- Cocoercive Operator and an Application for Solving Generalized Variational Inclusions

and Applied Analysis 3 monotone and Lipschitz continuous mappings are cocoercive, and it follows that cocoercivity is an intermediate concept that lies between simple and strong monotonicity. Definition 2.3. A multivalued mapping M : X → 2 is said to be cocoercive if there exists a constant μ′′ > 0 such that 〈 u − v, x − y〉 ≥ μ′′‖u − v‖, ∀x, y ∈ X, u ∈ M x , v ∈ M(y). 2.7 Definition 2.4. Amapping T : X → X is said to be relaxed cocoercive if there exists a constant γ ′ > 0 such that 〈 T x − T(y), x − y〉 ≥ (−γ ′)∥∥T x − T(y)∥∥2, ∀x, y ∈ X. 2.8 Definition 2.5. Let H : X ×X → X and A,B : X → X be the mappings. i H A, · is said to be cocoercive with respect toA if there exists a constant μ > 0 such that 〈 H Ax, u −H(Ay, u), x − y〉 ≥ μ∥∥Ax −Ay∥∥2, ∀x, y ∈ X; 2.9 ii H ·, B is said to be relaxed cocoercive with respect to B if there exists a constant γ > 0 such that 〈 H u, Bx −H(u, By), x − y〉 ≥ (−γ)∥∥Bx − By∥∥2, ∀x, y ∈ X; 2.10 iii H A, · is said to be r1-Lipschitz continuous with respect to A if there exists a constant r1 > 0 such that ∥∥H Ax, · −H(Ay, ·)∥∥ ≤ r1∥∥x − y∥∥, ∀x, y ∈ X; 2.11 iv H ·, B is said to be r2-Lipschitz continuous with respect to B if there exists a constant r2 > 0 such that ∥∥H ·, Bx −H(·, By)∥∥ ≤ r2∥∥x − y∥∥, ∀x, y ∈ X. 2.12 Example 2.6. Let X R2 with usual inner product. Let A,B : R2 → R2 be defined by Ax 2x1 − 2x2,−2x1 4x2 , By (−y1 y2,−y2), ∀ x1, x2 , (y1, y2) ∈ R2. 2.13 Suppose that H A,B : R2 × R2 → R2 is defined by H ( Ax,By ) Ax By, ∀x, y ∈ R2. 2.14 4 Abstract and Applied Analysis ThenH A,B is 1/6 -cocoercive with respect toA and 1/2 -relaxed cocoercive with respect to B since 〈 H Ax, u −H(Ay, u), x − y〉 〈Ax −Ay, x − y〉 〈 2x1 − 2x2,−2x1 4x2 − ( 2y1 − 2y2,−2y1 4y2 ) , ( x1 − y1, x2 − y2 )〉 〈( 2 ( x1 − y1 ) − 2(x2 − y2),−2(x1 − y1) 4(x2 − y2)), ( x1 − y1, x2 − y2 )〉 2 ( x1 − y1 )2 4(x2 − y2)2 − 4(x1 − y1)(x2 − y2), ∥∥Ax −Ay∥∥2 〈( 2x1 − 2x2,−2x1 4x2 − (2y1 − 2y2,−2y1 4y2)), ( 2x1 − 2x2,−2x1 4x2 − ( 2y1 − 2y2,−2y1 4y2 ))〉 8 ( x1 − y1 )2 20(x2 − y2)2 − 24(x1 − y1)(x2 − y2) ≤ 12(x1 − y1)2 24(x2 − y2)2 − 24(x1 − y1)(x2 − y2) 6 { 2 ( x1 − y1 )2 4(x2 − y2)2 − 4(x1 − y1)(x2 − y2)} 6 {〈 H u,Ax −H(u,Ay), x − y〉}, 2.15 which implies that 〈 H Ax, u −H(Ay, u), x − y〉 ≥ 1 6 ∥∥Ax −Ay∥∥2, 2.16 That is, H A,B is 1/6 -cocoercive with respect to A. 〈 H u, Bx −H(u, By), x − y〉 〈Bx − By, x − y〉 〈 −x1 x2,−x2 − (−y1 y2,−y2), (x1 − y1, x2 − y2)〉 〈(−(x1 − y1) (x2 − y2),−(x2 − y2)), (x1 − y1, x2 − y2)〉 −(x1 − y1)2 − (x2 − y2)2 (x1 − y1)(x2 − y2) − {( x1 − y1 )2 (x2 − y2)2 − (x1 − y1)(x2 − y2)}, ∥∥Bx − By∥∥2 〈(−(x1 − y1) (x2 − y2),−(x2 − y2)), (−(x1 − y1) (x2 − y2),−(x2 − y2)〉 Abstract and Applied Analysis 5 ( x1 − y1 )2 2(x2 − y2)2 − 2(x1 − y1)(x2 − y2) ≤ 2 {( x1 − y1 )2 (x2 − y2)2 − (x1 − y1)(x2 − y2)} 2 −1 〈H Bx, u −H(By, u), x − y〉 2.17and Applied Analysis 5 ( x1 − y1 )2 2(x2 − y2)2 − 2(x1 − y1)(x2 − y2) ≤ 2 {( x1 − y1 )2 (x2 − y2)2 − (x1 − y1)(x2 − y2)} 2 −1 〈H Bx, u −H(By, u), x − y〉 2.17 which implies that 〈 H u, Bx −H(u, By), x − y〉 ≥ − 2 ∥∥Bx − By∥∥2, 2.18 that is, H A,B is 1/2 -relaxed cocoercive with respect to B. 3. H ·, · -Cocoercive Operator In this section, we define a newH ·, · -cocoercive operator and discuss some of its properties. Definition 3.1. Let A,B : X → X,H : X × X → X be three single-valued mappings. Let M : X → 2 be a set-valued mapping. M is said to be H ·, · -cocoercive with respect to mappings A and B or simply H ·, · -cocoercive in the sequel if M is cocoercive and H A,B λM X X, for every λ > 0. Example 3.2. Let X, A, B, and H be the same as in Example 2.6, and let M : R2 → R2 be define by M x1, x2 0, x2 , ∀ x1, x2 ∈ R2. Then it is easy to check that M is cocoercive and H A,B λM R2 R2, ∀λ > 0, that is, M isH ·, · -cocoercive with respect to A and B. Remark 3.3. Since cocoercive operators include monotone operators, hence our definition is more general than definition of H ·, · -monotone operator 10 . It is easy to check that H ·, · -cocoercive operators provide a unified framework for the existing H ·, · -monotone, H-monotone operators inHilbert space andH ·, · -accretive,H-accretive operators in Banach spaces. SinceH ·, · -cocoercive operators are more general thanmaximal monotone operators, we give the following characterization ofH ·, · -cocoercive operators. Proposition 3.4. Let H A,B be μ-cocoercive with respect to A, γ-relaxed cocoercive with respect to B, A is α-expansive, B is β-Lipschitz continuous, and μ > γ , α > β. Let M : X → 2 be H ·, · -cocoercive operator. If the following inequality 〈 x − y, u − v〉 ≥ 0 3.1 holds for all v, y ∈ Graph M , then x ∈ Mu, where Graph M { x, u ∈ X ×X : u ∈ M x }. 3.2 6 Abstract and Applied Analysis Proof. Suppose that there exists some u0, x0 such that 〈 x0 − y, u0 − v 〉 ≥ 0, ∀(v, y) ∈ Graph M . 3.3 Since M is H ·, · -cocoercive, we know that H A,B λM X X holds for every λ > 0, and so there exists u1, x1 ∈ Graph M such that H Au1, Bu1 λx1 H Au0, Bu0 λx0 ∈ X. 3.4 It follows from 3.3 and 3.4 that 0 ≤ 〈λx0 H Au0, Bu0 − λx1 −H Au1, Bu1 , u0 − u1〉, 0 ≤ λ〈x0 − x1, u0 − u1〉 −〈H Au0, Bu0 −H Au1, Bu1 , u0 − u1〉 −〈H Au0, Bu0 −H Au1, Bu0 , u0 − u1〉 −〈H Au1, Bu0 −H Au1, Bu1 , u0 − u1〉 ≤ −μ‖Au0 −Au1‖ γ‖Bu0 − Bu1‖ ≤ −μα‖u0 − u1‖ γβ‖u0 − u1‖ −(μα2 − γβ2)‖u0 − u1‖ ≤ 0, 3.5 which gives u1 u0 since μ > γ, α > β. By 3.4 , we have x1 x0. Hence u0, x0 u1, x1 ∈ Graph M and so x0 ∈ Mu0. Theorem 3.5. Let X be a Hilbert space and M : X → 2 a maximal monotone operator. Suppose that H : X × X → X is a bounded cocoercive and semicontinuous with respect to A and B. Let H : X × X → X be also μ-cocoercive with respect to A and γ-relaxed cocoercive with respect to B. The mapping A is α-expansive, and B is β-Lipschitz continuous. If μ > γ and α > β, then M is H ·, · -cocoercive with respect to A and B. Proof. For the proof we refer to 10 . Theorem 3.6. LetH A,B be a μ-cocoercive with respect to A and γ-relaxed cocoercive with respect to B, A is α-expansive, and B is β-Lipschitz continuous, μ > γ and α > β. Let M be an H ·, · cocoercive operator with respect to A and B. Then the operator H A,B λM −1 is single-valued. Proof. For any given u ∈ X, let x, y ∈ H A,B λM −1 u . It follows that −H Ax,Bx u ∈ λMx, −H(Ay,By) u ∈ λMy. 3.6 Abstract and Applied Analysis 7 AsM is cocoercive thus monotone , we have 0 ≤ 〈−H Ax,Bx u − (−H(Ay,By) u) , x − y〉 −〈H Ax,Bx −H(Ay,By), x − y〉 −〈H Ax,Bx −H(Ay,Bx) H(Ay,Bx) −H(Ay,By), x − y〉 −〈H Ax,Bx −H(Ay,Bx), x − y〉 − 〈H(Ay,Bx) −H(Ay,By), x − y〉. 3.7and Applied Analysis 7 AsM is cocoercive thus monotone , we have 0 ≤ 〈−H Ax,Bx u − (−H(Ay,By) u) , x − y〉 −〈H Ax,Bx −H(Ay,By), x − y〉 −〈H Ax,Bx −H(Ay,Bx) H(Ay,Bx) −H(Ay,By), x − y〉 −〈H Ax,Bx −H(Ay,Bx), x − y〉 − 〈H(Ay,Bx) −H(Ay,By), x − y〉. 3.7 Since H is μ-cocoercive with respect to A and γ-relaxed cocoercive with respect to B, A is α-expansive and B is β-Lipschitz continuous, thus 3.7 becomes 0 ≤ −μα2∥∥x − y∥∥2 γβ2∥∥x − y∥∥2 −(μα2 − γβ2)∥∥x − y∥∥2 ≤ 0 3.8 since μ > γ, α > β. Thus, we have x y and so H A,B λM −1 is single-valued. Definition 3.7. Let H A,B be μ-cocoercive with respect to A and γ-relaxed cocoercive with respect to B, A is α-expansive, B is β-Lipschitz continuous, and μ > γ , α > β. Let M be an H ·, · -cocoercive operator with respect to A and B. The resolvent operator R ·,· λ,M : X → X is defined by R H ·,· λ,M u H A,B λM −1 u , ∀u ∈ X. 3.9 Now, we prove the Lipschitz continuity of resolvent operator defined by 3.9 and estimate its Lipschitz constant. Theorem 3.8. Let H A,B be μ-cocoercive with respect to A, γ-relaxed cocoercive with respect to B, A is α-expansive, B is β-Lipschitz continuous, and μ > γ , α > β. Let M be an H ·, · -cocoercive operator with respect to A and B. Then the resolvent operator R ·,· λ,M : X → X is 1/μα2 − γβ2Lipschitz continuous, that is, ∥∥∥RH ·,· λ,M u − R ·,· λ,M v ∥∥∥ ≤ 1 μα2 − γβ2 ‖u − v‖, ∀u, v ∈ X. 3.10 Proof. Let u and v be any given points in X. It follows from 3.9 that R H ·,· λ,M u H A,B λM −1 u , R H ·,· λ,M v H A,B λM −1 v . 3.11 8 Abstract and Applied Analysis This implies that 1 λ ( u −H ( A ( R H ·,· λ,M u ) , B ( R H ·,· λ,M u ))) ∈ M ( R H ·,· λ,M u ) , 1 λ ( v −H ( A ( R H ·,· λ,M v ) , B ( R H ·,· λ,M v ))) ∈ M ( R H ·,· λ,M v ) . 3.12 For the sake of clarity, we take Pu R ·,· λ,M u , Pv R H ·,· λ,M v . 3.13 Since M is cocoercive hence monotone , we have 1 λ 〈u −H A Pu , B Pu − v −H A Pv , B Pv , Pu − Pv〉 ≥ 0, 1 λ 〈u − v −H A Pu , B Pu H A Pv , B Pv , Pu − Pv〉 ≥ 0, 3.14


Introduction
Various concepts of generalized monotone mappings have been introduced in the literature.Cocoercive mappings which are generalized form of monotone mappings are defined by Tseng 1 , Magnanti and Perakis 2 , and Zhu and Marcotte 3 .The resolvent operator techniques are important to study the existence of solutions and to develop iterative schemes for different kinds of variational inequalities and their generalizations, which are providing mathematical models to some problems arising in optimization and control, economics, and engineering sciences.In order to study various variational inequalities and variational inclusions, Fang and Huang, Lan, Cho, and Verma investigated many generalized operators such as H-monotone 4 , H-accretive 5 , H, η -accretive 6 , H, η -monotone 7, 8 , A, ηaccretive mappings 9 .Recently, Zou and Huang 10 introduced and studied H •, •accretive operators and Xu and Wang 11 introduced and studied H •, • , η -monotone operators.

Preliminaries
Throughout the paper, we suppose that X is a real Hilbert space endowed with a norm • and an inner product •, • , d is the metric induced by the norm • , 2 X resp., CB X is the family of all nonempty resp., closed and bounded subsets of X, and D •, • is the Hausdorff metric on CB X defined by iii strongly monotone if there exists a constant ξ > 0 such that Definition 2.2.A mapping T : X → X is said to be cocoercive if there exists a constant μ > 0 such that

2.6
Note 1. Clearly T is 1/μ -Lipschitz continuous and also monotone but not necessarily strongly monotone and Lipschitz continuous consider a constant mapping .Conversely, strongly Abstract and Applied Analysis 3 monotone and Lipschitz continuous mappings are cocoercive, and it follows that cocoercivity is an intermediate concept that lies between simple and strong monotonicity.
Definition 2.4.A mapping T : X → X is said to be relaxed cocoercive if there exists a constant γ > 0 such that

2.12
Example 2.6.Let X R 2 with usual inner product.Let A, B : R 2 → R 2 be defined by

2.13
Suppose that H A, B : R 2 × R 2 → R 2 is defined by H Ax, By Ax By, ∀x, y ∈ R 2 .

2.14
Then H A, B is 1/6 -cocoercive with respect to A and 1/2 -relaxed cocoercive with respect to B since

H •, • -Cocoercive Operator
In this section, we define a new H •, • -cocoercive operator and discuss some of its properties.

Proposition 3.4. Let H A, B be μ-cocoercive with respect to A, γ-relaxed cocoercive with respect to
holds for all v, y ∈ Graph M , then x ∈ Mu, where Proof.Suppose that there exists some u 0 , x 0 such that Since M is H •, • -cocoercive, we know that H A, B λM X X holds for every λ > 0, and so there exists It follows from 3.3 and 3.4 that which gives u 1 u 0 since μ > γ, α > β.By 3.4 , we have x 1 x 0 .Hence u 0 , x 0 u 1 , x 1 ∈ Graph M and so x 0 ∈ Mu 0 .Theorem 3.5.Let X be a Hilbert space and M : X → 2 X a maximal monotone operator.Suppose that H : X × X → X is a bounded cocoercive and semicontinuous with respect to A and B. Let H : X × X → X be also μ-cocoercive with respect to A and γ-relaxed cocoercive with respect to B. The mapping A is α-expansive, and B is β-Lipschitz continuous.If μ > γ and α > β, then M is H •, • -cocoercive with respect to A and B.
Proof.For the proof we refer to 10 .

Theorem 3.6. Let H A, B be a μ-cocoercive with respect to A and γ-relaxed cocoercive with respect to B, A is α-expansive, and B is β-Lipschitz continuous, μ > γ and α > β. Let M be an H •, •cocoercive operator with respect to A and B. Then the operator H A, B λM
3.9 Now, we prove the Lipschitz continuity of resolvent operator defined by 3.9 and estimate its Lipschitz constant.

Theorem 3.8. Let H A, B be μ-cocoercive with respect to A, γ-relaxed cocoercive with respect to B, A is α-expansive, B is β-Lipschitz continuous, and μ > γ, α > β. Let M be an H •, • -cocoercive operator with respect to A and B. Then the resolvent operator
3.10 Proof.Let u and v be any given points in X.It follows from 3.9 that

8 Abstract and Applied Analysis
This implies that

3.12
For the sake of clarity, we take Since M is cocoercive hence monotone , we have

3.15
Further, we have and so

3.18
This completes the proof.

Application of H •, • -Cocoercive Operators for Solving Variational Inclusions
We apply H •, • -cocoercive operators for solving a generalized variational inclusion problem.
We consider the problem of finding u ∈ X and w ∈ T u such that 0 ∈ w M g u , 4.1 where g : X → X, M : X → 2 X , and T : X → CB X are the mappings.Problem 4.1 is introduced and studied by Huang 12 in the setting of Banach spaces.Based on 4.2 , we construct the following algorithm.
Theorem 4.3.Let X be a real Hilbert space and A, B, g : X → X, H : X ×X → X the single-valued mappings.Let T : X → CB X be a multi-valued mapping and M : X → 2 X the multi-valued Then the generalized variational inclusion problem 4.1 has a solution u, w with u ∈ X, w ∈ T u , and the iterative sequences {u n } and {w n } generated by Algorithm 4.2 converge strongly to u and w, respectively.
Proof.Since T is δ-Lipschitz continuous, it follows from Algorithm 4.2 that for n 0, 1, 2, . ... Using the ξ-strong monotonicity of g, we have 4.6 Now we estimate g u n 1 − g u n by using the Lipschitz continuity of R H

4.7
Since H A, B is r 1 -Lipschitz continuous with respect to A and r 2 -Lipschitz continuous with respect to B, g is λ g -Lipschitz continuous and using 4. We know that θ n → θ and n → ∞.From assumption vii , it is easy to see that θ < 1.Therefore, it follows from 4.10 that {u n } is a Cauchy sequence in X.Since X is a Hilbert space, there exists u ∈ X such that u n → u as n → ∞.From 4.4 , we know that {w n } is also a Cauchy sequence in X, thus there exists w ∈ X such that w n → w and n → ∞.By the continuity of g, R

Lemma 4 . 1 .
The u, w , where u ∈ X, w ∈ T u , is a solution of the problem 4.1 , if and only if u, w is a solution of the following:g u R H •,• λ,M H A gu , B gu − λw , 4.2where λ > 0 is a constant.Proof.By using the definition of resolvent operator R H •,• λ,M , the conclusion follows directly.
i T is δ-Lipschitz continuous in the Hausdorff metric D •, • ;ii H A, B is μ-cocoercive with respect to A and γ-relaxed cocoercive with respect to B;