Weak convergence theorem for a class of split variational inequality problems and applications in a Hilbert space

In this paper, we consider the algorithm proposed in recent years by Censor, Gibali and Reich, which solves split variational inequality problem, and Korpelevich’s extragradient method, which solves variational inequality problems. As our main result, we propose an iterative method for finding an element to solve a class of split variational inequality problems under weaker conditions and get a weak convergence theorem. As applications, we obtain some new weak convergence theorems by using our weak convergence result to solve related problems in nonlinear analysis and optimization.


Introduction
The variational inequality problem (VIP) is generated from the method of mathematical physics and nonlinear programming. It has considerable applications in many fields, such as physics, mechanics, engineering, economic decision, control theory and so on. Variational inequality is actually a system of partial differential equations. In , Stampacchia [] first introduced the VIP for modeling in the mechanics problem. The VIP was generated from mathematical physics equations early on because the Lax-Milgram theorem was extended from the Hilbert space to its nonempty closed convex subset, so we got the first existence and uniqueness theorem of VIP. In the s, the VIP became more and more important in nonlinear analysis and optimization problem.
The VIP is to find an element x * ∈ C satisfying the inequality where C is a nonempty closed convex subset of a real Hilbert space H and A is a mapping of C into H. The set of solutions of VIP (.) is denoted by VI(C, A).
We can easily show that x * ∈ VI(C, A) is equivalent to A simple iterative method algorithm for solving VIP (.) is the projection method x n+ = P C (I -λA)x n (.) for each n ∈ N, where P C is the metric projection of H into C and λ is a positive real number. Indeed, if A is η-strongly monotone and L-Lipschitz continuous,  < λ < η L  , then there exists a unique point in VI(C, A), and the sequence {x n } generated by (.) converges strongly to this point. If A is α-inverse strongly monotone, the solution of VIP (.) does not always exist. Assume that VI(C, A) is nonempty,  < λ < α, then VI(C, A) is a closed and convex subset of H. The sequence {x n } generated by (.) converges weakly to one of the points in VI(C, A). But if A is monotone and Lipschitz continuous, the sequence {x n } generated by (.) is not always convergent. So, we cannot use algorithm (.) to solve VIP (.).
In , Korpelevich [] introduced the following so-called extragradient method for solving VIP (.) when A is monotone and k-Lipschitz continuous in the finitedimensional Euclidean space R n : ⎧ ⎨ ⎩ y n = P C (x n -λAx n ), x n+ = P C (x n -λAy n ), (.) for each n ∈ N, where λ ∈ (,  k ). The sequence {x n } converges to a point in VI(C, A). The split feasibility problem (SFP) is also important in nonlinear analysis and optimization. In , Censor and Elfving [] first proposed it for modeling in medical image reconstruction. Recently, the SFP has been widely used in intensity-modulation therapy treatment planning.
The SFP is to find a point x * satisfying the conditions where C and Q are nonempty closed convex of real Hilbert spaces H  and H  , respectively, and A is a bounded linear operator of H  into H  . Censor and Elfving used their algorithm to solve the SFP in the finite-dimensional Euclidean space R n . In , Byrne [] improved and extended Censor and Elfving's algorithm and introduced a new method called CQ algorithm for solving the SFP (.) for each n ∈ N, where  < γ <  A  and A * is the adjoint operator of A. The sequence {x n } generated by (.) converges weakly to a point which solves SFP (.).
Very recently, Censor et al.
[] combined the variational inequality problem and the split feasibility problem and proposed a new problem called split variational inequality problem (SVIP). The SVIP is to find a point x * satisfying x * ∈ VI(C, f ) and Ax * ∈ VI(Q, g), where C and Q are nonempty closed convex of the real Hilbert spaces H  and H  , respectively, and A is a bounded linear operator of H  into H  . For solving SVIP (.), Censor, Gibali and Reich introduced the following algorithm: for each n ∈ N. They obtained the following result.
with L being the spectral radius of the operator A * A, λ ∈ (, α) and suppose that for all x * solving SVIP (.), Then the sequence {x n } generated by (.) converges weakly to a solution of SVIP (.).
In this paper, based on the work by Censor et al. combined with Korpelevich's extragradient method and Byrne's CQ algorithm, we propose an iterative method for finding an element to solve a class of split variational inequality problems under weaker conditions and get a weak convergence theorem. As applications, we obtain some new weak convergence theorems by using our weak convergence result to solve related problems in nonlinear analysis and optimization.

Preliminaries
In this section, we introduce some mathematical symbols, definitions, lemmas and corollaries which can be used in the proofs of our main results.
Let N and R be the sets of positive integers and real numbers, respectively. Let H be a real Hilbert space with the inner product ·, · and the norm · . Let {x n } be a sequence in H, we denote the sequence {x n } converging weakly to x by 'x n x' and the sequence {x n } converging strongly to x by 'x n → x' . z is called a weak cluster point of {x n } if there exists a subsequence {x n i } of {x n } converging weakly to z. The set of all weak cluster points of {x n } is denoted by ω w (x n ). A fixed point of the mapping T : H → H is a point x ∈ H such that Tx = x. The set of all fixed points of the mapping T is denoted by Fix(T).
We introduce some useful operators.
We can easily show that if T is a nonexpansive mapping and assume that Fix(T) is nonempty, then Fix(T) is a closed convex subset of H.
We need the propositions of averaged mappings and inverse strongly monotone mappings.

and T  are averaged and have a common fixed point, then
Let C be a nonempty closed convex subset of H. For each x ∈ H, there exists a unique nearest point in C, denoted by P C x, such that x -P C x ≤ xy , ∀y ∈ C. P C is called the metric projection of H into C. We know that P C is firmly nonexpansive.

Lemma . ([]) Let C be a nonempty closed convex subset of a real Hilbert space H. Given x ∈ H and z ∈ C. Then z = P C x if and only if there holds the inequality
Lemma . ([]) Let C be a nonempty closed convex subset of a real Hilbert space H. Given x ∈ H and z ∈ C. Then z = P C x if and only if there holds the inequality We introduce some definitions and propositions about set-valued mappings.
A monotone mapping B is called maximal if its graph is not properly contained in the graph of any other monotone mappings on D(B).
In fact, the definition of the maximal monotone mapping is not very convenient to use. We usually use a property of the maximal monotone mapping: A monotone mapping B is maximal if and only if for ( For a maximal monotone mapping B, we define its resolvent J r by where r > . We know that J r is firmly nonexpansive and Fix(J r ) = B -  for each r > . For a nonempty closed convex subset C, the normal cone to C at x ∈ C is defined by It can be easily shown that N C is a maximal monotone mapping and its resolvent is P C .

Lemma . ([]) Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-Lipschitz continuous mapping of C into H. Define
Then B is maximal monotone and  ∈ Bv if and only if v ∈ VI(C, A).
Then the following are equivalent: Corollary . Let H  and H  be real Hilbert spaces. Let C be a nonempty closed convex subset of H  . Let T : H  → H  be a nonexpansive mapping, and let A : Then the following are equivalent: We also need the following lemmas.

Main results
In this section, based on Censor, Gibali and Reich's recent work and combining it with Byrne's CQ algorithm and Korpelevich's extragradient method, we propose a new iterative method for solving a class of the SVIP under weaker conditions in a Hilbert space.
First, we consider a class of the generalized split feasibility problems (GSFP): finding an element that solves a variational inequality problem such that its image under a given bounded linear operator is in a fixed point set of a nonexpansive mapping, i.e., find x * satisfying x * ∈ VI(C, f ) and Ax * ∈ Fix(T). (.) t n = P C (y nλ n f (y n )), . Then the sequence {x n } converges weakly to a point z ∈ , where z = lim n→∞ P x n .
Proof From Lemma .(ii), (iii), (iv) and Lemma ., we can easily know that P C (Iγ n A * (I -T)A) is +γ n A   -averaged. So, y n can be rewritten as where α n = +γ n A   and V n is a nonexpansive mapping for each n ∈ N. Let u ∈ , we have So, we obtain that On the other hand, from Lemma ., we have Then, from Lemma ., we obtain that Therefore, there exists and the sequence {x n } is bounded. From (.), we also get From (.) and (.), we have Notice that P C is firmly nonexpansive, so We obtain x n+t n → , n → ∞. Since {x n } is bounded, for each z ∈ ω w (x n ), there exists a subsequence {x n i } of {x n } converging weakly to z. Without loss of generality, the subsequence {γ n i } of {γ n } converges to a pointγ ∈ (,  A  ). Since A * (I -T)A is inverse strongly monotone, we know that {A * (I -T)Ax n i } is bounded. Since P C is firmly nonexpansive, From γ n i →γ , we have From (.), we have we have From Lemma ., we obtain that From Corollary ., we obtain Now, we show that z ∈ VI(C, f ). From (.), (.) and (.), we have y n i z, t n i z and From Lemma ., we know that B is maximal monotone and  ∈ Bv if and only if v ∈ VI(C, f ). For each (v, w) ∈ G(B), we have Hence On the other hand, from v ∈ C and x n+ = P C y nλ n f (t n ) , we get Therefore, from (.) and (.), we obtain that Since B is maximal monotone, we have  ∈ Bz and hence z ∈ VI(C, f ). So, we obtain that From Lemma ., we get t n = P C (y nλ n f (y n )), . Then the sequence {x n } converges weakly to a point z ∈ , where z = lim n→∞ P x n .
Proof From Lemma ., we can easily know that z ∈ VI(Q, g) if and only if z = P Q (Iμg)z for μ > , and for μ ∈ (, α), P Q (Iμg) is nonexpansive. Putting T = P Q (Iμg) in Theorem ., we get the desired result.

Application
In this part, some applications which are useful in nonlinear analysis and optimization problems in a Hilbert space will be introduced. This section deals with some weak convergence theorems for some generalized split feasibility problems (GSFP) and some related problems by Theorem . and Theorem .. Let C be a nonempty closed convex subset of a real Hilbert space H, and let F : C × C → R be a bifunction. Consider the equilibrium problem, i.e., find x * satisfying F x * , y ≥ , ∀y ∈ C.
(  .  ) We denote the set of solutions of equilibrium problem (.) by EP(F). For solving equilibrium problem (.), we assume that F satisfies the following properties: (A) F is monotone, i.e., F(x, y) + F(y, x) ≤  for all x, y ∈ C; (A) for each x, y, z ∈ C, lim t↓ F(tz + (t)x, y) ≤ F(x, y); (A) for each x ∈ C, y → F(x, y) is convex and lower semicontinuous. Then we have the following lemmas.
Lemma . ([]) Let C be a nonempty closed convex subset of a real Hilbert space H, and let F be a bifunction of C × C into R satisfying the properties (A)-(A). Let r be a positive real number and x ∈ H. Then there exists z ∈ C such that Lemma . ([]) Assume that F : C × C → R is a bifunction satisfying the properties (A)-(A). For r >  and x ∈ H, define the resolvent T r : H → C of F by Then the following hold: T r x -T r y  ≤ T r x -T r y, xy , ∀x, y ∈ H; (iii) Fix(T r ) = EP(F); (iv) EP(F) is closed and convex.
Applying Theorem ., Lemma . and Lemma ., we get the following results. For a nonempty closed convex subset C, the constrained convex minimization problem is to find x * ∈ C such that where C is a nonempty closed convex subset of a real Hilbert space H and φ is a real-valued convex function. Proof Let z be a solution of (.). For each x ∈ C, z + λ(xz) ∈ C, ∀λ ∈ (, ). Since φ is differentiable, we have Conversely, if z ∈ VI(C, ∇φ), i.e., ∇φ(z), xz ≥ , ∀x ∈ C. Since φ is convex, we have Hence z is a solution of (.).
Applying Theorem . and Lemma ., we obtain the following result.
t n = P C (y nλ n f (y n )), x n+ = P C (y nλ n f (t n )), Proof Putting g = ∇φ in Theorem ., we get the desired result by Lemma ..
Applying Theorem . and Lemma ., we obtain the following result. The problem to solve in the following theorem is called split minimization problem (SMP). It is also important in nonlinear analysis and optimization.
Theorem . Let H  and H  be real Hilbert spaces. Let C be a nonempty closed convex subset of H  and Q be a nonempty closed convex subset of H  . Let A : H  → H  be a bounded linear operator such that A = , φ  and φ  be differentiable convex functions of H  into R and H  into R, respectively. Suppose that ∇φ  is k-Lipschitz continuous and ∇φ  is αism. Setting = {z ∈ arg min x∈C φ  (x) : Az ∈ arg min y∈Q φ  (y)}, assume that = ∅. Let the sequences {x n }, {y n } and {t n } be generated by x  = x ∈ C and ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y n = P C (x nγ n A * (I -P Q (I -μ∇φ  ))Ax n ), t n = P C (y nλ n ∇φ  (y n )), x n+ = P C (y nλ n ∇φ  (t n )), (.) for each n ∈ N, where {γ n } ⊂ [a, b] for some a, b ∈ (,  A  ) and {λ n } ⊂ [c, d] for some c, d ∈ (,  k ), μ ∈ (, α). Then the sequence {x n } converges weakly to a point z ∈ , where z = lim n→∞ P x n .
Proof Since φ  is convex, we can easily obtain that ∇φ  is monotone. Putting f = ∇φ  and g = ∇φ  in Theorem ., we obtain the desired result by Lemma ..

Conclusion
It should be pointed out that the variational inequality problem and the split feasibility problem are important in nonlinear analysis and optimization. Censor et al. have recently introduced an algorithm which solves the split feasibility problem generated from the variational inequality problem and the split feasibility problem. The base of this paper is the work done by Censor et al. combined with Byrne's CQ algorithm for solving the split feasibility problem and Korpelevich's extragradient method for solving the variational inequality problem with a monotone mapping. The main aim of this paper is to propose an iterative method to find an element for solving a class of split feasibility problems under weaker conditions. As applications, we obtain some new weak convergence theorems by using our weak convergence result to solve the related problems in a Hilbert space.
Theorem . improves and extends Theorem . in the following ways: (i) The inverse strongly monotone mapping f is extended to the case of a monotone and Lipschitz continuous mapping. (ii) The fixed coefficient γ is extended to the case of a sequence {γ n }. (iii) (.) is not necessary in Theorem ..