ITERATIVE ALGORITHM FOR SOLVING THE NEW SYSTEM OF GENERALIZED VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper, we introduce an iterative method to approximate a common solution of a new general system of variational inequalities, a mixed equilibrium problem and a fixed point problem for a nonexpansive mapping in real Hilbert spaces. We prove that the iterative sequence converges strongly to a common solution of the three problems in the framework of Hilbert spaces. Our main results extend and improve some results in the literature. AMS Subject Classification: 47H10, 49J40, 47H05 47H09, 46B20


Introduction
The study of variational inequality problem is an interesting and fascinating branch of applicable mathematics with a wide range of applications industry, finance, economics, optimization, social, regional, pure and applied science.A closely related subject of current interest is the problem of finding common elements in the fixed point set of nonlinear operators and in the solution set of monotone variational inequalities; see [1,2,3] and the references therein.For the past years, many existence results and iterative algorithms for various variational inequality and variational inclusion problems have been extended and generalized in various directions using and innovative techniques; see [4,5,6,7] and the references therein.
Motivated by recent work going in this direction.In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of a new general system of variational inequalities, the set of solutions of a mixed equilibrium problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space.Furthermore, we prove that the sequence generated by the iterative scheme converges strongly to a common element of those three sets under some control conditions.The results presented in this paper extend and improve the corresponding results of [6] and many others.

Preliminaries
Let H be a real Hilbert space with inner product ., .and let C be a nonempty closed convex subset of H.A mapping T : C → C is said to be nonexpansive mapping if T x − T y ≤ x − y for all x, y ∈ C. The fixed point set of T is denoted by F (T ) := {x ∈ C : T x = x}.A mapping A : C → H is called α-inverse-strongly monotone, if there exists a positive real number α > 0 such that Ax − Ay, x − y ≥ α Ax − Ay 2 , ∀x, y ∈ C.
Let A i : C → H for all i = 1, 2, 3 be three mappings, then we consider the new general system of variational inequalities of finding (x * , y * , z where λ i > 0 for all i = 1, 2, 3.

Some special cases:
(I) If A 3 = 0 and z * = x * , then problem (2.1) reduces to find (x * , y * ) ∈ C×C such that which is called a general system of variational inequalities and defined by the authors in [6].The set of solutions of problem (2.2) denoted by which is called the new system of variational inequalities, and defined by the author in [7].
which is called the variational inequality problem.
Let ϕ : C → R {+∞} be a proper extended real-valued function and F be a bifunction from C × C to R, where R is the set of real numbers.In 2008, Ceng and Yao [8], introduced the mixed equilibrium problem which is to find The set of solution of problem (2.4) is denoted by which is called the equilibrium problem.The set of solution of (2.5) is denoted by EP (F ).In recent yeas, the equilibrium problem has been intensively studied by many authors (see, for example [1,9,10] and references therein).We recall the well-known results and give some useful lemmas that are used in the next section.
For every point x ∈ H, there exists a unique nearest point in C, denoted by P C x, such that x − P C x ≤ x − y , ∀y ∈ C. P C is called the metric projection of H onto C. It is well known that P C is a nonexpansive mapping of H onto C and satisfies x − y, P C x − P C y ≥ P C x − P C y 2 , ∀x, y ∈ H. (2.6) Obviously, this immediately implies that Recall that, P C x is characterized by the following properties: for all x ∈ H and y ∈ C.
For solving the mixed equilibrium problem, let us assume the following assumptions for the bifunction F, ϕ and the set C: (B2) C is a bounded set.
In the sequel we shall need to use the following lemma.
Lemma 2.1.( [11]) Let C be a nonempty closed convex subset of H. Let F be a bifunction from C ×C to R satisfying (A1)-(A5) and let ϕ : C → R {+∞} be a proper lower semicontinuous and convex function.Assume that either (B1) or (B2) holds.For r > 0 and x ∈ H, define a mapping T r : H → C as follows.
Lemma 2.2.( [12]) Let H be an inner product space.Then, for all x, y, z ∈ H and α, β, γ ∈ [0, 1] with α + β + γ = 1, we have Lemma 2.3.In a real Hilbert space H, there holds the inequality Lemma 2.6.( [15]) Demi-closedness principle.Assume that T is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H.If T has a fixed point, then I − T is demi-closed: that is, whenever {x n } is a sequence in C converging weakly to some x ∈ C (for short, x n ⇀ x ∈ C), and the sequence {(I − T )x n } converges strongly to some y (for short, (I − T )x n → y), it follows that (I − T )x = y.Here I is the identity operator of H. Lemma 2.7.( [16]) Let C be a nonempty closed and convex subset of a real Hilbert space H and A i : C → H be three possibly nonlinear mappings, for i = 1, 2, 3. Define a mapping G : C → C as follows: Throughout this paper, the set of fixed points of the mapping G is denoted by GV I(C, A 1 , A 2 , A 3 ).

Main Results
In this section, we prove our strong convergence theorem.The next lemma is crucial for proving the main theorem.Lemma 3.1.Let C be a nonempty closed and convex subset of a real Hilbert space H and let A i : C → H be α i -inverse-strongly monotone mappings, for i = 1, 2, 3.
where G is the mapping defined as in Lemma 2.7.
Proof.For all x, y ∈ C, we have It is well known that if A : C → H be α-inverse-strongly monotone, then I −λA is nonexpansive for all λ ∈ (0, 2α].By our assumption, we obtain , we obtain immediately that the mapping G is nonexpansive. Theorem 3.2.Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C × C to R satisfying (A1)-(A5) and ϕ : C → R {+∞} be a proper lower semicontinuous and convex function.Let the mappings A i : C → H be α i -inverse-strongly monotone, for all i = 1, 2, 3 and T be a nonexpansive self-mapping of C such that Ω = F (T ) GV I(C, A 1 , A 2 , A 3 ) M EP (F, ϕ) = Ø.Assume that either (B1) or (B2) holds and that v is an arbitrary point in C. Let x 1 ∈ C and {x n }, {y n }, {z n }, {u n } be the sequences generated by where λ i ∈ (0, 2α i ), for all i = 1, 2, 3 and {a n }, {b n } are two sequences in [0, 1] and Then {x n } converges strongly to x = P Ω v and (x, y, z) is a solution of problem (2.1), where y = P C (z − λ 2 A 2 z) and z = P C (x − λ 3 A 3 x).
Proof.Step 1.We claim that {x n } is bounded.Let x * ∈ Ω and {T rn } be a sequence of mappings defined as in Lemma 2.1.It follows from Lemma 2.7 that By nonexpansiveness of I − λ i (i = 1, 2, 3), we have which implies that Thus, {x n } is bounded.Consequently, the sequences {y n }, {z n }, {t n }, {A 1 y n }, {A 2 z n }, {A 3 u n } and {T t n } are also bounded.

.22)
Step4.We claim that lim sup n→∞ v − x, x n − x ≤ 0, where x = P Ω v. Indeed, since {t n } and {T t n } are two bounded sequences in C, we can choose a subsequence {t n Since lim n→∞ T t n − t n = 0, we obtain that T t n i ⇀ z as i → ∞.
Next, we show that z ∈ Ω.Since t n i ⇀ z and T t n − t n → 0, we obtain by Lemma 2.6 that z ∈ F (T ).From (3.22) and (3.10), we obtain Furthermore, by Lemma 3.1, we have G : C → C is nonexpansive.Then, we have hence lim n→∞ t n − G(t n ) = 0. Again by Lemma 2.6, we have z ∈ GV I(C, A 1 , A 2 , A 3 ).Since t n i ⇀ z and x n −t n → 0, we obtain that x n i ⇀ z.From u n −x n → 0, we also obtain that u n i ⇀ z.By using the same argument as that in the proof of [11,Theorem 3.1,pp.1825], we can show that z ∈ M EP (F, ϕ).Therefore z ∈ Ω.
On the other hand, it follows from (2.8), (3.10) and T t n i ⇀ z as i → ∞ that lim sup (3.23) Step 5. We claim that x n → x as n → ∞.Since which implies that This together with (C1) and (3.23), we have by Lemma 2.4 that {x n } converges strongly to x.This completes the proof.
y) is lower semicontinuous; (B1) For each x ∈ H and r > 0, there exist a bounded subset D x ⊆ C and y x ∈ C such that for any z ∈ C \ D x , F (z, y x ) + ϕ(y x ) + 1 r y x − z, z − x < ϕ(z).