A New Iterative Construction for Approximating Solutions of a Split Common Fixed Point Problem

In this paper, we aim to construct a new strong convergence algorithm for a split common ﬁxed point problem involving the demicontractive operators. It is proved that the vector sequence generated via the Halpern-like algorithm converges to a solution of the split common ﬁxed point problem in norm. The main convergence results presented in this paper extend and improve some corresponding results announced recently. The highlights of this paper shed on the novel algorithm and the new analysis techniques.


Introduction
Let H 1 and H 2 be the Hilbert spaces and C and Q be nonempty closed and convex subsets of H 1 and H 2 , respectively. e split feasibility problem (SFP) is known to find where A: H 1 ⟶ H 2 is a linear bounded operator.
In [1], the split feasibility problem (SFP) in the finite-dimensional Hilbert spaces was introduced by Censor and Elfving. is problem is equivalent to a number of nonlinear optimization problems and finds numerous real applications, such as signal processing and medical imaging (see, e.g., [2][3][4][5][6][7]).
For this split problem, simultaneous multiprojections algorithm was employed by Censor and Elfving in the finitedimensional space R n to obtain the algorithm as follows: where both C and Q are convex and closed subsets of R n , the linear bounded operator A of R n is an n × n matrix, and P Q is the orthogonal projection operator onto the sets Q.
e above algorithm (2) involves the matrix A − 1 (one always assumes the existence of A − 1 ) at every iterative step. Calculating A − 1 is very much time-consuming, if the dimensions are large scale, in particular, and thus it does not become popular.
In order to overcome the fault, Byrne [2,8] proposed the following novel algorithm CQ, which is under the spotlight of recent research x n+1 � P C x n − cA * I − P Q Ax n , n ≥ 0, where P C and P Q are the orthogonal projection operators onto the sets C and Q, respectively, and 0 < c < (2/ρ) with ρ being the spectral radius of the composite mapping A * A. But, the CQ algorithm's step-size is fixed, and it is related to spectral radius of A * A. On the other hand, the orthogonal projection onto the subsets C and Q in Hilbert space H 1 is not easily calculated generally except the special cases, such as balls and polyhedrals. With the real applications (intensitymodulated radiation therapy and medical imaging) of the SFP in signal processing, the SFP has obtained much attention. Now, the approximate solutions of the SFP have been studied extensively by scholars and engineers (see, e.g., [9][10][11][12][13]).
In (1), if C and Q are the intersections of fixed point sets of finite many nonlinear operators, the SFP becomes the split common fixed point problem (SCFPP). e SCFPP was studied first by Censor and Segal [14] in 2009, which consists of finding an element x ∈ H 1 with where Fix(T i ) denotes the fixed point set of T i : H 1 ⟶ H 1 and Fix(S j ) denotes the fixed point sets of S i : H 2 ⟶ H 2 , respectively.
In particular, if m � n � 1, then and T: H 1 ⟶ H 1 , S: H 2 ⟶ H 2 , and Fix(T) denotes the fixed point set of T, and Fix(S) denotes the fixed point set of S. e SCFPP becomes a specific case of SFP and closely related to SFP. To solve this problem, the original algorithm for the directed operator was introduced by Censor and Segal [14] in 2009 as follows: where ρ satisfies the constraint condition 0 < ρ < (2/‖A‖ 2 ), and the authors got the weak convergence of the sequence x n for solving the SCFPP (5) if the SCFPP consists, that is, its solution set is nonempty.
Recently, Cui and Wang [15] studied the following algorithm, and they got the weak convergence of the sequence x n for solving the SCFPP (5): where U λ � (1 − λ)I + λU and ρ n is given in the following pattern: e step-size of this algorithm ρ n does not depend on the norm of the operator A and searches automatically.
In 2015, Boikanyo [16] extended the main results of Cui and Wang [15] and constructed the Halpern-type algorithm for demicontractive operators that converge to a solution of the SCFPP (5) strongly: where ρ n is given as (8). In this result, the resolvent I − ρ n A * (I − T)A plays an important role. Indeed, the techniques of resolvents is quite popular, and it acts as a bridge between fixed point problems and a number of optimization problems (see, e.g., [17][18][19][20][21] and the references therein). Motivated by the above results, we propose a novel algorithm on demicontractive operators for approximating a solution of the SCFPP (5): x n+1 � 1 − α n 1 − ξ n I + ξ n U 1 − η n I + η n U u n + α n u, where ρ n is also obtained by (8). Our algorithm is also based on the Halpern iteration. Indeed, it is a core for many algorithms in split problems (see, e.g., [22][23][24][25][26]). We get the strong convergence of the iterative sequence x n generated by (10) for solving the SCFPP (5). Our main results are in two folds. First, we construct a novel iterative algorithm to solve the split common fixed point problem for the demicontractive operators. Second, we permit step-size to be selected self-adaptively by the self-adaptive method, which avoids to depend on the norm of the nonlinear operator A. Our results extend and improve some results of Boikanyo [16], Cui and Wang [15], Yao et al. [27], and many others.

Preliminaries
In this section, we will present some lemmas, which are useful to prove our main results as follows.
Let H be a Hilbert space, which is endowed with the inner product 〈·, ·〉, norm ‖ · ‖.
en, the following inequalities hold: Definition 1. Let T: H ⟶ H be an operator, then I − T called demiclosed at zero, if the following implication holds for any x n in H: Note that the nonexpansive operator is demiclosed at zero [28].
Lemma 1 (see [29]). Let a n be a sequence of real nonnegative numbers with a n+1 ≤ 1 − c n a n + δ n , (14) where c n is a sequence in (0, 1) and δ n is a real sequence such that |δ n | < ∞ Then, lim n⟶∞ a n � 0.
Lemma 2 (see [15]). Let A: H 1 ⟶ H 2 be a linear bounded operator and T: Lemma 3 (see [30]). Let H be a Hilbert space and let T be an L-Lipschitzian mapping defined on H with the module L ≥ 1. Set , then the following conclusions hold: ator, then the operator K is quasi-non-expansive Lemma 4 (see [31]). Let s k be a real numbers sequence that does not decrease at infinity in the sense that there exists a subsequence s k j of s k such that s k j < s k j+1 for all j ≥ 0.
Define an integer sequence m k k ≥ k 0 by Then, m k ⟶ ∞ as k ⟶ ∞ and for all k ≥ k 0 .

Some Nonlinear Operators
for all x, y ∈ C.
Definition 6. An operator T: H ⟶ H is said to be firmly quasi-non-expansive if and only if Fix(T) ≠ ∅ and Note that T is pseudocontractive if and only if the operator I − T is monotone. ere is also an alternative definition for pseudocontractive operators, that is, T is said to be pseudocontractive if and only if ∀x ∈ H, ∀x * ∈ Fix(T).
It is easy to obtain that (29) is equivalent to Journal of Mathematics Remark 1. e classes of k-demicontrative mappings, directed mappings, quasi-non-expansive mappings, and nonexpansive mappings are closely related. By the above definitions, we obtain the following conclusion relations easily (see Figures 1-7).

Main Results
In this section, some assumptions are as follows: (1) H 1 and H 2 are two Hilbert spaces, A: H 1 ⟶ H 2 is a linear bounded operator, and A * is the adjoint of A Now, we give the new algorithm to find x * ∈ S.where A is a bounded and linear mapping, A * is the adjoint of operator A, and ρ n is obtained as follows:

Algorithm 1. H 1 is a real Hilbert space, and Fix(U) ≠ ∅.
Take an initial point x 0 ∈ H 1 arbitrarily, and fix u ∈ H 1 and θ n ⊂ (0, 1). If the n− th iteration x n is available, then the (n + 1)− th iteration is constructed via the following formula:

Non-expansive mapping
Firmly non-expansive mapping
Proof. Since z ∈ Fix(U), we get from (30) that

Proof.
is proof is split into three parts as follows.

□
Step 1. Prove that x n is a bounded sequence. Take p ∈ S. From eorem 1, we know that U μ n ,] n is quasi-non-expansive. From (32), we have By induction, we get us, x n is bounded.
Step 2 where x � P S u.

Journal of Mathematics
Hence, Step 3. Prove that x n ⟶ x as n ⟶ ∞. is step is divided into two cases. Denote s n ≔ ‖x n − x‖ 2 . Case 1. Assume there exists a positive integer n 0 and the sequence s n is decreasing for any n ≥ n 0 . en, s n converges to some point strongly by the monotonic bounded principle.
First, we show that Using the choice (33) of the step-size ρ n , (32), (34), (35), and (11), we get So, where L is a nonnegative real constant such that sup n∈N 2〈f(x n ) − x, x n+1 − x〉 ≤ L. Based on the fact that s n is convergent, we have Moreover, Hence, ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ is a fixed point in R 3 , and the initial point ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ and x n � a n b n c n ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ is generated by the algorithm (10). We plot the numbers of iterations and ‖x n+1 − x n ‖ 2 in the following graphs (Figures 8 and 9), the numbers of iterations and x n � a n , b n , c n .

Conclusion
In this paper, we proposed a new iteration algorithm (10) and we obtained the strong convergence of the sequence x n for split common fixed point problems (5). e main result is an extension of the related results announced in [15,16,27]. e research highlights of this paper are novel algorithms and their analysis techniques. e improvement on the extension of the operator, such as the demicontractive mappings, the directed operators, the quasi-non-expansive operators, and quasi-pseudo-contractive operators will be of interest for further research in the future.

Data Availability
e data used to support the findings of this study are included within the article.  Journal of Mathematics 11