Convergence of iterative algorithms for a generalized variational inequality and a nonexpansive mapping

(1)

Available online at http://scik.org

Engineering Mathematics Letters, 1 (2012), No. 1, 44-57 ISSN 2049-9337

CONVERGENCE OF ITERATIVE ALGORITHMS FOR A GENERALIZED VARIATIONAL INEQUALITY AND A

NONEXPANSIVE MAPPING

SONGTAO LV1_{, AND CHANGQUN WU}2,∗

1_{Department of Mathematics, Shangqiu Normal University, Shangqiu 476000, China} 2_{School of Business and Administration, Henan University, Kaifeng 475000, China}

Abstract. In this paper, fixed point problems of nonexpansive mappings and solution problems of generalized variational inequalities are investigated based on a viscosity approximate iterative algorithm. Strong convergence theorems for common elements which lie in the fixed point set and in the solution set are established in the framework of Hilbert spaces.

Keywords: nonexpansive mapping; fixed point; relaxed cocoercive mapping; variational inequality. 2000 AMS Subject Classification: 47H05, 47H09, 47J25

1. Introduction

Variational inequalities introduced in the early seventies have witnessed an explosive growth in theoretical advances, algorithmic development and applications across all the discipline of pure and applied sciences; see [1-24] and the references therein. It combines novel theoretical and algorithmic advances with new domain of applications. Analysis of these problems requires a blend of techniques from convex analysis, functional analysis and numerical analysis. Now, we have a variety of techniques to suggest and analyze various numerical methods including projection technique and its variant forms, auxiliary

∗_{Corresponding author}

Received May 3, 2012

(2)

principle, Wiener-Hopf equations and so on. Recently, some classes of generalized varia-tional inequalities involving two or three three nonlinear operators has been studied by many authors; see [1-6] and the references therein. The generalized variational inequali-ties are useful and important extension and generalizations of the variational inequaliinequali-ties with a wide range of applications in industry, mathematical finance, economics, decision sciences, ecology, mathematical and engineering sciences. For solution problems of the generalized variational inequalities, projection methods which link solution problem of variational inequalities and fixed point problems of nonlinear operators are efficient and popular. Viscosity approximation method which was first introduced by Moudafi [18] has been studied iterative solutions of variational inequalities and fixed points of nonexpansive mappings. In this paper, fixed point problems of a nonexpansive mapping and solution problems of a generalized variational inequality are investigated based on a composite ap-proximate iterative algorithm. Strong convergence theorems for common elements which lie in the fixed point set and in the solution set are established in the framework of Hilbert spaces.

2. Preliminaries

Throughout this paper, we assume thatH is a real Hilbert space, whose inner product and norm are denoted byh·,·i and k · k. Let C be a nonempty closed and convex subset of H and A:C →H a nonlinear mapping. Recall the following definitions:

(a) A is said to be monotone if

hAx−Ay, x−yi ≥0, ∀x, y ∈C.

(b) A is said to be ν-strongly monotone if there exists a positive real number ν > 0 such that

(3)

46 SONGTAO LV , AND CHANGQUN WU

(c) A is said to be relaxed µ-cocoercive if there exists a positive real number µ > 0 such that

hAx−Ay, x−yi ≥(−µ)kAx−Ayk2_, _∀_{x, y} _∈_C.

(d) Ais said to berelaxed (µ, ν)-cocoerciveif there exist positive real numbersµ, ν >0 such that

hAx−Ay, x−yi ≥(−µ)kAx−Ayk2+νkx−yk2, ∀x, y ∈C.

Next, we consider the following generalized variational inequality problem. Give non-linear mappings T1 :C →H and T2 :C →H, find an u∈C such that

hu−λ1T1u+λT2u, v−ui ≥0, ∀v ∈C, (2.1)

whereλ1, and λ2 are constants. In this paper, we useV I(C, T1, T2) to denote the solution

set of the variational inequality problem (2.1).

It is easy to see that an element u ∈ C is a solution to the problem (2.1) if and only if u ∈ C is a fixed point of the mapping PC(λ1T1 −λT2), where PC denotes the metric

projection from H ontoC.

If T1 =I, the identity mapping, and λ1 = 1, then the problem (2.1) is reduced to the

following. Find u∈C such that

hT2u, v−ui ≥0, ∀v ∈C. (2.2)

The variational inequality (2.2) was introduced by Stampacchia [23] in 1964. The problem (2.2) has emerged as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure and applied sciences. In this paper, we useV I(C, T2) to denote the solution

set of the variational inequality problem (2.2).

LetS :C →C be a mapping. Recall thatS is said to be nonexpansive if kSx−Syk ≤ kx−yk, ∀x, y ∈C.

(4)

Recently, many authors studied the problem of approximating a common element in the solution set of variational inequalities (2.1), (2.2), and in the fixed point set of nonex-pansive mappings. Motivated by the research work going on in this direction, fixed point problems of nonexpansive mappings and solution problems of generalized variational in-equalities are investigated based on a composite approximate iterative algorithm in this paper. Strong convergence theorems for common elements which lie in the fixed point set and in the solution set are established in the framework of Hilbert spaces.

In order to prove our main results, we also need the following lemmas.

Lemma 2.1 ([25]). Let C be a nonempty closed and convex subset of a real Hilbert space H. Let S1 : C → C and S2 : C → C be nonexpansive mappings. Suppose that

F(S1)∩F(S2) is nonempty. Define a mapping S :C →C by

Sx=aS1x+ (1−a)S2x, ∀x∈C.

Then S is nonexpansive with F(S) =F(S1)∩F(S2).

Lemma 2.2([26]). Let C be a nonempty closed and convex subset of a real Hilbert space

H and S :C →C a nonexpansive mapping. Then I−S is demi-closed at zero.

Lemma 2.3([27]). Let {xn} and{yn} be bounded sequences in a Hilbert space H and let

{βn} be a sequence in (0,1) with

0<lim inf

n→∞ βn≤lim sup_n_→∞ βn<1.

Suppose that xn+1 = (1−βn)yn+βnxn for all integers n≥0 and

lim sup

n→∞

(kyn+1−ynk − kxn+1−xnk)≤0.

Then limn→∞kyn−xnk= 0.

Lemma 2.4 ([28]). Assume that {αn} is a sequence of nonnegative real numbers such

that

αn+1 ≤(1−γn)αn+δn,

where {γn} is a sequence in (0,1) and {δn} is a sequence such that

(a) limn→∞γn= 0, P∞

(5)

(b) lim sup_n_→∞δn/γn ≤0 or P∞

n=1|δn|<∞.

Then limn→∞αn = 0.

3. Main results

Theorem 3.1. Let C be a nonempty closed and convex subset of a real Hilbert space

H. Let f : C → C be a contractive mapping with the contractive constant κ, and S :

C → C a nonexpansive mapping with fixed points. Let T(m,1) : C → H be a relaxed

(µ(m,1), ν(m,1))-cocoercive and L(m,1)-Lipschitz continuous mapping and T(m,2) : C → H

be a relaxed (µ(m,2), ν(m,2))-cocoercive and L(m,2)-Lipschitz continuous mapping for each

positive integer m ∈ [1, N], where N ≥ m is some positive integer. Assume that F = ∩N

m=1GV I(C, T(m,1), T(m,2))

T

F(S)6=∅. Let {xn}be a sequence generated by the following

algorithm



     

     

x1 ∈C,

yn=δnSxn+ (1−δn) PN

m=1η(m,n)PC(λ(m,1)T(m,1)xn−λ(m,2)T(m,2)xn),

xn+1 =αnf(xn) +βnxn+γnyn, n ≥1,

where {αn}, {βn}, {γn}, {δn}, and {η(m,n)} are sequences in(0,1)satisfying the following

restrictions:

(a) αn+βn+γn= 1, ∀n≥1;

(b) limn→∞αn= 0, and P∞

n=1α=∞;

(c) 0<lim infn→∞βn ≤lim supn→∞βn <1;

(d) PN

m=1η(m,n)= 1, and limn→∞η(m,n) =ηm ∈(0,1);

(e) limn→∞δn=δ∈(0,1),

and {λ(m,1)} and {λ(m,2)} are sequences such that

(f) q1−2λ(m,1)ν(m,1)+ 2λ(m,1)µ(m,1)L2₍_m,₁₎+λ2₍_m,₁₎L2₍_m,₁₎

(6)

Then the sequence {xn} generated by the algorithm converges strongly to x¯, wherex¯∈ F,

and solves the following variational inequality: find some point y such that

hf(y)−y, y−xi ≥0, ∀x∈ F.

Proof. First, we prove that the mapping PC(λ(m,1)T(m,1) −λ(m,2)T(m,2)) is nonexpansive

for each m ∈ {1,2, . . . , N}. For each x, y ∈C, we have

kPC(λ(m,1)T(m,1)−λ(m,2)T(m,2))x−PC(λ(m,1)T(m,1)−λ(m,2)T(m,2))yk

≤ k(λ(m,1)T(m,1)−λ(m,2)T(m,2))x−(λ(m,1)T(m,1)−λ(m,2)T(m,2))yk

≤ k(x−y)−λ(m,1)(T(m,1)x−T(m,1)y)k+k(x−y)−λ(m,2)(T(m,2)x−T(m,2)y)k.

(3.1)

It follows from the assumption that T(m,1) : C → H is relaxed (µ(m,1), ν(m,1))-cocoercive

and L(m,1)-Lipschitz that

k(x−y)−λ(m,1)(T(m,1)x−T(m,1)y)k2

=kx−yk2₋₂_λ

(m,1)hT(m,1)x−T(m,1)y, x−yi+λ2(m,1)kT(m,1)x−T(m,1)yk2

≤ kx−yk2−2λ(m,1)(−µ(m,1)kT1x−T1yk2+ν(m,1)kx−yk2) +λ2(m,1)L 2

(m,1)kx−yk 2

≤θ₍2_m,₁₎kx−yk2_,

where θ(m,1) =

q

1−2λ(m,1)ν(m,1)+ 2λ(m,1)µ(m,1)L2₍_m,₁₎+λ2₍_m,₁₎L2₍_m,₁₎. That is,

k(x−y)−λ(m,1)(T(m,1)x−T(m,1)y)k ≤θ(m,1)kx−yk. (3.2)

On the other hand, by the the assumption that T(m,2) :C →H is relaxed (µ(m,2), ν(m,2)

)-cocoercive and L(m,2)-Lipschitz, we arrive at

k(x−y)−λ(m,2)(T(m,2)x−T(m,2)y)k2

=kx−yk2₋₂_λ

(m,2)hT(m,2)x−T(m,2)y, x−yi+λ2(m,2)kT(m,2)x−T(m,2)yk2

≤ kx−yk2−2λ(m,2)(−µ(m,2)kT2x−T2yk2+ν(m,2)kx−yk2) +λ2(m,2)L 2

(m,2)kx−yk 2

(7)

where θ(m,2) =

q

1−2λ(m,2)ν(m,2)+ 2λ(m,2)µ(m,2)L2₍_m,₂₎+λ2₍_m,₂₎L2₍_m,₂₎. This implies that

k(x−y)−λ(m,2)(T(m,2)x−T(m,2)y)k ≤θ(m,2)kx−yk. (3.3)

Substituting (3.2) and (3.3) into (3.1), we see from the restriction (f) that

kPC(λ(m,1)T(m,1) −λ(m,2)T(m,2))x−PC(λ(m,1)T(m,1)−λ(m,2)T(m,2))k ≤ kx−yk.

This shows thatPC(λ(m,1)T(m,1)−λ(m,2)T(m,2)) is nonexpansive for eachm∈ {1,2, . . . , N}.

Fixp∈ F and Put

zn = N X

m=1

η(m,n)PC(λ(m,1)T(m,1)xn−λ(m,2)T(m,2)xn).

It follows that kzn−pk ≤ kxn−pk. This in turn implies that

kyn−pk=kδnSxn+ (1−δn)zn−pk

≤δnkSxn−Spk+ (1−δn)kzn−pk

≤ kxn−pk.

It follows that

kxn+1−pk ≤αnkf(xn)−pk+βnkxn−pk+γnkyn−pk

≤αnκkxn−pk+αnkf(p)−pk+βnkxn−pk+γnkxn−pk

= 1−αn(1−κ)

kxn−pk+αnkf(p)−pk.

By mathematical inductions, we arrive at kxn−pk ≤max{

kf(p)−pk

1−κ ,kx1−pk}, ∀n ≥1.

This completes the proof that the sequence{xn}is bounded. Since the mappingPC(λ(m,1)T(m,1)−

λ(m,2)T(m,2)) is nonexpansive for each m∈ {1,2, . . . , N}, we see that that

kzn+1−znk=k N X

m=1

η(m,n+1)PC(λ(m,1)T(m,1)xn+1−λ(m,2)T(m,2)xn+1)

−

N X

m=1

η(m,n)PC(λ(m,1)T(m,1)xn−λ(m,2)T(m,2)xn)k

≤ kxn+1−xnk+M1

N X

m=1

|η(m,n+1)−η(m,n)|,

(8)

where

M1 = max{sup

n≥1

kPC(λ(m,1)T(m,1)xn−λ(m,2)T(m,2)xn)k,∀1≤m ≤N}.

This in turn implies that

kyn+1−ynk=kδn+1Sxn+1+ (1−δn+1)zn+1−δnSxn−(1−δn)znk

≤δn+1kSxn+1−Sxnk+ (1−δn+1)kzn+1−znk+|δn+1−δn|kSxn−znk

≤ kxn+1−xnk+M1

N X

m=1

|η(m,n+1)−η(m,n)|+|δn+1−δn|kSxn−znk.

(3.5) Putln= xn+1

−βnxn

1−βn . It follows that

xn+1 = (1−βn)ln+βnxn, ∀n ≥1. (3.6)

Now, we estimate kln+1−lnk.In view of

ln+1−ln

= αn+1 1−βn+1

f(xn+1) +

1−βn+1−αn+1

1−βn+1

yn+1−

αn

1−βn

f(xn)−

1−βn−αn

1−βn

yn

= αn+1 1−βn+1

(f(xn+1)−yn+1) +

αn

1−βn

(yn−f(xn)) +yn+1−yn,

we obtain that kln+1−lnk ≤

αn+1

1−βn+1

kf(xn+1)−yn+1k+

αn

1−βn

kyn−f(xn)k+kyn+1−ynk. (3.7)

Substituting (3.5) into (3.7), we obtain that kln+1−lnk − kxn+1−xnk ≤

αn+1

1−βn+1

kf(xn+1)−yn+1k+

αn

1−βn

kyn−f(xn)k

+M1

N X

m=1

|η(m,n+1)−η(m,n)|+|δn+1−δn|kSxn−znk

It follows from the restrictions (b), (c), (d) and (e) that lim sup

n→∞

(kln+1−lnk − kxn+1−xn+1k)<0.

In view of Lemma 2.3, we see that lim

(9)

Thanks to (3.6), we obtain that

xn+1−xn= (1−βn)(ln−xn).

It follows from (3.8) that

lim

n→∞kxn+1−xnk= 0. (3.9)

On the other hand, we have

kyn−xnk ≤

1

γn

kxn+1−xnk+

αn

γn

kxn−f(xn)k.

From the conditions (b) and (c), we obtain that

lim

n→∞kyn−xnk= 0. (3.10)

Define a mapping R :C→C by

Rx =δSx+ (1−δ)

N X

m=1

η(m,n)PC(λ(m,1)T(m,1)x−λ(m,2)T(m,2)x), ∀x∈C,

where δ = limn→∞δn. From Lemma 2.1, we see that R is nonexpansive with F(R) =

F ∩N

m=1PC(λ(m,1)T(m,1)−λ(m,2)T(m,2))TF(S)

=F. Next, we show that Rxn−xn→0 as n→ ∞. Note that

kRxn−xnk ≤ kδSxn+ (1−δ) N X

m=1

η(m,n)PC(λ(m,1)T(m,1)xn−λ(m,2)T(m,2)xn)−ynk+kyn−xnk

≤ |δ−δn|M2+kyn−xnk,

where

M2 = max{sup

n≥1

kSxn− N X

m=1

η(m,n)PC(λ(m,1)T(m,1)xn−λ(m,2)T(m,2)xn)k,∀1≤m≤N}.

In view of the restriction (e), we see from (3.10) that

lim

n→∞kRxn−xnk= 0. (3.11)

(10)

Next, we show that lim sup_n_→∞hf(¯x)−x, x¯ n −x¯i ≤ 0. To show it, we can choose a

sequence {xni} of {xn} such that

lim sup

n→∞

hf(¯x)−x, x¯ n−x¯i= lim

i→∞hf(¯x)−x, x¯ xni −x¯i. (3.12)

Since{xni}is bounded, there exists a subsequence{xn_ij}of{xni}which converges weakly

to b. Without loss of generality, we may assume that xni * ρ. From Lemma 2.1 and

Lemma 2.2, we see that

ρ∈F(R) = F ∩N

m=1PC(λ(m,1)T(m,1)−λ(m,2)T(m,2))

\

F(S)=F.

This completes the proof of (3.12).

Finally, we show thatxn →x¯ asn → ∞.Note that

kxn+1−x¯k2

=αnhf(xn)−f(¯x), xn+1−x¯i+αnhf(¯x)−x, x¯ n+1−x¯i+βnhxn−x, x¯ n+1−x¯i

+γnhyn−x, x¯ n+1−x¯i

≤αnκkxn−x¯kkxn+1−x¯k+αnhf(¯x)−x, x¯ n+1−x¯i+βnkxn−x¯kkxn+1−x¯k

+γnkyn−x¯kkxn+1−x¯k

≤ 1−αn(1−κ)

2 (kxn−x¯k

2₊_k_x

n+1−x¯k2) +αnhf(¯x)−x, x¯ n+1−x¯i.

This in turn implies that

kxn+1−x¯k2 ≤ 1−αn(1−κ)

kxn−x¯k2+ 2αnhf(¯x)−x, x¯ n+1−x¯i.

In view of the restriction (b), and (3.12), we find from Lemma 2.4 that This completes the proof.

If T(m,1) =I, whereI denotes the identity mapping, and λ(m,1) = 1, then we find from

Theorem 2.1 the following.

Corollary 2.2. Let C be a nonempty closed and convex subset of a real Hilbert space H.

Letf :C→C be a contractive mapping with the contractive constant κ, andS :C→C a nonexpansive mapping with fixed points. LetTm :C →H be a relaxed(µm, νm)-cocoercive

(11)

is some positive integer. Assume that F = ∩N

m=1V I(C, Tm)TF(S) 6= ∅. Let {xn} be a

sequence generated by the following algorithm



     

     

x1 ∈C,

yn=δnSxn+ (1−δn)PN_m₌₁η(m,n)PC(xn−λmTmxn),

xn+1 =αnf(xn) +βnxn+γnyn, n ≥1,

restrictions:

(a) αn+βn+γn= 1, ∀n≥1;

n=1α=∞;

(d) PN

m=1η(m,n)= 1, and limn→∞η(m,n) =ηm ∈(0,1);

(e) limn→∞δn=δ∈(0,1),

and {λm} is a sequence such that

(f) λm ≤ 2νm

−2µmL2m L2

m .

hf(y)−y, y−xi ≥0, ∀x∈ F.

If N = 1, and f(x) = u, where u is a fixed element in C, for all x ∈ C, then we have the following.

Corollary 2.3. Let C be a nonempty closed and convex subset of a real Hilbert space

H. Let S : C → C be a nonexpansive mapping with fixed points. Let T1 : C → H

be a relaxed (µ1, ν1)-cocoercive and L1-Lipschitz continuous mapping and T2 : C → H

(12)

GV I(C, T1, T1)TF(S)6=∅. Let {xn} be a sequence generated by the following algorithm 

     

     

x1 ∈C,

yn=δnSxn+ (1−δn)PC(λ1T1xn−λ2T2xn),

xn+1 =αnu+βnxn+γnyn, n≥1,

restrictions:

(a) αn+βn+γn= 1, ∀n≥1;

n=1α=∞;

(d) limn→∞δn=δ∈(0,1),

and λ1 and λ2 are two constants such that

(e) p1−2λ1ν1+ 2λ1µ1L21+λ21L21+

p

1−2λ2ν2+ 2λ2µ2L22+λ22L22 ≤1.

hf(y)−y, y−xi ≥0, ∀x∈ F.

References

[1] M.A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004) 199-277.

[2] X. Qin, M.A. Noor, GeneralWiener-Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces, Appl. Math. Comput. 201 (2008) 716-722.

[3] M.A. Noor, On general quasi-variational inequalities, J. King Saud University-Science, 24 (2012) 81-88.

[4] M.A. Noor, Diffierentiable non-convex functions and general variational inequalities, Appl. Math. Comput. 199 (2008) 623-630.

(13)

[6] Y.J. Cho, X. Qin, Systems of generalized nonlinear variational inequalities and its projection meth-ods, Nonlinear Anal. 69 (2008) 4443-4451.

[7] S.Y. Cho, S.M. Kang, Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 24 (2011) 224-228.

[8] D.R. Sahu, J.C. Yao, The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces. J. Optim. Theory Appl. 151 (2011) 641-655.

[9] O. G¨uller, On the convergence of the proximal point algorithmfor convex minimization. SIAM J. Control Optim. 29 (1991) 403-419.

[10] L.J. Lin, Y.J. Huang, Generalized vector quasi-equilibrium problems with applications to common fixed point theorems and optimization problems. Nonlinear Anal 66 (2007) 1275-1289.

[11] J. Eckstein, Approximate iterations in Bregman-function-based proximal algorithms. Math. Pro-gram. 83 (1998) 113-123.

[12] S.S. Chang, H.W.J. Lee, C.K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. 70 (2009) 3307-3319.

[13] S. Husain, S. Gupta, A resolvent operator technique for solving generalized system of nonlinear relaxed cocoercive mixed variational inequalities. Adv. Fixed Point Theory 2 (2012) 18-28.

[14] S.M. Kang, S.Y. Cho, Z. Liu, Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010 (2010) 827082.

[15] J.K. Kim, S.Y. Cho, X. Qin, Hybrid projection algorithms for generalized equilibrium problems and strictly pseudocontractive mappings. J. Inequal. Appl. 2010 (2010) Article ID 312602.

[16] J.K. Kim, Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-φ-nonexpansive mappings, Fixed Point Theory Appl. 2011 2011:10.

[17] J. Ye, J. Huang, Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces, Journal of Mathematical and Computational Science, 1 (2011) 1-18.

[18] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46-55.

[19] N.K. Mathato, C. Nahak, Equilibrium problem under various types of convexities in Banach space, J. Math. Comput. Sci. 1 (2011) 77-88.

(14)

[21] L. Yu, M. Liang, Convergence theorems of solutions of a generalized variational inequality, Fixed Point Theory Appl. 2011 2011:19.

[22] X. Qin, Y.J. Cho, S.M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009) 20-30.

[23] Stampacchia G: Formes bilineaires coercitives sur les ensembles convexes. CR Acad Sci Paris 1964, 258: 4413-4416.

[24] R.U. Verma, Rockafellar’s celebrated theorem based on A-maximal monotonicity design, Appl. Math. Lett. 21 (2008) 355-360.

[25] R.E. Bruck, Properties of fixed point sets of nonexpansive mappings in Banach spaces, Tras. Amer. Math. Soc. 179 (1973) 251-262.

[26] F.E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Pure. Math. 18 (1976) 78-81.

[27] T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter non-expansive semigroups without Bochne integrals, J. Math. Anal. Appl. 305 (2005) 227-239.