Inertial Iteration for Split Common Fixed-Point Problem for Quasi-Nonexpansive Operators
Abstract
Inspired by the note on split common fixed-point problem for quasi-nonexpansive operators presented by Moudafi (2011), based on the very recent work by Dang et al. (2012), in this paper, we propose an inertial iterative algorithm for solving the split common fixed-point problem for quasi-nonexpansive operators in the Hilbert space. We also prove the asymptotical convergence of the algorithm under some suitable conditions. The results improve and develop previously discussed feasibility problems and related algorithms.
1. Introduction
The split common fixed-point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP); see [11]. Our main purpose here is to give an extension of the results developed in [12] to the split common fixed-point problem for quasi-nonexpansive operators, and we will introduce weak symposium convergence result of the algorithm under some suitable conditions. This will be done in the context of general Hilbert spaces.
The paper is organized as follows. In Section 2, we recall some preliminaries. In Section 3, we present an inertial CQ algorithm and show its convergence.
2. Preliminaries
Throughout the rest of the paper, I denotes the identity operator and Fix (T) denotes the set of the fixed points of an operator T, that is, Fix (T): = {x∣x = T(x)}.
Usually, the convergence of fixed-point algorithms requires some additional smoothness properties of the mapping T such as demiclosedness.
Definition 1. A mapping T is said to be demiclosed if for any sequence {xk} which weakly converges to y and if the sequence {T(xk)} strongly converges to z, then T(y) = z.
In what follows, only the particular case of demiclosedness at zero will be used, which is the particular case when z = 0.
The following lemmas will be needed in the proof of the convergence of the algorithm.
Lemma 2. Let T be a quasi-nonexpansive mapping. Set Tα : = (1 − α)I + αT. Then, it is immediate that for all (x, q) ∈ H × Fix (T):
- (1)
〈x − T(x), x − q〉≥(1/2)∥x−T(x)∥2 and 〈x − T(x), q − T(x)〉≤(1/2)∥x−T(x)∥2;
- (2)
;
- (3)
〈x − Tα(x), x − q〉≥(α/2)∥x−T(x)∥2.
Lemma 3 (see [8].)Assume φk ∈ [0, ∞) and δk ∈ [0, ∞) satisfy
- (1)
φk+1 − φk ≤ θk(φk − φk−1) + δk,
- (2)
,
- (3)
{θk}⊂[0, θ], where θ ∈ [0,1).
3. The Inertial Algorithm and Its Asymptotic Convergence
3.1. The Inertial Algorithm
Algorithm 4.
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Initialization: Let x0 ∈ H1 be arbitrary.
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Iterative step: For k ∈ N, set u = I + γηA*(T − I)A, and let
()
3.2. Asymptotic Convergence of the Inertial Algorithm
In this subsection, we establish the asymptotic convergence of Algorithm 4.
Lemma 5 (Opial [16]). Let H be a Hilbert space and let {xk} be a sequence in H such that there exists a nonempty set S ⊂ H satisfying
- (1)
for every x*, lim k∥xk − x*∥ exists,
- (2)
any weak cluster point of the sequence {xk} belongs to S. Then, there exists z ∈ S such that {xk} weakly converges to z.
Theorem 6. Given a bounded linear operator A : H1 → H2, let U : H1 → H1 be a quasi-nonexpansive operator with nonempty Fix (U) = C and let T : H2 → H2 be a quasi-nonexpansive operator with nonempty Fix (T) = Q. Assume that U − I and T − I are demiclosed at 0. If Γ ≠ ∅, then any sequence {xk} generated by Algorithm 4 weakly converges to a split common fixed point, provided that we choose θk satisfying with , θ ∈ [0,1). γ ∈ (0,1/(λη)) and αk ∈ (δ, 1 − δ) for a small enough δ > 0.
Proof. Taking z ∈ Γ, and using (2) in Lemma 2, we obtain
Hence,
Since there is no more than one weak-cluster point, the weak convergence of the whole sequence {xk} follows by applying Lemma 5 with S = Γ.
Remark 7. Since the current value of ∥xk − xk−1∥ is known before choosing the parameter θk, then θk is well-defined in Theorem 6. In fact, from the process of proof for Theorem 6, we can get the following assert: the convergence result of Theorem 6 always holds provided that we take θk ∈ [0, θ], θ ∈ [0,1), for all k ≥ 0, with
To conclude, we have proposed an algorithm for solving the SCFP in the wide class of quasi-nonexpansive operators and proved its convergence in general Hilbert spaces. Next, we will improve the algorithm to assure the strong convergence in infinite Hilbert spaces.
Acknowledgments
This work was supported by the National Science Foundation of China (under Grant no. 11171221), Shanghai Municipal Committee of Science and Technology (under Grant no. 10550500800), Shanghai Leading Academic Discipline (under Grant no. XTKX 2012), Basic and Frontier Research Program of the Science and Technology Department of Henan Province (under Grant nos. 112300410277 and 082300440150), and China Coal Industry Association Scientific and Technical Guidance to Project (under Grant no. MTKJ-2011-403).
