A Generalized Mixed Variational Inclusion Involving ( H ( · , · ) , η )-Monotone Operators in Banach Spaces

In this paper, we introduce a new class of monotone operators − (H(·, ·), η)monotone operators, which generalize many existing monotone operators. The resolvent operator associated with an (H(·, ·), η)monotone operator is defined and its Lipschitz continuity is presented. As an application, we also consider a new generalized mixed variational inclusion involving (H(·, ·), η)-monotone operators and construct a new algorithm for solving the generalized mixed variational inclusion. Under some suitable conditions, we prove the convergence of the iterative sequences generated by the algorithm. These results improve and generalize many corresponding results in recently literatures.


Introduction
The resolvent operator method is an important and useful tool to study approximation solvability of nonlinear variational inequalities and variational inclusions, which are providing mathematical models to some problems arising in optimization and control, economics and engineering science.In order to study various variational inequalities and variational inclusions, Ding(2000), Huang and Fang (2003), Fang and Huang (2003), Fang et al.(2005), Verma (2006), Zhang (2007), Sun et al.(2008), Xia and Huang (2007), Feng and Ding (2009) and He et al.(2008) have introduced the concepts of η-subdifferential operators, maximal η-monotone operators, H-monotone operators, (H, η)-monotone operators, A-monotone operators, (A, η)-monotone operators, G-η-monotone operators, M-monotone operators in Hilbert spaces, H-monotone operators, A-monotone operators and H-η-monotone operators in Banach spaces and their resolvent operators, respectively.Further, by using the resolvent operator technique, a number of nonlinear variational inclusions and many systems of variational inequalities and variational inclusions have been studied by some authors in recent years (for example Lan (2007), Ding and Feng (2008), Peng and Zhu (2007), Zeng (2007), Ding and Wang (2009)).
Motivated and inspired by the above works, we introduce a new class of monotone operators: (H(•, •), η)monotone operators, which provide a unifying framework for maximal monotone operators, η-subdifferential operators, maximal ηmonotone operators, H-monotone operators, (H, η)-monotone operators, A-monotone mappings, (A, η)-monotone operators, G-η-monotone operators, M-monotone operators, H-monotone operators, A-monotone operators and H-η-monotone operators.The resolvent operator associated with an (H(•, •), η)monotone operator is defined and its Lipschitz continuity is presented.We also consider a new generalized mixed variational inclusion involving (H(•, •), η)-monotone operators and construct a new algorithm for solving the generalized mixed variational inclusion.Under some suitable conditions, we prove the convergence of iterative sequences generated by the algorithm.These results improve and generalize many known corresponding results.

Preliminaries
Let E be a real Banach space with dual space E * , and the norm and the dual pair between E and E * are denoted by • and •, • respectively.CB(E) denotes the family of all bounded closed subsets of E. The Hausdorff metric on CB(E) is defined by The normalized duality mapping J : E → 2 E * on E is defined by If E = H is a Hilbert space, then J is the identity mapping on H.
Lemma 2.1 (Petryshyn(1970)) Let E be a real Banach space and J : E → 2 E * be the normalized duality mapping.Then for all x, y ∈ E, (1) T is said to be η-monotone if T (x) − T (y), η(x, y) ≥ 0; (2) T is said to be strictly η-monotone if T is η-monotone and if and only if x = y; (3) H(A, •) is said to be α-strongly η-monotone with respect to A if there exists a constant α > 0 such that (4) H(•, B) is said to be β-relaxed η-monotone with respect to B if there exists a constant β > 0 such that (5) H(•, •) is said to be λ-Lipschitz continuous with respect to A if there exists a constant λ > 0 such that (6) T is said to be -Lipschitz continuous if there exists a constant > 0 such that (7) η is said to be τ-Lipschitz continuous if there exists a constant τ > 0 such that η(x, y) ≤ τ x − y , ∀ x, y ∈ E.
Definition 2.2 (Lou et al.(2008)) Let M : E → 2 E * be a multi-valued mapping, H : E → E * and η : E × E → E be single-valued mappings.M is said to be (3) strictly η-monotone if M is η-monotone and equality holds if and only if x = y; (4) r-strongly η-monotone if there exists a constant r > 0 such that (5) m-relaxed η-monotone if there exists a constant m > 0 such that (6) maximal monotone, if M is monotone and has no a proper monotone extension in E, i.e., for all u, v 0 ∈ E, x ∈ M(u), x − y 0 , u − v 0 ≥ 0 implies y 0 ∈ M(v 0 ); when E is a reflexive Banach space, M is maximal monotone if and only if M is monotone and (J + λM)E = E * for all λ > 0; (7) maximal η-monotone, if M is η-monotone and has no a proper η-monotone extension in E, when E is a reflexive Banach space, M is maximal η-monotone if and only if M is η-monotone and (J + λM)E = E * for all λ > 0; (8) H-monotone, if M is monotone and (H + λM)E = E * for all λ > 0; (9) (H, η)-monotone, if M is η-monotone and (H + λM)E = E * for all λ > 0; (10) H-η-monotone, if M is m-η-relaxed monotone and (H + λM)E = E * for all λ > 0.
Definition 2.3 Let T : E → E be a single-valued mapping.T is said to be δ-strongly accretive, if there exists a constant δ > 0 and j(x − y) ∈ J(x − y) such that (2) If E = H is a Hilbert space, m = 0 and η(x, y) = x − y, ∀ x, y ∈ H, then Definition 3.1 reduces to the definition of M-monotone operators (Sun et al. (2008)).
Proof Since M is (H(•, •), η)-monotone with respect to A and B, we know that (H(A, B) + ρM)(E) = E * holds for all ρ > 0 and so there exists (u 1 , x 1 ) ∈ Graph(M) such that Since H(A, B) is α-strongly η-monotone with respect to A, β-relaxed monotone with respect to B and α > β, we have

Proof. For any given u
-monotone operator with respect to A and B and H(A, B) is α-strongly η-monotone with respect to A, β-relaxed η-monotone with respect to B and α > β, we have This show that Remark 3.2 Theorem 3.1 and Theorem 3.2 improve the similar conclusions (see Sun et al.(2008), Huang and Fang (2003), Zhang (2007), Feng and Ding (2009)).
uous and H(A, B) be α-strongly η-monotone with respect to A, β-relaxed η-monotone with respect to B and α > β.Let M : E → 2 E * be an (H(•, •), η)-monotone operator with respect to A and B. Then the resolvent operator For the sake of brevity, let From the above inequality and the conditions in the Theorem 3.3, we have This completes the proof. 2 4. An application for solving a generalized mixed variational inclusion In this section, we shall study a new generalized mixed variational inclusion involving (H(•, •), η)-monotone operators in Banach spaces and construct an iterative algorithm for approximating the solution of this variational inclusion by using the resolvent operator technique.
Throughout the rest of the paper, unless otherwise stated, let E be a real Banach space with the dual space E * and the norm • , •, • be the dual pair between E and E * , CB(E) be the family of all bounded closed subsets of E, J : E → 2 E * be the normalized duality mapping on E defined by and J * : E * → E * * be the normalized duality mapping on E * defined by where E * * is a dual space of E * .
We observe that E = H is a Hilbert space, then J and J * are the identity mappings on H.In the sequel, j and j * denote a selection of J and J * , respectively.
Let G : E → CB(E), S : E → CB(E) and T : E → CB(E) be set-valued mappings, and let N : ( If M is H-monotone in the first argument, then the problem (2) was introduced and studied by Zeng (2007).
From Definition 3.2, we can obtain the following conclusion.
Lemma 4.1 Let E, ω, F, N, T , S , p, g, H, η, G and M be same as in the problem (1).Then (u, x, y, z) is a solution of the problem (1) if and only if (u, x, y, z, ) satisfies the following relation −1 and ρ > 0 are constants.Remark 4.1 The equality (3) can be written as , where ω ∈ E * is any given element and ρ > 0 is a constant.By Nadler (1969), we know that this formulation enables us to suggest the following iterative algorithm.
Step 2. Let Step 3. Choose (5) Step 4. Choose errors {e n } ⊂ E to take into account a possible inexact computation such that Step 5.If (1) ξ 1 -Lipschitz continuous in the first argument if there exists some constant ξ 1 > 0 such that (2) ξ 2 -Lipschitz continuous in the second argument if there exists some constant ξ 2 > 0 such that (3) ξ 3 -Lipschitz continuous in the third argument if there exists some constant ξ 3 > 0 such that Definition 4.3 Let N : E × E 2 → E * be a single-valued mapping.N is said to be (1) ᾱ-Lipschitz in the first argument if there exists some constant ᾱ > 0 such that (2) β-Lipschitz in the second argument if there exists some constant β > 0 such that and T : E → CB(E) be set-valued mappings, and let N : mapping with respect to A and B, and (g − p)(E) dom(M(•, z)) ∅.Furthermore, suppose the following conditions are satisfied: (i) H(A, B) is α-strongly η-monotone with respect to A, β-relaxed η-monotone with respect to B and α > β, and 1 -Lipschitz continuous with respect to A and 2 -Lipschitz continuous with respect to B; (ii) S , T and G are H-Lipschitz continuous with constants l 1 , l 2 and l 3 , respectively; (iii) η : E × E → E is τ-Lipschitz continuous and N is ᾱ-Lipschitz continuous in the first argument and β-Lipschitz continuous in the second argument; (iv) g − p is s-Lipschitz continuous, and g − p − I is ς-strongly accretive, where I denotes the identity mapping on E; (v) F is ξ j -Lipschitz continuous in the j-th argument for j = 1, 2, 3; In addition, if there are constants μ > 0 such that and there exist constants 0 < ρ < r m , where r = α − β, such that Then the iterative sequences {u n }, {x n }, {y n } and {z n } generated by Algorithm 4.1 converge strongly to u * , x * , y * and z * , respectively, and (u * , x * , y * , z * ) is a solution of the problem (1).