General Split Feasibility Problems in Hilbert Spaces

Abstract

Introducing a general split feasibility problem in the setting of infinite-dimensional Hilbert spaces, we prove that the sequence generated by the purposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.

1. Introduction

Let 𝐻 and 𝐾 be infinite-dimensional real Hilbert spaces, and let 𝐴∶𝐻→𝐾 be a bounded linear operator. Let {𝐶_𝑖}^𝑝_𝑖=1 and {𝑄_𝑖}^𝑟_𝑖=1 be the families of nonempty closed convex subsets of 𝐻 and 𝐾, respectively.

(a) The convex feasibility problem (CFP) is formulated as the problem of finding a point 𝑥^⋆ with the property: 𝑥^⋆∈^𝑝_𝑖=1𝐶_𝑖.(1)

(b) The split feasibility problem (SEP) is formulated as the problem of finding a point 𝑥^⋆ with the property: 𝑥^⋆∈𝐶,𝐴𝑥^⋆∈𝑄,(2) where 𝐶 and 𝑄 are nonempty closed convex subsets of 𝐻 and 𝐾, respectively.

(c) The multiple-set split feasibility problem (MSSFP) is formulated as the problem of finding a point 𝑥^⋆ with the property: 𝑥^⋆∈^𝑝_𝑖=1𝐶_𝑖,𝐴𝑥^⋆∈^𝑟_𝑖=1𝑄_𝑖.(3) Note that (MSSFP) reduces to (SEP) if we take 𝑝=𝑟=1.

There is a considerable investigation on CFP in view of its applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [1]. The split feasibility problem SFP in the setting of finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [2] for modelling inverse problems which arise from phase retrievals and in medical image reconstruction [3]. Since then, a lot of work has been done on finding a solution of SFP and MSSFP; see, for example, [2–25]. Recently, it is found that the SFP can also be applied to study the intensity-modulated radiation therapy; see, for example, [6, 16] and the references therein. Very recently, Xu [8] considered the SFP in the setting of infinite-dimensional Hilbert spaces.

The original algorithm given in [2] involves the computation of the inverse 𝐴⁻¹ provided it exists. In [8], Xu studied some algorithm and its convergence. In particular, he applied Mann’s algorithm to the SFP and purposed an algorithm which is proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained. In [7], Wang and Xu purposed the following cyclic algorithm to solve MSSFP: 𝑥_𝑛+1=𝑃_𝐶[𝑛]𝑥_𝑛+𝛾𝐴^∗𝑃_𝑄[𝑛]−𝐼𝐴𝑥_𝑛,(4) where [𝑛]∶=𝑛(mod𝑝), (mod function take values in {1,2,…,𝑝}), and 𝛾∈(0,2/‖𝐴‖²). They show that the sequence {𝑥_𝑛} convergence weakly to a solution of MSSFP provided the solution exists. To study strong convergence to a solution of MSSFP, first we introduce a general form of the split feasibility problem for infinite families as follows.

(d) General split feasibility problem (GSFP) is to find a point 𝑥^∗ such that 𝑥^⋆∈^∞_𝑖=1𝐶_𝑖,𝐴𝑥^⋆∈^∞_𝑖=1𝑄_𝑖.(5) We denote by Ω the solution set of GSFP.

In this paper, using viscosity iterative method defined by Moudafi [21], we propose an algorithm for finding the solutions of the general split feasibility problem in a Hilbert space. We establish the strong convergence of the proposed algorithm to a solution of GSFP.

2. Preliminaries

Throughout the paper, we denote by 𝐻 a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Let {𝑥_𝑛} be a sequence in 𝐻 and 𝑥∈𝐻. Weak convergence of {𝑥_𝑛} to 𝑥 is denoted by 𝑥_𝑛⇀𝑥, and strong convergence by 𝑥_𝑛→𝑥. Let 𝐶 be a closed and a convex subset of 𝐻. For every point 𝑥∈𝐻, there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥. This point satisfies 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑦∈𝐶.(6) The operator 𝑃_𝐶 is called the metric projection or the nearest point mapping of 𝐻 onto 𝐶. The metric projection 𝑃_𝐶 is characterized by the fact that 𝑃_𝐶(𝑥)∈𝐶 and 𝑦−𝑃_𝐶(𝑥),𝑥−𝑃_𝐶(𝑥)≤0,∀𝑥∈𝐻,𝑦∈𝐶.(7) Recall that a mapping 𝑇∶𝐶→𝐶 is called nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶.(8) It is well known that 𝑃_𝐶 is a nonexpansive mapping. It is also known that 𝐻 satisfies Opial’s condition, that is, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥, the inequality liminf_𝑛→∞𝑥_𝑛−𝑥<liminf_𝑛→∞𝑥_𝑛−𝑦(9) holds for every 𝑦∈𝐻 with 𝑦≠𝑥.

Lemma 1. Let 𝐻 be a Hilbert space. Then, for all 𝑥,𝑦∈𝐻𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦.(10)

Lemma 2 (see [22]). Let 𝐻 be a Hilbert space, and let {𝑥_𝑛} be a sequence in 𝐻. Then, for any given sequence {𝜆_𝑛}^∞_𝑛=1⊂(0,1) with ∑^∞_𝑛=1𝜆_𝑛=1 and for any positive integer 𝑖,𝑗 with 𝑖<𝑗, ^∞_𝑛=1𝜆_𝑛𝑥_𝑛²≤^∞_𝑛=1𝜆_𝑛𝑥_𝑛²−𝜆_𝑖𝜆_𝑗𝑥_𝑖−𝑥_𝑗².(11)

Lemma 3 (see [23]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛾_𝑛𝛿_𝑛+𝛽_𝑛,𝑛≥0,(12) where {𝛾_𝑛}, {𝛽_𝑛}, and {𝛿_𝑛} satisfy the following conditions: (i)𝛾_𝑛⊂[0,1], ∑^∞_𝑛=1𝛾_𝑛=∞,(ii)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞_𝑛=1|𝛾_𝑛𝛿_𝑛|<∞,(iii)𝛽_𝑛≥0 for all 𝑛≥0 with ∑^∞_𝑛=0𝛽_𝑛<∞.Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 4 (see [24]). Let {𝑡_𝑛} be a sequence of real numbers such that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑡_𝑛𝑖<𝑡_𝑛𝑖+1 for all 𝑖∈ℕ. Then, there exists a nondecreasing sequence {𝜏(𝑛)}⊂ℕ such that 𝜏(𝑛)→∞, and the following properties are satisfied by all (sufficiently large) numbers 𝑛∈ℕ: 𝑡_𝜏(𝑛)≤𝑡_𝜏(𝑛)+1,𝑡_𝑛≤𝑡_𝜏(𝑛)+1.(13) In fact 𝜏(𝑛)=max𝑘≤𝑛∶𝑡_𝑘<𝑡_𝑘+1.(14)

Lemma 5 (demiclosedness principle [25]). Let 𝐶 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping such that Fix(𝑇)≠∅. Then, 𝑇 is demiclosed on 𝐶, that is, if 𝑦_𝑛⇀𝑧∈𝐶, and (𝑦_𝑛−𝑇𝑦_𝑛)→𝑦, then (𝐼−𝑇)𝑧=𝑦.

3. Main Result

In the following result, we propose an algorithm and prove that the sequence generated by the proposed method converges strongly to a solution of the GSFP.

Theorem 6. Let 𝐻 and 𝐾 be real Hilbert spaces, and let 𝐴∶𝐻→𝐾 be a bounded linear operator. Let {𝐶_𝑖}^∞_𝑖=1 and {𝑄_𝑖}^∞_𝑖=1 be the families of nonempty closed convex subsets of 𝐻 and 𝐾, respectively. Assume that GSFP (5) has a nonempty solution set Ω. Suppose that 𝑓 is a self 𝑘-contraction mapping of 𝐻, and let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐻 as 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑓𝑥_𝑛+^∞_𝑖=1𝛾_𝑛,𝑖𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛,𝑛≥0,(15) where 𝛼_𝑛+𝛽_𝑛+∑^∞_𝑖=1𝛾_𝑛,𝑖=1. If the sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛,𝑖}, and {𝜆_𝑛,𝑖} satisfy the following conditions: (i)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞,(ii)for each 𝑖∈ℕ, liminf_𝑛𝛼_𝑛𝛾_𝑛,𝑖>0, (iii)for each 𝑖∈ℕ, {𝜆_𝑛,𝑖}⊂(0,2/‖𝐴‖²) and 0<liminf_𝑛→∞𝜆_𝑛,𝑖≤limsup_𝑛→∞𝜆_𝑛,𝑖<2/‖𝐴‖²,then, the sequence {𝑥_𝑛} converges strongly to 𝑥^⋆∈Ω, where 𝑥^⋆=𝑃_Ω𝑓(𝑥^⋆).

Proof. First, we show that {𝑥_𝑛} is bounded. In fact, let 𝑧∈Ω. Since {𝜆_𝑛,𝑖}⊂(0,2/‖𝐴‖²), the operators 𝑃_𝐶𝑖(𝐼−𝜆_𝑛,𝑖𝐴^∗(𝐼−𝑃_𝑄𝑖)𝐴) are nonexpansive, and hence we have 𝑥_𝑛+1=𝛼−𝑧_𝑛𝑥_𝑛+𝛽_𝑛𝑓𝑥_𝑛+^∞_𝑖=1𝛾_𝑛,𝑖𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑧≤𝛼_𝑛𝑥_𝑛−𝑧+𝛽_𝑛𝑓𝑥_𝑛+−𝑧_∞𝑖=1𝛾_𝑛,𝑖𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑧≤𝛼_𝑛𝑥_𝑛−𝑧+𝛽_𝑛𝑓𝑥_𝑛+−𝑧_∞𝑖=1𝛾_𝑛,𝑖𝑥_𝑛≤−𝑧1−𝛽_𝑛𝑥_𝑛−𝑧+𝛽_𝑛𝑓𝑥_𝑛≤−𝑧1−𝛽_𝑛𝑥_𝑛−𝑧+𝛽_𝑛𝑓𝑥_𝑛−𝑓(𝑧)+𝛽_𝑛𝑓(𝑧)≤−𝑧1−𝛽_𝑛𝑥_𝑛−𝑧+𝛽_𝑛𝑘𝑥_𝑛−𝑧+𝛽_𝑛𝑓(𝑧)≤((𝛽−𝑧1−1−𝑘))_𝑛𝑥_𝑛+()𝛽−𝑧1−𝑘^𝑛𝑓(𝑧)𝑥1−𝑘−𝑧≤max_𝑛,1−𝑧𝑓(𝑧)⋮𝑥1−𝑘−𝑧≤max₀,1−𝑧𝑓(𝑧),1−𝑘−𝑧(16) which implies that {𝑥_𝑛} is bounded, and we also obtain that {𝑓(𝑥_𝑛)} is bounded. Next, we show that for each 𝑖∈ℕ, lim_𝑛→∞𝑥_𝑛−𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛=0.(17) By using Lemma 2, for every 𝑧∈Ω and 𝑖∈ℕ, we have that 𝑥_𝑛+1−𝑧²=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑓𝑥_𝑛+^∞_𝑗=1𝛾_𝑛,𝑗𝑃_𝐶𝑗𝐼−𝜆_𝑛,𝑗𝐴^∗𝐼−𝑃_𝑄𝑗𝐴𝑥_𝑛−𝑧²≤𝛼_𝑛𝑥_𝑛−𝑧²+𝛽_𝑛𝑓𝑥_𝑛−𝑧²+^∞_𝑗=1𝛾_𝑛,𝑗𝑃_𝐶𝑗𝐼−𝜆_𝑛,𝑗𝐴^∗𝐼−𝑃_𝑄𝑗𝐴𝑥_𝑛−𝑧²−𝛼_𝑛𝛾_𝑛,𝑖𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥_𝑛²≤𝛼_𝑛𝑥_𝑛−𝑧²+𝛽_𝑛𝑓𝑥_𝑛−𝑧²+^∞_𝑗=1𝛾_𝑛,𝑗𝑥_𝑛−𝑧²−𝛼_𝑛𝛾_𝑛,𝑖𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥_𝑛²≤1−𝛽_𝑛𝑥_𝑛−𝑧²+𝛽_𝑛𝑓𝑥_𝑛−𝑧²−𝛼_𝑛𝛾_𝑛,𝑖𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥_𝑛².(18) Hence, for each 𝑖∈ℕ, we have 𝛼_𝑛𝛾_𝑛,𝑖𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑧²−𝑥_𝑛+1−𝑧²+𝛽_𝑛𝑓𝑥_𝑛−𝑧².(19)
Next, we show that there exists a unique 𝑥^⋆∈Ω such that 𝑥^⋆=𝑃_Ω𝑓(𝑥^⋆). We observe that for each 𝑛≥0, 𝑥^⋆∈Ω solves the GSFP (5) if and only if 𝑥^⋆ solves the fixed point equation 𝑥^⋆=𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥^⋆,𝑖∈ℕ,(20) that is, the solution sets of fixed point equation (20) and GSFP (5) are the same (see for details [8]). Note that if {𝜆_𝑛,𝑖}⊂(0,2/‖𝐴‖²), then the operators 𝑃_𝐶𝑖(𝐼−𝜆_𝑛,𝑖𝐴^∗(𝐼−𝑃_𝑄𝑖)𝐴) are nonexpansive. Since the fixed point set of nonexpansive operators is closed and convex, the projection onto the solution set Ω is well defined whenever Ω≠∅. We observe that 𝑃_Ω(𝑓) is a contraction of 𝐻 into itself. Indeed, since 𝑃_Ω is nonexpansive, 𝑃_Ω𝑓(𝑥)−𝑃_Ω𝑓𝑦≤𝑓(𝑥)−𝑓𝑦≤𝑘𝑥−𝑦.(21) Hence, there exists a unique element 𝑥^⋆∈Ω such that 𝑥^⋆=𝑃_Ω𝑓(𝑥^⋆).
In order to prove that 𝑥_𝑛→𝑥^⋆ as 𝑛→∞, we consider two possible cases.
Case 1. Assume that {‖𝑥_𝑛−𝑥^⋆‖} is a monotone sequence. In other words, for 𝑛₀ large enough, {‖𝑥_𝑛−𝑥^⋆‖}_𝑛≥𝑛0 is either nondecreasing or nonincreasing. Since ‖𝑥_𝑛−𝑥^⋆‖ is bounded we have ‖𝑥_𝑛−𝑥^⋆‖ is convergent. Since lim_𝑛→∞𝛽_𝑛=0 and {𝑓(𝑥_𝑛)} is bounded, from (19) we get that lim_𝑛→∞𝛼_𝑛𝛾_𝑛,𝑖𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥_𝑛²=0.(22) By assuming that liminf_𝑛𝛼_𝑛𝛾_𝑛,𝑖>0, we obtain lim_𝑛→∞𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥_𝑛=0,∀𝑖∈ℕ.(23) Now, we show that limsup_𝑛→∞𝑓𝑥^⋆−𝑥^⋆,𝑥_𝑛−𝑥^⋆≤0.(24) To show this inequality, we choose a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that lim_𝑘→∞⟨𝑓𝑥^⋆−𝑥^⋆,𝑥_𝑛𝑘−𝑥^⋆⟩=limsup_𝑛→∞⟨𝑓𝑥^⋆−𝑥^⋆,𝑥_𝑛−𝑥^⋆⟩.(25) Since {𝑥_𝑛𝑘} is bounded, there exists a subsequence {𝑥_𝑛𝑘𝑗} of {𝑥_𝑛𝑘} which converges weakly to 𝑤. Without loss of generality, we can assume that 𝑥_𝑛𝑘⇀𝑤 and 𝜆_𝑛,𝑖→𝜆_𝑖∈(0,2/‖𝐴‖²) for each 𝑖∈ℕ. From (23), we have 𝑃_𝐶𝑖𝐼−𝜆_𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥_𝑛≤𝑃_𝐶𝑖𝐼−𝜆_𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛+𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥_𝑛≤𝐼−𝜆_𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛+𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥_𝑛≤𝜆_𝑖−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛+𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥_𝑛→0as𝑛→∞.(26) Notice that for each 𝑖∈ℕ, 𝑃_𝐶𝑖(𝐼−𝜆_𝑖𝐴^∗(𝐼−𝑃_𝑄𝑖)𝐴) is nonexpansive. Thus, from Lemma 5, we have 𝑤∈Ω. Therefore, it follows that limsup_𝑛→∞𝑓𝑥^⋆−𝑥^⋆,𝑥_𝑛−𝑥^⋆=lim_𝑘→∞𝑓𝑥^⋆−𝑥^⋆,𝑥_𝑛𝑘−𝑥^⋆=𝑓𝑥^⋆−𝑥^⋆,𝑤−𝑥^⋆≤0.(27) Finally, we show that 𝑥_𝑛→𝑃_Ω𝑓(𝑥^⋆). Applying Lemma 1, we have that 𝑥_𝑛+1−𝑥^⋆2=𝛼_𝑛𝑥_𝑛−𝑥^⋆+^∞_𝑖=1𝛾_𝑛,𝑖𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛−𝑥^⋆2+2𝛽_𝑛𝑓𝑥_𝑛−𝑥^⋆,𝑥_𝑛+1−𝑥^⋆≤1−𝛽_𝑛²𝑥_𝑛−𝑥^⋆2+2𝛽_𝑛𝑓𝑥_𝑛−𝑓𝑥^⋆,𝑥_𝑛+1−𝑥^⋆+2𝛽_𝑛𝑓𝑥^⋆−𝑥^⋆,𝑥_𝑛+1−𝑥^⋆≤1−𝛽_𝑛²𝑥_𝑛−𝑥^⋆2+2𝛽_𝑛𝑘𝑥_𝑛−𝑥^⋆𝑥_𝑛+1−𝑥^⋆+2𝛽_𝑛𝑓𝑥^⋆−𝑥^⋆,𝑥_𝑛+1−𝑥^⋆≤1−𝛽_𝑛²𝑥_𝑛−𝑥^⋆2+𝛽_𝑛𝑘𝑥_𝑛−𝑥^⋆2+𝑥_𝑛+1−𝑥^⋆2+2𝛽_𝑛𝑓𝑥^⋆−𝑥^⋆,𝑥_𝑛+1−𝑥^⋆.(28) This implies that 𝑥_𝑛+1−𝑥^⋆2≤1−𝛽_𝑛²+𝛽_𝑛𝑘1−𝛽_𝑛𝑘𝑥_𝑛−𝑥^⋆2+2𝛽_𝑛1−𝛽_𝑛𝑘𝑓𝑥^⋆−𝑥^⋆,𝑥_𝑛+1−𝑥^⋆=1−2𝛽_𝑛+𝛽_𝑛𝑘1−𝛽_𝑛𝑘𝑥_𝑛−𝑥^⋆2+𝛽²_𝑛1−𝛽_𝑛𝑘𝑥_𝑛−𝑥^⋆2+2𝛽_𝑛1−𝛽_𝑛𝑘𝑓(𝑧)−𝑥^⋆,𝑥_𝑛+1−𝑥^⋆≤2()𝛽1−1−𝑘_𝑛1−𝛽_𝑛𝑘𝑥_𝑛−𝑥^⋆2+2(1−𝑘)𝛽_𝑛1−𝛽_𝑛𝑘𝛽_𝑛𝑀2()+11−𝑘𝑓𝑥1−𝑘^⋆−𝑥^⋆,𝑥_𝑛+1−𝑥^⋆≤1−𝜂_𝑛𝑥_𝑛−𝑥^⋆2+𝜂_𝑛𝛿_𝑛,(29) where 𝛿_𝑛=𝛽_𝑛𝑀2()+11−𝑘𝑓𝑥1−𝑘^⋆−𝑥^⋆,𝑥_𝑛+1−𝑥^⋆,(30)𝑀=sup{‖𝑥_𝑛−𝑥^⋆‖²∶𝑛≥0} and 𝜂_𝑛=2(1−𝑘)𝛽_𝑛/(1−𝛽_𝑛𝑘). It is easy to see that 𝜂_𝑛→0, ∑^∞_𝑛=1𝜂_𝑛=∞ and limsup_𝑛→∞𝛿_𝑛≤0. Hence, by Lemma 3, the sequence {𝑥_𝑛} converges strongly to 𝑥^⋆=𝑃_Ω𝑓(𝑥^⋆).
Case 2. Assume that {‖𝑥_𝑛−𝑥^⋆‖} is not a monotone sequence. Then, we can define an integer sequence {𝜏(𝑛)} for all 𝑛≥𝑛₀ (for some 𝑛₀ large enough) by 𝜏(𝑛)=max𝑘∈ℕ;𝑘≤𝑛∶𝑥_𝑘−𝑥^⋆<𝑥_𝑘+1−𝑥^⋆.(31) Clearly, 𝜏(𝑛) is a nondecreasing sequence such that 𝜏(𝑛)→∞ as 𝑛→∞ and for all 𝑛≥𝑛₀, 𝑥_𝜏(𝑛)−𝑥^⋆<𝑥_𝜏(𝑛)+1−𝑥^⋆.(32) From (19), we obtain that lim_𝑛→∞𝑃_𝐶𝑖𝐼−𝜆_{𝜏(𝑛),𝑖}𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝜏(𝑛)−𝑥_𝜏(𝑛)=0.(33) Following an argument similar to that in Case 1, we have limsup_𝑛→∞𝑓𝑥^⋆−𝑥^⋆,𝑥_𝜏(𝑛)+1−𝑥^⋆≤0.(34) And by similar argument, we have 𝑥_𝜏(𝑛)+1−𝑥^⋆2≤1−𝜂_𝜏(𝑛)𝑥_𝜏(𝑛)−𝑥^⋆2+𝜂_𝜏(𝑛)𝛿_𝜏(𝑛),(35) where 𝜂_𝜏(𝑛)→0, ∑^∞_𝑛=1𝜂_𝜏(𝑛)=∞ and limsup_𝑛→∞𝛿_𝜏(𝑛)≤0. Hence, by Lemma 3, we obtain lim_𝑛→∞‖𝑥_𝜏(𝑛)−𝑥^⋆‖=0 and lim_𝑛→∞‖𝑥_𝜏(𝑛)+1−𝑥^⋆‖=0. Now, from Lemma 4, we have 0≤𝑥_𝑛−𝑥^⋆≤max𝑥_𝜏(𝑛)−𝑥^⋆,𝑥_𝑛−𝑥^⋆≤𝑥_𝜏(𝑛)+1−𝑥^⋆.(36) Therefore, {𝑥_𝑛} converges strongly to 𝑥^⋆=𝑃_Ω𝑓(𝑥^⋆).

For finite collections we have the following consequence of Theorem 6.

Theorem 7. Let 𝐻 and 𝐾 be real Hilbert spaces, and let 𝐴∶𝐻→𝐾 be a bounded linear operator. Let{𝐶_𝑖}^𝑝_𝑖=1 be a family of nonempty closed convex subsets in 𝐻, and let {𝑄_𝑖}^𝑝_𝑖=1 be a family of nonempty closed convex subsets in 𝐾. Assume that MSSFP has a nonempty solution set Ω. Let 𝑢 be an arbitrary element in 𝐻, and let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐻 and 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+𝛽_𝑛𝑢+^𝑝_𝑖=1𝛾_𝑛,𝑖𝑃_𝐶𝑖𝐼−𝜆_𝑛,𝑖𝐴^∗𝐼−𝑃_𝑄𝑖𝐴𝑥_𝑛,𝑛≥0,(37) where 𝛼_𝑛+𝛽_𝑛+∑^𝑝_𝑖=1𝛾_𝑛,𝑖=1. If the sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛,𝑖}, and {𝜆_𝑛,𝑖} satisfy the following conditions: (i)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞,(ii)for all 𝑖∈{1,2,…,𝑝}, liminf_𝑛𝛼_𝑛𝛾_𝑛,𝑖>0, (iii)for all 𝑖∈{1,2,…,𝑝}, {𝜆_𝑛,𝑖}⊂(0,2/‖𝐴‖²) and 0<liminf_𝑛→∞𝜆_𝑛,𝑖≤limsup_𝑛→∞𝜆_𝑛,𝑖<2‖𝐴‖²,(38)then the sequence {𝑥_𝑛} converges strongly to 𝑥^⋆∈Ω, where 𝑥^⋆=𝑃_Ω𝑢.

Acknowledgment

This paper is funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors therefore, acknowledge with thanks the DSR for technical and financial support.