Resolvent operator technique for solving a system of ... - PMF-a

Filomat 26:5 (2012), 897–908 DOI 10.2298/FIL1205897A

Published by Faculty of Sciences and Mathematics, University of Niˇs, Serbia Available at: http://www.pmf.ni.ac.rs/filomat

Resolvent operator technique for solving a system of generalized variational-like inclusions in Banach spaces Rais Ahmada , Mohammad Dilshada , Mohammad Akrama a Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

Abstract. In this paper, we apply H(·, ·)-η-cocoercive operator introduced in [2] for solving a system of generalized variational-like inclusions in q-uniformly smooth Banach spaces. By using the approach of resolvent operator associated with H(·, ·)-η-cocoercive operator, an iterative algorithm for solving a system of generalized variational-like inclusions is constructed. We prove the existence of solutions of system of generalized variational-like inclusions and convergence of iterative sequences generated by the algorithm. An example through Matlab programming is constructed.

1. Introduction Variational inequalities and variational inclusions are the mathematical models of some problems arising in economics, mechanics, and engineering sciences. Ding [8], Huang and Fang [14], Fang and Huang [11], Verma [22], Lan et al. [16], Fang and Huang [12], Huang and Fang [10], Huang [13], Fang et al. [9] and Lan et al. [17] introduced the concept of η-subdifferential operators, maximal η-monotone operators, generalized monotone operators, A-monotone operators, (A, η)-monotone operators, (H, η)-monotone operators in Hilbert spaces, H-accretive operators, generalized m-accretive operators and (H, η)-accretive operators and (A, η)-accretive operators in Banach spaces and their resolvent operators, respectively. Recently, Zou and Huang [27] introduced and studied H(·, ·)-accretive operators and Xu and Wang [25] introduced and studied H(·, ·)-η-monotone operators. The cocoercive operators which are the generalized form of monotone operators are defined by Tseng [21], Magananti and Perakis [18] and Zhu and Marcotte [26]. The concept of η-cocoercivity, η-strong monotonicity and η-strong convexity of a mapping was introduced and studied by Ansari and Yao [6]. Very recently, Ahmad et al. [1, 2] introduced H(·, ·)-cocoercive and H(·, ·)-η-cocoercive operators and apply them to solve variational inclusion problems. Motivated by the above excellent work, in this paper, we apply H(·, ·)-η-cocoercive operator and its resolvent operator to solve a system of generalized variational-like inclusions in q-uniformly smooth Banach spaces. We note that the system of variational inequalities is a powerful tool to study the Nash equilibrium problem [19, 20]; See, for example, [4, 5, 7] and the references therein. 2010 Mathematics Subject Classification. Primary 49J40; Secondary 47H19 Keywords. H(·, ·)-η-cocoercive operator, iterative algorithm, solution, convergence, Banach spaces Received: 26 November 2011; Accepted: 09 February 2012 Communicated by Qamrul Hasan Ansari and Ljubiˇsa D.R. Koˇcinac This work is supported by Department of Science and Technology, Government of India under grant no. SR/S4/MS: 577/09 Email addresses: [email protected] (Rais Ahmad), [email protected] (Mohammad Dilshad), akramkhan [email protected] (Mohammad Akram)

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2. Preliminaries Let X is a real Banach space with the dual space X⋆ , ⟨·, ·⟩ be the duality pairing between X and X⋆ , ∥ · ∥ be a norm on X and 2X (respectively CB(X)) be the family of all the non-empty (respectively, closed and bounded) subsets of X, and D(·, ·) be the Hausdorff metric on CB(X) defined for P, Q ∈ CB(X) by D(P, Q) = max{ sup d′ (x, Q), sup d′ (P, y) }, x∈P

y∈Q

where d′ (x, Q) = inf d′ (x, y) and d′ (P, y) = inf d′ (x, y), d′ is the metric induced by the norm ∥ · ∥. The x∈P

y∈Q

⋆

generalized duality mapping Jq : X → 2X is defined, by { } Jq (x) = f ⋆ ∈ X⋆ : ⟨x, f ⋆ ⟩ = ∥x∥q , ∥ f ⋆ ∥ = ∥x∥q−1 , ∀ x ∈ X, where q > 1 is a constant. The modulus of smoothness of X is the function ρX : [0, ∞) → [0, ∞) defined by ρX (t) = sup

{1 } (∥x + y∥ + ∥x − y∥) − 1 : ∥x∥ ≤ 1, ∥y∥ ≤ t . 2

A Banach space X is said to be uniformly smooth, if lim t→0

ρX (t) = 0. t

X is said to be q-uniformly smooth, if there exists a constant C > 0 such that ρX (t) ≤ Ctq , q > 1. Note that Jq is single-valued, if X is uniformly smooth. For further details, we refer to [3] and the references therein. We need the following definitions and results for the presentation of the paper. Lemma 2.1. ([24]) Let X be a real uniformly smooth Banach space. Then X is q-uniformly smooth if and only if there exists a constant Cq > 0 such that, for all x, y ∈ X, ∥x + y∥q ≤ ∥x∥q + q⟨y, Jq (x)⟩ + Cq ∥y∥q . Definition 2.2. Let X be a q-uniformly smooth Banach space. Let A, B : X → X, η : X × X → X be the ⋆ mappings and let Jq : X → 2X be the generalized duality mapping. Then A, B and η are said to be, respectively: (i) η-cocoercive, if there exists a constant µ1 > 0 such that ⟨Ax − Ay, Jq (η(x, y))⟩ ≥ µ1 ∥Ax − Ay∥q , ∀ x, y ∈ X; (ii) η-relaxed cocoercive, if there exists a constant γ1 > 0 such that ⟨Ax − Ay, Jq (η(x, y))⟩ ≥ (−γ1 )∥Ax − Ay∥q , ∀ x, y ∈ X; (iii) η-accretive, if

⟨Ax − Ay, Jq (η(x, y))⟩ ≥ 0, ∀ x, y ∈ X;

(iv) η-strongly accretive, if there exists a constant β1 > 0 such that ⟨Ax − Ay, Jq (η(x, y))⟩ ≥ β1 ∥x − y∥q , ∀ x, y ∈ X; (v) α-expansive, if there exists a constant α > 0 such that ∥Ax − Ay∥ ≥ α∥x − y∥, ∀ x, y ∈ X;

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(vi) β-Lipschitz continuous, if there exists a constant β > 0 such that ∥Bx − By∥ ≤ β∥x − y∥, ∀ x, y ∈ X; (vii) Lipschitz continuous, if there exists a constant τ > 0 such that ∥η(x, y)∥ ≤ τ∥x − y∥, ∀ x, y ∈ X. Definition 2.3. Let X be a q-uniformly smooth Banach space. Let A, B : X → X, H : X×X → X, η : X×X → X ⋆ be three single-valued mappings and Jq : X → 2X be the generalized duality mapping. Then (i) H(A, ·) is said to be η-cocoercive with respect to A, if there exists a constant µ > 0 such that ⟨H(Ax, u) − H(Ay, u), Jq (η(x, y))⟩ ≥ µ∥Ax − Ay∥q , ∀ x, y, u ∈ X; (ii) H(·, B) is said to η-relaxed cocoercive with respect to B, if there exists a constant γ > 0 such that ⟨H(u, Bx) − H(u, By), Jq (η(x, y))⟩ ≥ (−γ)∥Bx − By∥q , ∀ x, y, u ∈ X; (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, u) − H(Ay, u)∥ ≤ r1 ∥x − y∥, ∀ x, y, u ∈ X; (iv) H(·, B) is said to be r2 -Lipschitz continuous with respect to B, if there exists a constant r2 > 0 such that ∥H(u, Bx) − H(u, By)∥ ≤ r2 ∥x − y∥, ∀ x, y, u ∈ X. Definition 2.4. Let X be a q-uniformly smooth Banach space. A multi-valued mapping M : X → 2X is said to be η-cocoercive, if there exists a constant µ2 > 0 such that ⟨u − v, Jq (η(x, y))⟩ ≥ µ2 ∥u − v∥q , ∀ x, y ∈ X, u ∈ Mx, v ∈ My. Definition 2.5. Let X be a q-uniformly smooth Banach space. Let S, T : X × X → X, p, d : X → X be the mappings, then (i) S is said to be δS -η-strongly accretive with respect to p, if there exists a constant δS > 0 such that ⟨S(p(xn ), vn ) − S(p(xn−1 ), vn ), Jq (η(xn , xn−1 ))⟩ ≥ δS ∥xn − xn−1 ∥q , ∀ xn , xn−1 ∈ X; (ii) S is said to be Lipschitz continuous with respect to p, if there exists a constant λSp > 0 such that ∥S(p(xn ), vn ) − S(p(xn−1 ), vn )∥ ≤ λSp ∥xn − xn−1 ∥, ∀ xn , xn−1 ∈ X; (iii) T is said to be δT -η-strongly accretive with respect to d, if there exists a constant δT > 0 such that ⟨T(un , d(yn )) − T(un , d(yn−1 )), Jq (η(yn , yn−1 ))⟩ ≥ δT ∥yn − yn−1 ∥q , ∀ yn , yn−1 ∈ X; (iv) T is said to be Lipschitz continuous with respect to d, if there exists a constant λTd > 0 such that ∥T(un , d(yn )) − T(un , d(yn−1 ))∥ ≤ λTd ∥yn − yn−1 ∥, ∀ yn , yn−1 ∈ X. We apply the following concepts and results to prove the main results. Definition 2.6. ([2]) Let X be a q-uniformly smooth Banach space. Let A, B : X → X, H : X × X → X, η : X × X → X be the single-valued mappings. Then a set-valued mapping M : X → 2X is said to be H(·, ·)-η-cocoercive with respect to the mappings A and B, if M is η-cocoercive and (H(A, B) + λM)(X) = X, for all λ > 0.

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Theorem 2.7. ([2]) Let X be a q-uniformly smooth Banach space. Let H(A, B) be η-cocoercive with respect to A with constant µ > 0 and η-relaxed cocoercive with respect to B with constant γ > 0, A be α-expansive and B be β-Lipschitz continuous, µ > γ and α > β. Let M be H(·, ·)-η-cocoercive operator with respect to A and B. Then the operator (H(A, B) + λM)−1 is single-valued. Proof. For any u ∈ X, let x, y ∈ (H(A, B) + λM)−1 (u). It follows that −H(Ax, Bx) + u ∈ λMx and

−H(Ay, By) + u ∈ λMy

as M is η-cocoercive (thus η-accretive), we have 0 ≤ ⟨−H(Ax, Bx) + u − (−H(Ay, By) + u), Jq (η(x, y))⟩ = −⟨H(Ax, Bx) − H(Ay, By), Jq (η(x, y))⟩ = −⟨H(Ax, Bx) − H(Ay, Bx) + H(Ay, Bx) − H(Ay, By), Jq (η(x, y))⟩ = −⟨H(Ax, Bx) − H(Ay, Bx), Jq (η(x, y))⟩ − ⟨H(Ay, Bx) − H(Ay, By), Jq (η(x, y))⟩. Since H is η-cocoercive with respect to A with constant µ and η-relaxed cocoercive with respect to B with constant γ, A is α-expansive and B is β-Lipschitz continuous, thus (2.1) becomes 0 ≤ −µαq ∥x − y∥q + γβq ∥x − y∥q = −(µαq − γβq )∥x − y∥q ≤ 0,

(2.2)

since µ > γ and α > β. Thus, we have x = y and so (H(A, B) + λM)−1 is single-valued. Definition 2.8. ([2]) Let X be a q-uniformly smooth Banach space. Let H(A, B) be η-cocoercive with respect to A with constant µ > 0 and η-relaxed cocoercive with respect to B with constant γ > 0, A be α-expansive, B be β-Lipschitz continuous and η is τ-Lipschitz continuous and µ > γ, α > β. Let M be an H(·, ·)-η-cocoercive H(·,·)−η operator with respect to A and B. Then the resolvent operator Rλ,M : X → X is defined by H(·,·)−η

Rλ,M

(u) = (H(A, B) + λM)−1 (u), ∀ u ∈ X.

(2.3)

Now, we show the Lipschitz continuity of the resolvent operator and calculate its Lipschitz constant. Theorem 2.9. ([2]) Let X be a q-uniformly smooth Banach space. Let H(A, B) be η-cocoercive with respect to A with constant µ > 0 and η-relaxed cocoercive with respect to B with constant γ > 0, A be α-expansive, B be β-Lipschitz continuous and η be τ-Lipschitz continuous and µ > γ, α > β. Let M be an H(·, ·)-η-cocoercive operator with respect τq−1 H(·,·)−η to A and B. Then the resolvent operator Rλ,M : X → X is -Lipschitz continuous, that is µαq − γβq H(·,·)−η

∥Rλ,M

H(·,·)−η

(u) − Rλ,M

(v)∥ ≤

τq−1 ∥u − v∥, ∀ u, v ∈ X. µαq − γβq

Proof. Let u and v be any given points in X. It follows from (2.3) that H(·,·)−η

Rλ,M and

H(·,·)−η

Rλ,M

(u) = (H(A, B) + λM)−1 (u),

(v) = (H(A, B) + λM)−1 (v).

This implies that 1 H(·,·)−η H(·,·)−η H(·,·)−η (u − H(A(Rλ,M (u)), B(Rλ,M (u)))) ∈ M(Rλ,M (u)), λ

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and

1 H(·,·)−η H(·,·)−η H(·,·)−η (v − H(A(Rλ,M (v)), B(Rλ,M (v)))) ∈ M(Rλ,M (v)). λ For the sake of clarity, we take H(·,·)−η

Pu = Rλ,M

H(·,·)−η

(u) and Pv = Rλ,M

(v).

Since M is η-cocoercive (thus η-accretive), we have 1 ⟨u − H(A(Pu), B(Pu)) − (v − H(A(Pv), B(Pv))), Jq (η(Pu, Pv))⟩ ≥ 0, λ 1 ⟨u − v − (H(A(Pu), B(Pu)) − H(A(Pv), B(Pv))), Jq (η(Pu, Pv))⟩ ≥ 0. λ Therefore we have ⟨u − v, Jq (η(Pu, Pv))⟩ ≥ ⟨H(A(Pu), B(Pu)) − H(A(Pv), B(Pv)), Jq (η(Pu, Pv))⟩. It follows that ∥u − v∥∥η(Pu, Pv)∥q−1 ≥ ⟨u − v, Jq (η(Pu, Pv))⟩ ≥ ⟨H(A(Pu), B(Pu)) − H(A(Pv), B(Pu)), Jq (η(Pu, Pv))⟩ + ⟨H(A(Pv), B(Pu)) − H(A(Pv), B(Pv)), Jq (η(Pu, Pv))⟩ ≥ µ∥A(Pu) − A(Pv)∥q − γ∥B(Pu) − B(Pv)∥q ≥ µαq ∥Pu − Pv∥q − γβq ∥Pu − Pv∥q and so

|u − v∥∥η(Pu, Pv)∥q−1 ≥ (µαq − γβq )∥Pu − Pv∥q , (µαq − γβq )∥Pu − Pv∥q ≤ ∥u − v∥∥η(Pu, Pv)∥q−1 ≤ ∥u − v∥τq−1 ∥Pu − Pv∥q−1 ∥Pu − Pv∥ ≤ H(·,·)−η

i.e. ∥Rλ,M

H(·,·)−η

(u) − Rλ,M

τq−1 ∥u − v∥, − γβq

µαq

(v)∥ ≤

τq−1 ∥u − v∥, ∀ u, v ∈ X. − γβq

µαq

This completes the proof. The following Matlab programme shows that H(·, ·) is 13 -η-cocoercive with respect to A and 12 -η-relaxed cocoercive with respect to B. Example 2.10. Let X = R2 and A, B : R2 → R2 be defined by A(x1 , x2 ) = (x1 , 3x2 ), B(y1 , y2 ) = (−y1 , −y1 − y2 ), ∀ (x1 , x2 ), (y1 , y2 ) ∈ R2 . Let H(A, B), η : R2 × R2 → R2 be defined by H(Ax, By) = Ax + By, η(x, y) = x − y, ∀ x, y ∈ R2 . x1 = input (′ enter the vector x1 :′ ); x2 = input (′ enter the vector x2 :′ ); y1 = input (′ enter the vector y1 :′ ); y2 = input(′ enterthevector y2 :′ ); Assign values: T1 = x1 ; T2 = 3. ∗ x2 ; P1 = y1 ; P2 = 3. ∗ y2 ; u1 = x1 − y1 ; u2 = x2 − y2 ;

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compute inner product ⟨H(Ax, u) − H(Ay, u), η(x, y)⟩ = W, where W1 = u1 .2 ; W2 = 3. ∗ u2 .2 ; W = W1 + W2 compute square of norm ∥Ax − Ay∥2 = M1 , where L1 = u1 .2 ; L2 = 9. ∗ u2 .2 ; M1 = L1 + L2 ; M = (1/3). ∗ M1 then it is easy to check that H is 13 -η-cocoercive with respect to A, i.e. W ≥ M. Next compute inner product ⟨H(u, Bx) − H(u, By), η(x, y)⟩ = V. V = −(u1 .2 + u1 . ∗ 3. ∗ u2 + 3. ∗ u2 .2 ) compute square of norm ∥Bx − By∥2 = B1 , where B1 = (2. ∗ (u1 .2 ) + (3. ∗ u2 ).2 + (2. ∗ u1 ). ∗ (3. ∗ u2 )); Z = −(1/2). ∗ B1 then it is easy to check that H is 21 -η-relaxed cocoercive with respect to B, i.e. V ≥ Z. 3. An application for solving a system of generalized variational-like inclusions In this section, we show that H(·, ·)-η-cocoercive operator plays an important role for solving a system of generalized variational-like inclusions. Let X1 and X2 be two q-uniformly smooth Banach spaces. Let f : X1 → X1 , 1 : X2 → X2 , S : X1 ×X2 → X1 , T : X1 × X2 → X2 , p : X1 → X1 , d : X2 → X2 , A1 , B1 : X1 → X1 , A2 , B2 : X2 → X2 , H1 : X1 × X1 → X1 , H2 : X2 ×X2 → X2 , η1 : X1 ×X1 → X1 , η2 : X2 ×X2 → X2 be the single-valued mappings. Let E : X1 → CB(X1 ), F : X2 → CB(X2 ), M : X1 → 2X1 and N : X2 → 2X2 be the multi-valued mappings such that M is H1 (A1 , B1 )η1 -cocoercive mapping and N is H2 (A2 , B2 )-η2 -cocoercive mapping. We consider the problem of finding (x, y) ∈ X1 × X2 , u ∈ E(x), v ∈ F(y) such that   0 ∈ S(p(x), v) + M( f (x)),    (3.1)     0 ∈ T(u, d(y)) + N(1(y)). Problem (3.1) is called the system of generalized variational-like inclusions. If X1 , X2 are real Hilbert spaces, A1 , B1 , A2 , B2 , H1 , H2 , η1 , η2 are all identity mappings and M and N are nonlinear mappings, then system (3.1) is introduced and studied by Lan et al. [15]. For suitable choices of operators involved in the formulation of system (3.1), many systems of variational inclusions can be obtained from (3.1), which exist in literatures. Lemma 3.1. Let X1 and X2 be two q-uniformly smooth Banach spaces. Let f : X1 → X1 , 1 : X2 → X2 , S : X1 ×X2 → X1 , T : X1 × X2 → X2 , p : X1 → X1 , d : X2 → X2 , A1 , B1 : X1 → X1 , A2 , B2 : X2 → X2 , H1 : X1 × X1 → X1 , H2 : X2 × X2 → X2 , η1 : X1 × X1 → X1 , η2 : X2 × X2 → X2 be the single-valued mappings. Let H1 (A1 , B1 ) be µ1 -η1 -cocoercive with respect to A1 and γ1 -η1 -relaxed cocoercive with respect to B1 , A1 be α1 -expansive and B1 be β1 -Lipschitz continuous, η1 be τ1 -Lipschitz continuous, µ1 > γ1 and α1 > β1 and H2 (A2 , B2 ) be µ2 -η2 -cocoercive with respect to A2 and γ2 -η2 -relaxed cocoercive with respect to B2 , A2 be α2 -expansive and B2 be β2 -Lipschitz continuous, η2 be τ2 -Lipschitz continuous, µ2 > γ2 and α2 > β2 . Let E : X1 → CB(X1 ) and F : X2 → CB(X2 ), M : X1 → 2X1 and N : X2 → 2X2 be the multi-valued mappings such that M is H1 (A1 , B1 )-η1 -cocoercive mapping and N is H2 (A2 , B2 )η2 -cocoercive mapping. Then for any (x, y) ∈ X1 × X2 , u ∈ E(x), v ∈ F(y), (x, y, u, v) is a solution of problem (3.1) if and only if (x, y, u, v) satisfies [ ]  H1 (·,·)−η1   f (x) = R H (A ( f (x)), B ( f (x))) − λS(p(x), v) , 1 1 1  λ,M   (3.2)   [ ]   1(y) = RH2 (·,·)−η2 H (A (1(y)), B (1(y))) − ρT(u, d(y)) ,  2 2 2 ρ,N where λ > 0 and ρ > 0 are two constants.

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903 H (·,·)−η1

1 Proof. It can be proved easily by the application of definitions of resolvent operators Rλ,M

H (·,·)−η2

2 and Rρ,N

.

Remark 3.2. The equalities (3.2) can be written as, respectively, [ ]  H1 (·,·)−η1   [H1 (A1 ( f (x)), B1 ( f (x))) − λS(p(x), v)] , x = (1 − t1 )x + t1 x − f (x) + Rλ,M      [ ]    y = (1 − t2 )y + t2 y − 1(y) + RH2 (·,·)−η2 [H2 (A2 (1(y)), B2 (1(y))) − ρT(u, d(y))] , ρ,N where 0 < t1 , t2 ≤ 1 are two parameters and λ, ρ > 0 are constants. These fixed point formulations enable us to suggest the following iterative algorithm. Algorithm 3.3. Let X1 , X2 , f , 1, S, T, p, d, A1 , B1 , A2 , B2 , H1 , H2 , η1 , η2 , E, F, M and N all be same as in problem (3.1). For any given (x0 , y0 ) ∈ X1 × X2 , u0 ∈ E(x0 ), v0 ∈ F(y0 ), we suggest the following iterative schemes: [ ] H1 (·,·)−η1 xn+1 = (1 − t1 )xn + t1 xn − f (xn ) + Rλ,M [H1 (A1 ( f (xn )), B1 ( f (xn ))) − λS(p(xn ), vn )] , (3.3) [ ] H2 (·,·)−η2 yn+1 = (1 − t2 )yn + t2 yn − 1(yn ) + Rρ,N [H2 (A2 (1(yn )), B2 (1(yn ))) − ρT(un , d(yn ))] , where 0 < t1 , t2 ≤ 1 are two parameters and λ, ρ > 0 are constants, n = 0, 1, 2, ..... and we choose un+1 ∈ E(xn+1 ), vn+1 ∈ F(yn+1 ) such that    ∥un+1 − un ∥ ≤ D(E(xn+1 ), E(xn )),       ∥vn+1 − vn ∥ ≤ D(F(yn+1 ), F(yn )). Theorem 3.4. Let X1 and X2 be two q-uniformly smooth Banach spaces. Let A1 , B1 , p : X1 → X1 , A2 , B2 , d : X2 → X2 , H1 : X1 × X2 → X1 , H2 : X2 × X1 → X2 , be the mappings such that H1 (A1 , B1 ) is η1 -cocoercive with respect to A1 with constant µ1 and η1 -relaxed cocoercive with respect to B1 with constant γ1 , A1 is α1 -expansive, B1 be β1 -Lipschitz continuous , α1 > β1 and µ1 > γ1 ; H2 (A2 , B2 ) is η2 -cocoercive with respect to A2 with constant µ2 and η2 -relaxed cocoercive with respect to B2 with constant γ2 , A2 is α2 -expansive, B2 is β2 -Lipschitz continuous, α2 > β2 and µ2 > γ2 . Assume that η1 : X1 × X1 → X1 is τ1 -Lipschitz continuous, η2 : X2 × X2 → X2 is τ2 -Lipschitz continuous, f : X1 → X1 is strongly accretive with constant δ1 and λ f -Lipschitz continuous and 1 : X2 → X2 is strongly accretive with constant δ2 and λ1 -Lipschitz continuous. Let S : X1 × X2 → X1 be η1 -strongly accretive with respect to p with constant δS and λSp -Lipschitz continuous with respect to p and λS2 -Lipschitz continuous in the second argument. Suppose that T : X1 × X2 → X2 is η2 -strongly accretive with constant δT with respect to d and λTd -Lipschitz continuous with respect to d and λT1 -Lipschitz continuous in the first argument. Let E : X1 → CB(X1 ) be D-Lipschitz continuous with constant λDE and F : X2 → CB(X2 ) be D-Lipschitz continuous with constant λDF . Let H1 (A1 , B1 ) be r1 -Lipschitz continuous with respect to A1 and r2 -Lipschitz continuous with respect to B1 and H2 (A2 , B2 ) be r3 -Lipschitz continuous with respect to A2 and r4 -Lipschitz continuous with respect to B2 . Suppose that M : X1 → 2X1 is H1 (A1 , B1 )-η1 -cocoercive and N : X2 → 2X2 is H2 (A2 , B2 )-η2 -cocoercive. If there exist positive constants ρ and λ such that:  √  t2 τ2 q−1 ρλT1 λDE t1 τ1 q−1  q q ′   (1 − t1 ) + t1 1 − qδ1 + Cq λ f + < 1,  qθ + q q q   µ1 α1 − γ1 β1 µ2 α2 − γ2 β2       √   q q−1 q1 q q ′ q q−1 q  where θ = (r1 + r2 ) λ f − qλδS + qλλsp (r1 + r2 ) λ f + qλλsp τ1 + Cq λ λsp ,      (3.4)    √ q−1 q−1   t τ λλ λ t τ 1 1 S D 2 2  2 F q ′′  (1 − t2 ) + t2 q 1 − qδ2 + Cq λ1 + < 1,  qθ + q q   µ2 α2 q − γ2 β2 µ1 α1 − γ1 β1       √    q q 1−1 q−1 q ′′ q q−1 q  where θ = r3 + r4 ) λ1 − qρδT + qρλTd (r3 + r4 ) λ1 + qρλTd τ + Cq ρ λ .  2

Td

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Then the iterative sequences {xn }, {yn }, {un } and {vn } generated by Algorithm 3.3 converge strongly to x, y, u and v, respectively and (x, y, u, v) is a solution of the system of generalized variational-like inclusions (3.1). Proof. Using Algorithm 3.3 and the Lipschitz continuity of the resolvent operator, we have H (·,·)−η1

1 ∥xn+1 − xn ∥ = ∥(1 − t1 )xn + t1 [xn − f (xn ) + Rλ,M

[H1 (A1 ( f (xn )), B1 ( f (xn )))

− λS(p(xn ), vn )] − {(1 − t1 )xn−1 + t1 [xn−1 − f (xn−1 ) H (·,·)−η1

1 + Rλ,M

[H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 ))) − λS(p(xn−1 ), vn−1 )]}∥

≤ (1 − t1 )∥xn − xn−1 ∥ + t1 ∥xn − xn−1 − ( f (xn ) − f (xn−1 ))∥ H (·,·)−η1

1 + t1 ∥Rλ,M

−

[H1 (A1 ( f (xn )), B1 ( f (xn ))) − λS(p(xn ), vn )]

H1 (·,·)−η1 Rλ,M [H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 )))

− λS(p(xn−1 ), vn−1 )]∥

≤ (1 − t1 )∥xn − xn−1 ∥ + t1 ∥xn − xn−1 − ( f (xn ) − f (xn−1 ))∥ q−1

+

t1 τ1 q

q ∥[H1 (A1 ( f (xn )), B1 ( f (xn )))

µ1 α1 − γ1 β1

− H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 )))]

− λ[S(p(xn ), vn ) − S(p(xn−1 ), vn−1 )]∥ ≤ (1 − t1 )∥xn − xn−1 ∥ + t1 ∥xn − xn−1 − ( f (xn ) − f (xn−1 ))∥ q−1

+

t1 τ1 q

q ∥[H1 (A1 ( f (xn )), B1 ( f (xn )))

µ1 α1 − γ1 β1

− H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 )))]

− λ[S(p(xn ), vn ) − S(p(xn−1 ), vn )] q−1

+

t1 τ1 λ q

q ∥S(p(xn−1 ), vn )

µ1 α1 − γ1 β1

− S(p(xn−1 ), vn−1 )∥.

(3.5)

Since f is strongly accretive with constant δ1 and Lipschitz continuous with constant λ f , we have ∥xn − xn−1 − ( f (xn ) − f (xn−1 ))∥q ≤ ∥xn − xn−1 ∥q − q⟨ f (xn ) − f (xn−1 ), Jq (xn − xn−1 )⟩ + Cq ∥ f (xn ) − f (xn−1 )∥q q

≤ ∥xn − xn−1 ∥q − qδ1 ∥xn − xn−1 ∥q + Cq λ f ∥xn − xn−1 ∥q q

= (1 − qδ1 + Cq λ f )∥xn − xn−1 ∥q , thus ∥xn − xn−1 − ( f (xn ) − f (xn−1 ))∥ ≤

√ q

q

1 − qδ1 + Cq λ f ∥xn − xn−1 ∥.

(3.6)

As H1 (·, ·) is r1 -Lipschitz continuous with respect to A1 and r2 -Lipschitz continuous with respect to B1 and f is λ f -Lipschitz continuous, we have ∥H1 (A1 ( f (xn )), B1 ( f (xn ))) − H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 )))∥ = ∥H1 (A1 ( f (xn )), B1 ( f (xn ))) − H1 (A1 ( f (xn−1 )), B1 ( f (xn ))) + H1 (A1 ( f (xn−1 )), B1 ( f (xn ))) − H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 )))∥ = ∥H1 (A1 ( f (xn )), B1 ( f (xn ))) − H1 (A1 ( f (xn−1 )), B1 ( f (xn )))∥ + ∥H1 (A1 ( f (xn−1 )), B1 ( f (xn ))) − H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 )))∥ ≤ r1 ∥ f (xn ) − f (xn−1 )∥ + r2 ∥ f (xn ) − f (xn−1 )∥ ≤ (r1 + r2 )λ f ∥xn − xn−1 ∥.

(3.7)

Making use of (3.7) and the facts that S is δS -η1 -strongly accretive and λSp -Lipschitz continuous with respect to p and η1 is τ1 -Lipschitz continuous, we obtain

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∥H1 (A1 ( f (xn )), B1 ( f (xn ))) − H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 ))) − λ[S(p(xn ), vn ) − S(p(xn−1 ), vn )]∥q ≤ ∥H1 (A1 ( f (xn )), B1 ( f (xn ))) − H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 )))∥q − qλ⟨S(p(xn ), vn ) − S(p(xn−1 ), vn ), Jq (η1 (xn , xn−1 ))⟩ − qλ⟨S(p(xn ), vn ) − S(p(xn−1 ), vn ), Jq [H1 (A1 ( f (xn )), B1 ( f (xn ))) − H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 )))] − Jq (η1 (xn , xn−1 ))⟩ + Cq λq ∥S(p(xn ), vn ) − S(p(xn−1 ), vn )∥q [ q ≤ (r1 + r2 )q λ f ∥xn − xn−1 ∥q − qλδS ∥xn − xn−1 ∥q + qλ∥S(p(xn ), vn ) − S(p(xn−1 ), vn )∥ × ∥H1 (A1 ( f (xn )), B1 ( f (xn ))) ] q − H1 (A1 ( f (xn−1 )), B1 ( f (xn−1 )))∥q−1 + ∥η1 (xn , xn−1 )∥q−1 + Cq λq λSp ∥xn − xn−1 ∥q [ q q−1 ≤ (r1 + r2 )q λ f ∥xn − xn−1 ∥q − qλδS ∥xn − xn−1 ∥q + qλλSp ∥xn − xn−1 ∥ (r1 + r2 )q−1 λ f ∥xn − xn−1 ∥q−1 ] q−1 q + τ1 ∥xn − xn−1 ∥q−1 + Cq λq λSp ∥xn − xn−1 ∥q ] [ q q−1 q−1 q (3.8) = (r1 + r2 )q λ f − qλδS + qλλSp (r1 + r2 )q−1 λ f + qλλSp τ1 + Cq λq λSp ∥xn − xn−1 ∥q . As S is λS2 -Lipschitz continuous in the second argument and F is D-Lipschitz continuous with constant λDF , we have ∥S(p(xn−1 ), vn ) − S(p(xn−1 ), vn−1 )∥ ≤ λS2 ∥vn − vn−1 ∥ ≤ λS2 D(F(yn ), F(yn−1 )) ≤ λS2 λDF ∥yn − yn−1 ∥.

(3.9)

Using (3.6), (3.8) and (3.9), (3.5) becomes √ q ∥xn+1 − xn ∥ ≤ (1 − t1 )∥xn − xn−1 ∥ + t1 q 1 − qδ1 + Cq λ f ∥xn − xn−1 ∥ q−1

+

t1 τ1 q

q−1

qθ

′

µ1 α1 − γ1 β1

∥xn − xn−1 ∥ +

t1 τ1 λ q

q λS2 λDF ∥yn

µ1 α1 − γ1 β1

− yn−1 ∥

= θ1 ∥xn − xn−1 ∥ + θ2 ∥yn − yn−1 ∥. where

√

q−1

θ1 = (1 − t1 ) + t1 1 − qδ1 + q

and θ′ =

(3.10)

q Cq λ f

+

t1 τ1 q

q−1

qθ

µ1 α1 − γ1 β1

′

, θ2 =

t1 τ1 λ q

q λS2 λDF

µ1 α1 − γ1 β1

√ q q−1 q−1 q q (r1 + r2 )q λ f − qλδS + qλλSp (r1 + r2 )q−1 λ f + qλλSp τ1 + Cq λq λSp .

In a similar way, we estimate H (·,·)−η

2 2 ∥yn+1 − yn ∥ = ∥(1 − t2 )yn + t2 [yn − 1(yn ) + Rρ,N [H2 (A2 (1(yn )), B2 (1(yn ))) { − ρT(un , d(yn ))] − (1 − t2 )yn−1 + t2 [yn−1 − 1(yn−1 ) } H2 (·,·)−η2 [H2 (A2 (1(yn−1 )), B2 (1(yn−1 ))) − ρT(un−1 , d(yn−1 ))] ∥ + Rρ,N

≤ (1 − t2 )∥yn − yn−1 ∥ + t2 ∥yn − yn−1 − (1(yn ) − 1(yn−1 ))∥ q−1

+

t2 τ2 q

q ∥H2 (A2 (1(yn )), B2 (1(yn )))

µ2 α2 − γ2 β2

− H2 (A2 (1(yn−1 )), B2 (1(yn−1 ))) − ρ[T(un , d(yn )) − T(un , d(yn−1 ))] − ρ[T(un , d(yn−1 )) − T(un−1 , d(yn−1 ))]∥

(3.11)

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906

≤ (1 − t2 )∥yn − yn−1 ∥ + t2 ∥yn − yn−1 − (1(yn ) − 1(yn−1 ))∥ q−1

+

t2 τ2 q

q ∥H2 (A2 (1(yn )), B2 (1(yn )))

µ2 α2 − γ2 β2

− H2 (A2 (1(yn−1 )), B2 (1(yn−1 )))

− ρ[T(un , d(un )) − T(un , d(yn−1 ))]∥ q−1

+

t2 τ2 ρ q

q ∥T(un , d(yn−1 ))

µ2 α2 − γ2 β2

− T(un−1 , d(yn−1 ))∥.

(3.12)

Since 1 is strongly accretive with constant δ2 and Lipschitz continuous with constant λ1 , using the same arguments as for (3.6), we have √ q q ∥yn − yn−1 − (1(yn ) − 1(yn−1 ))∥ ≤ 1 − qδ2 + Cq λ1 ∥yn − yn−1 ∥. (3.13) As H2 (·, ·) is r3 -Lipschitz continuous with respect to A2 and r4 -Lipschitz continuous with respect to B2 and 1 is λ1 -Lipschitz continuous, we obtain ∥H2 (A2 (1(yn )), B2 (1(yn ))) − H2 (A2 (1(yn−1 )), B2 (1(yn−1 )))∥ ≤ (r3 + r4 )λ1 ∥yn − yn−1 ∥.

(3.14)

Making use of (3.13) and the facts that T is δT -η2 -strongly accretive with respect to d, λTd -Lipschitz continuous with respect to d and η2 is τ2 -Lipschitz continuous, we obtain ∥H2 (A2 (1(yn )), B2 (1(yn ))) − H2 (A2 (1(yn−1 )), B2 (1(yn−1 ))) − ρ[T(un , d(yn )) − T(un , d(yn−1 ))]∥q ≤ ∥H2 (A2 (1(yn )), B2 (1(yn ))) − H2 (A2 (1(yn−1 )), B2 (1(yn−1 )))∥q − qρ⟨T(un , d(yn )) − T(un , d(yn−1 )), Jq (η2 (yn , yn−1 ))⟩ − qρ⟨T(un , d(yn )) − T(un , d(yn−1 )), Jq [H2 (A2 (1(yn )), B2 (1(yn ))) − H2 (A2 (1(yn−1 )), B2 (1(yn−1 )))] − Jq (η2 (yn , yn−1 ))⟩ + Cq ρq ∥T(un , d(yn )) − T(un , d(yn−1 ))∥q

[ q ≤ (r3 + r4 )q λ1 ∥yn − yn−1 ∥q − qρδT ∥yn − yn−1 ∥q + qρ∥T(un , d(yn )) − T(un , d(yn−1 ))∥ × ∥H2 (A2 (1(yn )), B2 (1(yn ))) ] q − H2 (A2 (1(yn−1 )), B2 (1(yn−1 )))∥q−1 + ∥η2 (yn , yn−1 )∥q−1 + Cq ρq λTd ∥yn − yn−1 ∥q [ q q−1 ≤ (r3 + r4 )q λ1 ∥yn − yn−1 ∥q − qρδT ∥yn − yn−1 ∥q + qρλTd ∥yn − yn−1 ∥ (r3 + r4 )q−1 λ1 ∥yn − yn−1 ∥q−1 ] q−1 q + τ2 ∥yn − yn−1 ∥q−1 + Cq ρq λTd ∥yn − yn−1 ∥q [ ] q q−1 q−1 q (3.15) = (r3 + r4 )q λ1 − qρδT + qρλTd (r3 + r4 )q−1 λ1 + qρλTd τ2 + Cq ρq λTd ∥yn − yn−1 ∥q . As T is λT1 -Lipschitz continuous in the first argument and E is D-Lipschitz continuous with constant λDE , we have ∥T(un , d(yn−1 )) − T(un−1 , d(yn−1 ))∥ ≤ λT1 ∥un − un−1 ∥ ≤ λT1 D(E(xn ), E(xn−1 )) ≤ λT1 λDE ∥xn − xn−1 ∥.

(3.16)

Using (3.13), (3.15) and (3.16), (3.12) becomes √ q q ∥yn+1 − yn ∥ ≤ (1 − t2 )∥yn − yn−1 ∥ + t2 1 − qδ2 + Cq λ1 ∥yn − yn−1 ∥ q−1

+

q−1

t2 τ2

t2 τ2 ρ ′′ q q θ ∥yn − yn−1 ∥ + q q λT1 λDE ∥xn − xn−1 ∥ µ2 α2 − γ2 β2 µ2 α2 − γ2 β2

= θ3 ∥yn − yn−1 ∥ + θ4 ∥xn − xn−1 ∥, where

√ q q θ3 = (1 − t2 ) + t2 1 − qδ2 + Cq λ1 +

q−1

t2 τ2

(3.17) q−1

t2 τ2 ρ ′′ q q θ , θ4 = q q λT1 λDE µ2 α2 − γ2 β2 µ2 α2 − γ2 β2

(3.18)

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√ q q q−1 q−1 q and θ′′ = (r3 + r4 )q λ1 − qρδT + qρλTd (r3 + r4 )q−1 λ1 + qρλTd τ2 + Cq ρq λTd . Combining (3.10) and (3.17), we obtain ∥xn+1 − xn ∥ + ∥yn+1 − yn ∥ ≤ θ1 ∥xn − xn−1 ∥ + θ2 ∥yn − yn−1 ∥ + θ3 ∥yn − yn−1 ∥ + θ4 ∥xn − xn−1 ∥ = (θ1 + θ4 )∥xn − xn−1 ∥ + (θ2 + θ3 )∥yn − yn−1 ∥, which implies that

[ ] ∥xn+1 − xn ∥ + ∥yn+1 − yn ∥ ≤ θ ∥xn − xn−1 ∥ + ∥yn − yn−1 ∥ ,

(3.19)

where θ = max{(θ1 + θ4 ), (θ2 + θ3 )}, where θ1 and θ2 are defined in (3.11) and θ3 and θ4 are defined in (3.18). By (3.4), θ < 1 and (3.19) implies that {xn } and {yn } are both Cauchy sequences. Thus there exist x ∈ X1 and y ∈ X2 such that xn → x and yn → y as n → ∞. From Algorithm 3.3, we have ∥un+1 − un ∥ ≤ D(E(xn+1 ), E(xn )) ≤ λDE ∥xn+1 − xn ∥, ∥vn+1 − vn ∥ ≤ D(F(yn+1 ), F(yn )) ≤ λDF ∥yn+1 − yn ∥, it follows that {un } and {vn } are both Cauchy sequences, thus un → u ∈ E(x) and vn → v ∈ F(y). Further d′ (u, E(x)) ≤ ∥u − un ∥ + d′ (un , E(x)) ≤ ∥u − un ∥ + D(E(xn ), E(x)) ≤ ∥u − un ∥ + λDE ∥xn − x∥ → 0, as n → ∞, which implies that d′ (u, E(x)) = 0. Since E(x) ∈ CB(X1 ), it follows that u ∈ E(x). Similarly, we can show that v ∈ F(y). By the continuity, (3.3) and Algorithm 3.3, it is easy to see that [ ] H1 (·,·)−η1 f (x) = Rλ,M H1 (A1 ( f (x)), B1 ( f (x))) − λS(p(x), v) , H (·,·)−η2

2 1(y) = Rρ,N

[

] H2 (A2 (1(y)), B2 (1(y))) − ρT(u, d(y)) .

Thus by Lemma (3.1), the required result follows. References [1] R. Ahmad, M. Dilshad, M-M. Wong, J.C. Yao, H(·, ·)-cocoercive operators and an application for solving generalized variational inclusions, Abstr. Appl. Anal. 2011 (2011), Article ID 261534, 12 pages. [2] R. Ahmad, M. Dilshad, M-M. Wong, J.C. Yao, Generalized set-valued variational-like inclusions involving H(·, ·)-η-cocoercive operators in Banach spaces, Preprint. [3] Q.H. Ansari, Topics in Nonlinear Analysis and Optimization, World Education, Delhi, 2012. [4] Q.H. Ansari, J.C. Yao, A fixed point theorem and its applications to the system of variational inequalities, Bull. Austral. Math. Soc. 59 (1999) 433–442. [5] Q.H. Ansari, J.C. Yao, System of generalized variational inequalities and their applications, Appl. Anal. 76 (3-4) (2000) 203–217. [6] Q.H. Ansari, J.C. Yao, Iterative schemes for solving mixed variational-like inequalities, J. Optim. Theory Appl. 108 (2001) 527–541. [7] J. P. Aubin, Mathematical Methods of Game Theory and Economic, North-Holland, Amsterdam, 1982. [8] X.P. Ding, Existence and algorithm of solutions for generalized mixed implicit quasi-variational inequalities, Appl. Math. Comput. 113 (2000) 67–80. [9] Y.P. Fang, Y.J. Cho, J.K. Kim, (H, η)-accretive operators and approximating solutions for systems of variational inclusions in Banach spaces, Reprint (2004). [10] Y.P. Fang, N.J. Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (2004) 647–653. [11] Y.P. Fang, N.J. Huang, H-monotone operator and resolvent operator technique for variational inclusions, Appl. Math. Comput. 145 (2003) 795–803.

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