An Inertial Method for Split Common Fixed Point Problems in Hilbert Spaces

In this paper, we consider the split common fixed point problem in Hilbert spaces. By using the inertial technique, we propose a new algorithm for solving the problem. Under some mild conditions, we establish two weak convergence theorems of the proposed algorithm. Moreover, the stepsize in our algorithm is independent of the norm of the given linear mapping, which can further improve the performance of the algorithm.


Introduction
In recent years, there has been growing interest in the study of the split common fixed point problem because of its various applications in signal processing and image reconstruction [1][2][3]. More specifically, the problem consists in finding x ∈ H 1 satisfying where F(U) and F(T) stand for the fixed point sets of mappings U: H 1 ⟶ H 1 and T: H 2 ⟶ H 2 , respectively, and A: H 1 ⟶ H 2 is a bounded linear mapping. Here, H 1 and H 2 are two Hilbert spaces. In particular, if we let the mappings in (1) be the projections, then it is reduced to the well-known split feasibility problem (SFP): find x ∈ H 1 such that where C ⊆ H 1 and Q ⊆ H 2 are two nonempty closed convex subsets and A: H 1 ⟶ H 2 is a bounded linear mapping; see, e.g., [1,[4][5][6][7].
ere are several algorithms for solving the split common fixed point problem. Among them, Censor and Segal [8] introduced an algorithm as where I stands for the identity mapping, A * is the adjoint mapping of A, and the stepsize τ is a constant in (0, 2‖A‖ − 2 ). In particular, when U � P C and T � P Q , then the above algorithm is reduced to the well-known CQ algorithm for solving the split feasibility problem [4]. Note that this choice of the stepsize requires the exact value or estimation of the norm ‖A‖. To avoid the calculation of ‖A‖, Cui and Wang [9] proposed a variable stepsize as It is readily seen that the above choice of the stepsize does not need any prior knowledge of the linear operator. Recently, Wang [10] introduced a new method for solving (1) as where the stepsize is set as Recently, the above algorithms were further extended to the general case; see, e.g., [2,[10][11][12][13][14][15][16][17]. e inertial method was first introduced in [18], and now, it has been successfully applied to solving various optimization problems arising from some applied sciences [19,20]. In particular, this method was also applied for solving the split feasibility problem [21,22]. By applying the inertial technique, Dang et al. [21] recently proposed the inertial relaxed CQ algorithm, which is defined as where 0 ≤ θ k < θ < 1 and 0 < τ < (2/‖A‖ 2 ). It is clear that the constant stepsize requires the estimation of the norm ‖A‖. To avoid the estimation of the norm, Gibali et al. [23] modified the above stepsize as with 0 < ρ k < 4. It is shown that the inertial relaxed CQ algorithm converges weakly toward a solution of the SFP provided that ∞ k�1 θ k ‖x k − x k− 1 ‖ 2 < ∞. e main advantage of the inertial method is that it can indeed speed up the convergence of the original algorithm. It is thus natural to extend it to the split common fixed point problem. Recently, Cui et al. [24] proposed a modified algorithm of (3) as where 0 ≤ θ k < θ < 1 and τ k is defined as in (6). It was shown that algorithm (9) converges weakly to a solution of the problem provided that ∞ k�1 θ k ‖x k − x k− 1 ‖ 2 < ∞. In this paper, we aim to continue the study of the split common fixed point problem in Hilbert spaces. Motivated by the inertial method, we propose a new algorithm for solving the split common fixed point problem that greatly improves the performance of the original algorithm. Moreover, the stepsize in our algorithm is independent of the norm ‖A‖. Under some mild conditions, we establish two weak convergence theorems of the proposed algorithm.

Preliminary
In the following, we shall assume that problem (1) is consistent, that is, its solution set denoted by f is nonempty. e notation " ⟶ " stands for strong convergence, "⇀" weak convergence, and ω w x n the set of weak cluster points of a sequence x n . Let C be a nonempty closed convex subset. For a mapping T defined on C, we let F(T) � x ∈ C: Tx � x { } be its fixed point set and T ′ � I − T be its complement.
en, T ′ is said to be demiclosed at 0 if, for any x k in C, there holds the following implication: It is well known that if T is a nonexpansive mapping, then T ′ is demiclosed at 0; see [25].
Lemma 2 (see [25]). Assume that x k is a sequence in H such that (i) For each z ∈ C, the limit of ‖x k − z‖ exists (ii) Any weak cluster point of x k belongs to C en, x k is weakly convergent to an element in C.
Lemma 4 (see [25]). Let s, t ∈ R and x, y ∈ H. It then follows that

The Proposed Algorithm
Algorithm 1. Let x 0 , x 1 be arbitrary. Given x k , x k− 1 , choose θ k ∈ [0, 1], and set If ‖U ′ w k + A * T ′ Aw k ‖ � 0, then stop; otherwise, update the next iteration via 2 Journal of Mathematics where Remark 1. In comparison, our stepsize (18) is independent of the norm ‖A‖ so that the calculation or estimation of ‖A‖ is avoided.

Remark 2.
If ‖U ′ w k + A * T ′ Aw k ‖ � 0 for some k ∈ N, then w k is a solution of the problem. To see this, let z ∈ f. It then follows from Lemma 1 that ‖U ′ w k ‖ 2 ≤ 2〈U ′ w k , w k − z〉, and Combining these inequalities yields is yields ‖U ′ w k ‖ � ‖T ′ Aw k ‖ � 0, which implies w k ∈ f.
If we let θ k ≡ 0 in (16), then we get a new algorithm for problem (1). Algorithm 2. Let x 0 be arbitrary. Given x k , if ‖U ′ x k + A * T ′ Ax k ‖ � 0, then stop; otherwise, update the next iteration via where

Convergence Analysis
In this section, we shall establish the convergence of the proposed algorithm. By Remark 2, we may assume that Algorithm 1 generates an infinite iterative sequence. To proceed, we first prove the following lemma.
Lemma 5. Let x k and w k be the sequences generated by Algorithm 1. Let δ k � (1/(4(1 + ‖A‖ 2 )))(‖U ′ w k ‖ 2 + ‖T ′ Aw k ‖ 2 ). en, for any z ∈ S, it follows that Proof. Since U is quasi-nonexpansive, we have In view of (18), we have To finish the proof, it suffices to note that is completes the proof. □ Theorem 1. Assume that U is quasi-nonexpansive such that U ′ is demiclosed at 0, and T is quasi-nonexpansive such that T ′ is demiclosed at 0. If, for each k ∈ N, θ k ≤ θ < 1 such that (c1) ∞ k�1 θ k ‖x k − x k− 1 ‖ 2 < ∞, then the sequence x k generated by Algorithm 1 converges weakly to an element in f.

Proof.
We first show that the sequence ‖x k − z‖ is convergent for any z ∈ f. From Lemma 4, we deduce By Lemma 5, this yields

Journal of Mathematics
Let ϕ k : � ‖x k − z‖ 2 . en, the above inequality can be rewritten as By condition (c1), we then apply Lemma 3 to deduce that ϕ k is convergent, and so is the sequence ‖x k − z‖ .
We next show that each weak cluster point of x k belongs to f. Since ϕ k is convergent, this implies that ϕ k − ϕ k+1 converges to 0 as n ⟶ ∞. It then follows from (29) that Note that lim k θ k ‖x k − x k− 1 ‖ 2 � 0 by condition (c1). By passing to the limit in the above inequality, we have δ k converging to 0 so that Moreover, it is clear that x k is bounded; thus, the set ω w (x n ) is nonempty. Now, take any x ∈ ω w (x k ), and take a subsequence x k l such that it weakly converges to x. On the contrary, we deduce from (c1) that so that w k l also weakly converges to x and Aw k l weakly converges to Ax. Since U ′ and T ′ are both demiclosed at 0, this together with (31) indicates x ∈ F(U) and Ax ∈ F(T); that is, x is an element in f. Finally, by Lemma 2, the sequence x k converges weakly to a solution of problem (1). □ Remark 3. We now construct a sequence satisfying condition (c1). For each k ∈ N, let θ k � min 0.5, 1 We next study the convergence of Algorithm 1 under another condition. To proceed, we need the following lemma. Lemma 6. Let x k and w k be the sequences generated by Algorithm 1. For any z ∈ f, let ϕ k � ‖x k − z‖ 2 − θ k ‖x k− 1 − z‖ 2 + (θ k /2)(3 + θ k )‖x k − x k− 1 ‖ 2 . If θ k is nondecreasing, then where δ k is defined as in Lemma 5.
Proof. In view of (17) and (18), we get It then follows from inequality (25) that