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