GENERAL SYSTEM OF STRONGLY PSEUDOMONOTONE NONLINEAR VARIATIONAL INEQUALITIES BASED ON PROJECTION SYSTEMS

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Strongly Pseudomonotone Nonlinear Variational

Inequalities R. U. Verma vol. 8, iss. 1, art. 6, 2007

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GENERAL SYSTEM OF STRONGLY

PSEUDOMONOTONE NONLINEAR VARIATIONAL INEQUALITIES BASED ON PROJECTION SYSTEMS

R. U. VERMA

Department of Mathematics, University of Toledo Toledo, Ohio 43606, USA

EMail:verma99@msn.com

Received: 26 February, 2006

Accepted: 11 December, 2006

Communicated by: R.N. Mohapatra 2000 AMS Sub. Class.: 49J40, 65B05, 47H20.

Key words: Strongly pseudomonotone mappings, Approximation solvability, Projection methods, System of nonlinear variational inequalities.

Abstract: Let K1 and K2, respectively, be non empty closed convex subsets of real Hilbert spacesH1 andH2.TheApproximation−solvabilityof a generalized system of nonlinear variational inequality(SN V I)problems based on the convergence of projection methods is discussed. The SNVI problem is stated as follows: find an element (x^∗, y^∗)∈K1×K2such that

hρS(x^∗, y^∗), x−x^∗i ≥0, ∀x∈K1and forρ >0, hηT(x^∗, y^∗), y−y^∗i ≥0, ∀y∈K2and forη >0, whereS :K1×K2→H1andT :K1×K2→H2are nonlinear mappings.

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1. Introduction

Projection-like methods in general have been one of the most fundamental tech- niques for establishing the convergence analysis for solutions of problems arising from several fields, such as complementarity theory, convex quadratic programming, and variational problems. There exists a vast literature on approximation-solvability of several classes of variational/hemivariational inequalities in different space settings. The author [6,7] introduced and studied a new system of nonlinear variational inequalities in Hilbert space settings. This class encompasses several classes of nonlinear variational inequality problems. In this paper we intend to explore, based on a general system of projection-like methods, the approximation-solvability of a system of nonlinear strongly pseudomonotone variational inequalities in Hilbert spaces. The obtained results extend/generalize the results in [1], [5] – [7] to the case of strongly pseudomonotone system of nonlinear variational inequalities. Approximation solvability of this system can also be established using the resolvent operator technique but in the more relaxed setting of Hilbert spaces. For more details, we refer the reader to [1] – [10].

LetH₁ andH₂ be two real Hilbert spaces with the inner producth·,·iand norm k · k. Let S : K1 × K2 → H1 and T : K1 × K2 → H2 be any mappings on K1 × K2, where K1 and K2 are nonempty closed convex subsets of H1 and H2, respectively. We consider a system of nonlinear variational inequality (abbreviated as SNVI) problems: determine an element(x^∗, y^∗)∈K₁ ×K₂ such that

(1.1) hρS(x^∗, y^∗), x−x^∗i ≥0∀x∈K₁

(1.2) hηT(x^∗, y^∗), y −y^∗i ≥0∀y∈K₂, whereρ, η >0.

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The SNVI(1.1)−(1.2)problem is equivalent to the following projection formulas x^∗ =P_k[x^∗−ρS(x^∗, y^∗)] forρ >0

y^∗ =Q_k[y^∗ −ηT(x^∗, y^∗)] forη >0,

wherePk is the projection ofH1ontoK1 andQK is the projection ofH2 ontoK2. We note that the SNVI(1.1)−(1.2)problem extends the NVI problem: determine an elementx^∗ ∈K₁ such that

(1.3) hS(x^∗), x−x^∗i ≥0, ∀x∈K₁.

Also, we note that the SNVI (1.1)− (1.2) problem is equivalent to a system of nonlinear complementarities (abbreviated as SNC): find an element(x^∗, y^∗)∈K₁× K₂such thatS(x^∗, y^∗)∈K₁^∗, T(x^∗, y^∗)∈K₂^∗, and

(1.4) hρS(x^∗, y^∗), x^∗i= 0 for ρ >0,

(1.5) hηT(x^∗, y^∗), y^∗i= 0 for η >0,

whereK₁^∗ andK₂^∗, respectively, are polar cones toK₁andK₂defined by K₁^∗ ={f ∈H1 :hf, xi ≥0, ∀x∈K1}.

K₂^∗ ={g ∈H₂ :hg, yi ≥0, ∀g ∈K₂}.

Now, we recall some auxiliary results and notions crucial to the problem on hand.

Lemma 1.1. For an elementz ∈H, we have

x∈K and hx−z, y−xi ≥0, ∀y∈K if and only if x=P_k(z).

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Lemma 1.2 ([3]). Let {α^k},{β^k}, and {γ^k}be three nonnegative sequences such that

α^k+1 ≤(1−t^k)α^k+β^k+γ^k for k = 0,1,2, ..., wheret^k ∈ [0,1],P∞

k=0t^k = ∞, β^k =o(t^k),andP∞

k=0γ^k < ∞.Thenα^k → 0as k → ∞.

A mapping T : H → H from a Hilbert space H into H is called monotone if hT(x)−T(y), x−yi ≥0for all x, y ∈H.The mappingT is(r)−strongly monotone if for eachx, y ∈H,we have

hT(x)−T(y), x−yi ≥r||x−y||² for a constantr >0.

This implies that kT(x) − T(y)k ≥ rkx − yk, that is, T is (r)-expansive, and when r = 1, it is expansive. The mapping T is called (s)-Lipschitz continuous (or Lipschitzian) if there exists a constant s ≥ 0 such thatkT(x)−T(y)k ≤ skx−yk, ∀x, y ∈H. T is called(µ)-cocoercive if for eachx, y ∈H,we have

hT(x)−T(y), x−yi ≥µ||T(x)−T(y)||² for a constantµ >0.

Clearly, every(µ)-cocoercive mappingT is(¹_µ)-Lipschitz continuous. We can eas- ily see that the following implications on monotonicity, strong monotonicity and expansiveness hold:

strong monotonicity

⇓ monotonicity

⇒expansiveness

T is called relaxed(γ)-cocoercive if there exists a constantγ >0such that hT(x)−T(y), x−yi ≥(−γ)kT(x)−T(y)k², ∀x, y ∈H.

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T is said to be(r)-strongly pseudomonotone if there exists a positive constantrsuch that

hT(y), x−yi ≥0⇒ hT(x), x−yi ≥rkx−yk², ∀x, y ∈H.

T is said to be relaxed(γ, r)-cocoercive if there exist constantsγ,r >0such that hT(x)−T(y), x−yi ≥(−γ)kT(x)−T(y)k²+rkx−yk².

Clearly, it implies that

hT(x)−T(y), x−yi ≥(−γ)kT(x)−T(y)k², that is,T is relaxed(γ)-cocoercive.

T is said to be relaxed(γ, r)-pseudococoercive if there exist positive constantsγ andrsuch that

hT(y), x−yi ≥0⇒ hT(x), x−yi ≥(−γ)kT(x)−T(y)k²+rkx−yk², ∀x, y ∈H.

Thus, we have following implications:

(r)-strong monotonicity

⇓

relaxed(γ, r)-cocoercivity

⇓

relaxed(γ, r)-pseudococoercivity

⇒ strong(r)-pseudomonotonicity

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2. General Projection Methods

This section deals with the convergence of projection methods in the context of the approximation-solvability of the SNVI(1.1)−(1.2)problem.

Algorithm 1. For an arbitrarily chosen initial point(x⁰, y⁰) ∈ K₁ ×K₂,compute the sequences{x^k}and{y^k}such that

x^k+1 = (1−a^k−b^k)x^k+a^kP_k[x^k−ρS(x^k, y^k)] +b^ku^k y^k+1 = (1−α^k−β^k)y^k+α^kQK[y^k−ηT(x^k, y^k)] +β^kv^k,

whereP_K is the projection of H₁ ontoK₁, Q_K is the projection ofH₂ ontoK₂, ρ, η > 0 are constants, S : K₁ ×K₂ → H₁ and T : K₁ ×K₂ → H₂ are any two mappings, andu^k andv^k,respectively, are bounded sequences in K₁ andK₂.The sequences{a^k},{b^k},{α^k},and {β^k}are in[0,1]with(k ≥0)

0≤a^k+b^k ≤1, 0≤α^k+β^k≤1.

x^k+1 = (1−a^k−b^k)x^k+a^kP_k[x^k−ρS(x^k, y^k)] +b^ku^k y^k+1 = (1−a^k−b^k)y^k+a^kQ_K[y^k−ηT(x^k, y^k)] +b^kv^k,

whereP_K is the projection of H₁ ontoK₁, Q_K is the projection ofH₂ ontoK₂, ρ, η > 0 are constants, S : K₁ ×K₂ → H₁ and T : K₁ ×K₂ → H₂ are any two mappings, andu^k andv^k,respectively, are bounded sequences in K₁ andK₂.The sequences{a^k}and{b^k},are in[0,1]with(k≥0)

0≤a^k+b^k ≤1.

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x^k+1 = (1−a^k)x^k+a^kPk[x^k−ρS(x^k, y^k)]

y^k+1 = (1−a^k)y^k+a^kQ_K[y^k−ηT(x^k, y^k)],

whereP_K is the projection of H₁ ontoK₁, Q_K is the projection ofH₂ ontoK₂, ρ, η > 0 are constants, S : K₁ ×K₂ → H₁ and T : K₁ ×K₂ → H₂ are any two mappings. The sequence{a^k} ∈[0,1]fork ≥0.

We consider, based on Algorithm 2, the approximation solvability of the SNVI (1.1)−(1.2)problem involving strongly pseudomonotone and Lipschitz continuous mappings in Hilbert space settings.

Theorem 2.1. LetH₁ andH₂ be two real Hilbert spaces and, K₁ andK₂, respec- tively, be nonempty closed convex subsets ofH₁ andH₂.LetS :K₁ ×K₂ →H₁ be strongly (r)− pseudomonotone and (µ)−Lipschitz continuous in the first variable and letS be(ν)−Lipschitz continuous in the second variable. LetT : K₁×K₂ → H₂ be strongly (s)−pseudomonotone and (β)−Lipschitz continuous in the second variable and letT be(τ)−Lipschitz continuous in the first variable. Letk · k^∗denote the norm onH1×H2 defined by

k(x, y)k^∗ = (kxk+kyk)∀(x, y)∈H₁×H₂. In addition, let

θ = s

1−2ρr+ρ+ ρµ²

2

+ρ²µ² +ητ <1

σ= s

1−2ηr+η+ ηβ²

2

+η²β²+ρν <1,

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let(x^∗, y^∗) ∈ K₁×K₂ form a solution to the SNVI(1.1)−(1.2)problem, and let sequences{x^k},and{y^k}be generated by Algorithm2. Furthermore, let

(i) hS(x^∗, y^k), x^k−x^∗i ≥0;

(ii) hT(x^k, y^∗), y^k−y^∗i ≥0;

(iii) 0≤a^k+b^k ≤1;

(iv) P∞

k=0a^k =∞,andP∞

k=0b^k <∞;

(v) 0< ρ < _µ^2r2 and0< η < ^2s

β².

Then the sequence{x^k, y^k}converges to(x^∗, y^∗).

Proof. Since(x^∗, y^∗)∈K₁×K₂forms a solution to the SNVI(1.1)−(1.2)problem, it follows that

x^∗ =P_K[x^∗−ρS(x^∗, y^∗)] and y^∗ =Q_K[x^∗−ηT(x^∗, y^∗)].

Applying Algorithm2, we have kx^k+1−x^∗k

(2.1)

=k(1−a^k−b^k)x^k+a^kPK[x^k−ρS(x^k, y^k)] +b^ku^k

−(1−a^k−b^k)x^∗ −a^kPK[x^∗−ρS(x^∗, y^∗)]−b^kx^∗k

≤(1−a^k−b^k)kx^k−x^∗k

+a^kkPK[x^k−ρS(x^k, y^k)]−PK[x^∗−ρS(x^∗, y^∗)]k+M b^k

≤(1−a^k)kx^k−x^∗k+a^kkx^k−x^∗−ρ[S(x^k, y^k)−S(x^∗, y^k) +S(x^∗, y^k)−S(x^∗, y^∗)]k+M b^k

≤(1−a^k)kx^k−x^∗k+a^kkx^k−x^∗−ρ[S(x^k, y^k)−S(x^∗, y^k)]k +ρk[S(x^∗, y^k)−S(x^∗, y^∗)]k+M b^k,

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where

M = max{supku^k−x^∗k,supkv^k−y^∗k}<∞.

SinceSis strongly(r)−pseudomonotone and(µ)−Lipschitz continuous in the first variable, andSis(ν)−Lipschitz continuous in the second variable, we have in light of (i) that

kx^k−x^∗−ρ[S(x^k, y^k)−S(x^∗, y^k)]k² (2.2)

=kx^k−x^∗k²−2ρhS(x^k, y^k)−S(x^∗, y^k), x^k−x^∗i +ρ²kS(x^k, y^k)−S(x^∗, y^k)k²

=kx^k−x^∗k²−2ρhS(x^k, y^k), x^k−x^∗i+ 2ρhS(x^∗, y^k), x^k−x^∗i +ρ²kS(x^k, y^k)−S(x^∗, y^k)k²

≤ kx^k−x^∗k²−2ρrkx^k−x^∗k²+ρ²µ²kx^k−x^∗k² + 2ρhS(x^∗, y^k), x^k−x^∗i

= [1−2ρr+ρ²µ²]kx^k−x^∗k²+ 2ρhS(x^∗, y^k), x^k−x^∗i.

On the other hand, we have

2ρhS(x^∗, y^k), x^k−x^∗i ≤ρ[kS(x^∗, y^k)k²+kx^k−x^∗k²] and

kS(x^∗, y^k)k² (2.3)

= 1 2

kS(x^∗, y^k)−S(x^k, y^k)k²−2[kS(x^k, y^k)k²

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−1

2kS(x^k, y^k) +S(x^∗, y^k)k²

≤ 1 2

kS(x^∗, y^k)−S(x^k, y^k)k²−2[kS(x^k, y^k)k²

−1

2 | kS(x^k, y^k)−S(x^∗, y^k)k |²]

≤ 1

2kS(x^∗, y^k)−S(x^k, y^k)k²

≤ µ²

2 kx^k−x^∗k², where

kS(x^k, y^k)k²− 1

2 | kS(x^k, y^k)−S(x^∗, y^k)k |²>0.

Therefore, we get

2ρhS(x^∗, y^k), x^k−x^∗i ≤

ρ+ ρµ²

2

kx^k−x^∗k².

It follows that

kx^k−x^∗−ρ[S(x^k, y^k)−S(x^∗, y^k)]k² ≤

1−2ρr+ρ+ ρµ²

2

+ (ρµ)²

kx^k−x^∗k².

As a result, we have

(2.4) kx^k+1−x^∗k ≤(1−a^k)kx^k−x^∗k+a^kθkx^k−x^∗k+a^kρνky^k−y^∗k+M b^k,

whereθ = r

1−2ρr+ρ+

ρµ² 2

+ρ²µ².

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Similarly, we have (2.5)

y^k+1−y^∗

≤(1−a^k)ky^k−y^∗k+a^kσky^k−y^∗k+a^kητkx^k−x^∗k+M b^k, whereσ =

r

1−2ηr+η+

ηβ² 2

+η²β². It follows from(2.4)and(2.5)that

kx^k+1−x^∗k+ky^k+1−y^∗k (2.6)

≤(1−a^k)kx^k−x^∗k+a^kθkx^k−x^∗k+a^kητkx^k−x^∗k+M b^k

+ (1−a^k)ky^k−y^∗k+a^kσky^k−y^∗k+a^kρνky^k−y^∗k+M b^k

= [1−(1−δ)a^k](kx^k−x^∗k+ky^k−y^∗k) + 2M b^k,

whereδ = max{θ+ητ , σ+ρν}andH₁ ×H₂ is a Banach space under the norm k · k^∗.

If we set

α^k=kx^k−x^∗k+ky^k−y^∗k, t^k = (1−δ)a^k, β^k = 2M b^k fork = 0,1,2, ...,

in Lemma1.2, and apply(iii)and(iv),we conclude that kx^k−x^∗k+ky^k−y^∗k →0 ask → ∞.

Hence,

kx^k+1−x^∗k+ky^k+1−y^∗k →0.

Consequently, the sequence {(x^k, y^k)}converges strongly to(x^∗, y^∗),a solution to the SNVI(1.1)−(1.2)problem. This completes the proof.

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Note that the proof of the following theorem follows rather directly without using Lemma1.2.

Theorem 2.2. LetH1 andH2 be two real Hilbert spaces and, K1 andK2, respec- tively, be nonempty closed convex subsets ofH₁ andH₂.LetS :K₁ ×K₂ →H₁ be strongly(r)− pseudomonotone and(µ)−Lipschitz continuous in the first variable and letS be(ν)−Lipschitz continuous in the second variable. LetT : K₁×K₂ → H₂ be strongly (s)−pseudomonotone and (β)−Lipschitz continuous in the second variable and letT be(τ)−Lipschitz continuous in the first variable. Letk · k^∗denote the norm onH₁×H₂ defined by

k(x, y)k^∗ = (kxk+kyk) ∀(x, y)∈H₁×H₂. In addition, let

θ= s

1−2ρr+ρ+ ρµ²

2

+ρ²µ²+ητ <1,

σ= s

1−2ηr+η+ ηβ²

2

+η²β²+ρν <1,

let(x^∗, y^∗) ∈ K₁×K₂ form a solution to the SNVI(1.1)−(1.2)problem, and let sequences{x^k},and{y^k}be generated by Algorithm3. Furthermore, let

(i) hS(x^∗, y^k), x^k−x^∗i ≥0 (ii) hT(x^k, y^∗), y^k−y^∗i ≥0 (iii) 0≤a^k ≤1

(iv) P∞

k=0a^k =∞

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(v) 0< ρ < _µ^2r2 and0< η < ^2s

β².

Then the sequence{x^k, y^k)}converges strongly to(x^∗, y^∗).

Proof. Since(x^∗, y^∗)∈K₁×K₂forms a solution to the SNVI(1.1)−(1.2)problem, it follows that

x^∗ =PK[x^∗−ρS(x^∗, y^∗)] and y^∗ =QK[x^∗−ηT(x^∗, y^∗)].

Applying Algorithm3, we have kx^k+1−x^∗k

(2.7)

=k(1−a^k)x^k+a^kP_K[x^k−ρS(x^k, y^k)]−(1−a^k)x^∗

−a^kP_K[x^∗−ρS(x^∗, y^∗)]k

≤(1−a^k)kx^k−x^∗k+a^kkP_K[x^k−ρS(x^k, y^k)]−P_K[x^∗−ρS(x^∗, y^∗)]k

≤(1−a^k)kx^k−x^∗k

+a^kkx^k−x^∗−ρ[S(x^k, y^k)−S(x^∗, y^k) +S(x^∗, y^k)−S(x^∗, y^∗)]k

≤(1−a^k)kx^k−x^∗k+a^kkx^k−x^∗−ρ[S(x^k, y^k)−S(x^∗, y^k)]k +ρk[S(x^∗, y^k)−S(x^∗, y^∗)]k.

SinceSis strongly(r)−pseudomonotone and(µ)−Lipschitz continuous in the first variable, andSis(ν)−Lipschitz continuous in the second variable, we have in light of(i)that

kx^k−x^∗ −ρ[S(x^k, y^k)−S(x^∗, y^k)]k² (2.8)

=kx−x^∗k² −2ρhS(x^k, y^k)−S(x^∗, y^k), x^k−x^∗i +ρ²kS(x^k, y^k)−S(x^∗, y^k)k²

=kx−x^∗k² −2ρhS(x^k, y^k), x^k−x^∗i+ 2ρhS(x^∗, y^k), x^k−x^∗i

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+ρ²kS(x^k, y^k)−S(x^∗, y^k)k²

≤ kx^k−x^∗k²−2ρrkx^k−x^∗k²+ρ²µ²kx^k−x^∗k² + 2ρhS(x^∗, y^k), x^k−x^∗i

= [1−2ρr+ρ²µ²]kx^k−x^∗k²+ 2ρhS(x^∗, y^k), x^k−x^∗i.

On the other hand, we have

2ρhS(x^∗, y^k), x^k−x^∗i ≤ρ[kS(x^∗, y^k)k²+kx^k−x^∗k²] and

kS(x^∗, y^k)k² (2.9)

= 1 2

(

kS(x^∗, y^k)−S(x^k, y^k)k²

−2

2kS(x^k, y^k) +S(x^∗, y^k)k² )

≤ 1 2

(

kS(x^∗, y^k)−S(x^k, y^k)k²

−2

2 | kS(x^k, y^k)−S(x^∗, y^k)k |² )

≤ 1

2kS(x^∗, y^k)−S(x^k, y^k)k² ≤ µ²

2 kx^k−x^∗k²,

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where

kS(x^k, y^k)k²− 1 2

S(x^k, y^k)−S(x^∗, y^k)

2 >0.

Therefore, we get

2ρhS(x^∗, y^k), x^k−x^∗i ≤

ρ+ ρµ²

2

kx^k−x^∗k².

It follows that

kx^k−x^∗−ρ[S(x^k, y^k)−S(x^∗, y^k)]k² ≤

1−2ρr+ρ+ ρµ²

2

+ (ρµ)²

kx^k−x^∗k².

As a result, we have

(2.10) kx^k+1−x^∗|| ≤(1−a^k)kx^k−x^∗k+a^kθkx^k−x^∗k+a^kρνky^k−y^∗k,

whereθ = r

1−2ρr+ρ+

ρµ² 2

+ρ²µ². Similarly, we have

(2.11) ky^k+1−y^∗k ≤(1−a^k)ky^k−y^∗k+a^kσky^k−y^∗k+a^kητkx^k−x^∗k,

whereσ = r

1−2ηr+η+

ηβ² 2

+η²β². It follows from(2.9)and(2.10)that

kx^k+1−x^∗k+ky^k+1−y^∗k (2.12)

≤(1−a^k)kx^k−x^∗k+a^kθkx^k−x^∗k+a^kητkx^k−x^∗k + (1−a^k)ky^k−y^∗k+a^kσky^k−y^∗k+a^kρνky^k−y^∗k

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= [1−(1−δ)a^k](kx^k−x^∗k+ky^k−y^∗k)

≤

k

Y

j=0

[1−(1−δ)a^j](kx⁰−x^∗k+ky⁰−y^∗k),

whereδ = max{θ+ητ , σ+ρν}andH₁ ×H₂ is a Banach space under the norm k · k^∗.

Sinceδ <1andP∞

k=0a^kis divergent, it follows that

k→∞lim

k

Y

j=0

[1−(1−δ)a^j] = 0 as k→ ∞.

Therefore,

kx^k+1−x^∗k+ky^k+1−y^∗k →0,

and consequently, the sequence{(x^k, y^k)}converges strongly to(x^∗, y^∗),a solution to theSN V I(1.1)−(1.2)problem. This completes the proof.

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References

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[9] E. ZEIDLER, Nonlinear Functional Analysis and its Applications I, Springer- Verlag, New York, New York, 1986.

[10] E. ZEIDLER, Nonlinear Functional Analysis and its Applications II/B, Springer-Verlag, New York, New York, 1990.