Strong Convergence Theorems for the Split Common Fixed Point Problem for Countable Family of Nonexpansive Operators

We introduce a new iterative algorithm for solving the split common fixed point problem for countable family of nonexpansive operators. Under suitable assumptions, we prove that the iterative algorithm strongly converges to a solution of the problem.


Introduction
Let H 1 and H 2 be two real Hilbert spaces and let A : H 1 → H 2 be a bounded linear operator.The split feasibility problem SFP , see 1 , is to find a point x * with the property: where C and Q are nonempty closed convex subsets of H 1 and H 2 , respectively.A more general form of the SFP is the so-called multiple-set split feasibility problem MSSFP which was recently introduced by Censor et al. 2 .Given integers p, r ≥ 1, the MSSFP is to find a point x * with the property: where {C i } p i 1 and {Q j } r j 1 are nonempty closed convex subsets of H 1 and H 2 , respectively.The SFP 1.1 and the MSSFP 1.2 serve as a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in this operator's ranges.Recently, the SFP 1.1 and the MSSFP 1.2 are widely applied in the image reconstructions 1, 3 , the intensity-modulated radiation therapy 4, 5 , and many other areas.The problems have been investigated by many researchers, for instance, 6-13 .The SFP 1.1 can be viewed as a special case of the convex feasibility problem CFP since the SFP 1.1 can be rewritten as However, the methods for study the SFP 1.1 are actually different from those for the CFP in order to avoid the usage of the inverse A −1 .Byrne 6 introduced a so-called CQ algorithm: where the operator A −1 is not relevant.
Censor and Segal in 14 firstly introduced the concept of the split common fixed point problem SCFPP in finite-dimensional Hilbert spaces.The SCFPP is a generalization of the convex feasibility problem CFP and the split feasibility problem SFP .The SCFPP considers to find a common fixed point of a family of operators in H 1 such that its image under a linear transformation A is a common fixed point of another family of operators in H 2 .That is, the SCFPP is to find a point x * with the property: where U i : H 1 → H 1 i 1, 2, . . ., p and T j : H 2 → H 2 j 1, 2, . . ., r are nonlinear operators.If p r 1, the problem 1.5 deduces to the so-called two-set SCFPP, which is to find a point x * such that where U : H 1 → H 1 and T : H 2 → H 2 are nonlinear operators.
Censor and Segal in 14 considered the following iterative algorithm for the SCFPP 1.6 for Class-operators in finite-dimensional Hilbert spaces: where x 0 ∈ H 1 , 0 < γ < 2/ A 2 and I is the identity operator.
Recently, in the infinite-dimensional Hilbert space, Wang and Xu 15 studied the SCFPP 1.5 and introduced the following iterative algorithm for Class-operators: where n n mod p and p r.Under some mild conditions, they proved that {x n } converges weakly to a solution of the SCFPP 1.5 , extended and improved Censor and Segal's results.Moreover, they proved that the SCFPP 1.5 for the Class-operators is equivalent to a common fixed point problem.This is also a classical method.Many problems eventually converted to a common fixed point problem, see 16-18 .Very recently, the split common fixed point problems for various types of operators were studied in 19-21 .
The above-mentioned results are about a finite number of operators; that is, the constraints are finite imposed on the solutions.In this paper, we consider the constraints are infinite, but countable.That is, we consider the generalized case of SCFPP for two countable families of operators denoted GSCFPP , which is to find a point x * such that Fix T j . 1.9 Of course, the GSCFPP is more general and widely used than the SCFPP.This is a novelty of this paper.At the same time, we consider the nonexpansive operator.The nonexpansive operator is important because it includes many types of nonlinear operator arising in applied mathematics.For instance, the projection and the identity operator are nonexpansive.We prove that the GSCFPP 1.9 for the nonexpansive operators is equivalent to a common fixed point problem.Very recently, Gu et al. 22 introduced a new iterative method for dealing with the countable family of operators.They studied the following iterative algorithm: where S and {T i } ∞ i 1 are nonexpansive, α 0 1, {α n } is strictly decreasing sequence in 0, 1 , and {β n } is a sequence in 0, 1 .Under some certain conditions on parameters, they proved that the sequence {x n } converges strongly to x * ∈ ∞ i 1 F T i .On the other hand, from weakly convergence to strongly convergence, the viscosity approximation method is also one of the classical methods, see 22-24 .Motivated and inspired by the above results, we introduce the following algorithm: Under some certain conditions, we prove that the sequence {x n } generated by 1.11 converges strongly to the solution of the GSCFPP 1.9 .

Preliminaries
Throughout this paper, we write x n x and x n → x to indicate that {x n } converges weakly to x and converges strongly to x, respectively.
Let H be a real Hilbert space.An operator T : H → H is said to be nonexpansive if Tx − Ty ≤ x − y for all x, y ∈ H.The set of fixed points of T is denoted by Fix T .It is known that Fix T is closed and convex, see 25 where {γ n } is a sequence in 0, 1 and

Main Results
Now we state and prove our main results of this paper.Theorem 3.1.Let {U n } and {T n } be sequences of nonexpansive operators on real Hilbert spaces H 1 and H 2 , respectively.Let f : H 1 → H 1 be a contraction with coefficient ρ ∈ 0, 1 .Suppose that the solution set Ω of GSCFPP 1.9 is nonempty.Let x 1 ∈ H 1 and 0 < γ < 2/ A 2 .Set α 0 1, and let {α n } ⊂ 0, 1 be a strictly decreasing sequence satisfying the following conditions: Then the sequence {x n } generated by 1.11 converges strongly to w ∈ Ω, where w P Ω f w .
Proof.We proceed with the following steps.
Step 1.First show that there exists w ∈ Ω such that w P Ω f w .
In fact, since f is a contraction with coefficient ρ, we have for every x, y ∈ H 1 .Hence P Ω f is also a contraction of H 1 into itself.Therefore, there exists a unique w ∈ H 1 such that w P Ω f w .At the same time, we note that w ∈ Ω.
Step 2. Now we show that {x n } is bounded.For simplicity, we set V i I γA * T i − I A. Then we can rewrite 1.11 to Observe that for all x, y ∈ H 1 .Thus it follows that

3.4
For 0 < γ < 1/ A 2 , we can immediately obtain that V i is a nonexpansive operator for every i ∈ N.
Let p ∈ Ω, then U i p p and T i Ap Ap for every i ≥ 1.Thus T i − I Ap 0, which implies that V i p p. Since n i 1 α i−1 − α i 1 − α n , we have

3.5
Then it follows that for every n ∈ N.This shows that {x n } and {U n V n x n−1 } is bounded.Hence, {f x n } is also bounded.
Step 3. We show lim n → ∞ x n 1 − x n 0. From 3.2 , we have where M is a constant such that

3.8
From i , ii , iii , and Lemma 2.2, it follows that lim Using 3.2 and 3.10 , we deduce

3.11
Journal of Applied Mathematics 7 Noting that lim n → ∞ x n − x n 1 0 and lim n → ∞ α n 0, then we immediately obtain Since {α n } is strictly decreasing, it follows that

3.13
Next we show lim n → ∞ T i Ax n − Ax n 0, for every i ∈ N. Note for every i ∈ N,

which follows that
for every i ∈ N. From 3.2 , we have

3.16
By 3.15 , it follows that 3.17 Thus,

3.18
Using lim n → ∞ α n 0 and lim n → ∞ x n 1 − x n 0, we have Since {α n } is strictly decreasing, we obtain

3.21
Last we show lim n → ∞ U i x n − x n 0 for every i ∈ N. In fact, we note that for every i ∈ N,

3.22
Then by 3.13 and 3.21 , we obtain

3.23
Step 5. Show lim sup n → ∞ f w − w, x n − w ≤ 0, where w P Ω f w .
Since {x n } is bounded, there exist a point v ∈ H 1 and a subsequence {x Step 6. Show x n → w P Ω f w .Since w ∈ Ω, we have U i w w and T i Aw Aw for every i ∈ N. It follows that V i w w.Using 3.2 , we have for every n ∈ N. Consequently, according to 3.25 , ρ ∈ 0, 1 , and Lemma 2.2, we deduce that {x n } converges strongly to w P Ω w .This completes the proof.
Remark 3.2.If we set α n 1/n and f x u for all x ∈ H 1 , where u is an arbitrary point in H 1 , it is easily seen that our conditions are satisfied.Corollary 3.3.Let U : H 1 → H 1 and T : H 2 → H 2 be nonexpansive operators.Let f : H 1 → H 1 be a contraction with coefficient ρ ∈ 0, 1 .Suppose that the solution set Ω of SCFPP 1.6 is nonempty.Let x 1 ∈ H 1 and define a sequence {x n } by the following algorithm: x n 1 α n f x n 1 − α n U x n γA * T − I Ax n , 3.28 where 0 < γ < 1/ A 2 , α 0 1 and {α n } ⊂ 0, 1 is a strictly decreasing sequence satisfying the following conditions: Then {x n } converges strongly to w ∈ Ω, where w P Ω f w .
Proof.Set {U n } and {T n } to be sequences of operators defined by U n U and T n T for all n ∈ N in Theorem 3.1.Then by Theorem 3.1 we obtain the desired result.Remark 3.4.By adding more operators to the families {U n } and {T n } by setting U i I for i ≥ p 1 and T j I for j ≥ r 1, the SCFPP 1.5 can be viewed as a special case of the GSCFPP 1.9 .
. An operator f : H → H is called contraction if there exists a constant ρ ∈ 0, 1 such that f x − f y ≤ ρ x − y for all x, y ∈ H. 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 ≤ x − y for every y ∈ C. P C is called the metric projection of H onto C. It is known that, for each x ∈ H, In order to prove our main results, we collect the following lemmas in this section.Lemma 2.1 see 26 .Let H be a Hilbert space, C a closed convex subset of H, and T : C → C a nonexpansive operator with Fix T / ∅.If {x n } is a sequence in C weakly converging to x ∈ C and { I − T x n } converges strongly to y ∈ C, then I − T x y.In particular, if y 0, then x ∈ Fix T .
Lemma 2.2 see 23 .Assume {a n } is a sequence of nonnegative real numbers such that Since A is a bounded linear operator, we have Ax n k Av.Now applying 3.21 , 3.23 , and Lemma 2.1, we conclude that v ∈ Fix U i and Av ∈ Fix T i for every i.Hence, v ∈ Ω.Since Ω is closed and convex, by 2.1 , we get lim sup