have

yn−y∗ _≤p

2yn−y∗

ΦT2r2 p2

ΦT3r3 p3

1−p3

xn−x∗_{, 3.24}

that is,

yn−y∗ ≤

ΦT2r2 p2

ΦT3r3 p3

1−p2

1−p3

xn−x∗. 3.25

By3.8and3.25, we obtain

yn−y∗_−r

1

T1yn−T1y∗≤

ΦT1r1

ΦT2r2 p2

ΦT3r3 p3

1−p2

1−p3

xn−x∗_{. 3.26}

On the other hand, sinceg1isλ1-strongly monotone andδ1-Lipschitz mapping, we can show

that

xn−x∗_−g

1xn−g1x∗

≤p1xn−x∗, 3.27

yn−y∗−g1

yn−g1

y∗ _≤p

1yn−y∗. 3.28

Substituting3.25into3.28yields that

yn−y∗−g1

yn−g1

y∗_≤ p1

ΦT2r2 p2

ΦT3r3 p3

1−p2

1−p3

xn−x∗. 3.29

Writing

♦

ΦT2r2 p2

ΦT3r3 p3

1−p2

1−p3

3.30

and substituting3.26,3.27, and3.29into3.7, we will get

xn1−x∗ ≤

1−αn1−τp1p1♦ ΦT1r1♦

Table1

μi νi ⎡ ⎢ ⎣0,μi−

μ2 i−ν2i 2μi

⎞ ⎟ ⎠∪

⎡ ⎢ ⎣μi

μ2 i−ν2i 2μi ,1

⎞ ⎟ ⎠

⎛ ⎜ ⎝νi−

ν2

i−μ2i4pi1−pi

μ2 i

,νi

ν2

i−μ2i4pi1−pi

μ2 i

⎞ ⎟ ⎠

T1 1

2 1

2 0,1 0,4 :R1

T2 1

4 1

4 0,1 0,8 :R2

T3 1

2 1 4

0,2−

√ 3 4 ∪ 2√3

4 ,1

7−√22 7 ,

7√22 7

:R3

Notice that, by conditionsiandii, we have

3

i1

_Φ

Tiri pi

1−pi <1. 3.32

This implies that

♦< 1−p1

ΦT1r1 p1

, 3.33

that is,

Δ :p1p1♦ ΦT1r1♦<1. 3.34

Put

anxn−x∗_,

λnαn1−τΔ.

3.35

By conditioniii, in view of3.32and 3.34, we see thatτΔ ∈ 0,1; this implies

λn ∈0,1. Meanwhile, from conditioniv, we also have∞n0λn∞. Hence, all conditions

ofLemma 2.4are satisfied, and we can conclude thatxn → x∗ asn → ∞. Consequently,

from3.23and3.25, we know thatzn → z∗ andyn → y∗asn → ∞, respectively. This completes the proof.

Example 3.3. LetH 0,1andC 0,1/2. Fori1,2,3, letTi, gi :H → Hbe mappings

which are defined byT1x x/2,T2x x/4,T3x x2/4,g1x x, andg2x g3x

27/28x. Then, one can show thatp10 andp2p31/28. Consequently, we haveTable 1.

Now letγ∈1,∞be a fixed positive real number andα∈0,1/γ _i3₁ΦTiripi/1−pi.

IfS:H → His a mapping which is defined by

Sx αxγ_{, ∀x∈H. 3.36}

Then we know that the conditionsiiandiiiofTheorem 3.2are satisfied. In fact, we have

0,0,0∈SGVIDΞ,Λ, Cand 0∈FS.

Applying ourTheorem 3.2, the following results are obtained immediately.

Corollary 3.4. LetC be a closed convex subset of a real Hilbert space H. LetTi : H → H be

νi-strongly monotone and μi-Lipschitz mapping, and let g : H → H be λ-strongly monotone

and δ-Lipschitz mapping for i 1,2,3. Let S : H → H be a τ-Lipschitz mapping such that

SGVIDΞ, g, C₁∩FS/∅. Letr1,r2, andr3be positive real numbers that generate the problem

2.7. For arbitrary chosen initialx0 ∈H, compute the sequences{xn},{yn}, and{zn}such that

gzn PCgxn−r3T3xn,

gynPCgzn−r2T2zn,

xn1 1−αnxnαnS

xn−gxn PCgyn−r1T1yn

.

3.37

Putp√1δ2_{−2λ. If the following control conditions are satisfied:}

ip∈0,μi−

μ2

i −νi2/2μi∪μi

μ2

i −ν2i/2μi,1, for eachi1,2,3,

ii|ri−νi/μ2_i|<

ν2

i −μ2i4p1−p/μ2i, for eachi1,2,3,

iiiτ 3_i1ΦTiri p/1−p<1,

iv∞_n0αn∞,

then the sequences{xn},{yn}, and {zn} generated by3.37converge strongly tox∗,y∗, and z∗,

respectively, such thatx∗, y∗, z∗∈SGVIDΞ, g, Candx∗∈FS.

Corollary 3.5. LetCbe a closed convex subset of a real Hilbert spaceH. LetT : H → H beν

-strongly monotone andμ-Lipschitz continuous mapping, and letg :H → Hbeλ-strongly monotone

andδ-Lipschitz mapping. LetS:H → Hbe aτ-Lipschitz mapping such thatSGVIT, g, C₁∩

FS/∅. Letr1,r2, and r3 be positive real numbers that generate the problem 2.8. For arbitrary

chosen initialx0∈H, compute the sequences{xn},{yn}, and{zn}such that

gzn PCgxn−r3Txn,

gynPCgzn−r2Tzn

,

xn1 1−αnxnαnSxn−gxn PCgyn−r1Tyn.

3.38

If the following control conditions are satisfied:

ii|r−ν/μ2_|<ν2_−μ2_4p1−p/μ2_{, wherermax{r}

1, r2, r3},

iiiτ 3_i1ΦTri p/1−p<1,

iv∞_n0αn∞,

then the sequences{xn},{yn}, and {zn} generated by3.38converge strongly tox∗,y∗, and z∗,

respectively, such thatx∗, y∗, z∗∈SGVIT, g, Candx∗∈FS.

Corollary 3.6. LetC be a closed convex subset of a real Hilbert spaceH. LetTi : H → H be

νi-strongly monotone andμi-Lipschitz continuous mapping fori 1,2,3. LetS : C → Cbe aτ

-Lipschitz mapping such thatSVIDΞ, C₁∩FS/∅. Letr1,r2, andr3be positive real numbers that

generate the problem2.9. For arbitrary chosen initialx0 ∈ H, compute the sequences{xn},{yn},

and{zn}such that

znPCxn−r3T3xn,

ynPCzn−r2T2zn,

xn1 1−αnxnαnSPCyn−r1T1yn.

3.39

If the following control conditions are satisfied:

iri ∈0,2νi/μ2i, for eachi1,2,3,

iiτ 3_i1ΦTri<1,

iii∞_n0αn∞,

then the sequences{xn},{yn}, and {zn} generated by3.39converge strongly tox∗,y∗, and z∗,

respectively, such thatx∗, y∗, z∗∈SVIDΞ, Candx∗∈FS.

Proof. Since the identity mapping is 1-strongly monotone and 1-Lipschitz mapping, it follows

that the numberp, defined inCorollary 3.4, is identically zero. Hence, the required result can be obtained immediately.

Corollary 3.7. LetCbe a closed convex subset of a real Hilbert spaceH. LetT : H → H beν

-strongly monotone andμ-Lipschitz mapping. LetS : C → Cbe aτ-Lipschitz mapping such that

SVIT, C₁∩FS/∅. Letr1,r2, andr3be positive real numbers that generate the problem2.10.

For arbitrary chosen initialx0∈H, compute the sequences{xn},{yn}, and{zn}such that

znPCxn−r3Txn,

ynPCzn−r2Tzn,

xn1 1−αnxnαnSPCyn−r1Tyn.

3.40

If the following control conditions are satisfied:

iri ∈0,2ν/μ2_{, for eachi1,2,3,}

iiτ 3_i1ΦTri<1,

then the sequences{xn},{yn}, and {zn} generated by3.40converge strongly tox∗,y∗, and z∗,

respectively, such thatx∗, y∗, z∗∈SVIT, Candx∗∈FS.

Remark 3.8. Corollary 3.9mainly improves and extends the results of Verma20.

Corollary 3.9. Let Cbe a closed convex subset of a real Hilbert space H. Let T : H → H be

ν-strongly monotone andμ-Lipschitz mapping, and letg :H → Hbeδ-strongly monotone andλ

-Lipschitz mapping. LetS:H → Hbe aτ-Lipschitz mapping such thatGVIT, g, C∩FS/∅. Put

r ν/μ2_{be a fixed positive real number. For arbitrary chosen initialx}

0 ∈H, compute the sequence

{xn}such that

gzn PCgxn−rTxn,

gynPCgzn−rTzn,

xn1 1−αnxnαnS

gxn−xnPCgyn−rTyn.

3.41

If the following control conditions are satisfied:

ip∈0,μ−μ2_−ν2_/2μ∪μμ2_−ν2_{/2μ,1, wherep}√_1δ2_−2λ,

iiτ ∈0, μ1−p/μp

μ2_−ν2_,

iii∞_n0αn∞,

then the sequences {xn} generated by 3.41 converges strongly to x∗, such that x∗ ∈

GVIT, C!FS.

Proof. Notice that ΦT0 1 and ΦTν/μ2

μ2_−ν2_{/μ. Consequently, condition ii}

implies that

τ

p ΦTν/μ2

1−p

<1. 3.42

Moreover, by setting r2 r3 0, we see that the problem SGVIT, g, C is reduced to

the problem GVIT, g, C. Using these observations, one can easily see that the required conclusion is followed immediately from theCorollary 3.5.

Remark 3.10. Corollary 3.9extends the results in24in some extent.

In light of Corollaries3.6and3.9, we obtain the following result immediately.

Corollary 3.11. LetC be a closed convex subset of a real Hilbert spaceH. LetT : H → H be

VIT, C∩FS/∅. Letrν/μ2_{be a fixed positive real number. For arbitrary chosen initialx} 0∈H,

compute the sequence{xn}such that

znPCxn−rTxn,

ynPCzn−rTzn,

xn1 1−αnxnαnSPC

yn−rTyn.

3.43

If the following control conditions are satisfied:

iτ ∈0, μ/

μ2_−ν2_,

ii∞_n0αn∞.

then the sequences{xn}generated by3.43converges strongly tox∗, such thatx∗∈VIT, C∩FS.

Remark 3.12. Corollary 3.11extends and improves the main result announced by Noor and

Huang26, from a class nonexpansive mappings to a class of any Lipschitzian mappings.

Remark 3.13. The choicer ν/μ2_{is a possible sharp for applying Corollaries3.9and3.11to}

a wide class of Lipschitz mappings. Indeed, notice that

ΦT _ν

μ2

μ2−ν2

μ r∈inf0,∞{ΦTr}.

3.44

Since both Corollaries3.9and3.11are special cases ofCorollary 3.5, thus, based on condition

iiiofCorollary 3.5, our remark is asserted.

Now we show an application ofTheorem 3.2. Recall that a mappingQ :H → His said to beasymptotically strict pseudocontractionif there exists a constantλ∈0,1satisfying

Qn_x−Qn_y2_≤1γ

nx−y2λI−Qnx−I−Qny2 3.45

for allx, y ∈Hand all integern≥1, whereγn ≥0 for alln≥1 such thatγn → 0 asn → ∞. In this case, we also sayQis an asymptoticallyλ-strict pseudocontraction.

Lemma 3.14see29. LetQ:H → Hbe an asymptoticallyλ-strict pseudocontraction. Then, for

eachn≥1,Qnsatisfies the Lipschitz condition

Qn_x−Qn_{y ≤L}

nx−y, ∀x, y∈H, 3.46

whereLn λ"1γn1−λ/1−λ.

For each i 1,2,3, let Ti : H → H be a νi-strongly monotone and μi-Lipschitz

mapping, and letgi:H → Hbe aδi-strongly monotone andλi-Lipschitz mapping. Put

ξ3

i1

_Φ

Tiri pi

wherepiis defined as inTheorem 3.2, for eachi1,2,3, andr1,r2,r3are positive real numbers

that generate the problem2.5. Notice that, ifξ∈0,1−λ/1λ, then there exists a natural numberj such thatLj <1/ξ, sinceLn ↓1λ/1−λasn → ∞. Using this observation, we can applyTheorem 3.2to obtain the following result.

Example 3.15. LetHbe a real Hilbert space. For eachi1,2,3, letTi:H → Hbe aνi-strongly

monotone andμi-Lipschitz mapping, and letgi:H → Hbe aδi-strongly monotone andλi -Lipschitz mapping. Assume that the problem2.5is generated by the positive real numbers

r1,r2, andr3such that the conditionsiandiiinTheorem 3.2are satisfied. LetQ:H → H

be an asymptoticallyλ-strict pseudocontraction satisfyingξ∈0,1−λ/1λ, and letj∈N be a natural number such thatLj<1/ξ, whereLjis defined as inLemma 3.14. Let{xn},{yn}, and{zn}be three sequences generated byAlgorithm 1withS:Qj.

IfSQVIDΞ,Λ, C₁∩FQ/∅and∞_n0αn ∞, then the sequences{xn},{yn}, and

{zn}converge strongly tox∗,y∗, andz∗, respectively, such thatx∗, y∗, z∗∈SGVIDΞ,Λ, C andx∗ ∈ FQ. Indeed, letx∗, y∗, z∗ ∈SQVIDΞ,Λ, Cbe such thatx∗ ∈FQ. It follows that x∗ ∈ FQn for all n ∈ N. Using this one together with the fact that ξLj < 1, as an

application ofTheorem 3.2, we know that{xn},{yn}, and{zn}converge strongly tox∗,y∗, andz∗, respectively.

Remark 3.16. Ifλ0, thenQis fallen to a class of mappings as asymptotically nonexpansive

mapping. Hence,Example 3.15can be viewed as an extension of the main result announced by Cho and Qin25in some aspects.

Remark 3.17. Recall that a mappingT :H → His said to be

iμ-cocoerciveif there exists a constantμ >0 such that

Tx−Ty, x−y ≥μTx−Ty2_{, ∀x, y∈H, 3.48}

iirelaxedμ-cocoerciveif there exists a constantμ >0 such that

Tx−Ty, x−y ≥−μTx−Ty2_{, ∀x, y∈H, 3.49}

iiirelaxedμ, ν-cocoerciveif there exist constantsμ, ν >0 such that

Tx−Ty, x−y ≥−μTx−Ty2_νx−y2_{, ∀x, y∈H. 3.50}

Obviously, the class of the relaxedμ, ν-cocoercive mappings is the most general one, of course, larger than the class of strongly monotone mappings. However, it is worth noting that, if the mappingTis relaxedμ, ν-cocoercive andτ-Lipschitz mapping such thatν−μτ2> 0,T must be aν−μτ2_{-strongly monotone. Hence, the results that appeared in this paper}

Acknowledgments

The authors wish to express their gratitude to the referees for a careful reading of the paper and helpful suggestions. The project was supported by the Faculty of Science, Naresuan University and the Centre of Excellence in Mathematics under the Commission on Higher Education, Ministry of Education, Thailand.

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