Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces

The purpose of this paper is to introduce and study a general split variational inclusion problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split variational inclusion problem. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces.


Introduction
Let C and Q be nonempty closed convex subsets of real Hilbert spaces H  and H  , respectively. The split feasibility problem (SFP) is formulated as to find x * ∈ C and Ax * ∈ Q, (.) where A : H  → H  is a bounded linear operator. In , Censor and Elfving [] first introduced the SFP in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction []. It has been found that the SFP can also be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning [-]. The SFP in an infinite-dimensional real Hilbert space can be found in [, , -]. For comprehensive literature, bibliography and a survey on SFP, we refer to [].
Assuming that the SFP is consistent, it is not hard to see that x * ∈ C solves SFP if and only if it solves the fixed point equation where P C and P Q are the metric projection from H  onto C and from H  onto Q, respectively, γ >  is a positive constant, and A * is the adjoint of A. http://www.fixedpointtheoryandapplications.com/content/2014/1/171 A popular algorithm to be used to solves the SFP (.) is due to Byrne's CQ-algorithm []: where γ k ∈ (, /λ) with λ being the spectral radius of the operator A * A.
On the other hand, let H be a real Hilbert space, and B be a set-valued mapping with domain D(B) := {x ∈ H : B(x) = ∅}. Recall that B is called monotone, if uv, x, xy ≥  for any u ∈ Bx and v ∈ By; B is maximal monotone, if its graph {(x, y) : x ∈ D(B), y ∈ Bx} is not properly contained in the graph of any other monotone mapping. An important problem for set-valued monotone mappings is to find x * ∈ H such that  ∈ B(x * ). Here, x * is called a zero point of B. A well-known method for approximating a zero point of a maximal monotone mapping defined in a real Hilbert space H is the proximal point algorithm first introduced by Martinet [] and generated by Rockafellar []. This is an iterative procedure, which generates {x n } by x  = x ∈ H and where {β n } ⊂ (, ∞), B is a maximal monotone mapping in a real Hilbert space, and J B r is the resolvent mapping of B defined by J B r = (I + rB) - for each r > . Rockafellar [] proved that if the solution set B - () is nonempty and lim inf n→∞ β n > , then the sequence {x n } in (.) converges weakly to an element of B - (). In particular, if B is the sub-differential ∂f of a proper convex and lower semicontinuous function f : H → R, then (.) is reduced to The following examples are special cases of (GSVIP) (.).
Classical split variational inclusion problem.  to find x * ∈ H  such that  ∈ ∂ f x * and  ∈ ∂ g Ax * .
(  .  ) Split feasibility problem. As in (.), let C and Q be two nonempty closed convex subsets of real Hilbert spaces H  and H  , respectively and A be the same as above. The split feasibility problem is to find x * ∈ C such Ax * ∈ Q.
(  .  ) It is well known that this kind of problems was first introduced by Censor and Elfving [] for modeling inverse problems arising from phase retrievals and in medical image reconstruction []. Also it can be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning. Let i C (i Q ) be the indicator function of C (Q), i.e., Then i C and i Q both are proper convex and lower semicontinuous functions, and its subdifferentials ∂i C and ∂i Q are maximal monotone operators. Consequently problem (.) is equivalent to the following 'split optimization problem' and 'Moudafi's classical split variational inclusion problem' , i.e., For solving (GSVIP) (.), in our paper we propose the following iterative algorithms:

Preliminaries
Throughout the paper, we denote by H a real Hilbert space, C be a nonempty closed and convex subset of H. F(T) denote by the set of fixed points of a mapping T. Let {x n } be a sequence in H and x ∈ H. Strong convergence of {x n } to x is denoted by x n → x, and weak convergence of {x n } to x is denoted by x n x. For every point x ∈ H, there exists a unique nearest point in C, denoted by P C x. This point satisfies.
The operator P C is called the metric projection. The metric projection P C is characterized by the fact that P C x ∈ C and Recall that a mapping T : C → H is said to be nonexpansive, if Tx -Ty ≤ xy for every x, y ∈ C. T is said to be quasi-nonexpansive, if F(T) = ∅ and Txp ≤ xp for every x ∈ C and p ∈ F(T). It is easy to see that F(T) is a closed convex subset of C if T is a quasi-nonexpansive mapping. Besides, T is said to be a firmly nonexpansive, if then lim n→∞ a n = .

Lemma . []
Let {a n } be a sequence of real numbers such that there exists a subsequence {n i } of {n} such that a n i < a n i + for all i ∈ N. Then there exists a nondecreasing sequence β is a single-valued and firmly nonexpansive mapping; Proof By Lemma .(iii), the mapping (I -J B β ) is firmly nonexpansive, hence the conclusions (i) and (ii) are obvious. Now we prove the conclusion (iii). In fact, for any x, y ∈ H  , it follows from the conclusions (i) and (ii) that This completes the proof of Lemma .. http://www.fixedpointtheoryandapplications.com/content/2014/1/171

Main results
The following lemma will be used in proving our main results.
. . , be two families of set-valued maximal monotone mappings, and let β >  and γ > . If = ∅ (the solution set of (GSVIP) (.)), then x * ∈ H  is a solution of (GSVIP) (.) if and only if for each i ≥ , for each γ >  and for each β >  Proof Indeed, if x * is a solution of (GSVIP) (.), then for each i ≥ , γ >  and β > , Hence we have On the other hand, by Lemma .(v) again Adding up (.) and (.), we have Simplifying it, we have By the assumption that = ∅. Taking w ∈ , hence for each i ≥  w ∈ B - i () and Aw ∈ K - i (). In (.), taking y = w and v = Aw, then we have This implies that Ax * = J (i) lim n→∞ ξ n = , and ∞ n= ξ n = ∞; Proof (I) First we prove that {x n } is bounded.
In fact, letting z ∈ , by Lemma ., for each i ≥ , Hence it follows from Lemma .(iii) that for each i ≥  and each n ≥  we have By induction, we can prove that This implies that {x n } is bounded, so is {f (x n )}.
(II) Now we prove that for each j ≥  α n γ n,j x n -J Indeed, it follows from Lemma . that for any positive j ≥  Simplifying it, (.) is proved. By the assumption that = ∅, and it is easy to prove that is closed and convex. This implies that P is well defined. Again since P f : H  → is a contraction mapping with contractive constant k ∈ (, ), there exists a unique x * ∈ such that x * = P fx * . Since x * ∈ , it solves (GSVIP) (.). By Lemma ., (.) (III) Now we prove that x n → x * . In order to prove that x n → x * (as n → ∞), we consider two cases. Case . Assume that { x nx * } is a monotone sequence. In other words, for n  large enough, { x nx * } n≥n  is either nondecreasing or non-increasing. Since { x nx * } is bounded, { x nx * } is convergence. Again since lim n→∞ ξ n = , and {f (x n )} is bounded, from (.) we get By condition (ii), we obtain To show this inequality, we choose a subsequence {x n k } of {x n } such that x n k w, λ n k ,i → λ i ∈ (,  A  ) for each i ≥ , and (  .   ) http://www.fixedpointtheoryandapplications.com/content/2014/1/171 It follows from (.) that (IV) Finally, we prove that x n → P f (x * ). In fact, from Lemma . we have Simplifying it, we have where δ n = ξ n M (-k) +  -k f (x * )x * , x n+x * , M = sup n≥ x nx *  , and η n = (-k)ξ n -ξ n k . It is easy to see that η n → , ∞ n= η n = ∞, and lim sup n→∞ δ n ≤ . Hence by Lemma ., the sequence {x n } converges strongly to x * = P f (x * ).
Case . Assume that { x nx * } is not a monotone sequence. Then, by Lemma ., we can define a sequence of positive integers: {τ (n)}, n ≥ n  (where n  large enough) by Clearly {τ (n)} is a nondecreasing sequence such that τ (n) → ∞ as n → ∞, and for all n ≥ n  Therefore { x τ (n)x * } is a nondecreasing sequence. According to Case (), lim n→∞ x τ (n)x * =  and lim n→∞ x τ (n)+x * = . Hence we have This implies that x n → x * and x * = P f (x * ) is a solution of (GSVIP) (.). This completes the proof of Theorem ..

Applications
In this section we shall utilize the results presented in Theorem . and Theorem . to study some problems.

Application to split optimization problem
Let H  and H  be two real Hilbert spaces. Let h : H  → R and g : H  → R be two proper, convex and lower semicontinuous functions, and A : H  → H  be a linear and bounded operators. The so-called split optimization problem (SOP) is to find x * ∈ H  such that h x * = min y∈H  h(y) and g Ax * = min z∈H  g(z).

Application to split feasibility problem
Let C ⊂ H  and Q ⊂ H  be two nonempty closed convex subsets and A : H  → H  be a bounded linear operator. Now we consider the following split feasibility problem, i.e.: to find x * ∈ C such that Ax * ∈ Q.
(  .  ) Let i C and i Q be the indicator functions of C and Q defined by (.). Let N C (u) be the normal cone at u ∈ H  defined by N C (u) = z ∈ H  : z, vu ≤ , ∀v ∈ C .
Since i C and i Q both are proper convex and lower semicontinuous functions on H  and H  , respectively, and the subdifferential ∂i C of i C (resp. ∂i Q of i Q ) is a maximal monotone operator, we can define the resolvents J where β > . By definition, we know that ∂i C (x) = z ∈ H  : i C (x) + z, yx ≤ i C (y), ∀y ∈ H  = z ∈ H  : z, yx ≤ , ∀y ∈ C = N C (x), x ∈ C.
This implies that J ∂i C β = P C . Similarly J ∂i Q β = P Q . Taking h(x) = i C (x) and g(x) = i Q (x) in (.), then the (SFP) (.) is equivalent to the following split optimization problem: to find x * ∈ H  such that i C x * = min x n+ = α n x n + ξ n f (x n ) + γ n P C x nλ n A * (I -P Q )Ax n , ∀n ≥ . (.) If the solution set of the split optimization problem (.)  = ∅, and the following conditions are satisfied: