AN APPROXIMATION OF SOLUTIONS OF VARIATIONAL INEQUALITIES

It is known (see [5, 6]) that when K is a closed convex cone, problems NCP( f ,K) and VI( f ,K) are equivalent. To study the existence of solutions of the NCP( f ,K) and VI( f ,K) problems, many authors have used the techniques of KKM mappings, and the Fan-KKM theorem from fixed point theory (see [1, 5, 6, 7, 8, 9, 10]). In case B is a Hilbert space, Isac and other authors have used the notion of “exceptional family of elements” (EFE) and the LeraySchauder alternative theorem (see [5, 6]). In [1, 2], Alber generalized the metric projection operator PK to a generalized projection operator πK : B∗ → K from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces and Alber used this operator to study VI( f ,K) problems and to

approximate the solutions by an iteration sequence. In [7], the author used the generalized projection operator and a Mann-type iteration sequence to approximate the solutions of the VI( f ,K) problems.
In case B is a uniformly convex and uniformly smooth Banach space, the continuity property of the metric projection operator P K has been studied by Goebel, Reich, Roach, and Xu (see [4,12,13]). In this paper, we use the operator P K and a Mann-type iteration scheme to approximate the solutions of NCP( f ,K) problems.

Preliminaries
Let (X, · ) be a normed linear space and let K be a nonempty subset of X. For every x ∈ X, the distance between a point x and the set K is denoted by d(x,K) and is defined by the following minimum equation (2.1) The metric projection operator (or the nearest-point projection operator) P K defined on X is a mapping from X to 2 K : (2.2) If P K (x) = ∅, for every x ∈ X, then K is called proximal. If P K (x) is a singleton for every x ∈ X, then K is said to be a Chebyshev set. Since uniformly convex and uniformly smooth Banach spaces are reflexive and strictly convex, the above theorem implies that if (B, · ) is a uniformly convex and uniformly smooth Banach space, then every nonempty closed convex subset K ⊂ B is a Chebyshev set.
Let T be a uniformly convex Banach space. Its modulus of convexity is denoted by δ and is defined by It follows that δ is a strictly increasing, convex, and continuous function from (0,2] to [0,1], and it is known that δ( )/ is nondecreasing on (0,2]. If B is uniformly smooth, its modulus of smoothness is denoted by ρ(τ) and is defined by It can be shown that ρ is a convex and continuous function from [0, ∞) to [0,∞) with the properties that ρ(τ)/τ is nondecreasing, ρ(τ) ≤ τ for all τ ≥ 0, and lim τ→0 + ρ(τ)/τ = 0. For the details of the properties of δ and ρ, the reader is refereed to [10,11].
Theorem 2.2 (Xu and Roach [13]). Let M be a convex Chebyshev set of a uniformly convex and uniformly smooth Banach spaces X and let P;X → M be the metric projection. Then (i) P is Lipschitz continuous mod M; namely, there exists a constant k > 0 such that for any x ∈ X and z ∈ M, (2.5) (ii) P is uniformly continuous on every bounded subset of X and, furthermore, there exist positive constants k r for every B r := {x ∈ X : x ≤ r} such that where ψ is defined by Theorem 2.3 (Xu and Roach [13]). If X = L p , p , or W p m (1 < p < ∞) in Theorem 2.2, then the metric projection P is Hölder continuous on every bounded subset of X, and, moreover, there exist positive k r for every B r such that (2,p) , for any x, y ∈ B r . (2.8) The normalized duality mapping J : B → 2 B * is defined by Clearly, j(x) is the B * -norm of j(x) and x is its B-norm. It is known that if B is uniformly convex and uniformly smooth, then J is a single-valued, strictly monotone, homogeneous, and uniformly continuous operator on each bounded set. Furthermore, J is the identity in Hilbert spaces; that is, J = I H .
The following theorem provides a tool to solve a variational inequality by finding a fixed point of a certain operator.
Theorem 2.4 (Li [8]). Let (B, · ) be a reflexive and smooth Banach space and K ⊂ B a nonempty closed convex subset. For any given (2.10) Let F : K → B be a mapping. The locality variational inequality defined by the mapping F and the set K is LVI(F,K) : find x * ∈ K and j F x * ∈ J F x * such that j F x * , y − x * ≥ 0, for every y ∈ K. (2.11)

Solutions of variational inequalities
The next theorem follows from Theorem 2.4.
Theorem 2.5 (Li [8]). Let (B, · ) be a reflexive and smooth Banach space and K ⊂ B a nonempty closed convex subset. Let F : B → B be a mapping. Then an element x * ∈ KE is a solution of LVI(F,K) if and only if x * ∈ P K (x * − F(x * )).

The compact case
Theorem 3.1. Let (B, · ) be a uniformly convex and uniformly smooth Banach space and K a nonempty compact convex subset of B. Let F : K → B be a continuous mapping. Suppose that LVI(F,K) has a solution x * ∈ K and F satisfies the following condition: where k r is the positive constant given in Theorem 2.2 that depends on the bounded subset K. For any x 0 ∈ K, define the Mann iteration scheme as follows: Proof. Since B is uniformly convex and uniformly smooth, there exists a continuous strictly increasing and convex function g : R + → R + such that g(0) = 0 and, for all x, y ∈ B r (0) := {x ∈ E : x ≤ r} and for any α ∈ [0,1], we have where r is a positive number such that K ⊆ B r (0) (see [3]). Since x * is a solution of LVI(F,K), from Theorems 2.1 and 2.5, x * = P K (x * − F(x * )). Using Theorem 2.2, Since ρ(τ)/τ is nondecreasing and lim τ→0 (3.5) Applying condition (3.1) and the above inequality, we obtain For any positive integer m, taking the sum for n = 1,2,3,...,m, we have From the condition α n (1 − α n ) = ∞, there exists a subsequence {n(i)} ⊆ {n} such that g( P K (x n(i) − F(x n(i) )) − x n(i) ) → 0 as i → ∞. Since g is continuous and strictly 382 Solutions of variational inequalities increasing such that g(0) = 0, we obtain (3.10) From the compactness of K, there exists a subsequence of {x n(i) } which, without loss of generality, we may assume is the sequence {n(i)}, and an element x ∈ K such that x n(i) → x and i → ∞. From the continuity of P K and F, we have P K (x n(i) − F(x n(i) )) → P K (x − F(x )) as i → ∞. Statement (3.10) implies that P K (x − F(x )) = x . Applying Theorem 2.5, x is a solution of the LVI(F,K) problem.

Corollary 3.2. Theorem 3.1 is still true if condition (3.1) is replaced by the following condition:
Proof. From the property ρ(τ) ≤ τ for all τ ≥ 0, and the nondecreasing property of δ (and δ −1 ), condition (3.11) implies condition (3.1). The conclusion of the corollary then follows immediately from Theorem 3.1.
One of the most important types of variational inequalities and complementarily problems deals with completely continuous field mappings. This type of variational inequality and complementarily problem has been studied by many authors in Hilbert spaces (see, e.g., [5,6]). Recently, the first author and Isac have studied the existence of solutions of this type problem in uniformly convex and uniformly smooth Banach spaces.
Recall that a mapping T : B → B is completely continuous if T is continuous, and for any bounded set D ⊂ B, we have that T(D) is relatively compact. A mapping F : B → B has a completely continuous field if F has a representation F(x) = x − T(X) for all x ∈ B, where T : B → B is a completely continuous mapping.
As an application of Theorem 3.1 we have the following corollary.
where k r is the positive constant given in Theorem  It is well known that L p , p , and W p m (1 < p < ∞) are special uniformly convex and uniformly smooth Banach spaces. In [2], Alber and Notik provided formulas for the calculation of the modulus of convexity δ and the modulus of smoothness ρ for these spaces: (3.13) Applying the above formulas to Theorem 3.1, we can obtain more detailed applications and examples.
. Suppose that LVI(F,K) has a solution x * ∈ K and, for every x ∈ K,T satisfies the following conditions:

14)
where k r is the positive constant given in Theorem 2.

Then there exists a subsequence {n(i)} of the sequence defined by (3.2) such that {x n(i) } converges to a solution x of LVI(F,K).
Proof. Assume that 1 < p ≤ 2. From the inequality we obtain (3.16)

Solutions of variational inequalities
Noting that both δ and ρ are strictly increasing, and using the inequality ρ(τ) ≤ τ p / p, we have (3.17) The last inequality follows from the condition of this corollary. Then this case can be obtained by using Corollary 3.4. The case for 2 ≤ p < ∞ can be proved similarly. Proof. This corollary follows immediately from Theorem 3.5.
If we apply Theorem 2.3 to the special uniformly convex and uniformly smooth Banach spaces L p , p , and W p m (1 < p < ∞), and apply the techniques of the proof of Theorem 3.1, we obtain the following.
Theorem 3.7. Let B = L p , p , or W p m (1 < p < ∞) and K a nonempty compact convex subset of B. Let F : K → B be a continuous mapping. Suppose that LVI(F,K) as a solution x * ∈ K and F satisfies the following: The rest of the proof is similar to that of Theorem 3.1.  (3.2) converges to a solution of the LVI(F,K) problem.

The unbounded case
If K is unbounded, for example, if K is a closed convex cone, the following theorems are needed for estimation.
Theorem 4.1 (Xu and Roach [13]). Let M be a convex Chebyshev set of a uniformly convex and uniformly smooth Banach space X and P : X → M be the metric projection. Then, for every x, y ∈ X, P(x) − P(y) where C 1 is a fixed constant and ψ is as defined in Theorem 2.2.
Theorem 4.2. Let (B, · ) be a uniformly convex and uniformly smooth Banach space and K a nonempty closed convex subset of B. Let F : K → B be a continuous mapping such that the LVI(F,K) problem has a solution x * ∈ K. If there exist positive constants κ and λ satisfying the following conditions: (ii) t −1 δ −1 (t) ≤ λ ∀t; (iii) (κ + 4C 1 κλ) < 1, where C 1 is the constant given in Theorem 4.1, then the sequence {x n } defined by (3.2) converges to the solution x * of the LVI(F,K) problem.