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volume 7, issue 3, article 83, 2006.

Received 15 March, 2006;

accepted 17 March, 2006.

Communicated by:G. Anastassiou

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Journal of Inequalities in Pure and Applied Mathematics

GENERALIZED(A, η)−RESOLVENT OPERATOR TECHNIQUE AND SENSITIVITY ANALYSIS FOR RELAXED COCOERCIVE VARIATIONAL INCLUSIONS

RAM U. VERMA

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

EMail:verma99@msn.com

c

2000Victoria University ISSN (electronic): 1443-5756 080-06

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Generalized(A, η)−Resolvent Operator Technique and Sensitivity Analysis for Relaxed

Cocoercive Variational Inclusions Ram U. Verma

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Abstract

Sensitivity analysis for relaxed cocoercive variational inclusions based on the generalized resolvent operator technique is discussed The obtained results are general in nature.

2000 Mathematics Subject Classification:49J40, 47H10.

Key words: Sensitivity analysis, Quasivariational inclusions, Maximal relaxed mono- tone mapping,(A, η)−monotonemapping, Generalized resolvent operator technique.

1. Introduction

In [8] the author studied sensitivity analysis for quasivariational inclusions us- ing the resolvent operator technique. Resolvent operator techniques have been frequently applied to a broad range of problems arising from several fields, including equilibria problems in economics, optimization and control theory, op- erations research, and mathematical programming. In this paper we intend to present the sensitivity analysis for(A, η)−monotone quasivariational inclusions involving relaxed cocoercive mappings. The notion of(A, η)−monotonicity [8]

generalizes the notion of A− monotonicity in [12]. The obtained results gen- eralize a wide range of results on the sensitivity analysis for quasivariational inclusions, including [2] – [5] and others. For more details on nonlinear variational inclusions and related resolvent operator techniques, we recommend the reader [1] – [12].

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2. (A, η)-Monotonicity

In this section we explore some basic properties derived from the notion of (A, η)−monotonicity. Letη : X×X → X be(τ)−Lipschitz continuous, that is, there exists a positive constantτ >0such that

kη(u, v)k ≤τku−vk ∀u, v ∈X.

Definition 2.1. Let η : X × X → X be a single-valued mapping, and let M :X →2^X be a multivalued mapping onX.The mapM is said to be:

(i) (r, η)-strongly monotone if

hu^∗−v^∗, η(u, v)i ≥rku−vk² ∀(u, u^∗),(v, v^∗)∈Graph(M).

(ii) (r, η)-strongly pseudomonotone if

hv^∗, η(u, v)i ≥0 implies

hu^∗, η(u, v)i ≥rku−vk² ∀(u, u^∗),(v, v^∗)∈Graph(M).

(iii) (η)-pseudomonotone if

hv^∗, τ(u, v)i ≥0 implies

hu^∗, η(u, v)i ≥0 ∀(u, u^∗),(v, v^∗)∈Graph(M).

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(iii) (m, η)-relaxed monotone if there exists a positive constantmsuch that hu^∗−v^∗, η(u, v)i ≥(−m)ku−vk² ∀(u, u^∗),(v, v^∗)∈Graph(M).

Definition 2.2. A mapping M :X →2^X is said to be maximal(m, η)-relaxed monotone if

(i) M is(m, η)-relaxed monotone, (ii) For(u, u^∗)∈X×X,and

hu^∗−v^∗, η(u, v)i ≥(−m)ku−vk² ∀(v, v^∗)∈Graph(M), we haveu^∗ ∈M(u).

Definition 2.3. Let A : X → X andη : X×X → X be two single-valued mappings. The mapM :X →2^X is said to be(A, η)-monotone if

(i) M is(m, η)-relaxed monotone (ii) R(A+ρM) =X forρ >0.

Alternatively, we have

Definition 2.4. Let A : X → X andη : X×X → X be two single-valued mappings. The mapM :X →2^X is said to be(A, η)-monotone if

(i) M is(m, η)-relaxed monotone

(ii) A+ρM is(η)-pseudomonotone forρ >0.

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Proposition 2.1. LetA:X →Xbe an(r, η)-strongly monotone single-valued mapping and letM : X → 2^X be an(A), η)-monotone mapping. Then M is maximal(m, η)-relaxed monotone for0< ρ < _m^r.

Proposition 2.2. LetA:X →Xbe an(r, η)-strongly monotone single-valued mapping and letM :X →2^X be an(A, η)-monotone mapping. Then(A+ρM) is maximal(η)-monotone for0< ρ < _m^r.

Proof. Since A is (r, η)-strongly monotone and M is (A, η)-monotone, it implies thatA+ρM is(r−ρm, η)-strongly monotone. This in turn implies that A+ρM is(η)-pseudomonotone, and henceA+ρM is maximal(η)-monotone under the given conditions.

Proposition 2.3. Let A : X → X be an (r, η)-strongly monotone mapping and let M : X → 2^X be an (A, η)-monotone mapping. Then the operator (A+ρM)⁻¹is single-valued.

Definition 2.5. LetA:X →Xbe an(r, η)-strongly monotone mapping and let M :X →2^X be an(A, η)-monotone mapping. Then the generalized resolvent operatorJ_ρ,A^M :X →X is defined by

J_ρ,A^M (u) = (A+ρM)⁻¹(u).

Furthermore, we upgrade the notions of the monotonicity as well as strong monotonicity in the context of sensitivity analysis for nonlinear variational inclusion problems.

Definition 2.6. The mapT :X×X×L→Xis said to be:

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(i) Monotone with respect toAin the first argument if

hT(x, u, λ)−T(y, u, λ), A(x)−A(y)i ≥0

∀(x, y, u, λ)∈X×X×X×L.

(ii) (r)-strongly monotone with respect toAin the first argument if there exists a positive constantrsuch that

hT(x, u, λ)−T(y, u, λ), A(x)−A(y)i ≥(r)kx−yk²

∀(x, y, u, λ)∈X×X×X×L.

(iii) (γ, α)-relaxed cocoercive with respect to A in the first argument if there exist positive constantsγ andαsuch that

hT(x, u, λ)−T(y, u, λ), A(x)−A(y)i ≥ −γkT(x)−T(y)k²+αkx−yk²

∀(x, y, u, λ)∈X×X×X×L.

(iv) (γ)-relaxed cocoercive with respect toAin the first argument if there exists a positive constantγsuch that

hT(x, u, λ)−T(y, u, λ), A(x)−A(y)i ≥ −γkT(x)−T(y)k²

∀(x, y, u, λ)∈X×X×X×L.

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3. Results On Sensitivity Analysis

LetXdenote a real Hilbert space with the normk·kand inner producth·,·i.Let N :X×X×L→Xbe a nonlinear mapping andM :X×X×L→2^X be an A-monotone mapping with respect to the first variable, whereLis a nonempty open subset ofX. Furthermore, letη : X×X → X be a nonlinear mapping.

Then the problem of finding an elementu∈Xfor a given elementf ∈X such that

(3.1) f ∈N(u, u, λ) +M(u, u, λ),

where λ ∈ L is the perturbation parameter, is called a class of generalized strongly monotone mixed quasivariational inclusion (abbreviated SMMQVI) problems.

The solvability of theSM M QV I problem(3.1)depends on the equivalence between(3.1)and the problem of finding the fixed point of the associated generalized resolvent operator.

Note that if M is(A, η)-monotone, then the corresponding generalized resolvent operatorJ_ρ,A^M in first argument is defined by

(3.2) J_ρ,A^M^(·,y)(u) = (A+ρM(·, y))⁻¹(u) ∀u∈X, whereρ >0andAis an(r, η)-strongly monotone mapping.

Lemma 3.1. LetXbe a real Hilbert space, and letη :X×X →X be a(τ)- Lipschitz continuous nonlinear mapping. Let A : X → X be (r, η)-strongly monotone, and let M : X ×X ×L → 2^X be (A, η)-monotone in the first

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variable. Then the generalized resolvent operator associated with M(·, y, λ) for a fixedy∈Xand defined by

J_ρ,A^M^(·,y,λ)(u) = (A+ρM(·, y, λ))⁻¹(u) ∀u∈X, is(_r−ρm^τ )-Lipschitz continuous.

Proof. By the definition of the generalized resolvent operator, we have 1

ρ

u−A

J_ρ,A^M(·,y,λ)(u)

∈M

J_ρ,A^M^(·,y,λ)(u) , and

1 ρ

v−A

J_ρ,A^M(·,y,λ)(v)

∈M

J_ρ,A^M(·,y,λ)(v)

∀u, v ∈X.

GivenM is(m, η)-relaxed monotone, we find 1

ρ D

u−v− A

−A

J_ρ,A^M^(·,y,λ)(v)

, η

J_ρ,A^M(·,y,λ)(u), J_ρ,A^M(·,y,λ)(v)E

≥(−m)

J_ρ,A^M^(·,y,λ)(u)−J_ρ,A^M(·,y,λ)(v)

2

. Therefore,

τku−vk

J_ρ,A^M^(·,y,λ)(u)−J_ρ,A^M^(·,y,λ)(v)

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≥D

u−v, η

J_ρ,A^M(·,y,λ)(u), J_ρ,A^M^(·,y,λ)(v) E

≥D A

−A

J_ρ,A^M(·,y,λ)(v) , η

J_ρ,A^M^(·,y,λ)(u), J_ρ,A^M^(·,y,λ)(v)E

−(ρm)

J_ρ,A^M^(·,y,λ)(u)−J_ρ,A^M^(·,y,λ)(v)

2

≥(r−ρm)

J_ρ,A^M^(·,y,λ)(u)−J_ρ,A^M(·,y,λ)(v)

2

. This completes the proof.

Lemma 3.2. LetXbe a real Hilbert space, letA :X →Xbe(r, η)−strongly monotone, and let M : X ×X ×L → 2^X be (A), η−monotone in the first variable. Let η : X ×X → X be a (τ) −Lipschitz continuous nonlinear mapping. Then the following statements are mutually equivalent:

(i) An elementu∈X is a solution to(3.1).

(ii) The mapG:X×L→Xdefined by

G(u, λ) = J_ρ,A^M(·,u,λ)(A(u)−ρN(u, u, λ) +ρf) has a fixed point.

Theorem 3.3. LetXbe a real Hilbert space, and letη :X×X →Xbe a(τ)- Lipschitz continuous nonlinear mapping. Let A : X → X be (r, η)-strongly monotone and (s)-Lipschitz continuous, and let M : X × X ×L → 2^X be (A, η)-monotone in the first variable. Let N : X ×X ×L → X be (γ, α)- relaxed cocoercive (with respect toA) and(β)-Lipschitz continuous in the first

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variable, and let N be (µ)-Lipschitz continuous in the second variable. If, in addition,

(3.3)

J_ρ,A^M^(·,u,λ)(w)−J_ρ,A^M^(·,v,λ)(w)

≤δku−vk ∀(u, v, λ)∈X×X×L, then

(3.4) kG(u, λ)−G(v, λ)k ≤θku−vk ∀(u, v, λ)∈X×X×L, where

θ = τ r−ρm

hps²−2ρα+ 2ρβ²γ+ρ²β²+ρµi

+δ <1,

ρ− (α−γβ²)τ²−r[µτ +m(1−δ)](1−δ) β²−(µτ +m(1−δ))²

<

p[(α−γβ²)τ²−r(µτ +m(1−δ))(1−η)]²−B β²−(µτ −m(1−δ))² , B = [β² −(µτ +m(1−δ))²](s²τ²−r²(1−δ)²),

for

α(α−γβ²)τ² > r(µτ +m(1−δ))(1−δ) +√ B, β > µτ +m(1−δ), 0< δ <1.

Consequently, for each λ ∈ L, the mapping G(u, λ) in light of (3.4) has a unique fixed pointz(λ).Hence, in light of Lemma3.2,z(λ)is a unique solution to(3.1).Thus, we have

G(z(λ), λ) =z(λ).

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Proof. For any element(u, v, λ)∈X×X×L,we have G(u, λ) = J_ρ,A^M(·,u,λ)(A(u)−ρN(u, u, λ) +ρf),

G(v, λ) = J_ρ,A^M^(·,v,λ)(A(v)−ρN(v, v, λ) +ρf).

It follows that

kG(u, λ)−G(v, λ)k

=

J_ρ,A^M^(·,u,λ)(A(u)−ρN(u, u, λ) +ρf)−J_ρ,A^M^(·,v,λ)(A(v)−ρN(v, v, λ) +ρf)

≤

J_ρ,A^M(·,u,λ)(A(u)−ρN(u, u, λ) +ρf)−J_ρ,A^M(·,u,λ)(A(v)−ρN(v, v, λ) +ρf)

+

J_ρ,A^M^(·,u,λ)(A(v)−ρN(v, v, λ) +ρf)−J_ρ,A^M(·,v,λ)(A(v)−ρN(v, v, λ) +ρf)

≤ τ

r−ρmkA(u)−A(v)−ρ(N(u, u, λ)−N(v, v, λ))k+δku−vk

≤ τ

r−ρm[kA(u)−A(v)−ρ(N(u, u, λ)−N(v, u, λ))k +kρ(N(v, u, λ)−N(v, v, λ))k] +δku−vk.

The (γ, α)-relaxed cocoercivity and (β)-Lipschitz continuity of N in the first argument imply that

kA(u)−A(v)−ρ(N(u, u, λ)−N(v, u, λ))k²

=kA(u)−A(v)k²−2ρhN(u, u, λ)−N(v, u, λ), A(u)−A(v)i +ρ²kN(u, u, λ)−N(v, u, λ)k²

≤(s²−2ρα+ 2ρβ²γ+ρ²β²)ku−vk².

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On the other hand, the (µ)-Lipschitz continuity of N in the second argument results

k(N(v, u, λ))−N(v, v, λ))k ≤µku−vk.

In light of above arguments, we infer that

(3.5) kG(u, λ)−G(v, λ)k ≤θku−vk, where

θ= τ

(r−ρm)

hps²−2ρα+ 2ρβ²γ+ρ²β²+ρµ i

+δ <1.

Sinceθ <1,it concludes the proof.

Theorem 3.4. LetXbe a real Hilbert space, letA:X →X be(r, η)-strongly monotone and (s)-Lipschitz continuous, and let M : X × X ×L → 2^X be (A, η)-monotone in the first variable. LetN :X×X×L→Xbe(γ, α)-relaxed cocoercive (with respect toA) and(β)-Lipschitz continuous in the first variable, and letN be(µ)-Lipschitz continuous in the second variable. Furthermore, let η :X×X →X be(τ)-Lipschitz continuous. In addition, if

J_ρ,A^M^(·,u,λ)(w)−J_ρ,A^M(·,v,λ)(w)

≤δku−vk ∀(u, v, λ)∈X×X×L, then

(3.6) kG(u, λ)−G(v, λ)k ≤θku−vk ∀(u, v, λ)∈X×X×L, where

θ = τ r−ρm

hps²−2ρα+ 2ρβ²γ+ρ²β²+ρµi

+δ <1,

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ρ− (α−γβ²)τ²−r[µτ +m(1−δ)](1−δ) β²−(µτ +m(1−δ))²

<

p[(α−γβ²)τ²−r(µτ +m(1−δ))(1−η)]²−B β²−(µτ −m(1−δ))² , B = [β² −(µτ +m(1−δ))²](s²τ²−r²(1−δ)²),

for

(α−γβ²)τ² > r(µτ +m(1−δ))(1−δ) +√ B, β > µτ +m(1−δ), 0< δ <1.

If the mappings λ → N(u, v, λ) and λ → J_ρ,A^M(·,u,λ)(w) both are continuous (or Lipschitz continuous) from L to X, then the solution z(λ) of(3.1)is continuous (or Lipschitz continuous) fromLtoX.

Proof. From the hypotheses of the theorem, for anyλ, λ^∗ ∈L,we have kz(λ)−z(λ^∗)k

=kG(z(λ), λ)−G(z(λ^∗), λ^∗)k

≤ kG(z(λ), λ)−G(z(λ^∗), λ)k+kG(z(λ^∗), λ)−G(z(λ^∗), λ^∗)k

≤θkz(λ)−z(λ^∗)k+kG(z(λ^∗), λ)−G(z(λ^∗), λ^∗)k.

It follows that

kG(z(λ^∗), λ)−G(z(λ^∗), λ^∗)k

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=

J_ρ,A^M^(·,z(λ^∗^),λ)(A(z(λ^∗))−ρN(z(λ^∗), z(λ^∗), λ))

−J_ρ,A^M(·,z(λ^∗^),λ^∗⁾(A(z(λ^∗))−ρN(z(λ^∗), z(λ^∗), λ^∗))

≤

J_ρ,A^M^(·,z(λ^∗^),λ)(A(z(λ^∗))−ρN(z(λ^∗), z(λ^∗), λ))

−

J_ρ,A^M^(·,z(λ^∗^),λ)(A(z(λ^∗))−ρN(z(λ^∗), z(λ^∗), λ^∗))

+

J_ρ,A^M(·,z(λ^∗^),λ)(A(z(λ^∗))−ρN(z(λ^∗), z(λ^∗), λ^∗))

− J_ρ,A^M(·,z(λ^∗^),λ^∗⁾(A(z(λ^∗))−ρN(z(λ^∗), z(λ^∗), λ^∗))

≤ ρτ

r−ρmkN(z(λ^∗), z(λ^∗), λ)−N(z(λ^∗), z(λ^∗), λ^∗)k +

J_ρ,A^M(·,z(λ^∗^),λ)(z(λ^∗)−ρN(z(λ^∗), z(λ^∗), λ^∗))

− J_ρ,A^M(·,z(λ^∗^),λ^∗⁾(z(λ^∗)−ρN(z(λ^∗), z(λ^∗), λ^∗)) . Hence, we have

kz(λ)−z(λ^∗)k ≤ ρτ

(r−ρm)(1−θ)kN(z(λ^∗), z(λ^∗), λ)−N(z(λ^∗), z(λ^∗), λ^∗)k

+ 1

1−θ

J_ρ,A^M^(·,z(λ^∗^),λ)(z(λ^∗)−ρN(z(λ^∗), z(λ^∗), λ^∗))

−J_ρ,A^M(·,z(λ^∗^),λ^∗⁾(z(λ^∗)−ρN(z(λ^∗), z(λ^∗), λ^∗)) . This completes the proof.

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References

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[7] A. MOUDAFI, Mixed equilibrium problems: Sensitivity analysis and al- gorithmic aspect, Comp. and Math. with Applics., 44 (2002), 1099–1108.

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[8] R.U. VERMA, Sensitivity analysis for generalized strongly monotone variational inclusions based on the (A, η)-resolvent operator technique Appl. Math. Lett., (2006) (in press).

[9] R.U. VERMA, New class of nonlinear A− monotone mixed variational inclusion problems and resolvent operator technique, J. Comput. Anal. and Applics., 8(3) (2006), 275–285.

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