A Strongly Convergent Method for the Split Feasibility Problem

Abstract

The purpose of this paper is to introduce and analyze a strongly convergent method which combined regularized method, with extragradient method for solving the split feasibility problem in the setting of infinite-dimensional Hilbert spaces. Note that the strong convergence point is the minimum norm solution of the split feasibility problem.

1. Introduction

In 1994, Censor and Elfving [1] first introduced the split feasibility problem, (SFP) in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. A number of image reconstruction problems can be formulated as the SFP; see, for example, [2] and the references therein. Recently, it is found that the SFP can also be applied to study the intensity-modulated radiation therapy; see, for example, [3–5] and the references therein. Very recently, Xu [6] considered the SFP in the framework of infinite-dimensional Hilbert spaces. In this setting, the SFP is formulated as finding a point 𝑥^∗ with the property 𝑥^∗∈𝐶,𝐴𝑥^∗∈𝑄, (1.1) where 𝐶 and 𝑄 are two closed convex subsets of two Hilbert spaces 𝐻₁ and 𝐻₂, respectively, and 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator. We use Γ to denote the solution set of the (SFP), that is, Γ={𝑥∈𝐶∶𝐴𝑥∈𝑄}. (1.2) Assume that the (SFP) is consistent. A special case of the (SFP) is the convexly constrained linear inverse problem [7] in the finite-dimensional Hilbert spaces 𝑥^∗∈𝐶,𝐴𝑥^∗=𝑏, (1.3) which has extensively been investigated by using the Landweber iterative method [8]: with𝑥₀arbitraryand𝑛=0,1,…,let𝑥_𝑛+1=𝑥_𝑛+𝛾𝐴^𝑇𝑏−𝐴𝑥_𝑛. (1.4) Comparatively, the SFP has received much less attention so far, due to the complexity resulted from the set 𝑄. Therefore, whether various versions of the projected Landweber iterative method can be extended to solve the SFP remains an interesting open topic. For example, it is yet not clear if the dual approach to (1.2) of [9] can be extended to the SFP.

The original algorithm introduced in [1] involves the computation of the inverse 𝐴⁻¹: 𝑥_𝑘+1=𝐴⁻¹𝑃_𝑄𝑃_𝐴(𝐶)𝐴𝑥_𝑘,𝑘≥0, (1.5) where 𝐶,𝑄⊂𝑅^𝑛 are closed convex sets, 𝐴 a full rank 𝑛×𝑛 matrix, and 𝐴(𝐶)={𝑦∈𝑅^𝑛∣𝑦=𝐴𝑥,𝑥∈𝐶}, and thus does not become popular. A more popular algorithm that solves the (SFP) seems to be the 𝐶𝑄 algorithm of Byrne ([2, 10]). The 𝐶𝑄 algorithm only involves the computations of the projections 𝑃_𝐶 and 𝑃_𝑄 onto the sets 𝐶 and 𝑄, respectively, and is therefore implementable in the case where 𝑃_𝐶 and 𝑃_𝑄 have closed-form expressions (e.g., 𝐶 and 𝑄 are the closed balls or half-spaces). There are a large number of references on the 𝐶𝑄 method for the (SFP) in the literature, see, for instance, [11–24]. It remains, however, a challenge how to implement the 𝐶𝑄 algorithm in the case where the projections 𝑃_𝐶 and/or 𝑃_𝑄 fail to have closed-form expressions though theoretically we can prove (weak) convergence of the algorithm.

Very recently, Xu [6] gave a continuation of the study on the 𝐶𝑄 algorithm and its convergence. He applied Mann’s algorithm to the SFP and proposed an averaged 𝐶𝑄 algorithm, which was proved to be weakly convergent to a solution of the SFP. He derived a weak convergence result, which shows that for suitable choices of iterative parameters (including the regularization), the sequence of iterative solutions can converge weakly to an exact solution of the SFP.

Note that 𝑥∈Γ means that there is an 𝑥∈𝐶 such that 𝐴𝑥−𝑥^∗=0 for some 𝑥^∗∈𝑄. This motivates us to consider the distance function 𝑑(𝐴𝑥,𝑥^∗)=‖𝐴𝑥−𝑥^∗‖ and the minimization problem min_{𝑥∈𝐶,𝑥∗∈𝑄}12𝐴𝑥−𝑥^∗2. (1.6) Minimizing with respect to 𝑥^∗∈𝑄 first makes us consider the minimization: min_𝑥∈𝐶𝑓(𝑥)1∶=2𝐴𝑥−𝑃_𝑄𝐴𝑥². (1.7) However, (1.7) is, in general, ill-posed. So regularization is needed. We consider Tikhonov’s regularization min_𝑥∈𝐶𝑓_𝛼1∶=2(𝐼−𝑃_𝑄)𝐴𝑥²+12𝛼‖𝑥‖², (1.8) where 𝛼>0 is the regularization parameter. We can compute the gradient ∇𝑓_𝛼 of 𝑓_𝛼 as ∇𝑓_𝛼=∇𝑓(𝑥)+𝛼𝐼=𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼𝐼. (1.9) Define a Picard iteration 𝑥^𝛼_𝑛+1=𝑃_𝐶𝐼−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴+𝛼𝐼𝑥^𝛼_𝑛 (1.10) Xu [6] has has shown that if the (SFP) (1.1) is consistent, then as 𝑛→∞, 𝑥^𝛼_𝑛→𝑥_𝛼 and consequently the strong lim_𝛼→0𝑥_𝛼 exists and is the minimum-norm solution of the (SFP). Note that (1.10) is a double-step iteration. Xu [6] further suggested a single-step regularized method: 𝑥_𝑛+1=𝑃_𝐶𝐼−𝛾_𝑛∇𝑓_𝛼𝑛𝑥_𝑛=𝑃_𝐶1−𝛼_𝑛𝛾_𝑛𝑥_𝑛−𝛾_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛. (1.11) Xu proved that the sequence {𝑥_𝑛} generated by (1.11) converges in norm to the minimum-norm solution of the (SFP) provided the parameters {𝛼_𝑛} and {𝛾_𝑛} satisfy the following conditions:(i)𝛼_𝑛→0 and 0<𝛾_𝑛≤(𝛼_𝑛/(‖𝐴‖²+𝛼_𝑛));(ii)∑_𝑛𝛼_𝑛𝛾_𝑛=∞;(iii)(|𝛾_𝑛+1−𝛾_𝑛|+𝛾_𝑛|𝛼_𝑛+1−𝛼_𝑛|)/(𝛼_𝑛+1𝛾_𝑛+1)²→0.Motivated by the ideas of extragradient method and Xu’s regularization, Ceng et al. [25] presented the following extragradient method with regularization for finding a common element of the solution set of the split feasibility problem and the set Fix(𝑆) of fixed points of a nonexpansive mapping 𝑆: 𝑥₀=𝑥∈𝐻₁𝑦chosenarbitrarily,_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓𝑥_𝑛+𝛼_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛∇𝑓𝑦_𝑛+𝛼_𝑛𝑦_𝑛,𝑛≥0. (1.12) Ceng et al. only obtained the weak convergence of the algorithm (1.12).

The purpose of this paper is to further introduce and analyze a strongly convergent method, which combined regularized method with extragradient method for solving the split feasibility problem in the setting of infinite-dimensional Hilbert spaces. Note that the strong convergence point is the minimum norm solution of the split feasibility problem.

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.1) We will use Fix(𝑇) to denote the set of fixed points of 𝑇, that is, Fix(𝑇)={𝑥∈𝐶∶𝑥=𝑇𝑥}. A mapping 𝑇∶𝐶→𝐶 is said to be 𝜈-inverse strongly monotone (𝜈-ism) if there exists a constant 𝜈>0 such that 𝑥−𝑦,𝑇𝑥−𝑇𝑦≥𝜈𝑇𝑥−𝑇𝑦²,∀𝑥,𝑦∈𝐶. (2.2) Recall that the (nearest point or metric) projection from 𝐻 onto 𝐶, denoted 𝑃_𝐶, assigns, to each 𝑥∈𝐻, the unique point 𝑃_𝐶(𝑥)∈𝐶 with the property 𝑥−𝑃_𝐶(𝑥)=inf𝑥−𝑦∶𝑦∈𝐶. (2.3) It is well known that the metric projection 𝑃_𝐶 of 𝐻 onto 𝐶 has the following basic properties:(a)‖𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻;(b)⟨𝑥−𝑦,𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)⟩≥‖𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)‖² for every 𝑥,𝑦∈𝐻;(c)⟨𝑥−𝑃_𝐶(𝑥),𝑦−𝑃_𝐶(𝑥)⟩≤0 for all 𝑥∈𝐻, 𝑦∈𝐶.

Especially, 2𝑃_𝐶−𝐼 is nonexpansive.

Let 𝐾 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝐹∶𝐾→𝐻 be a monotone mapping. The variational inequality problem (VIP) is to find 𝑥∈𝐾 such that 𝐹𝑥,𝑦−𝑥≥0,∀𝑦∈𝐾. (2.4) The solution set of the VIP is denoted by VIP(𝐾,𝐹). It is well known that 𝑥∈VI(𝐾,𝐹)⟺𝑥=𝑃_𝐾(𝑥−𝜆𝐹𝑥),∀𝜆>0. (2.5) A set-valued mapping 𝑇∶𝐻→2^𝐻 is called monotone if for all 𝑥,𝑦∈𝐻, 𝑓∈𝑇𝑥, and 𝑔∈𝑇𝑦 imply 𝑥−𝑦,𝑓−𝑔≥0. (2.6) A monotone mapping 𝑇∶𝐻→2^𝐻 is called maximal if its graph 𝐺(𝑇) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping 𝑇 is maximal if and only if, for (𝑥,𝑓)∈𝐻×𝐻, ⟨𝑥−𝑦,𝑓−𝑔⟩≥0 for every (𝑦,𝑔)∈𝐺(𝑇) implies 𝑓∈𝑇𝑥. Let 𝐹∶𝐾→𝐻 be a monotone and 𝑘-Lipschitz continuous mapping, and let 𝑁_𝐾𝑣 be the normal cone to 𝐾 at 𝑣∈𝐾, that is, 𝑁_𝐾𝑣={𝑤∈𝐻∶⟨𝑣−𝑢,𝑤⟩≥0,∀𝑢∈𝐾}. (2.7) Define 𝑇𝑣=𝐹𝑣+𝑁_𝐾𝑣,if𝑣∈𝐾,∅,if𝑣∉𝐾. (2.8) Then, 𝑇 is maximal monotone and 0∈𝑇𝑣 if and only if 𝑣∈VI(𝐾,𝐹); see [21] for more details.

Next we adopt the following notation(i)𝑥_𝑛→𝑥 means that 𝑥_𝑛 converges strongly to 𝑥;(ii)𝑥_𝑛⇀𝑥 means that 𝑥_𝑛 converges weakly to 𝑥;(iii)𝜔_𝑤(𝑥_𝑛)∶={𝑥∶∃𝑥_𝑛𝑗⇀𝑥} is the weak 𝜔-limit set of the sequence {𝑥_𝑛}.

Lemma 2.1 (see [6]). We have the following assertions.(a)𝑇 is nonexpansive if and only if the complement 𝐼−𝑇 is (1/2)-ism.(b)If 𝑆 is 𝜈-ism, then for 𝛾>0, 𝛾𝑆 is (𝜈/𝛾)-ism.(c)𝑆 is averaged if and only if the complement 𝐼−𝑆 is 𝜈-ism for some 𝜈>(1/2).(d)If 𝑆 and 𝑇 are both averaged, then the product (composite) 𝑆𝑇 is averaged.

Lemma 2.2 (see [26]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋, and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. Suppose that 𝑥_𝑛+1=1−𝛽_𝑛𝑦_𝑛+𝛽_𝑛𝑥_𝑛 (2.9) for all 𝑛≥0 and limsup_𝑛→∞𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (2.10) Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

Lemma 2.3 (see [27]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛, (2.11) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that(1)∑^∞_𝑛=1𝛾_𝑛=∞; (2)limsup_𝑛→∞(𝛿_𝑛/𝛾_𝑛)≤0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞.Then lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

Let 𝐻₁ and 𝐻₂ be two infinite-dimensional Hilbert spaces. Let 𝐶 and 𝑄 be two nonempty closed and convex subset of 𝐻₁ and 𝐻₂, respectively. Let 𝐴∶𝐻₁→𝐻₂ be a bounded linear operator. In this section, we will devote to solve the SFP (1.1). First, we need the following propositions.

Proposition 3.1 (see [6, 25]). Given 𝑥^∗∈𝐻₁, then the following statements are equivalent.(i)𝑥^∗ solves the SFP;(ii)𝑥^∗ solves the fixed equation 𝑥^∗=𝑃_𝐶(𝐼−𝜆∇𝑓)𝑥^∗=𝑃_𝐶(𝐼−𝜆𝐴^∗(𝐼−𝑃_𝑄)𝐴)𝑥^∗;(iii)𝑥^∗ solves the variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that ∇𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (3.1) where ∇𝑓=𝐴^∗(𝐼−𝑃_𝑄)𝐴 and 𝐴^∗ is the adjoint of 𝐴.

Proposition 3.2 (see [28]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let the mapping 𝐵∶𝐶→𝐻 be 𝛼-inverse strongly monotone and 𝛾>0 a constant. Then, we have (𝐼−𝛾𝐵)𝑥−(𝐼−𝛾𝐵)𝑦²≤𝑥−𝑦²+𝛾𝛾−2𝛼𝐵𝑥−𝐵𝑦²,∀𝑥,𝑦∈𝐶. (3.2) In particular, if 0≤𝛾≤2𝛼, then 𝐼−𝛾𝐵 is nonexpansive.

Proposition 3.3 (see [6]). We have the following conclusions:(i)𝐴^∗(𝐼−𝑃_𝑄)𝐴 is Lipschitz continuous with Lipschitz constant ‖𝐴‖²;(ii)𝐴^∗(𝐼−𝑃_𝑄)𝐴 is (1/‖𝐴‖²)-ism,(iii)𝐼−𝛾𝐴^∗(𝐼−𝑃_𝑄)𝐴 is nonexpansive for all 𝛾∈(0,2/‖𝐴‖²).

Algorithm 3.4. For given 𝑥₀∈𝐶 arbitrarily, define a sequence {𝑥_𝑛} iteratively by 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝛼_𝑛𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+𝜇𝑦_𝑛−𝑥_𝑛,𝑛≥0, (3.3) where {𝛼_𝑛}⊂(0,1) is a sequence, 𝜆∈[𝑎,𝑏]⊂(0,2/‖𝐴‖²) and 𝜇∈(0,1) are two constants such that (𝜆/𝜇)≤(2/‖𝐴‖²).

Theorem 3.5. Suppose that Γ≠∅. Assume lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. Then the sequence {𝑥_𝑛} generated by (3.3) converges strongly to 𝑥∈𝑃_Γ(0), which is the minimum norm element in Γ.

Proof. Note that the conditions 𝛼_𝑛→0 and 𝜆∈(0,2/‖𝐴‖²). We deduce 𝛼_𝑛<1−(𝜆‖𝐴‖²/2) for enough large 𝑛. Without loss of generality, we may assume that, for all 𝑛∈ℕ, 𝛼_𝑛<1−(𝜆‖𝐴‖²/2), that is, 𝜆/(1−𝛼_𝑛)∈(0,2/‖𝐴‖²).
Pick up any 𝑥^∗∈Γ. From Proposition 3.1, we have 𝑥^∗=𝑃_𝐶[𝑥^∗−𝛿𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑥^∗] for any 𝛿>0. Thus, 𝑥^∗=𝑃_𝐶𝑥^∗−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗=𝑃_𝐶𝛼_𝑛𝑥^∗+1−𝛼_𝑛𝑥^∗−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗,∀𝑛≥0. (3.4)
From (3.3) and (3.4), we have 𝑦_𝑛−𝑥^∗=𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑥^∗=𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑃_𝐶𝛼_𝑛𝑥^∗+1−𝛼_𝑛𝑥^∗−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗≤𝛼_𝑛−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑥^∗−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗. (3.5) From Propositions 3.1 and 3.2, we get that 𝐼−(𝜆/(1−𝛼_𝑛))𝐴^∗(𝐼−𝑃_𝑄)𝐴 is nonexpansive. It follows that 𝑦_𝑛−𝑥^∗≤𝛼_𝑛𝑥^∗+1−𝛼_𝑛𝜆𝐼−1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝜆𝐼−1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗≤𝛼_𝑛𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥^∗. (3.6)
Thus, 𝑥_𝑛+1−𝑥^∗=𝑃_𝐶𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+𝜇𝑦_𝑛−𝑥_𝑛−𝑥^∗=𝑃_𝐶1−𝜇𝑥_𝑛+𝜇𝑦_𝑛−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛−𝑃_𝐶1−𝜇𝑥^∗+𝜇𝑥^∗−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗≤1−𝜇𝑥_𝑛−𝑥^∗+𝜇𝑦_𝑛−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛−𝑥^∗−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗≤1−𝜇𝑥_𝑛−𝑥^∗+𝜇𝑦_𝑛−𝑥^∗≤1−𝜇𝑥_𝑛−𝑥^∗+𝜇𝛼_𝑛𝑥^∗+𝜇1−𝛼_𝑛𝑥_𝑛−𝑥^∗=1−𝜇𝛼_𝑛𝑥_𝑛−𝑥^∗+𝜇𝛼_𝑛𝑥^∗≤max𝑥^∗,𝑥₀−𝑥^∗. (3.7) Hence, {𝑥_𝑛} is bounded.
Set 𝑆=2𝑃_𝐶−𝐼. Note that 𝑆 is nonexpansive. Thus, we can rewrite 𝑥_𝑛+1 in (3.3) as 𝑥_𝑛+1=𝐼+𝑆21−𝜇𝑥_𝑛+𝜇𝑦_𝑛−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛=1−𝜇2𝑥_𝑛+𝜇2𝑦_𝑛−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+12𝑆1−𝜇𝑥_𝑛+𝜇𝑦_𝑛−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛=1−𝜇2𝑥_𝑛+1+𝜇2𝑧_𝑛, (3.8) where 𝑧_𝑛=𝜇𝑦_𝑛−𝜆/𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+𝑆1−𝜇𝑥_𝑛+𝜇𝑦_𝑛−𝜆/𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛.1+𝜇 (3.9) It follows that 𝑧_𝑛+1−𝑧_𝑛=𝜇𝑦_𝑛+1−𝜆/𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+1+𝑆𝑥1+𝜇1−𝜇_𝑛+1+𝜇𝑦_𝑛+1−𝜆/𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+1−𝜇𝑦1+𝜇_𝑛−𝜆/𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+𝑆𝑥1+𝜇1−𝜇_𝑛+𝜇𝑦_𝑛−𝜆/𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛≤𝜇1+𝜇𝑦1+𝜇^𝑛+1−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+1−𝑦_𝑛−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+1𝑆𝑥1+𝜇1−𝜇_𝑛+1+𝜇𝑦_𝑛+1−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+1−𝑆1−𝜇𝑥_𝑛+𝜇𝑦_𝑛−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛≤𝜇𝑦1+𝜇^𝑛+1−𝑦_𝑛+1𝑥1+𝜇1−𝜇_𝑛+1−𝑥_𝑛+𝜇𝑦_𝑛+1−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛+1−𝑦_𝑛−𝜆𝜇𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛≤𝜇𝑦1+𝜇^𝑛+1−𝑦_𝑛+1−𝜇𝑥1+𝜇^𝑛+1−𝑥_𝑛+𝜇𝑦1+𝜇^𝑛+1−𝑦_𝑛≤2𝜇𝑦1+𝜇^𝑛+1−𝑦_𝑛+1−𝜇𝑥1+𝜇^𝑛+1−𝑥_𝑛. (3.10) By (3.3), we have 𝑦_𝑛+1−𝑦_𝑛=𝑃_𝐶1−𝛼_𝑛+1𝑥_𝑛+1−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛+1−𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛≤1−𝛼_𝑛+1𝑥_𝑛+1−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛+1−1−𝛼_𝑛𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛≤𝑥_𝑛+1−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛+1−𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛+𝛼_𝑛+1𝑥_𝑛+1+𝛼_𝑛𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1𝑥_𝑛+1+𝛼_𝑛𝑥_𝑛. (3.11) Hence, we deduce 𝑧_𝑛+1−𝑧_𝑛≤2𝜇𝑥1+𝜇^𝑛+1−𝑥_𝑛+1−𝜇𝑥1+𝜇^𝑛+1−𝑥_𝑛+𝛼_𝑛+1𝑥_𝑛+1+𝛼_𝑛𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1𝑥_𝑛+1+𝛼_𝑛𝑥_𝑛. (3.12) Therefore, limsup_𝑛→∞𝑧_𝑛+1−𝑧_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.13) By Lemma 2.2, we obtain lim_𝑛→∞𝑧_𝑛−𝑥_𝑛=0. (3.14) Hence, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1+𝜇2𝑧_𝑛−𝑥_𝑛=0. (3.15) From (3.5), (3.7), Proposition 3.2, and the convexity of the norm, we deduce 𝑥_𝑛+1−𝑥^∗2≤1−𝜇𝑥_𝑛−𝑥^∗2+𝜇𝑦_𝑛−𝑥^∗2≤1−𝜇𝑥_𝑛−𝑥^∗2+𝜇𝛼_𝑛−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑥^∗−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗2≤𝜇𝛼_𝑛𝑥^∗2+1−𝛼_𝑛×𝜆𝐼−1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝜆𝐼−1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗2+1−𝜇𝑥_𝑛−𝑥^∗2≤1−𝛼_𝑛𝜇×𝑥_𝑛−𝑥^∗2+𝜆1−𝛼_𝑛𝜆1−𝛼_𝑛−2‖𝐴‖²𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗2+1−𝜇𝑥_𝑛−𝑥^∗2+𝛼_𝑛𝜇𝑥^∗2≤𝛼_𝑛𝑥^∗2+𝑥_𝑛−𝑥^∗2𝑏+𝜇𝑎1−𝛼_𝑛−2‖𝐴‖²𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗2. (3.16) Therefore, we have 2𝜇𝑎‖𝐴‖²−𝑏1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗2≤𝛼_𝑛𝑥^∗2+𝑥_𝑛−𝑥^∗2−𝑥_𝑛+1−𝑥^∗2≤𝛼_𝑛𝑥^∗2+𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗×𝑥_𝑛−𝑥_𝑛+1. (3.17) Since 𝛼_𝑛→0 and ‖𝑥_𝑛−𝑥_𝑛+1‖→0 as 𝑛→∞, we obtain liminf_𝑛→∞𝜇𝑎((2/‖𝐴‖²)− (𝑏/(1−𝛼_𝑛)))>0. Thus, we have lim_𝑛→∞𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗=0. (3.18)
By the property (b) of the metric projection 𝑃_𝐶, we have 𝑦_𝑛−𝑥^∗2=𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑃_𝐶𝑥^∗−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗2≤1−𝛼_𝑛𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑥^∗−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗,𝑦_𝑛−𝑥^∗=12𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑥^∗−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗−𝛼_𝑛𝑥_𝑛²+𝑦_𝑛−𝑥^∗2−1−𝛼_𝑛𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑥^∗−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗−𝑦_𝑛−𝑥^∗2≤12𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑥^∗−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗2+2𝛼_𝑛𝑥_𝑛𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑥^∗−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗−𝛼_𝑛𝑥_𝑛+𝑦_𝑛−𝑥^∗2−𝑥_𝑛−𝑦_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗−𝛼_𝑛𝑥_𝑛²≤12(𝑥_𝑛−𝜆𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑥_𝑛)−(𝑥^∗−𝜆𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑥^∗)²+𝛼_𝑛𝑀+𝑦_𝑛−𝑥^∗2−𝑥_𝑛−𝑦_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗−𝛼_𝑛𝑥_𝑛²≤12𝑥_𝑛−𝑥^∗2+𝛼_𝑛𝑀+𝑦_𝑛−𝑥^∗2−𝑥_𝑛−𝑦_𝑛²+2𝛼_𝑛𝑥_𝑛,𝑥_𝑛−𝑦_𝑛+2𝜆𝑥_𝑛−𝑦_𝑛,𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗−𝜆(𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑥_𝑛−𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑥^∗)+𝛼_𝑛𝑥_𝑛²≤12𝑥_𝑛−𝑥^∗2+𝛼_𝑛𝑀+𝑦_𝑛−𝑥^∗2−𝑥_𝑛−𝑦_𝑛²+2𝛼_𝑛𝑥_𝑛𝑥_𝑛−𝑦_𝑛+2𝜆𝑥_𝑛−𝑦_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗, (3.19) where 𝑀>0 is some constant such that sup_𝑛2𝑥_𝑛𝑥_𝑛−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑥^∗−𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗−𝛼_𝑛𝑥_𝑛≤𝑀. (3.20) It follows that 𝑦_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2+𝛼_𝑛𝑀−𝑥_𝑛−𝑦_𝑛²+2𝛼_𝑛𝑥_𝑛𝑥_𝑛−𝑦_𝑛+2𝜆𝑥_𝑛−𝑦_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗, (3.21) and hence 𝑥_𝑛+1−𝑥^∗2≤1−𝜇𝑥_𝑛−𝑥^∗2+𝜇𝑦_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2+𝛼_𝑛𝑀−𝜇𝑥_𝑛−𝑦_𝑛²+2𝛼_𝑛𝑥_𝑛𝑥_𝑛−𝑦_𝑛+2𝜆𝑥_𝑛−𝑦_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗, (3.22) which implies that 𝜇𝑥_𝑛−𝑦_𝑛²≤𝑥_𝑛−𝑥^∗+𝑥_𝑛+1−𝑥^∗𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝑀+2𝛼_𝑛𝑥_𝑛𝑥_𝑛−𝑦_𝑛+2𝜆𝑥_𝑛−𝑦_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗. (3.23) Since 𝛼_𝑛→0,‖𝑥_𝑛−𝑥_𝑛+1‖→0, and ‖𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑥_𝑛−𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑥^∗‖→0, we derive lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (3.24)
Next we show that limsup_𝑛→∞𝑥,𝑥−𝑦_𝑛≤0, (3.25) where 𝑥=𝑃_Γ(0). To show it, we choose a subsequence {𝑦_𝑛𝑖} of {𝑦_𝑛} such that limsup_𝑛→∞𝑥,𝑥−𝑦_𝑛=lim_𝑖→∞𝑥,𝑥−𝑦_𝑛𝑖. (3.26) Since {𝑥_𝑛} is bounded, we have that {𝑦_𝑛} is also bounded. As {𝑦_𝑛𝑖} is bounded, we have that a subsequence {𝑦_𝑛𝑖𝑗} of {𝑦_𝑛𝑖} converges weakly to 𝑧.
Next we show that 𝑧∈Γ. We define a mapping 𝑇 by 𝐴𝑇𝑣=^∗𝐼−𝑃_𝑄𝐴𝑣+𝑁_𝐶𝑣,𝑣∈𝐶,∅,𝑣∉𝐶. (3.27) Then 𝑇 is maximal monotone. Let (𝑣,𝑤)∈𝐺(𝑇). Since 𝑤−𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑣∈𝑁_𝐶𝑣 and 𝑦_𝑛∈𝐶, we have ⟨𝑣−𝑦_𝑛,𝑤−𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑣⟩≥0. On the other hand, from 𝑦_𝑛=𝑃_𝐶[(1−𝛼_𝑛)𝑥_𝑛−𝜆𝐴^∗(𝐼−𝑃_𝑄)𝐴𝑥_𝑛], we have 𝑣−𝑦_𝑛,𝑦_𝑛−1−𝛼_𝑛𝑥_𝑛+𝜆𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛≥0, (3.28) that is, 𝑣−𝑦_𝑛,𝑦_𝑛−𝑥_𝑛𝜆+𝐴𝑥_𝑛+𝛼_𝑛𝜆𝑥_𝑛≥0. (3.29) Therefore, we have 𝑣−𝑦_𝑛𝑖,𝑤≥𝑣−𝑦_𝑛𝑖≥,𝐴𝑣𝑣−𝑦_𝑛𝑖,𝐴^∗𝐼−𝑃_𝑄−𝐴𝑣𝑣−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝜆+𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑖+𝛼_𝑛𝑖𝜆𝑥_𝑛𝑖=𝑣−𝑦_𝑛𝑖,𝐴^∗𝐼−𝑃_𝑄𝐴𝑣−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑖−𝑦_𝑛𝑖−𝑥_𝑛𝑖𝜆−𝛼_𝑛𝑖𝜆𝑥_𝑛𝑖=𝑣−𝑦_𝑛𝑖,𝐴^∗𝐼−𝑃_𝑄𝐴𝑣−𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛𝑖−𝑣−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝜆+𝛼_𝑛𝑖𝜆𝑥_𝑛𝑖+𝑣−𝑦_𝑛𝑖,𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛𝑖−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑖≥𝑣−𝑦_𝑛𝑖,𝐴^∗𝐼−𝑃_𝑄𝐴𝑦_𝑛𝑖−𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛𝑖−𝑣−𝑦_𝑛𝑖,𝑦_𝑛𝑖−𝑥_𝑛𝑖𝜆+𝛼_𝑛𝑖𝜆𝑥_𝑛𝑖. (3.30) Noting that 𝛼_𝑛𝑖→0, ‖𝑦_𝑛𝑖−𝑥_𝑛𝑖‖→0, and 𝐴^∗(𝐼−𝑃_𝑄)𝐴 is Lipschitz continuous, we deduce from above ⟨𝑣−𝑧,𝑤⟩≥0. (3.31) Since 𝑇 is maximal monotone, we have 𝑧∈𝑇⁻¹(0) and hence 𝑧∈VI(𝐶,𝐴^∗(𝐼−𝑃_𝑄)𝐴)=Γ. Therefore, limsup_𝑛→∞𝑥,𝑥−𝑦_𝑛=lim_𝑖→∞𝑥,𝑥−𝑦_𝑛𝑖=⟨𝑥,𝑥−𝑧⟩≤0. (3.32) By the property (b) of metric projection 𝑃_𝐶, we have 𝑦_𝑛−𝑥²=𝑃_𝐶1−𝛼_𝑛𝑥_𝑛−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝑃_𝐶𝛼_𝑛𝑥+1−𝛼_𝑛𝜆𝑥−1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥²≤𝛼_𝑛(−𝑥)+1−𝛼_𝑛𝑥_𝑛−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝜆𝑥−1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥,𝑦_𝑛−𝑥≤𝛼_𝑛𝑥,𝑥−𝑦_𝑛+1−𝛼_𝑛×𝑥_𝑛−𝜆1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛−𝜆𝑥−1−𝛼_𝑛𝐴^∗𝐼−𝑃_𝑄𝐴𝑥𝑦_𝑛−𝑥≤𝛼_𝑛𝑥,𝑥−𝑦_𝑛+1−𝛼_𝑛𝑥_𝑛−𝑥𝑦_𝑛−𝑥≤𝛼_𝑛𝑥,𝑥−𝑦_𝑛+1−𝛼_𝑛2𝑥_𝑛−𝑥²+𝑦_𝑛−𝑥². (3.33) Hence 𝑦_𝑛−𝑥²≤1−𝛼_𝑛𝑥_𝑛−𝑥²+2𝛼_𝑛𝑥,𝑥−𝑦_𝑛. (3.34) Therefore, 𝑥_𝑛+1−𝑥²≤1−𝜇𝑥_𝑛−𝑥²+𝜇𝑦_𝑛−𝑥²≤1−𝜇𝛼_𝑛𝑥_𝑛−𝑥²+2𝜇𝛼_𝑛𝑥,𝑥−𝑦_𝑛. (3.35) We apply Lemma 2.3 to the last inequality to deduce that 𝑥_𝑛→𝑥. This completes the proof.

Remark 3.6. Now it is wellknown that the Korpelevich’s extragradient method has only weak convergence in the setting of infinite-dimensional Hilbert spaces. But our algorithm (3.3) which is similar to the Korpelevich’s method, has strong convergence in the setting of infinite-dimensional Hilbert spaces.

Remark 3.7. Algorithm (1.12) has only weak convergence for solving SFP. Our algorithm with strong convergence solves SFP under some weaker assumptions.

Acknowledgment

The research of N. Shahzad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.