EXISTENCE OF SOLUTIONS FOR GENERALIZED NONLINEAR MIXED VARIATIONAL-LIKE INEQUALITIES IN BANACH SPACES

We introduce and study a new class of generalized nonlinear mixed variational-like inequalities in reflexive Banach spaces. By applying Ding’s technique we prove several existence and uniqueness theorems of solutions for the generalized nonlinear mixed variational-like inequality, extend the auxiliary problem technique to suggest and analyze an iterative method to compute the approximate solutions of the generalized nonlinear mixed variational-like inequality, and establish the convergence criteria of the iterative method. The results presented in this paper improve, extend, and unify many known results in this area.


Introduction
Variational inequality theory has appeared as an effective and powerful tool to study and investigate a wide class of problems arising in pure and applied sciences including elasticity, optimization, economics, transportation, and structural analysis, see, for example, [1][2][3][4][5][6][7][8] and the references therein.The classical variational inequality problem has been generalized in various directions for its own development and its applications.Mixed variational-like inequality is an important and useful generalization of the classical variational inequality.On the other hand, constructing effective numerical algorithm is an important and difficult problem in the variational inequality theory.Cohen [3] first extended the auxiliary variational inequality technique to suggest and analyze an innovative iterative method for solutions of mixed variational inequalities in reflexive Banach spaces.Ding [8] proved an existence-uniqueness theorem of solutions of a new class of nonlinear mixed variational-like inequalities in reflexive Banach spaces by making use of minimax inequality due to Ding and Tan [9].Following the idea of Cohen [3], Ding suggested and analyzed a quite general iterative method to compute the approximate solutions and proved the convergence of the iterative method.
Motivated and inspired by the research work going on in this field, we will introduce and study a new class of generalized nonlinear mixed variational-like inequalities in reflexive Banach spaces, which includes many variational inequalities and variational-like 2 Generalized nonlinear mixed variational-like inequalities inequalities as special cases, for details, see [1][2][3][4][5][6][7][8] and the references therein.Following the idea of Ding [8], we prove several existence and uniqueness theorems of solutions for the generalized nonlinear mixed variational-like inequality, extend the auxiliary problem technique to suggest and analyze an iterative method to compute the approximate solutions of the generalized nonlinear mixed variational-like inequality, and obtain the convergence criteria of the iterative method.The results presented in this paper improve, extend, and unify many known results in this area.

Preliminaries
Let D be a nonempty convex subset of a Banach space E with dual E * and let u,v be the dual pairing between u ∈ E * and v ∈ E. Let T,A,B,C : be a real-valued functional.Now we consider the following generalized nonlinear mixed variational-like inequality problem.For a given w * ∈ E * , find u ∈ D such that where the functional b is nondifferential and satisfies the following conditions: Special cases.(A) If M(Bu,Cu) = 0 for all u ∈ D, and −w * is given, then the problem (2.1) reduces to the following variational-like inequality problem: for a given w * ∈ E * , find u ∈ D such that which was introduced and studied by Ding [8].
which was introduced and considered by Liu and Tang [26].
which is originally considered by Cohen [3].
For the appropriate and suitable choices of mappings M, N, η, b, T, A, B, C, D, w * , and the space E, we can obtain many known and new classes of variational inequalities and variational-like inequalities as special cases of the problem (2.1), see [1-4, 6, 7, 16-19, 25] and the references therein.In brief, the problem (2.1) is more general and a unifying one, which is also one of the main motivations of this paper.Now, we recall the following definitions and some known results.
(2) η-relaxed Lipschitz if there exists a constant α ≥ 0 such that (3) Lipschitz continuous if there exists a constant L > 0 such that (2.9) (4) η-generalized pseudocontractive if there exists a constant α > 0 such that (2.10) (5) η-relaxed monotone if there exists a constant α > 0 such that (1) N is said to be Lipschitz continuous in the first argument if there exists a constant α > 0 such that (2.12) (2) N is said to be η-cocoercive in the first argument with respect to T if there exists a constant α > 0 such that (3) N is said to be η-strongly monotone in the first argument with respect to A if there exists a constant α > 0 such that (4) N is said to be η-relaxed monotone in the first argument with respect to A if there exists a constant α > 0 such that (5) N is said to be Lipschitz continuous in the first argument with respect to A if there exists a constant α > 0 such that (2.17) Similarly, the Lipschitz continuity, η-strong monotonicity, relaxed monotonicity, cocoercivity, and generalized pseudocontractivity of N can be defined in the second argument.( 7) N(Tu,Au) is said to be η-strongly monotone, if there exists a constant α > 0 such that (2.18) (8) N(Tu,Au) is said to be η-generalized pseudocontractive if there exists a constant α > 0 such that The following notion is a generalization of the corresponding notion introduced by Ding and Tarafdar [10].
Let D be a nonempty convex subset of a Banach space E with the dual space E * .Let T : D → E * and η : D × D → E be mappings such that η(u,u) = 0 for all u ∈ D. Suppose that for given u ∈ D and w * ∈ E * , the functional v → Tu + w * ,η(u,v) is concave.Then T and η have the 0-diagonally concave relation with respect to w * .Lemma 2.5 [8].Let D be a nonempty convex subset of a topological vector space and let ϕ : (a) for each x ∈ D, y → ϕ(x, y) is lower semicontinuous on each nonempty compact subset of D, (b) for each nonempty finite set {x 1 ,...,x m } ⊂ D and for each y = m i=1 λ i x i with λ i ≥ 0 and m i=1 λ i = 1, min 1≤i≤m ϕ(x i , y) ≤ 0, (c) there exist a nonempty compact convex subset X 0 of D and a nonempty compact subset K of D such that for each y ∈ D \ K, there is an x ∈ co(X 0 ∪ {y}) with ϕ(x, y) > 0. Then there exists a y ∈ K such that ϕ(x, y) ≤ 0 for all x ∈ D.

Existence and uniqueness theorems
We now prove the following results.Proof.We first prove that for each fixed u * ∈ D, there exists a unique w ∈ D such that Since b satisfies the conditions (c3) and (c4), it is easy to see that b also satisfies (3.9) Hence the condition (c) of Lemma 2.5 is also satisfied.By Lemma 2.5, there exists a w ∈ K such that ϕ(v, w) ≤ 0 for all v ∈ D, that is, Now we prove that w is a unique solution of the problem (3.1).Suppose that w 1 ,w 2 are arbitrary two solutions of the problem (3.1).It follows that for all v ∈ D, Taking v = w 2 in (3.12) and v = w 1 in (3.13), and adding these inequalities, we obtain that Since β + λ + ν < α + ξ, it follows that w 1 = w 2 and w is the unique solution of the problem (3.1).Thus we have proved that for each u ∈ D, there exists a unique solution w(u) satisfying (3.1).Define a mapping F : D → D by u → w(u) = Fu.We prove that the mapping F is a contraction mapping.Indeed, for any u 1 ,u 2 ∈ D, there exist w 1 = F(u 1 ), w 2 = F(u 2 ) such that for all v ∈ D, Taking v = w 2 in (3.16) and v = w 1 in (3.17) and adding these inequalities, and considering the assumption on b and η, we have which implies that F is a contraction mapping.Hence there exists a unique point u ∈ D such that u = F( u), that is, and u ∈ D is the unique solution of the generalized nonlinear mixed variational-like inequality problem (2.1).This completes the proof.
As in the proof of Theorem 3.1, we have the following result.

Iterative method and convergence
In this section, by using the auxiliary principle technique, we will give a general iterative method of approximate solutions of the generalized nonlinear mixed variational-like inequality problem (2.1) and convergence analysis of the iterative method.
Let D be a nonempty convex subset of a reflexive Banach space E with dual space E * .Let N : We consider an auxiliary differentiable convex functional K : D → (−∞,+∞] and a positive number ρ > 0. For given w * ∈ D and u * ∈ D, we introduce the following auxiliary minimizing problem: If the function w → N(Tu * ,Au * ) − M(Bu * ,Cu * ) + w * ,η(w,u * ) is convex, then the solution w of the auxiliary problem (4.1) can be characterized by the following auxiliary variational inequality problem: We note that if w = u * , then u * is a solution of the generalized nonlinear mixed variational-like inequality problem (2.1).Based on these observations, we suggest the following iterative method for computing the approximate solutions of the problem (2.1).
and b(u,v) = f (v) for all u,v ∈ D, where f : D → (−∞,+∞), then the problem (2.5) is equivalent to the following problem: find Yali Zhao et al. 3 u ∈ D such that

Definition 2 . 3 .
Let D be a nonempty convex subset of a Banach space E. Let T : D → E * and η : D × D → E be mappings.T and η are said to have 0-diagonally concave relation with respect to w

Theorem 3 . 1 .
Let D be a nonempty convex subset of a reflexive Banach space E with dual space E * .Let T,A,B,C :D → E * , M,N : E * × E * → E * , η : D × D → E be mappings, let w * ∈ E * begiven, and let b : D × D → (−∞,+∞] be a functional such that (a) the mapping u → N(Tu,Au) is continuous from the weak topology on E to the strong topology on E * such that N is η-strongly monotone in the first argument with respect to T with constant α > 0, and is η-relaxed monotone in the second argument with respect to A with constant λ > 0, (b) the mapping u → M(Bu,Cu) is continuous from the weak topology on E to the strong topology on E * such that M is η-generalized pseudocontractive in the first argument with respect to B with constant β > 0, and is η-relaxed Lipschitz in the second argument with respect to C with constant ξ > 0, (c) η is Lipschitz continuous with constant δ ≥ 0 and for each v ∈ D, u → η(u,v) is continuous from the weak topology to the weak topology and η(v,u) = −η(u, v) for all u,v ∈ D, (d) the mapping u → N(Tu,Au) − M(Bu,Cu) and η have the 0-diagonally concave relation with respect to w * , (e) b satisfies conditions (c1)-(c4) with ν + λ + β < α + ξ.Then the generalized nonlinear mixed variational-like inequality (2.1) has a unique solution u ∈ D.
) and hence for each fixed u ∈ D, b(u,•) is continuous on D. Notice that the mappings w → N(Tw,Aw) and w → M(Bw,Cw) are continuous from the weak topology to the strong 6 Generalized nonlinear mixed variational-like inequalities topology.For each w ∈ D and any sequence {w n } ⊂ D with w n w in the weak topology, we have lim n→∞ N(Tw n ,Aw n ) = N(Tw,Aw), lim n→∞ M(Bw n ,Cw n ) = M(Bw,Cw), and η(w n ,v) η(w,v) as n → ∞ in the weak topology and the sequence {η(w n ,v)} is bounded.It follows that for each v ∈ D, N Tw n ,Aw n − M Bw n ,Cw n + w * ,η w n ,v − N(Tw,Aw) − M(Bw,Cw) + w * ,η(w,v)

Algorithm 4 . 1 . 1 . 4 . 2 .
(a) At n = 0, start with some initial u 0 .(b) At step n, solve the auxiliary minimizing problem (4.1) or the auxiliary variational inequality problem (4.2) with u * = u n .Let u n+1 denote the solution of problem (4.1) or (4.2).(c) If, for given ε > 0, u n+1 − u n ≤ ε, stop.Otherwise, repeat (b).We now consider the convergence of Algorithm 4.Theorem Let D be a nonempty convex subset of a reflexive Banach space E with dual space E * .Let M,N : E * × E * → E * , T,A,B,C : D → E * and η : D × D → E be mappings.Let w * ∈ E * be given let b : D × D → (−∞,+∞] be a functional, and let K : E → (−∞,+∞] be a differentiable convex functional such that (a) the mapping w → N(Tw,Aw) is continuous from the weak topology to the strong topology, N is η-cocoercive in the first argument with respect to T with constant α > 0, and η-strongly monotone in the second argument with respect to A with constant ξ > 0, and is Lipschitz continuous in the first and second arguments with constants σ 1 and σ 2 , respectively, (b) the mapping w → M(Bw,Cw) is continuous from the weak topology to the strong topology, M is η-generalized pseudocontractive in the first argument with respect to B with constant β > 0, and η-relaxed Lipschitz in the second argument with respect to C with constants σ > 0 and β > σ, and is Lipschitz continuous in the first and second arguments with constants σ 3 > 0 and σ 4 > 0, respectively, (c) A,B,C : D → E * are Lipschitz continuous with constants σ A ,σ B ,σ C , respectively, (d) η is Lipschitz continuous with constant δ ≥ 0 such that 10 Generalized nonlinear mixed variational-like inequalities (1) for each v ∈ D, u → η(u,v) is continuous from the weak topology to the weak topology; (2) for all u,v,z ∈ D, η(u,v) = η(u,z) + η(z,v); (3) for each fixed u,w ∈ D, the function v → N(Tu,Au) − M(Bu,Cu) + w * , η(w,v) is concave, (e) b satisfies conditions (c1)-(c4) and β + ν < ξ + σ, (f) the derivative K of K is strongly convex with constant μ > 0 and the mapping w → K (w) is continuous from the weak topology to the strong topology.Then (A) the generalized nonlinear mixed variational-like inequality problem has a unique solution (2.1) u ∈ D; (B) for each ρ > 0, there exists a unique solution u n+1 ∈ D of Problems (4.1) or (4.2)with u n substituted for u

. 3 )
then the sequence {u n } defined by Algorithm 4.1 strongly converges to u. Proof.(A) In view of conditions (a) and (b), we get that the mapping u → N(Tu,Au) is ηstrongly monotone with constant ξ > 0 and the mapping u → M(Bu,Cu) is η-generalized pseudocontractive with constant β − σ, respectively.Hence the conditions (a) and (b) of Theorem 3.2 are satisfied.By (3) of (d), for each u ∈ D, the function v → (Tu,Au) − M(Bu,Cu) + w * ,η(u,v) is concave, it follows from Remark 2.4 that condition (d) of Theorem 3.1 holds.Noting that condition (2) of (d) implies that η(u,u)=0 and η(u,v)= −η(v, u) for all u,v ∈ D, it is easy to check that all conditions of Theorem 3.2 are satisfied.By Theorem 3.2, the conclusion (A) holds.(B) For each fixed ρ > 0 and u n ∈ D, define a functional ϕ : D × D → [−∞,+∞] by ϕ(v,w) = K u n − K (w),v − w − ρ N Tu n ,Au n − M Bu n ,Cu n + w * ,η(v,w) + ρb u n ,w − ρb u n ,v .

(4. 4 )
Since w → K (w) is continuous from the weak topology to the strong topology, it is easy to show that the function w → K (w),w is weakly continuous on D and for each v ∈ D, w → ϕ(v,w) is weak lower semicontinuous.By using similar argument as in the proof of Theorem 3.1, it is easy to show that ϕ satisfied all conditions of Lemma 2.5, and hence there exists w ∈ D such thatK ( w) − K u n ,v − w ≥ −ρ N Tu n ,Au n − M Bu n ,Cu n + w * ,η(v, w)+ ρb u n , w − ρb u n ,v , ∀v ∈ D.