Strong Convergence Theorems of Viscosity Iterative Algorithms for Split Common Fixed Point Problems

In this paper, we propose a viscosity approximation method to solve the split common fixed point problem and consider the bounded perturbation resilience of the proposed method in general Hilbert spaces. Under some mild conditions, we prove that our algorithms strongly converge to a solution of the split common fixed point problem, which is also the unique solution of the variational inequality problem. Finally, we show the convergence and effectiveness of the algorithms by two numerical examples.


Introduction
Let H 1 and H 2 be two real Hilbert spaces with the inner product •, • and the induced norm • .The split feasibility problem (SFP for short) is as follows: where C and Q are nonempty closed convex subsets of H 1 and H 2 , respectively, and A is a bounded linear operator of H 1 into H 2 .
If the set of solutions of the problem (1) is nonempty, then x * solving problem (1) is equivalent to where τ > 0 and P C denotes the metric projection of H 1 onto C and A * is the corresponding adjoint operator of A.
Recently, many problems in engineering and technology can be modeled by problem (1) and many authors have shown that the SFP has many applications in our real life such as image reconstruction, signal processing and intensity-modulated radiation therapy (see [1][2][3]).
In 1994, Censor and Elfving [4] used their algorithm to solve the SFP in the finite-dimensional Euclidean space.In 2002, Byrne [5] improved the algorithm of Censor and Elfving and presented a new method called the CQ algorithm for solving the SFP (1) as follows: x n+1 = P C (I − τ A * (I − P Q )A)x n , ∀n ∈ N. ( The split common fixed point problem (shortly, SCFPP) is formulated as follows: Find a point x * ∈ Fix(U) such that Ax * ∈ Fix(T), where U : H 1 → H 1 and T : H 2 → H 2 are nonlinear mappings; here, Fix(U) denotes the set of fixed points of the mapping U. We use S to denote the solution set of problem (4).Note that, since every closed convex subset of a Hilbert space is the fixed point set of its associating projection if U := P C and T := P Q , the SFP becomes a special case of the SCFPP.
In 2007, Censor and Segal [6] first studied the SCFPP and, to solve the SCFPP, they proposed the following iterative algorithm: where τ is a properly chosen stepsize.Algorithm (5) was originally designed to solve the problem (4) for directed mappings.
In 2010, Moudafi [7] proposed an iterative method to solve the SCFPP for quasi-nonexpansive mappings.In 2014, combining the Moudafi method with the Halpern iterative method, Kraikaew and Saejung [8] proposed a new iterative algorithm which does not involve the projection operator to solve the split common fixed point problem.More specifically, their algorithm generates a sequence {x n } via the recursions: where x 0 ∈ H is a fixed element, U and T are quasi-nonexpansive operators.
On the other hand, the bounded perturbation resilience and superiorization of iterative methods have been studied by some authors (see [18][19][20][21][22][23]). These problems have received much attention because of their applications in convex feasibility problems [24], image reconstruction [25] and inverse problems of radiation therapy [26] and so on.
Let P denote an algorithm operator.If the iteration x n+1 = Px n is replaced by where β n is a sequence of nonnegative real numbers and {ν n } is a sequence in H such that Then, the algorithm is still convergent and so the algorithm P is the bounded perturbation resilient [19].
In 2016, Jin, Censor and Jiang [21] introduced the projected scaled gradient method (PSG for short) with bounded perturbations for solving the following minimization problem: where f is a continuous differentiable, convex function.The method PSG generates a sequence {x n } defined by where D(x n ) is a diagonal scaling matrix and e(x n ) denotes the sequence of outer perturbations satisfying ∑ ∞ n=0 e(x n ) < ∞.
Recently, Xu [22] projected the superiorization techniques for the relaxed PSG as follows: where τ n is a sequence in [0, 1].
Recently, for solving minimization problem of the combination of two convex functions min x∈H f (x) + g(x), Guo and Cui [20] considered the modified proximal gradient method: and, under suitable conditions, they proved some strong convergence theorems of the method.The definition of proximal operator prox λϕ is as follows.
Definition 1 (see [27]).Let Γ 0 (H) be the space of functions on a real Hilbert space H that are proper, lower semicontinuous and convex.The proximal operator of ϕ ∈ Γ 0 (H) is defined by The proximal operator of ϕ of order λ > 0 is defined as the proximal operator of λϕ, that is, Now, we propose a viscosity method for the problem (4) as follows: If we treat the above algorithm as the basic algorithm P, the bounded perturbation of it is a sequence {x n } generated by the iterative process: In this paper, mainly based on the above works [6,20,22], we prove that our main iterative method (11) is the bounded perturbation resilient and, under some mild conditions, our algorithms strongly converge to a solution of the split common fixed point problem, which is also the unique solution of the variational inequality problem (13).Finally, we give two numerical examples to demonstrate the effectiveness of our iterative schemes.

Preliminaries
Let {x n } be a sequence in the real Hilbert space H.We adopt the following notations: (1) Denote {x n } converging weakly to x by x n x and {x n } converging strongly to x by x n → x. (2) Denote the weak ω-limit set of {x n } by ω w (x n ) := {x : ∃x n j x}.
Definition 2. A mapping F : H → H is said to be: In particular, if L = 1, then we say that F is nonexpansive, namely, where α ∈ [0, 1) and T : H → H is nonexpansive.
Using the Cauchy-Schwartz inequality, it is easy to deduce that B is 1 α −Lipschitz if it is α-ism.Now, we give the following lemmas and propositions needed in the proof of the main results.

Lemma 2 ([29]
).Let h : H → H be a ρ-contraction with ρ ∈ (0, 1) and T : H → H be a nonexpansive mapping.Then, (ii) If the mappings {T i } N i=1 are averaged and have a common fixed point, then we have T is averaged if and only if I − T is ν-ism for some ν > 1 2 .Indeed, for any 0 Lemma 3 ([31]).Let H be a real Hilbert space and T : H → H be a nonexpansive mapping with Fix(T) = ∅.
If {x n } is a sequence in H weakly converging to x and {(I − T)x n } converges strongly to y, then (I − T)x = y.
Lemma 4 ([32] or [33]).Assume that {s n } is a sequence of nonnegative real numbers such that where {γ n } is a sequence in (0, 1), {η n } is a sequence of nonnegative real numbers, {δ n } and {ϕ n } are two sequences in R such that To see the converse, let x ∈ H such that A * (I − T)Ax = 0. Take Az ∈ Fix(T).Since T is nonexpansive, we have Combine the above two formulas, we have This completes the proof.

The Main Results
In 2000, Moudafi [34] proposed the viscosity approximation method: which converges strongly to a fixed point x * of the nonexpansive mapping N (see [35,36]).In 2004, Xu [29] further proved that x * ∈ Fix(N) is also the unique solution of the following variational inequality problem: where h : H → H is a ρ-contraction.By Lemma 2, we get I − h is strongly monotone, hence the solution of problem ( 13) is unique.
In this section, we present a viscosity iterative method for solving problem (4).Meanwhile, the algorithm approximates the unique fixed point of variational inequality problem (13).
Putting e n := e(x n ), we can rewrite the iteration (11) as follows: where Since U is nonexpansive and h is contractive, it is easy to get Theorem 1.Let H 1 ,H 2 be two real Hilbert spaces and A : H 1 → H 2 be a bounded linear operator with L = A * A , where A * is the adjoint of A. Suppose that U : H 1 → H 1 and T : H 2 → H 2 are two averaged mappings with the coefficients γ 1 and γ 2 , respectively.Assume that the problem (4) is consistent (i.e., S = ∅).
Let h : H 1 → H 1 be a ρ-contraction with 0 ≤ ρ < 1.For any x 0 ∈ H 1 , define the sequence {x n } by (14).If the following conditions are satisfied: Then, the sequence {x n } converges strongly to a point x * ∈ S, which is also the unique solution of the variational inequality problem (13).
Step 1. Show that {x n } is bounded.For any z ∈ S, we have Note that the condition (iii) and (15) imply that ∑ ∞ n=0 ẽn < ∞ and, from the conditions (i), (iii) and α n > 0, it is easy to show that { ẽn /α n } is bounded.Therefore, there exists M 1 > 0, such that M 1 := sup n∈N { h(z) − z + ẽn /α n }.Thus, since the induction argument shows that it turns out that the sequence {x n } is bounded and so are {h(x n )}, {V τ n x n } and {A * (I − T)Ax n }.
Step 2. Show that, for any sequence {n k } ⊂ {n}, if η n k → 0, then lim k→∞ x n k − V τ n k x n k = 0. First, if z ∈ S, then we have where Second, we can rewrite V τ n as where w n = γ 1 + (1 − γ 1 )γ 2 τ n L and W n is nonexpansive.By the condition (ii), we get γ 1 < lim inf n→∞ w n ≤ lim sup n→∞ w n < 1.Thus, it follows from ( 14), ( 17) and ( 18) that Furthermore, set Using the condition (i), it is easy to get γ n → 0, Σ ∞ n=0 γ n = ∞ and ϕ n → 0. In order to complete the proof, from Lemma 4, it suffices to verify that η n k → 0 as k → ∞, which implies that lim sup the condition (iii).Thus, from (18), it follows that Step 3. Show that where ω w (x n k ) is the set of all weak cluster points of {x n k }.To see (21), we prove the following: Take x ∈ ω w {x n k } and assume that {x n k j } is a subsequence of {x n k } weakly converging to x.
Without loss of generality, we still use {x n k } to denote {x n k j }.Assume τ n k → τ.Then, we have Since τ n k → τ as k → ∞, it follows immediately from ( 22) that as k → ∞.Thus, we have Using Lemma 3, we get ω w (x n k ) ⊂ Fix(V).Since both U and T are averaged, it follows from Proposition 1 (ii) that Then, by Lemma 5, we obtain ω w (x n k ) ⊂ S immediately.Meanwhile, we have lim sup In addition, since x * is the unique solution of the variational inequality problem (13), we have lim sup together with (20) and hence lim sup k→∞ δ n k ≤ 0. This completes the proof.
Next, we consider the bounded perturbation of ( 14) generated by the following iterative process: Theorem 2. Assume that the sequences {β n } and {ν n } satisfy the condition (6).Let H 1 ,H 2 be two real Hilbert spaces and A : H 1 → H 2 be a bounded linear operator with L = A * A , where A * is the adjoint of A. Suppose that U : H 1 → H 1 and T : H 2 → H 2 are two averaged mappings with the coefficients γ 1 and γ 2 , respectively.Assume that problem (4) is consistent (i.e., S = ∅).Let h : For any x 0 ∈ H 1 , define the sequence {x n } by (25).If the following conditions are satisfied: Then, the sequence {x n } converges strongly to x * , where x * is a solution of the problem (4), which is also the unique solution of the variational inequality problem (13).
Then, Equation ( 25) can be rewritten as follows: In fact, by Proposition 1 (iii) and the nonexpansiveness of T, it is not hard to show that A * (I − T)A is 2L−Lipschitz.Thus, we have From the condition (iii) and condition (6), we have ∑ ∞ n=0 ẽn < ∞.Consequently, using Theorem 1, it follows that the algorithm ( 14) is bounded perturbation resilient.This completes the proof.

Numerical Results
In this section, we consider the following numerical examples to present the effectiveness, realization and convergence of Theorems 1 and 2: x and A = 2 0 0 3 .
Take U = P C and T = P Q , where C and Q are defined as follows: where y(i) denotes the ith element of y.We can compute the solution set S = {x : 3 ≤ x(1) ≤ 6, 2 ≤ x(2) ≤ 4, x 2 ≤ 4}.
Take the experiment parameters τ n = 1 L and α n = 1 n+1 in the following iterative algorithms and the stopping criteria is x n+1 − x n < error.According to the iterative process of Theorem 1, the sequence {x n } is generated by As n → ∞, we have x n → x * .Then, taking the random initial guess x 0 and using MATLAB software (MATLAB R2012a, MathWorks, Natick, MA, USA), we obtain the numerical experiment results in Table 1.Next, we consider the algorithm with bounded perturbation resilience.Choose the the bounded sequence {ν n } and the summarable nonnegative real sequence {β n } as follows: and for some c ∈ (0, 1), where the indicator function is the normal cone to C. The point d n is taken from N C (x n ).Setting c = 0.5, the numerical results can be seen in Table 2.As we have seen above, the accuracy of the solution is improved with the decrease of the stop criteria.In addition, the sequence {x n } converges to the point (3,2), which is a solution of the numerical example.Of course, it is also the unique solution of the variational inequality (I − h)x * , x − x * ≥ 0.
In addition, we contrast the approximate value of solution x * of Example 1 under the same parameter conditions, the same iterative numbers and the same initial value.The numerical results are reported in Tables 3 and 4, where {x n } and {x (2) n } denote the iterative sequences generated by the algorithm (14) in this paper and Theorem 3.2 in Ref. [8], respectively.
It is obvious that T is 1 2 − av and the set of fixed points Fix(T) = {y| (y(1), y(2), 0) T } is nonempty.Let U = P C and C = {x ∈ R 3 | x ≤ 1}.Then, we use the iterative algorithm of Theorem 1 to approximate a point x * ∈ C such that Ax * ∈ Fix(T).
Take the experiment parameters τ n = 1.9 * n (n+1)L and α n = 1 n+1 in the following iterative algorithms.Let F(x) = 1 2 (I − T)Ax 2 + I C (x) and the stopping criteria is F(x) < error.Then, taking the random initial guess x 0 and using MATLAB software, we obtain the numerical experiment results in Table 5. Next, we consider the bounded perturbation.The definitions of v n and β n are similar to the Example 1. Setting c = 0.8, the numerical results can be seen in Table 4.
As we have seen in Tables 5 and 6, the sequence {x n } approximates to the point (0, 0, 0) T , which is a solution of the numerical example.Of course, it is also the unique solution of the variational inequality (I − h)x * , x − x * ≥ 0.

Conclusions
The SCFPP is an inverse problem that consists in finding a point in a fixed point set such that its image under a bounded linear operator belongs to another fixed point set.Many iterative algorithms have been developed to solve these kinds of problems.In this paper, we have introduced a viscosity iterative sequence and obtained the strong convergence.We prove the main result using the weaker conditions than many existing similar methods-for example, Xu's algorithm [37] for the SFP.More specifically, his algorithm generates a sequence {x n } via the following recursions: where u is a a fixed element and {α n } ⊂ [0, 1] satisfies the assumptions: The second condition is not necessary in our theorems.We also consider the bounded perturbation resilience of the proposed method and get theoretical convergence results.Finally, numerical experiments have been presented to illustrate the effectiveness of the proposed algorithms.