Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 689478, ppt

Volume 2011, Article ID 689478, 17 pagesdoi:10.1155/2011/689478 Research Article System of General Variational Inequalities Involving Different Nonlinear Operators Related to Fixed Point

Volume 2011, Article ID 689478, 17 pages

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 Inchan1, 2 and Narin Petrot2, 3

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,narinp@nu.ac.th

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

Academic Editor: Qamrul Hasan Ansari

Copyrightq 2011 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

The originally variational inequality problem, introduced by Stampacchia1, in the early sixties, has had a great impact and influence in the development of almost all branches

of pure and applied sciences and has witnessed an explosive growth in theoretical advances, algorithmic development As a result of interaction between different branches

of mathematical and engineering sciences, we now have a variety of techniques to suggest and analyze various algorithms for solvinggeneralized variational inequalities and related optimization It is well known that the variational inequality problems are equivalent to the fixed point problems This alternative equivalent formulation is very important from the numerical analysis point of view and has played a significant part in several numerical methods for solving variational inequalities and complementarity; see2,3 In particular, the solution of the variational inequalities can be computed using the iterative projection

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, Noor 16, Petrot 17, Suantai and Petrot18, Verma 19,20, 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.13 and also extends the results of Huang and Noor21, Verma 20 to some extent

2 Preliminaries

Let C be a closed convex subset of real Hilbert H, whose inner product and norm are denoted

by·, · and  · , respectively.

We begin with some basic definitions and well-known results

exists a positive constant κ such that

Sx − Sy ≤ κx − y, ∀x, y ∈ H. 2.1

In the case κ  1, the mapping S is known as a nonexpansive mapping If S is a mapping,

we will denote by FS the set of fixed points of S, that is, FS  {x ∈ H : Sx  x}.

Let C be a nonempty closed convex subset of H It is well known that, for each z ∈ H, there exists a unique nearest point in C, denoted by P C z, such that

z − P C z ≤ z − y, ∀y ∈ C. 2.2

Such a mapping P C is called the metric projection of H onto C We know that P C is

nonexpansive Furthermore, for all z ∈ H and u ∈ C,

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

write GVIT, g, C is to find u ∈ H such that gu ∈ C and

Tu, gv − gu ≥ 0, ∀gv ∈ C. 2.4

The inequality of the type2.4 was introduced by Noor 22 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; see 22–

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 Let C be a closed convex set in H such that C ⊂ gH Then u ∈ H is a solution of the

problem2.4 if and only if gu  P C gu − ρTu, where ρ > 0 is 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

Let T i , g i : H → H be nonlinear operator, and let r i be a fixed positive real number,

for each i  1, 2, 3 Set Ξ  {T1, T2, T3} and Λ  {g1, g2, g3} The system of general variational

inequalities involving three different nonlinear operators generated by r1, r2, and r3 is defined as follows

Findx∗, y∗, z∗ ∈ H × H × H such that

r1T1y∗ g1x∗ − g1



y∗

, g1x − g1x∗ ≥ 0, ∀g1x ∈ C,

r2T2z∗ g2



y∗

− g2z∗, g2x − g2



y∗

 ≥ 0, ∀g2x ∈ C,



r3T3x∗ g3z∗ − g3x∗, g3x − g3z∗≥ 0, ∀g3x ∈ C.

2.5

We denote by SGVIDΞ, Λ, C the set of all solutions x∗, y∗, z∗ of the problem 2.5

By using2.3, we see that the problem 2.5 is equivalent to the following projection problem:

g1x∗  P C

g1



y∗

− r1T1y∗

,

g2

y∗

 P C

g2z∗ − r2T2z∗

,

g3z∗  P C

g3x∗ − r3T3x∗

,

2.6

provided C ⊂ g i H for each i  1, 2, 3.

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

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

x∗, y∗, z∗ ∈ H × H × H such that

r1T1y∗ gx∗ − gy∗

r2T2z∗ gy∗

− gz∗, gx − gy∗

 ≥ 0, ∀gx ∈ C,

r3T3x∗ gz∗ − gx∗, gx − gz∗ ≥ 0, ∀gx ∈ C.

2.7

We denote by SGVIDΞ, g, C the set of all solutions x∗, y∗, z∗ of the problem 2.7

ii If T1  T2  T3 T, then the system 2.7 reduces to the following system of general

variational inequalities , write SGVIT, g, C, for shot: find x∗, y∗, z∗∈ H such that

r1Ty∗ gx∗ − gy∗

r2Tz∗ gy∗

− gz∗, gx − gy∗

 ≥ 0, ∀gx ∈ C,

r3Tx∗ gz∗ − gx∗, gx − gz∗ ≥ 0, ∀gx ∈ C.

2.8

iii If g  I : the identity operator, then, from the problem 2.7, we have the

following system of variational inequalities involving three di fferent nonlinear operators

write SVIDΞ, C, for shot: find x∗, y∗, z∗ ∈ H × H × H such 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

iv If T1  T2  T3  T, then, from the problem 2.9, we have the following system of

variational inequalities write SVIT, C, for shot: find x∗, y∗, z∗ ∈ H × H × H such

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

v If r3  0, then the problem 2.10 reduces to the following problem: find x∗, y∗ ∈

H × H such that

r1Ty∗ x∗− y∗, x − x∗ ≥ 0, ∀x ∈ C,

The problem2.10 has been introduced and studied by Verma 20

vi If r2  0, then the problem 2.11 reduces to the following problem: find x∗ ∈ H

such that

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 27 and the references therein

Now we recall the definition of a class of mappings

constant ν > 0 such that



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

Lemma 2.4 see 28 Assume that {a n } is a sequence of nonnegative real numbers such that

a n1 ≤ 1 − λ n a n b n c n , ∀n ≥ n0, 2.14

where n0 is a nonnegative integer, {λ n } is a sequence in 0, 1 with Σ∞

n1 λ n  ∞, b n  ◦λ n , and

Σ∞

n1 c n < ∞, then lim n → ∞ a n  0.

x, y, z ∈ Ω}.

3 Main Results

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

Remark 3.1 If x∗, y∗, z∗ ∈ SGVIDΞ, Λ, C, by 2.6, we see that

x∗ x∗− g1x∗ P C

g1

y∗

− r1T1y∗

provided C ⊂ g1H Consequently, if S is a Lipschitz mapping such that x∗ ∈ FS, then it

follows that

x∗ Sx∗  Sx∗− g1x∗ P C

g1



y∗

− r1T1y∗

The formulation3.2 is used to suggest the following iterative method for finding common elements of two different sets, which are the solutions set of the problem 2.5 and the set of fixed points of a Lipschitz mapping Of course, since we hope to use the formulation

3.2 as an initial idea for constructing the iterative algorithm, hence, from now on, we will

assume that g i : H → H satisfies a condition C ⊂ g i H for each i  1, 2, 3 Now, in view of

the formulations2.6 and 3.2, we suggest the following algorithm

Algorithm 1 Let r1, r2, and r3be fixed positive real numbers For arbitrary chosen initial x0∈

H, compute the sequences {x n }, {y n }, and {z n} such that

g3z n   P C

g3x n  − r3T3x n

,

g2

y n

 P C

g2z n  − r2T2z n

,

x n1  1 − α n x n α n S

x n − g1x n  P C

g1



y n

− r1T1y n

,

3.3

where{α n } is a sequence in 0, 1 and S : H → H is a mapping.

In what follows, if T : H → H is a ν-strongly monotone and μ-Lipschitz continuous

mapping, then we define a function ΦT : 0, ∞ → −∞, ∞, associated with such a mapping T, by

ΦT r 1− 2rν r2μ2, ∀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 Let T i : H → H be

SGVIDΞ, Λ, C1∩ FS / ∅ Put

p i1 δ2

for each i  1, 2, 3 If

i p i ∈ 0, μ i−μ2

i /2μ i  ∪ μ iμ2

i /2μ i , 1, for each i  1, 2, 3,

ii |r i − ν i /μ2

i | <ν2

i 4p i 1 − p i /μ2

i , for each i  1, 2, 3,

iii τ 3

i1ΦT i r i  p i /1 − p i  < 1,

iv ∞n0 α n  ∞,

then the sequences {x n }, {y n }, and {z n } generated by Algorithm 1 converge strongly to x∗, y∗, and

z∗, respectively, such that x∗, y∗, z∗ ∈ SGVIDΞ, Λ, C and x∗∈ FS.

Proof Let x∗, y∗, z∗ ∈ SGVIDΞ, Λ, C be such that x∗∈ FS By 2.6 and 3.2, we have

g3z∗  P C

g3x∗ − r3T3x∗

,

g2

y∗

 P C

g2z∗ − r2T2z∗

,

x∗ 1 − α n x∗ α n S

x∗− g1x∗ P C

g1

y∗

− r1T1y∗

.

3.6

Consequently, by3.3, we obtain

x n1 − x∗  1 − α n x n α n S

x n − g1xn  P C

g1



y n

− r1T1y n

− x∗

≤ 1 − α n x n − x∗ α n n − g1x n  P C

g1

y n

− r1T1y n

−Sx∗− g1x∗ P C

g1

y∗

− r1T1y∗

≤ 1 − α n x n − x∗

α n τ

x n − x∗−g1x n  − g1x∗ n − y∗−g1

y n

− g1



y∗

n − y∗− r1



T1y n − T1y∗

3.7

By the assumption that T1is ν1-strongly monotone and μ1-Lipschitz mapping, we obtain

n − y∗− r1T1y n − T1y∗ 2 y n − y∗2− 2r1y n − y∗, T1y n − T1y∗ r2

1T1y n − T1y∗2

≤ y n − y∗2− 2r1ν1y n − y∗2 r2

1μ2

1y n − y∗2

1− 2r1ν1 r2

1μ2 1



y n − y∗2

 ΦT1r12y n − y∗2.

3.8 Notice that

y n − y∗  n − y∗−g2

y n

− g2



y∗

g2

y n

− g2



y∗

≤ n − y∗−g2

y n

− g2



y∗

 g2



y n

− g2



Now we consider,

n − y∗− g2y n  − g2y∗ 2 y n − y∗2− 2y n − y∗, g2y n − g2y∗ g2y n − g2y∗2

≤ y n − y∗2− 2λ2y n − y∗2

δ2

2y n − y∗2

1− 2λ2 δ2

2



y n − y∗2

p2

2

y n − y∗2,

3.10

since g2is λ2-strongly monotone and δ2-Lipschitz mapping And

g2



y n

− g2



y∗

  C

g2z n  − r2T2z n

− P C

g2z∗ − r2T2z∗

≤ 2z n  − g2z∗ − r2T2z n − T2z∗

≤ n − z∗−g2z n  − g2z∗ n − z∗− r2T2z n − T2z∗.

3.11

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

z n − z∗− r2T2z n − T2z∗2≤ ΦT2r22z n − z∗2, 3.12

n − z∗− g2z n  − g2z∗ 2 ≤p22

z n − z∗2, 3.13

respectively

Substituting3.12 and 3.13 into 3.11, we have

g2



y n

− g2



y∗

 ≤ΦT2r2 p2



z n − z∗. 3.14 Combining3.9, 3.10, and 3.14 yields that

y n − y∗ ≤ p2y n − y∗ ΦT2r2 p2



z n − z∗. 3.15 Observe that,

z n − z∗  n − z∗−g3z n  − g3z∗g3z n  − g3z∗

≤ n − z∗−g3z n  − g3z∗ 3z n  − g3z∗ 3.16

3z n  − g3z∗ n − x∗−g3x n  − g3x∗ n − x∗− r3T3x n − T3x∗. 3.17

Using the assumptions of T3and g3, we know that

x n − x∗− r3T3x n − T3x∗2≤ ΦT3r32x n − x∗2, 3.18

n − x∗− g3x n  − g3x∗ 2≤p3

2

x n − x∗2, 3.19

n − z∗−g3z n  − g3z∗ 3z n − z∗, 3.20 respectively Substituting3.18 and 3.19 into 3.17, we have

g3z n  − g3z∗ ≤ΦT3r3 p3



x n − x∗. 3.21 Combining3.16, 3.20, and 3.21 yields that

z n − z∗ ≤ p3z n − z∗ ΦT r3 p3



x n − x∗. 3.22

This implies that

z n − z∗ ≤



ΦT3r3 p3



1− p3 x n − x∗. 3.23 Substituting3.23 into 3.15, we have

y n − y∗ ≤ p2y n − y∗ ΦT2r2 p2

ΦT3r3 p3



1− p3 x n − x∗, 3.24 that is,

y n − y∗ ≤



ΦT2r2 p2



ΦT3r3 p3





1− p2



1− p3

 x n − x∗. 3.25

By3.8 and 3.25, we obtain



T1y n − T1y∗ T1r1ΦT2r2 p2



ΦT3r3 p3





1− p2



1− p3

 x n − x∗. 3.26

On the other hand, since g1is λ1-strongly monotone and δ1-Lipschitz mapping, we can show that

x n − x∗−g1x n  − g1x∗ ≤ p1x n − x∗, 3.27

y n − y∗−g1

y n

− g1



y∗

 ≤ p1y n − y∗. 3.28 Substituting3.25 into 3.28 yields that

y n

− g1





ΦT2r2 p2



ΦT3r3 p3





1− p2



1− p3

 x n − x∗. 3.29

Writing

♦ 



ΦT2r2 p2



ΦT3r3 p3





1− p2



1− p3

and substituting3.26, 3.27, and 3.29 into 3.7, we will get

x n1 − x∗ ≤1− α n

1− τp1 p1♦ ΦT r1♦x n − x∗. 3.31

Table 1

⎡

⎢

⎣0, μ i−



i − ν2

i

2μ i

⎞

⎟

⎠ ∪

⎡

⎢μ iμ2

i − ν2

i

2μ i , 1

⎞

⎟

⎛

⎜ν i−ν2

i − μ2

i 4p i 1 − p i

i

i − μ2

i 4p i 1 − p i

i

⎞

⎟

2

1

4

1

2

1

4



0,2−√3 4

∪



2 √3

4 , 1

7 −√22

7 ,7√22 7

: R3

Notice that, by conditionsi and ii, we have

3

i1

ΦT i r i  p i

This implies that

♦ < 1− p1

ΦT1r1 p1

that is,

Δ : p1 p1♦ ΦT1r1♦ < 1. 3.34 Put

a n  x n − x∗,

λ n  α n 1 − τΔ. 3.35

By conditioniii, in view of 3.32 and 3.34, we see that τΔ ∈ 0, 1; this implies

λ n ∈ 0, 1 Meanwhile, from condition iv, we also have ∞

n0 λ n ∞ Hence, all conditions

ofLemma 2.4are satisfied, and we can conclude that x n → x∗ as n → ∞ Consequently, from3.23 and 3.25, we know that z n → z∗ and y n → y∗as n → ∞, respectively This completes the proof

which are defined by T1x  x/2, T2x  x/4, T3x  x2/4, g1x  x, and g2x  g3x 

27/28x Then, one can show that p1 0 and p2 p3 1/28 Consequently, we haveTable 1

It follows that the conditioni of Theorem 3.2is satisfied Moreover, if for each i 

1, 2, 3 the real number r i belongs to R i, then we can check that 3

i1ΦT i r i  p i /1 − p i  < 1.