A Modified Halpern's Iterative Scheme for Solving Split Feasibility Problems

Abstract

The purpose of this paper is to introduce and study a modified Halpern’s iterative scheme for solving the split feasibility problem (SFP) in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions a strong convergence theorem is established. The main result presented in this paper improves and extends some recent results done by Xu (Iterative methods for the split feasibility problem in infinite-dimensional Hilbert space, Inverse Problem 26 (2010) 105018) and some others.

1. Introduction

Let 𝐶 and 𝑄 be nonempty-closed convex subsets of real Hilbert spaces 𝐻₁ and 𝐻₂, respectively. Let 𝐴 be a linear-bounded operator from 𝐻₁ to 𝐻₂. The split feasibility problem (SFP) is finding a point 𝑥 satisfying the following property: 𝑥∈𝐶,𝐴𝑥∈𝑄. (1.1)

The SFP was introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and medical image reconstruction [2], and very well-known iterative algorithms have been invented to solve it [2].

We use Γ to denote the solution set of SFP: Γ={𝑥∈𝐶∶𝐴𝑥∈𝑄}, (1.2) and assume that the SFP (1.1) is consistent (i.e., (1.1) has a solution) so that Γ is closed, convex, and nonempty, it is not hard to see that 𝑥∈𝐶 solves (1.1) if and only if it solves the following fixed point equation; 𝑥=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥,𝑥∈𝐶, (1.3) where 𝑃_𝐶 and 𝑃_𝑄 are the (orthogonal) projections onto 𝐶 and 𝑄, respectively, 𝛾>0 is any positive constant and 𝐴^∗ denotes the adjoint of 𝐴. Moreover, for sufficiently small 𝛾>0, the operator 𝑃_𝐶(𝐼−𝛾𝐴^∗(𝐼−𝑃_𝑄)𝐴) which defines the fixed point equation in (1.3) is nonexpansive.

To solve the SFP (1.1), Byrne [2] proposed his 𝐶𝑄 algorithm (see also [3]) which generates a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝑛≥0, (1.4) where 𝛾∈(0,2/𝜆) with 𝜆 being the spectral radius of the operator 𝐴^∗𝐴.

Very recently, Xu [4] has viewed the 𝐶𝑄 algorithm for averaged mappings and applied Mann's algorithm to solving the SFP, and he also proved that an averaged 𝐶𝑄 algorithm is weakly convergent to a solution of the SFP.

In this paper, we also regard the 𝐶𝑄 algorithm as a fixed point algorithm for averaged mappings and try to study the SFP by the following modified Halpern's iterative scheme;𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝐼−𝜉A^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝑛≥0, (1.5) where {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} are three sequences in [0,1] satisfying 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1. Furthermore, our result extends and improves the result of Xu [4] from weak to strong convergence theorems.

2. Preliminaries

Throughout the paper, we adopt the following notation.

Let 𝑥_𝑛 be a sequence and 𝑥 be a point in a normed space 𝑋. We use 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥 to denote strong and weak convergence to 𝑥 of the sequence {𝑥_𝑛}, respectively. In addition, we use 𝜔_𝑤(𝑥_𝑛) to denote the weak 𝜔-limit set of the sequence {𝑥_𝑛}; namely, 𝜔_w𝑥_𝑛∶=𝑥∶𝑥_𝑛𝑖⇀𝑥forsomesubsequence𝑥_𝑛𝑖of𝑥_𝑛. (2.1)

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively, and let 𝐾 be a nonempty-closed convex subset of 𝐻. For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐾, denoted by 𝑃_𝐾𝑥, such that 𝑥−𝑃_𝐾𝑥≤𝑥−𝑦,∀𝑦∈𝐾, (2.2)

𝑃_𝐾 is called the metric projection of 𝐻 onto 𝐾. It is well known that 𝑃_𝐾 is a nonexpansive mapping of 𝐻 onto 𝐾 and satisfies ⟨𝑥−𝑦,𝑃_𝐾𝑥−𝑃_𝐾𝑦⟩≥𝑃_𝐾𝑥−𝑃_𝐾𝑦², (2.3) for every 𝑥,𝑦∈𝐻. Moreover, 𝑃_𝐾𝑥 is characterized by the following properties: 𝑃_𝐾𝑥∈𝐾 and ⟨𝑥−𝑃_𝐾𝑥,𝑦−𝑃_𝐾𝑥⟩≤0,𝑥−𝑦²≥𝑥−𝑃_𝐾𝑥²+𝑦−𝑃_𝐾𝑥², (2.4) for all 𝑥∈𝐻,𝑦∈𝐾.

Some important properties of projections are gathered in the following proposition.

Proposition 2.1. Given 𝑥∈𝐻 and 𝑧∈𝐾. Then 𝑧=𝑃_𝐾𝑥 if and only if ⟨𝑥−𝑧,𝑦−𝑧⟩≤0,∀𝑦∈𝐾. (2.5) One also needs other sorts of nonlinear operators which are introduced below.
Let 𝑇,𝐴∶𝐻→𝐻 be the nonlinear operators. (1)𝑇is nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥, 𝑦∈𝐻. (2)𝑇 is firmly nonexpansive if 2𝑇−𝐼 is nonexpansive. Equivalent, 𝑇=(𝐼+𝑆)/2, where 𝑆∶𝐻→𝐻 is nonexpansive. Alternatively, 𝑇 is firmly nonexpansive if and only if ⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩≥𝑇𝑥−𝑇𝑦²,𝑥,𝑦∈𝐻. (2.6)(3)𝑇 is averaged if 𝑇=(1−𝛼)𝐼+𝛼𝑆, where 𝛼∈(0,1) and 𝑆∶𝐻→𝐻 is nonexpansive. In this case, one also says that 𝑇 is 𝛼-averaged. A firmly nonexpansive mapping is (1/2)-averaged.(4)𝐴 is monotone if ⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥0 for 𝑥,𝑦∈𝐻. (5)𝐴 is 𝛽-strongly monotone, with 𝛽>0, if ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥𝛽𝑥−𝑦²,𝑥,𝑦∈𝐻. (2.7)(6)𝐴 is 𝜈-inverse strongly monotone (𝜈-ism), with 𝜈>0, if ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥𝜈𝐴𝑥−𝐴𝑦²,𝑥,𝑦∈𝐻. (2.8)It is well known that both 𝑃_𝐾 and 𝐼−𝑃_𝐾 are firmly nonexpansive and (1/2)-ism.
Denote by Fix(𝑇) the set of fixed points of a self-mapping 𝑇 defined on 𝐻, (i.e., Fix(𝑇)=𝑥∈𝐻∶T𝑥=𝑥).

Proposition 2.2 (see [2, 5]). One has the following assertions.(1)𝑇 is nonexpansive if and only if the complement 𝐼−𝑇 is (1/2)-ism. (2) If 𝑇 is 𝜈-ism and 𝛾>0, then 𝛾𝑇 is (𝜈/𝛾)-ism. (3)𝑇 is averaged if and only if the complement 𝐼−𝑇 is 𝜈-ism, for some 𝜈>(1/2). Indeed, for 𝛼∈(0,1),𝑇 is 𝛼-averaged if and only if 𝐼−𝑇 is (1/2𝛼)-ism.(4) If 𝑇₁ is 𝛼₁-averaged and 𝑇₂ is 𝛼₂-averaged, where 𝛼₁,𝛼₂∈(0,1), then the composite 𝑇₁𝑇₂ is 𝛼-averaged, where 𝛼=𝛼₁+𝛼₂−𝛼₁𝛼₂.(5) If 𝑇₁ and 𝑇₂ are averaged and have a common fixed point, then Fix(𝑇₁𝑇₂)=Fix(𝑇₁)∩Fix(𝑇₂).

Lemma 2.3 (see [6]). Let 𝐾 be a nonempty-closed convex subset of a real Hilbert space 𝐻 and 𝑇 be nonexpansive mapping on 𝐾 with Fix(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐾 which converges weakly to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then 𝑦=(𝐼−𝑇)𝑥. In particular, if 𝑦=0, then 𝑥∈Fix(𝑇).

Lemma 2.4 (see [7]). Let (𝐸,⟨⋅,⋅⟩) be an inner product space. Then for all 𝑥,𝑦,𝑧∈𝐸 and 𝛼_𝑛,𝛽_𝑛,𝛾_𝑛∈[0,1] with 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1, one has 𝛼𝑥+𝛽𝑦+𝛾𝑧²=𝛼‖𝑥‖²+𝛽𝑦²+𝛾‖𝑧‖²−𝛼𝛽𝑥−𝑦²−𝛼𝛾‖𝑥−𝑧‖²−𝛽𝛾𝑦−𝑧². (2.9)

Lemma 2.5 (see [8]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛, (2.10) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that (1)∑^∞_𝑛=0𝛾_𝑛=∞; (2)limsup_𝑛→∞𝛿_𝑛/𝛾_𝑛≤0 or ∑^∞_𝑛=0|𝛿_𝑛|<∞.
Then, lim_𝑛→∞𝑎_𝑛=0.

3. Main Result

Let 𝐶 be a nonempty closed and convex subset of a Hilbert space 𝐻. For any 𝑢,𝑥₀∈𝐶, we define the sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑃_𝐶𝐼−𝜉𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝑛≥0, (3.1) where {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} are three sequences in [0,1] and satisfy 𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1.

Theorem 3.1. Suppose that the SFP is consistent and 0<𝜉<(2/‖𝐴‖²). Let {𝑥_𝑛} be a sequence defined as in (3.1). If the following assumptions are satisfied:(C1)lim_𝑛→∞𝛼_𝑛=0 but ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)limsup_n→∞𝛽_𝑛<1, (C3) the sums ∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|, ∑^∞_𝑛=1|𝛽_𝑛+1−𝛽_𝑛| and ∑^∞_𝑛=1|𝛾_𝑛+1−𝛾_𝑛| are finite. Then {𝑥_𝑛} converges strongly to a solution of the SFP (1.1).

Proof. We firstly show that the sequence {𝑥_𝑛} is bounded. For our convenience, we take 𝑇∶=𝑃_𝐶(𝐼−𝜉𝐴^∗(𝐼−𝑃_𝑄)𝐴). Then, for any 𝑥^∗∈Γ, we have 𝑇𝑥^∗=𝑥^∗. Now, we observe that 𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛−𝑥^∗≤𝛼_𝑛𝑢−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗+𝛾_𝑛𝑇𝑥_𝑛−𝑥^∗. (3.2) Now, we note that the condition 0<𝜉<(2/‖𝐴‖²) implies that the operator 𝑃_𝐶(𝐼−𝜉𝐴^∗(𝐼−𝑃_𝑄)𝐴) is averaged. Since 𝐼−𝑃_𝑄 is firmly nonexpansive mappings and so is (1/2)-average, which is 1-ism. Also observe that 𝐴^∗(𝐼−𝑃_𝑄)𝐴 is (1/‖𝐴‖²)-ism so that 𝜉𝐴^∗(𝐼−𝑃_𝑄)𝐴 is (1/𝜉‖𝐴‖²)-ism. Further, from the fact that 𝐼−𝜉𝐴^∗(𝐼−𝑃_𝑄)𝐴 is (𝜉‖𝐴‖²/2)-averaged and 𝑃_𝐶 is (1/2)-averaged, we may obtain that 𝑃_𝐶(𝐼−𝜉𝐴^∗(𝐼−𝑃_𝑄)𝐴) is 𝜒-averaged, where 1𝜒=2+𝜉‖𝐴‖²2−12⋅𝜉‖𝐴‖²2=2+𝜉‖𝐴‖²4. (3.3)
This implies that 𝑇=𝜒𝐼+(1−𝜒)𝑆, where 𝜒=(2+𝜉‖𝐴‖²/4)∈(0,1) for some nonexpansive mappings 𝑆. Note that 𝑇 is also nonexpansive mappings. Hence, we have 𝑇𝑥_𝑛−𝑥^∗=𝑇𝑥_𝑛−𝑇𝑥^∗≤𝑥_𝑛−𝑥^∗. (3.4) From the inequalities (3.2) and (3.4), we have 𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝑢−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥^∗≤max𝑢−𝑥^∗,𝑥_𝑛−𝑥^∗. (3.5) Continuing inductively, we may obtain that the inequality 𝑥_𝑛+1−𝑥^∗≤max𝑢−𝑥^∗,𝑥₀−𝑥^∗, (3.6) holds for all 𝑛≥0. So, {𝑥_𝑛} is bounded so does {𝑇𝑥_𝑛}.
Next, we will show that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. Observe that 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛−𝛼_𝑛−1𝑢+𝛽_𝑛−1𝑥_𝑛−1+𝛾_𝑛−1𝑇𝑥_𝑛−1≤𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛−𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑇𝑥_𝑛−1+𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛−1+𝛾_𝑛𝑇𝑥_𝑛−1−𝛼_𝑛−1𝑢+𝛽_𝑛−1𝑥_𝑛−1+𝛾_𝑛−1𝑇𝑥_𝑛−1≤1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1‖𝑢‖+𝛽_𝑛−𝛽_𝑛−1𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑇𝑥_𝑛−1. (3.7) Since {𝑥_𝑛} and {𝑇𝑥_𝑛} are bounded, there exists 𝑀=sup(‖𝑢‖,‖𝑥_𝑛+1‖,‖𝑇𝑥_𝑛−1‖)>0 such that 𝑥_𝑛+1−𝑥_𝑛≤1−𝛼_𝑛x_𝑛−𝑥_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝛾_𝑛−𝛾_𝑛−1. (3.8) According to Lemma 2.5 and the condition (C3), we have lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0.
We note that 𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+1−𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛=𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑢−𝑇𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑢−𝑇𝑥_𝑛+𝛽_𝑛𝑥_𝑛−𝑇𝑥_𝑛=11−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛1−𝛽_𝑛𝑢−𝑇𝑥_𝑛. (3.9) Consequently, by the condition (C1) and (C2), we also have lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0. Next, we will show that limsup_𝑛→∞𝑣−𝑧₀,𝑥_𝑛+1−𝑧₀≤0,where𝑧₀=𝑃_Γ𝑣. (3.10)
To show this, we can choose a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞⟨𝑣−𝑧₀,𝑇𝑥_𝑛−𝑧₀⟩=lim_𝑘→∞⟨𝑣−𝑧₀,𝑇𝑥_𝑛𝑘−𝑧₀⟩. (3.11)
As {𝑥_𝑛} is bounded, there exists a subsequence {𝑥_𝑛𝑘} which converges weakly to 𝑧. We may assume without loss of generality that 𝑥_𝑛𝑘⇀𝑧. Since ‖𝑇𝑥_𝑛−𝑥_𝑛‖→0, we obtain 𝑇𝑥_𝑛𝑘⇀𝑧 as 𝑘→∞. By Lemma 2.3, we obtain that 𝑧∈Fix(𝑇)=Γ.
Now from (2.4), observe that limsup_𝑛→∞𝑣−𝑧₀,𝑥_𝑛−𝑧₀=limsup_𝑛→∞⟨𝑣−𝑧₀,𝑇𝑥_𝑛−𝑧₀⟩=lim_𝑘→∞⟨𝑣−𝑧₀,𝑇𝑥_𝑛𝑘−𝑧₀⟩=⟨𝑣−𝑧₀,𝑧−𝑧₀⟩≤0. (3.12) Therefore, we compute 𝑥_𝑛+1−𝑧₀²=⟨𝛼_𝑛𝑣+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑇𝑥_𝑛−𝑧₀,𝑥_𝑛+1−𝑧₀⟩=𝛼_𝑛⟨𝑣−𝑧₀,𝑥_𝑛+1−𝑧₀⟩+𝛽_𝑛⟨𝑥_𝑛−𝑧₀,𝑥_𝑛+1−𝑧₀⟩+𝛾_𝑛⟨𝑇𝑥_𝑛−𝑧₀,𝑥_𝑛+1−𝑧₀⟩≤𝛼_𝑛⟨𝑣−𝑧₀,𝑥_𝑛+1−𝑧₀1⟩+2𝛽_𝑛𝑥_𝑛−𝑧₀²+𝑥_𝑛+1−z₀²+12𝛾_𝑛𝑥_𝑛−𝑧₀²+𝑥_𝑛+1−z₀²≤𝛼_𝑛⟨𝑣−𝑧₀,𝑥_𝑛+1−𝑧₀1⟩+21−𝛼_𝑛𝑥_𝑛−𝑧₀²+𝑥_𝑛+1−z₀² (3.13) which implies that 𝑥_𝑛+1−z₀²≤1−𝛼_𝑛𝑥_𝑛−𝑧₀²+2𝛼_𝑛⟨𝑣−𝑧₀,𝑥_𝑛+1−𝑧₀⟩. (3.14) Finally, by (3.12), (3.14), and Lemma 2.5, we conclude that {𝑥_𝑛} converges to 𝑧₀. This completes the proof.

Letting 𝛽_𝑛≡0 of iterative scheme (3.1) in Theorem 3.1, then we obtain the following corollary.

Corollary 3.2. For any 𝑢,𝑥₀∈𝐶, one defines the sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑃_𝐶𝐼−𝜉𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛,𝑛≥0, (3.15) where {𝛼_𝑛} is a sequence in [0,1]. Suppose that the SFP is consistent and 0<𝜉<(2/‖𝐴‖²).
Let {𝑥_𝑛} be defined as in (3.15). If the following assumptions are satisfied: (C1)lim_𝑛→∞𝛼_𝑛=0 but ∑^∞_𝑛=1𝛼_𝑛=∞, (C2)∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞. Then {𝑥_𝑛} converges to a solution of the SFP (1.1).

Remark 3.3. Theorem 3.1 and Corollary 3.2 extend and improve the result of Xu [4] from weak to strong convergence theorems by using the modified Halpern's iterative scheme.

Acknowledgment

The first author was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0033/2554).