Strong Convergence Theorems for an Implicit Iterative Algorithm for the Split Common Fixed Point Problem

The aim of this paper is to construct a novel implicit iterative algorithm for the split common fixed point problem for the demicontractive operators U, T, and xn = αnf(xn) + (1 − αn)Uλ(xn − ρnA∗(I − T)Axn), n ≥ 0, where Uλ = (1 − λ)I + λU, and we obtain the sequence which strongly converges to a solution ?̂? of this problem, and the solution ?̂? satisfies the variational inequality. ⟨?̂? − f(?̂?), ?̂? − z⟩ ≤ 0, ∀z ∈ S, where S denotes the set of all solutions of the split common fixed point problem.


Introduction
The split feasibility problem (SFP) is to find a point  ∈  such that  ∈ , where  is a nonempty closed convex subset of a Hilbert space  1 ,  is a nonempty closed convex subset of a Hilbert space  2 , and  :  1 →  2 is a bounded linear operator.This problem was proposed by Censor and Elfving [1] in 1994.
Since the SFP can extensively be applied in fields such as intensity-modulated radiation therapy, signal processing, and image reconstruction, then the SFP has received so much attention by so many scholars; see .
In 1994, Censor and Elfving [1] proposed the original algorithm in   , where  and  are nonempty closed convex subsets of   ,  in the finite-dimensional   is a  ×  matrix, and   is the projection operator from  2 onto .
As we know, the computation of the inverse  −1 is not easy if the inverse of  existed.So, the algorithm (2) does not become popular.
In 2002 and 2004, Byrne [2,3] gave the so-called  algorithm as follows: where 0 <  < 2/ with  taken as the largest eigenvalue of the operator  *  and   and   denote the projection operators from  1 and  2 onto the sets , , respectively.For the stepsize of algorithm (3) is fixed and closely related to spectral radius of  * , then the projection operators   and   are not easily calculated usually.
The split common fixed point problem (SCFP) is to find a point where  :  1 →  1 and  :  2 →  2 , and Fix() and Fix() denote the fixed point sets of  and .This problem was proposed by Censor and Segal [12] in 2009.Note that the SCFP is closely related to SFP and it is a particular case of SFP.
In order to overcome this disadvantage, Cui and Wang [26] proposed the following algorithm in 2014: where   = (1 − ) +  and the step size   is chosen by the following way: and they proved that the sequence {  } converges weakly to a solution of the SCFP (4).Note that the advantage of this algorithm is that the step size   searches automatically and does not depend on the norm of operator .
Motivated by the viscosity idea of [30], in this paper, we construct a novel algorithm for demicontractive operators to approximate the solution of the SCFP (4), that is, the following implicit iterative algorithm: where   = (1 − ) +  and the step size   is also chosen as (7).
The research highlight of this paper is that the strong convergence of the SCFP (4) is constructed; that is to say the sequence {  } generated by (9) converges strongly to a solution of the SCFP.

Preliminaries
Throughout this paper, we denote the set of all solutions of the SCFP (4) by .We use   ⇀  to indicate that {  } converges weakly to .Similarly,   →  symbolizes the sequence {  } which converges strongly to .
Let ,  1 , and  2 be Hilbert spaces endowed with the inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖, and  and  are nonempty closed convex subsets of  1 and  2 , respectively.Some concepts and lemmas are given in the following and they are useful in proving our main results.
(iv) -demicontractive with  < 1 if Note that ( 12) is equivalent to Definition 2. Let  :  →  be an operator, then  −  is said to be demiclosed at zero, if for any {  } in , the following implication holds As we know, the nonexpansive mappings are demiclosed at zero [31].Definition 3. Let  be a nonempty closed convex subset of a Hilbert space , the metric (nearest point) projection   from  to  is defined as follows: Given  ∈ ,    is the only point in  with the property Lemma 4 (see [32]).Let  be a nonempty closed convex subset of a Hilbert space ,   is a nonexpansive mapping from  onto  and is characterized as: Given  ∈ , there holds the inequality Lemma 5 (see [32]).Let  be a Hilbert space, then the following inequality holds, Lemma 6 (Cui and Wang [26]).Let  :  1 →  2 be a bounded linear operator and  :  2 →  2 a -demicontractive operator with  < 1.

Main Results
Proposition 8. Based on the definitions in preliminaries, the classes of -demicontractive operators, directed operators, quasi-nonexpansive operators, and nonexpansive operators have close relations.We can visually use the following Venn diagram (Figure 1) to denote their relations.
Proof.From Definition 1, the following conclusion is obtained easily.
Next, we give the novel implicit algorithm to solve the SCFP (4) for demicontractive operators.In the sequel, the assumptions are given as follows: (ii)  :  2 →  2 is a -demicontractive operator with  < 1.
(iii) Both  −  and  −  are demiclosed at zero.
Step 1.We show that {  } is bounded.
Step 2. We show that there exists a subsequence {   } ⊆ {  } such that    → x as  → ∞, and x ∈  solves the variational inequality (24).By the reflexivity of Hilbert space  1 and the boundedness of {  }, there exists a weakly convergence subsequence {   } ⊆ {  } such that    ⇀ x, as  → ∞.
First, we show that    → x, as  → ∞.
Next, we denote    by   .
(i) If    = 0. From ( 18) and ( 21 So For {  } ⇀ x as  → ∞, the above inequality implies that 18), ( 19) and ( 21), we have For the case    = 0, then it is clear we obtain From (37) and the demiclosedness of  −  at zero, then Since  is bounded linear operator, then  is weak continuity; then From (39) and the demiclosedness of  −  at zero, then Hence, x ∈  by (40) and (42).
To show that   → x as  → ∞, we only need to show that any subsequence of {  } converges strongly to x.
(58) Adding up the above variational inequality yields Thus x = x.This is contradicting with the assumption x ̸ = x, so {  } converges strongly to x.The proof is completed.

Applications
In this section, we consider some special cases as the applications of Theorem 10.
Based on the relations of -demicontractive operators, directed operators, and quasi-nonexpansive operators (Proposition 8), the following corollaries are obtained easily.
) if the solution of SCFP exists.But it is obvious that the choice of the step size  depends on the norm of operator, , which is the disadvantage of this algorithm.