A PROJECTION METHOD FOR SYSTEMS OF GENERALIZED VARIATIONAL INEQUALITIES WITH µ-LIPSCHITZ CONTINUOUS MAPPINGS

OF GENERALIZED VARIATIONAL INEQUALITIES WITH µ-LIPSCHITZ CONTINUOUS MAPPINGS

VASILE PREDA and MIRUNA BELDIMAN

We consider the problem of approximate solvability of a system of nonlinear varia- tional inequalities, introduced by Verma [11]. We discuss the problem on a closed convex subset of a Hilbert space, under relaxed (γ,r)-cocoercivity andµ-Lipschitz continuity in the second variable assumption. Also, the convergence of the pro- posed algorithm is studied.

AMS 2000 Subject Classification: 65K05, 47J20, 91B50.

Key words: system of variational inequalities, projection method, relaxed cocoer- civity,µ-Lipschitz continuity.

1. INTRODUCTION

Recently, Verma [9] introduced a new class of relaxed cocoercive map- pings. Based on the convergence of projection methods, he also discussed the approximate solvability of a system of nonlinear variational inequality prob- lems introduced by himself [11].

Projection methods, and their various forms, have been widely applied to many problems [4, 8, 12, 14]. The origin of the projection method can be traced back to Lions and Stampacchia [6]. For a good account on general variational inequality problems and related mappings see [1, 2, 3, 5, 13, 15].

Relative to the approximate solvability of a system of nonlinear varia- tional inequalities, some results were given by Nie, Liu, Kim and Kang [7], Verma [9, 10, 11], and others. We note that their results are obtained under the assumptions that the nonlinear mapping T involved is µ-Lipschitz in the first variable.

Based on the projection method, in this paper we consider the approxi- mate solvability of a system of nonlinear relaxed cocoercive variational inequal- ities in a Hilbert space setting, where the nonlinear mappingT :K×K→H is relaxed (γ, r)-cocoercive [9] andµ-Lipschitz in the second variable. We show that the sequences generated by the corresponding algorithm from [9] converge to the solution of the variational inequality.

MATH. REPORTS10(60),1 (2008), 97–103

2. PRELIMINARIES

LetH be a real Hilbert space with inner product h·,·i and normk · k.

Let T : K×K → H be a mapping on K×K, where K is a closed convex subset of H. As in Verma [9], we consider a system of nonlinear variational inequality (SNVI) as follows: find the elements x^∗, y^∗ ∈K such that

(2.1) hρT(y^∗, x^∗) +x^∗−y^∗, x−x^∗i ≥0, ∀x∈Kandρ >0, (2.2) hηT(x^∗, y^∗) +y^∗−x^∗, x−y^∗i ≥0, ∀x∈K andη >0.

This problem is equivalent to the projection formulas x^∗ =P_K[y^∗−ρT(y^∗, x^∗)], ρ >0, y^∗ =P_K[x^∗−ηT(x^∗, y^∗)], η >0, where P_K is the projection ofH onto K.

Also, Verma [9] considered the cases whereη= 0 andKis a closed convex cone of H. We see that, for η = 0,the SNVI problem (2.1)–(2.2) reduces to the following nonlinear variational inequality (NVI) problem: find an element x^∗∈K such that

hT(x^∗, x^∗), x−x^∗i ≥0, ∀x∈K.

In the case whereKis a closed convex cone ofH,the SNVI problem (2.1)–(2.2) is equivalent to the following system of nonlinear complementarities (SNC):

determine the elements x^∗, y^∗ ∈K such thatT(x^∗, y^∗), T(y^∗, x^∗)∈K^∗ and (2.3) hρT(y^∗, x^∗) +x^∗−y^∗, x^∗i= 0, ρ >0,

(2.4) hηT(x^∗, y^∗) +y^∗−x^∗, y^∗i= 0, η >0,

where K^∗={f ∈H : hf, xi ≥0, ∀x∈K} is the polar cone toK.

Now, in the context of approximate solvability of nonlinear variational inequality problems based on iterative procedures we recall a preliminary result and some definitions.

Lemma2.1 (see [5]). For an element z∈H,we have x∈K andhx−z, y−xi ≥0, ∀y∈K, if and only if x=P_K(x).

We now define some useful concepts in connection with our problem.

Definition 2.1. A mapping S : H → H is called monotone if for each x, y∈H we have

hS(x)−S(y), x−yi ≥0.

Definition 2.2. Let r >0 be a real constant. A mapping S :H →H is called r-strong monotone if for each x, y∈H we have

hS(x)−S(y), x−yi ≥rkx−yk². This implies that

kS(x)−S(y)k ≥rkx−yk,

that is, S is r-expansive. Whenr = 1,we say that S is expansive.

Definition 2.3. Let s > 0. A mapping S :H → H is called s-Lipschitz continuous (or Lipschitzian) if for each x, y∈H we have

kS(x)−S(y)k ≤skx−yk.

Definition 2.4 (see [10]). Let µ > 0. A mapping S : H → H is µ- cocoercive if for each x, y∈H we have

hS(x)−S(y), x−yi ≥µkS(x)−S(y)k²

Remark 2.1. We note that every µ-cocoercive mapping S is (1/µ)-Lip- schitz continuous. Also, every r-strong monotone mapping S is a monotone and expansive mapping.

Definition 2.5. Let γ > 0. A mapping S : H → H is called relaxed γ-cocoercive if for eachx, y∈H we have

hS(x)−S(y), x−yi ≥(−γ)kS(x)−S(y)k².

Definition 2.6 (see [9]). A mappingS :H → H is called relaxed (γ, r)- cocoercive if for each x, y∈H we have

hS(x)−S(y), x−yi ≥(−γ)kS(x)−S(y)k²+rkx−yk². We see that forγ = 0 the mappingS isr-strongly monotone.

3. VERMA’S ALGORITHM FOR SNVI. CONVERGENCE ANALYSIS AND THE MAIN RESULT

In the first part of this section we present the projection algorithms considered by Verma [9]. We study then convergence for projection methods in the frame of the approximate solvability of the SNVI problem (2.1)–(2.2) in the case where the mapping T is relaxed (γ, r)-cocoercive andµ-Lipschitz in the second variable.

Algorithm 3.1 (see [9]). Let x⁰, y⁰ ∈ K be two arbitrary initial points.

Generate sequences {x^k} and {y^k} satisfying x^k+1= (1−a_k)x^k+a_kP_K

y^k−ρT y^k, x^k , y^k =P_K

x^k−ηT x^k, y^k ,

where ρ and η >0 are real constants and 0≤a_k≤1 for each k= 0,1,2, . . . , and

∞

P

k=0

a_k=∞.

In the caseη= 0,Algorithm 3.1 reduces to an algorithm for solving the NVI problem.

Algorithm 3.2. For an arbitrarily chosen initial point x⁰ ∈K, generate the sequence{x^k}satisfying

x^k+1= (1−ak)x^k+akPK

x^k−ρT x^k, x^k ,

where ρ is a positive constant and 0 ≤ a_k ≤ 1 for each k = 0,1,2, . . . , and

∞

P

k=0

a_k= +∞.

Now, based on Algorithm 3.1, we consider the approximate solvabi- lity of the SNVI problem (2.1)–(2.2) involving mappings in a Hilbert space setting that are relaxed (γ, r)-cocoercive and µ-Lipschitz continuous in the second variable.

Theorem3.1. Let H be a real Hilbert space and K a nonempty closed convex subset of H. Suppose that the mapping T : K ×K → H is relaxed (γ, r)-cocoercive and µ-Lipschitz continuous in the second variable. Suppose that (x^∗, y^∗) ∈ K ×K is a solution to the SNVI problem (2.1)–(2.2), the sequences {x^k}and {y^k} are generated by Algorithm 3.1where η, r, γ, µand ρ all are positive numbers satisfying the conditions

(3.1) 2ηr <1, 2ηγµ²+η²µ² <1, µ²(2γ+η)<2r,

(3.2) ρ²µ²(1−2ηγµ²−η²µ²)+2ρ(2ηr²−r+γµ²−2ηγ²µ⁴−η²γµ⁴)<2ηr.

Then the sequences {x^k}and {y^k}converge to x^∗ and y^∗ respectively.

Proof. Using Lemma 2.1 and the assumption that (x^∗, y^∗) is a solution to the SNVI problem (2.1)–(2.2), we get

x^∗=P_K[y^∗−ρT(y^∗, x^∗)], y^∗ =P_K[x^∗−ηT(x^∗, y^∗)].

Since the sequences {x^∗} and {y^∗}are given by Algorithm 3.1, we have (3.3)

x^k+1−x^∗

≤(1−a_k)

x^k−x^∗ +ak

y^k−y^∗−ρ

T(y^k, x^k)−T(y^∗, x^∗) .

Also, since T is a relaxed (γ, r)-cocoercive mapping and a µ-Lipschitz continuous mapping in the second variable, we have

(3.4) ky^k−y^∗−ρ[T(y^k, x^k)−T(y^∗, x^∗)]k²≤ ky^k−y^∗k²+

+2ργkT(y^k, x^k)−T(y^∗, x^∗)k²−2ρrky^k−y^∗k²+ρ²µ²kx^k−x^∗k² ≤

≤ ky^k−y^∗k²+ 2ργµ²kx^k−x^∗k²−2ρrky^k−y^∗k²+ρ²µ²kx^k−x^∗k² =

= (1−2ρr)ky^k−y^∗k²+ (2γ+ρ)ρµ²kx^k−x^∗k². We also have

(3.5) ky^k−y^∗k² =kP_K[x^k−ηT(x^k, y^k)]−P_K[x^∗−ηT(x^∗, y^∗)]k² ≤

≤ kx^k−x^∗−η[T(x^k, y^k)−T(x^∗, y^∗)]k²=

=kx^k−x^∗k²−2ηhT(x^k, y^k)−T(x^∗, y^∗), x^k−x^∗i+η²kT(x^k, y^k)−T(x^∗, y^∗)k². Since T is relaxed (γ, r)-cocoercive and µ-Lipschitz continuous in the second variable, by (3.5) we have

ky^k−y^∗k² ≤ kx^k−x^∗k²−2η(−γkT(x^k, y^k)−T(x^∗, y^∗)k²+rkx^k−x^∗k²)+

+η²µ²ky^k−y^∗k²≤ kx^k−x^∗k²+ 2ηγµ²ky^k−y^∗k²+η²ky^k−y^∗k², hence

(3.6) ky^k−y^∗k²≤ (1−2ηr)

(1−2ηγµ²−η²µ²)kx^k−x^∗k². Using (3.4) and (3.6) we get

ky^k−y^∗−ρ[T(x^k, y^k)−T(x^∗, y^∗)]k² ≤

≤

(1−2ρr)(1−2ηr)

(1−2ηγµ²−η²µ²) + (2γ+ρ)ρµ²

kx^k−x^∗k², i.e.,

(3.7) ky^k−y^∗−ρ[T(x^k, y^k)−T(x^∗, y^∗)]k ≤ωkx^k−x^∗k, where

(3.8) ω=

(1−2ρr)(1−2ηr)

(1−2ηγµ²−η²µ²)+ (2γ+ρ)ρµ² 1/2

.

It follows from (3.3) and (3.7) that

kx^k+1−x^∗k ≤(1−ak)kx^k−x^∗k+akωkx^k−x^∗k= (3.9)

= [1−(1−ω)a_k]kx^k−x^∗k ≤

k

Y

j=0

[1−(1−ω)a_j]kx⁰−x^∗k,

whereωis given by (3.8). Inequality (3.2) implies 0< ω <1. By Algorithm 3.1 (see [5]), we have

(3.10) lim

k→∞

k

Y

j=0

[1−(1−ω)a_j] = 0.

Now (3.9) and (3.10) imply that the sequence {x^k} converges tox^∗.

Now, by (3.1) we have 0< _(1−2ηγµ^(1−2ηr)2−η²µ²) <1 and then (see [5]) we obtain ky^k−y^∗k ≤ kx^k−x^∗k.

This inequality and the fact that the sequence{x^k}converges tox^∗imply that the sequence{y^k} converges toy^∗. The proof is complete.

Remark 3.1. The system (3.1)–(3.2) is consistent. For example, if η = r =γ =µ= 10⁻³,we can take 0< ρ <(−3 +√

2·10⁶+ 9)10⁻³.

Corollary3.1. Let H be a real Hilbert space let K a nonempty closed convex subset of H. Let T : K ×K → H be r-strongly monotone and µ- Lipschitz continuous in the second variable. If (x^∗, y^∗) is a solution to the SNVI problem (2.1)–(2.2) and the sequences {x^k} and {y^k} are generated by Algorithm 3.1, where η, r, γ, µ, ρall are positive numbers such that

2ηr <1, ηµ <1, ηµ² <2r and

µ²(1−η²µ²)ρ²+ 2r(2ηr−1)ρ <2ηr,

then the sequences {x^k}and {y^k} converge to x^∗ and y^∗ respectively.

Corollary3.2. Let H be a real Hilbert space let K a nonempty closed convex subset of H. Let T :K×K → H be relaxed (γ, r)-cocoercive and µ- Lipschitz continuous in the second variable. Suppose that x^∗ ∈K is a solution to the NVI problem (2.3), the sequence {x^k} is generated by Algorithm 3.2 where ρ, r, γ, µ all are positive numbers such that µ²γ < r,(ρ+ 2γ)µ² <2r.

Then the sequence {x^k}converges to x^∗.

REFERENCES

[1] H. Brezis, Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam, 1973.

[2] R.W. Cottle, F. Giannessi and J.L. Lions, Variational Inequalities: Theory and Appli- cations. Wiley, New York, 1980.

[3] J.C. Dunn,Convexity, monotonicity and gradient processes in Hilbert spaces. J. Math.

Anal. Appl.53 (1976), 145–158.

[4] B.S. He,A new method for a class of linear variational inequalities. Math. Programing.

66 (1994), 137–144.

[5] D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities. Aca- demic Press, London, 1980.

[6] J.J. Lions and G. Stampacchia, Variational inequalities. Comm. Pure Appl. Math.20 (1967), 493–519.

[7] H. Nie, Z. Liu, K.H. Kim and S.M. Kang, A system of nonlinear variational inequali- tites involving strongly monotone and pseudocontractive mappings. Adv. Nonlinear Var.

Inequal.6(2003), 91–99.

[8] M.A. Noor and Th.M. Rassias,Projection methods for monotone variational inequalities.

J. Math. Anal. Appl.237(2) (1999), 405–412.

[9] R.U. Verma,Generalized system for relaxed cocoercive variational inequalities and pro- jection methods. J. Optim. Theory Appl.115(2002), 211–230.

[10] R.U. Verma,Projection methods and a new system of cocoercive variational inequality problems. Int. J. Differ. Equ. Appl.6(2002), 359–367.

[11] R.U. Verma,Projection methods, algorithms and a new system of nonlinear variational inequalities. Comput. Math. Appl.41(2001), 1025–1031.

[12] R.U. Verma,A class of quasivariational inequalities involving cocoercive mappings. Adv.

Nonlinear Var. Inequal.2(1999), 1–12.

[13] R. Wittmann, Approximation of fixed points of nonexpansive mappings. Arch. Math.

(Basel)58(1992), 486–491.

[14] N.H. Xiu and J.Z. Zhang, Local convergence analysis of projection type algorithms:

univied approach. J. Optim. Theory Appl.115 (2002), 230–251.

[15] E. Zeidler,Nonlinear Functional Analysis and Its Applications,II. Springer, New York, 1990.

Received 22 February 2007 University of Bucharest

Faculty of Mathematics and Computer Science St. Academiei 14

010014 Bucharest, Romania and

Romanian Academy Institute of Mathematical Statistics

and Applied Mathematics Calea 13 Septembrie no. 13 050711 Bucharest, Romania