Multiple-Set Split Feasibility Problems for Asymptotically Strict Pseudocontractions

Abstract

In this paper, we introduce an iterative method for solving the multiple-set split feasibility problems for asymptotically strict pseudocontractions in infinite-dimensional Hilbert spaces, and, by using the proposed iterative method, we improve and extend some recent results given by some authors.

1. Introduction

The split feasibility problem (SFP) in finite dimensional spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. Recently, it has been found that the SFP can also be used in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning [3–5].

The split feasibility problem in an infinite dimensional Hilbert space can be found in [2, 4, 6–8].

Throughout this paper, we always assume that 𝐻₁,𝐻₂ are real Hilbert spaces, “→”, “⇀” are denoted by strong and weak convergence, respectively.

The purpose of this paper is to introduce and study the following multiple-set split feasibility problem for asymptotically strict pseudocontraction (MSSFP) in the framework of infinite-dimensional Hilbert spaces. Find 𝑥^∗∈𝐶 such that𝐴𝑥^∗∈𝑄, (1.1) where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator, {𝑆_𝑖} and {𝑇_𝑖}, 𝑖=1,2,…,𝑀, are the families of mappings 𝑆_𝑖∶𝐻₁→𝐻₁ and 𝑇_𝑖∶𝐻₂→𝐻₂, respectively, 𝐶∶=⋂^𝑀_𝑖=1𝐹(𝑆_𝑖) and 𝑄∶=⋂^𝑀_𝑖=1𝐹(𝑇_𝑖), where 𝐹(𝑆_𝑖)={𝑥_𝑖∈𝐻₁∶𝑆_𝑖𝑥_𝑖=𝑥_𝑖} and 𝐹(𝑇_𝑖)={𝑦_𝑖∈𝐻₂∶𝑇_𝑖𝑦_𝑖=𝑦_𝑖} denote the sets of fixed points of 𝑆_𝑖 and 𝑇_𝑖, respectively. In the sequel, we use Γ to denote the set of solutions of the problem (MSSFP), that is,Γ={𝑥∈𝐶∶𝐴𝑥∈𝑄}. (1.2)

2. Preliminaries

We first recall some definitions, notations, and conclusions which will be needed in proving our main results.

Let 𝐸 be a Banach space. A mapping 𝑇∶𝐸→𝐸 is said to be demiclosed at origin if, for any sequence {𝑥_𝑛}⊂𝐸 with 𝑥_𝑛⇀𝑥^∗ and ‖(𝐼−𝑇)𝑥_𝑛‖→0, we have 𝑥^∗=𝑇𝑥^∗. A Banach space 𝐸 is said to have Opial’s property if, for any sequence {𝑥_𝑛} with 𝑥_𝑛⇀𝑥^∗, we haveliminf_𝑛→∞𝑥_𝑛−𝑥^∗<liminf_𝑛→∞𝑥_𝑛−𝑦 (2.1) for all 𝑦∈𝐸 with 𝑦≠𝑥^∗.

Remark 2.1. It is well known that each Hilbert space possesses Opial’s property.

Definition 2.2. Let 𝐻 be a real Hilbert space.
(1)A mapping 𝐺∶𝐻→𝐻 is called a (𝛾,{𝑘_𝑛})-asymptotically strict pseudocontraction if there exists a constant 𝛾∈[0,1) and a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 such that 𝐺^𝑛𝑥−𝐺^𝑛𝑦²≤𝑘_𝑛𝑥−𝑦²+𝛾(𝐼−𝐺^𝑛)𝑥−(𝐼−𝐺^𝑛)𝑦²,∀𝑥,𝑦∈𝐻. (2.2) Especially, if 𝑘_𝑛=1 for each 𝑛≥1 in (2.2) and there exists 𝛾∈[0,1) such that 𝐺𝑥−𝐺𝑦²≤𝑥−𝑦²+𝛾(𝐼−𝐺)𝑥−(𝐼−𝐺)𝑦²,∀𝑥,𝑦∈𝐻, (2.3) then 𝐺∶𝐻→𝐻 is called a 𝛾-strict pseudocontraction.(2)A mapping 𝐺∶𝐻→𝐻 is said to be uniformly 𝐿-Lipschitzian if there exists a constant 𝐿>0 such that 𝐺^𝑛𝑥−𝐺^𝑛𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐻,𝑛≥1. (2.4)(3)A mapping 𝐺∶𝐻→𝐻 is said to be semicompact if, for any bounded sequence {𝑥_𝑛}⊂𝐻 with lim_𝑛→∞‖𝑥_𝑛−𝐺𝑥_𝑛‖=0, there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖 converges strongly to a point 𝑥^∗∈𝐻.

Now, we give one example of the (𝛾,{𝑘_𝑛})-asymptotically strict pseudocontraction mapping.

Example 2.3. Let 𝐵 be the unit ball in a Hilbert space 𝑙², and define a mapping 𝑇∶𝐵→𝐵 by 𝑇=𝑥₁,𝑥₂,…=0,𝑥²₁,𝑎₂𝑥₂,𝑎₃𝑥₃,…, (2.5) where {𝑎_𝑖} is a sequence in (0,1) such that Π^∞_𝑖=2𝑎_𝑖=1/2. It is proved in Goebel and Kirk [9] that(a)‖𝑇𝑥=𝑇𝑦‖≤2‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐵,(b)‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤2Π^𝑛_𝑗=2𝑎_𝑗 for all 𝑛≥2 and 𝑥,𝑦∈𝐵.Denote by 𝑘₁^1/2=2, 𝑘_𝑛^1/2=2Π^𝑛_𝑗=2𝑎_𝑗(𝑛≥2) and 𝛾∈[0,1). Then, we have lim_𝑛→∞𝑘_𝑛=lim_𝑛→∞2^𝑛_𝑗=2𝑎_𝑗²𝑇=1,_𝑛𝑥−𝑇^𝑛𝑦²≤𝑘_𝑛𝑥−𝑦²+𝛾𝑥−𝑦−(𝑇^𝑛𝑥−𝑇^𝑛𝑦)²,∀𝑛≥1,𝑥,𝑦∈𝐵, (2.6) and so the mapping 𝑇 is a (𝛾,{𝑘_𝑛})-asymptotically strict pseudocontraction.

Remark 2.4. (1) If we put 𝛾=0 in (2.2), then the mapping 𝐺∶𝐻→𝐻 is asymptotically nonexpansive.
(2) If we put 𝛾=0 in (2.3), then the mapping 𝐺∶𝐻→𝐻 is nonexpansive.
(3) Each (𝛾,{𝑘_𝑛})-asymptotically strict pseudocontraction and each 𝛾-strictly pseudocontraction both are demiclosed at origin [10].

Proposition 2.5. Let 𝐺∶𝐻→𝐻 be a (𝛾,{𝑘_𝑛})- asymptotically strict pseudocontraction. If 𝐹(𝐺)≠∅, then, for any 𝑞∈𝐹(𝐺) and 𝑥∈𝐻, the following inequalities hold and they are equivalent: 𝐺^𝑛𝑥−𝑞²≤𝑘_𝑛𝑥−𝑞²+𝛾𝑥−𝐺^𝑛𝑥²,𝑥−𝐺^𝑛𝑥,𝑥−𝑞≥1−𝛾2𝑥−𝐺^𝑛𝑥²−𝑘_𝑛−12𝑥−𝑞²,⟨𝑥−𝐺^𝑛𝑥,𝑞−𝐺^𝑛𝑥⟩≤1+𝛾2𝑥−𝐺^𝑛𝑥²+𝑘_𝑛−12𝑥−𝑞². (2.7)(2.8)(2.9)

Lemma 2.6 (see [11]). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝛿_𝑛} be sequences of nonnegative real numbers satisfying 𝑎_𝑛+1≤1+𝛿_𝑛𝑎_𝑛+𝑏_𝑛,∀𝑛≥1. (2.10) If ∑^∞_𝑖=1𝛿_𝑛<∞ and ∑^∞_𝑖=1𝑏_𝑛<∞, then the limit lim_𝑛→∞𝑎_𝑛 exists.

3. Multiple-Set Split Feasibility Problem

For solving the multiple-set split feasibility problem (1.1), let us assume that the following conditions are satisfied:(C1)𝐻₁ and 𝐻₂ are two real Hilbert spaces, 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator;(C2)𝑆_𝑖∶𝐻₁→𝐻₁, 𝑖=1,2,…,𝑀, is a uniformly 𝐿_𝑖-Lipschitzian and (𝛽_𝑖,{𝑘_𝑖,𝑛})-asymptotically strict pseudocontraction, and 𝑇_𝑖∶𝐻₂→𝐻₂, 𝑖=1,2,…,𝑀, is a uniformly 𝐿_𝑖-Lipschitzian and (𝜇_𝑖,{𝑘_𝑖,𝑛})-asymptotically strict pseudocontraction satisfying the following conditions:(a)𝐶∶=⋂^𝑀_𝑖=1𝐹(𝑆_𝑖)≠∅ and 𝑄∶=⋂^𝑀_𝑖=1𝐹(𝑇_𝑖)≠∅,(b)𝛽=max_{1≤𝑖≤𝑀}𝛽_𝑖<1 and 𝜇=max_{1≤𝑖≤𝑀}𝜇_𝑖<1,(c)𝐿∶=max_{1≤𝑖≤𝑀}𝐿_𝑖<∞ and 𝐿∶=max_{1≤𝑖≤𝑀}𝐿_𝑖<∞,(d)𝑘_𝑛=max_{1≤𝑖≤𝑀}{𝑘_𝑖,𝑛,𝑘_𝑖,𝑛} and ∑^∞_𝑛=1(𝑘_𝑛−1)<∞.

We are now in a position to give the following result.

Theorem 3.1. Let 𝐻₁, 𝐻₂, 𝐴, {𝑆_𝑖}, {𝑇_𝑖}, 𝐶, 𝑄, 𝛽, 𝜇, 𝐿, 𝐿, and {𝑘_𝑛} be the same as above. Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝐻₁𝑥chosenarbitrarily,_𝑛+1=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑆^𝑛_𝑛𝑢_𝑛,𝑢_𝑛=𝑥_𝑛+𝛾𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛,∀𝑛≥1, (3.1) where 𝑆^𝑛_𝑛=𝑆^𝑛_{𝑛(mod𝑀)}, 𝑇^𝑛_𝑛=𝑇^𝑛_{𝑛(mod𝑀)} for all 𝑛≥1, {𝛼_𝑛} is a sequence in [0,1], and 𝛾>0 is a constant satisfying the following conditions.(e)𝛼_𝑛∈(𝛿,1−𝛽) for all 𝑛≥1 and 𝛾∈(0,(1−𝜇)/‖𝐴‖²), where 𝛿∈(0,1−𝛽) is a positive constant.(1) If Γ≠∅, then the sequence {𝑥_𝑛} converges weakly to a point 𝑥^∗∈Γ.(2) In addition, if there exists a positive integer 𝑗 such that 𝑆_𝑗 is semicompact, then the sequences {𝑥_𝑛} and {𝑢_𝑛} both converge strongly to a point 𝑥^∗∈Γ.

Proof. (1) The proof is divided into 5 steps as follows.
Step 1. We first prove that, for any 𝑝∈Γ, the limit lim_𝑛→∞𝑥_𝑛−𝑝 (3.2) exists. In fact, since 𝑝∈Γ, 𝑝∈𝐶∶=⋂^𝑀_𝑖=1𝐹(𝑆_𝑖), and 𝐴𝑝∈𝑄∶=⋂^𝑀_𝑖=1𝐹(𝑇_𝑖). From (3.1) and (2.8), it follows that 𝑥_𝑛+1−𝑝²=𝑢_𝑛−𝑝−𝛼_𝑛(𝑢_𝑛−𝑆^𝑛_𝑛𝑢_𝑛)²=𝑢_𝑛−𝑝²−2𝛼_𝑛⟨𝑢_𝑛−𝑝,𝑢_𝑛−𝑆^𝑛_𝑛𝑢_𝑛⟩+𝛼²_𝑛𝑢_𝑛−𝑆^𝑛_𝑛𝑢_𝑛²≤𝑢_𝑛−𝑝²−𝛼_𝑛1−𝛽𝑢_𝑛−𝑆^𝑛_𝑛𝑢_𝑛²−𝑘_𝑛−1𝑢_𝑛−𝑝²+𝛼²_𝑛𝑢_𝑛−𝑆^𝑛_𝑛𝑢_𝑛²=1+𝛼_𝑛𝑘_𝑛−1𝑢_𝑛−𝑝²−𝛼_𝑛1−𝛽−𝛼_𝑛𝑢_𝑛−𝑆^𝑛_𝑛𝑢_𝑛². (3.3)
On the other hand, since 𝑢_𝑛−𝑝²=𝑥_𝑛−𝑝+𝛾𝐴^∗(𝑇^𝑛_𝑛−𝐼)𝐴𝑥_𝑛²=𝑥_𝑛−𝑝²+𝛾²𝐴^∗(𝑇^𝑛_𝑛−𝐼)𝐴𝑥_𝑛²+2𝛾𝑥_𝑛−𝑝,𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛,𝛾²𝐴^∗(𝑇^𝑛_𝑛−𝐼)𝐴𝑥_𝑛²=𝛾²⟨𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛,𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛⟩=𝛾²⟨𝐴𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛,𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛⟩≤𝛾²‖𝐴‖²𝑇^𝑛_𝑛𝐴𝑥_𝑛−𝐴𝑥_𝑛²,2𝛾⟨𝑥_𝑛−𝑝,𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛⟩=2𝛾⟨𝐴𝑥_𝑛−𝐴𝑝,𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛⟩=2𝛾⟨𝐴𝑥_𝑛−𝐴𝑝+𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛−𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛,𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛⟩=2𝛾𝑇^𝑛_𝑛𝐴𝑥_𝑛−𝐴𝑝,𝑇^𝑛_𝑛𝐴𝑥_𝑛−𝐴𝑥_𝑛−𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛². (3.4)(3.5)(3.6) Further, letting 𝑥=𝐴𝑥_𝑛, 𝐺^𝑛=𝑇^𝑛_𝑛, 𝑞=𝐴𝑝, 𝛾=𝜇 in (2.9) and noting 𝐴𝑝∈𝐹(𝑇_𝑛), it follows that ⟨𝑇^𝑛_𝑛𝐴𝑥_𝑛−𝐴𝑝,𝑇^𝑛_𝑛𝐴𝑥_𝑛−𝐴𝑥_𝑛⟩≤1+𝜇2(𝑇^𝑛_𝑛−𝐼)𝐴𝑥_𝑛²+𝑘_𝑛−12𝐴𝑥_𝑛−𝐴𝑝²≤1+𝜇2(𝑇^𝑛_𝑛−𝐼)𝐴𝑥_𝑛²+𝑘_𝑛−1‖𝐴‖²2𝑥_𝑛−𝑝². (3.7) Substituting (3.7) into (3.6) and simplifying it, we have 2𝛾𝑥_𝑛−𝑝,𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛≤𝛾𝜇−1(𝑇^𝑛_𝑛−𝐼)𝐴𝑥_𝑛²+𝑘_𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑝². (3.8) Substituting (3.5) and (3.8) into (3.4) and simplifying it, we have 𝑢_𝑛−𝑝²≤𝑥_𝑛−𝑝²+𝛾²‖𝐴‖²𝑇^𝑛_𝑛𝐴𝑥_𝑛−𝐴𝑥_𝑛²+𝛾𝜇−1(𝑇^𝑛_𝑛−𝐼)𝐴𝑥_𝑛²+𝑘_𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑝²=𝑥_𝑛−𝑝²−𝛾1−𝜇−𝛾‖𝐴‖²𝑇^𝑛_𝑛𝐴𝑥_𝑛−𝐴𝑥_𝑛²+𝑘_𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑝². (3.9) Again, substituting (3.9) into (3.3) and simplifying it, we have 𝑥_𝑛+1−𝑝²≤1+𝛼_𝑛𝑘_𝑛×𝑥−1_𝑛−𝑝²−𝛾1−𝜇−𝛾‖𝐴‖²𝑇^𝑛_𝑛𝐴𝑥_𝑛−𝐴𝑥_𝑛²+𝑘_𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑝²−𝛼_𝑛1−𝛽−𝛼_𝑛𝑢_𝑛−𝑆^𝑛_𝑛𝑢_𝑛²≤1+𝛼_𝑛𝑘_𝑛−1𝑥_𝑛−𝑝²−𝛾1−𝜇−𝛾‖𝐴‖²𝑇^𝑛_𝑛𝐴𝑥_𝑛−𝐴𝑥_𝑛²+1+𝛼_𝑛𝑘_𝑛−1𝑘_𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑝²−𝛼_𝑛1−𝛽−𝛼_𝑛𝑢_𝑛−𝑆^𝑛_𝑛𝑢_𝑛². (3.10) By the condition (e), we have 𝑥_𝑛+1−𝑝²≤1+𝛼_𝑛𝑘_𝑛−1𝑥_𝑛−𝑝²+1+𝛼_𝑛𝑘_𝑛−1𝑘_𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑝²≤1+𝐾𝑘_𝑛−1𝑥_𝑛−𝑝², (3.11) where 𝐾=sup_𝑛≥1𝛼_𝑛+1+𝛼_𝑛𝑘_𝑛−1𝛾‖𝐴‖²<∞. (3.12) By the condition (d), ∑_𝑛=1(𝑘_𝑛−1)<∞; hence, from Lemma 2.6, we know that the following limit exists: lim_𝑛→∞𝑥_𝑛−𝑝. (3.13)
Step 2. We will now prove that, for each 𝑝∈Γ, the limit lim_𝑛→∞𝑢_𝑛−𝑝 (3.14) exists. In fact, from (3.10) and (3.13), it follows that 𝛾1−𝜇−𝛾‖𝐴‖²(𝑇^𝑛_𝑛−𝐼)𝐴𝑥_𝑛²+𝛼_𝑛1−𝛽−𝛼_𝑛𝑢_𝑛−𝑆^𝑛_𝑛𝑢_𝑛²≤𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²+𝐾𝑘_𝑛−1𝑥_𝑛−𝑝²⟶0(𝑛⟶∞). (3.15) This together with the condition (e) implies that lim_𝑛→∞𝑢_𝑛−𝑆^𝑛_𝑛𝑢_𝑛=0,lim_𝑛→∞𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛=0. (3.16)(3.17) Therefore, it follows from (3.4), (3.13), and (3.17) that the limit lim_𝑛→∞‖𝑢_𝑛−𝑝‖ exists.
Step 3. Now, we prove that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0,lim_𝑛→∞𝑢_𝑛+1−𝑢_𝑛=0. (3.18)
In fact, it follows from (3.1) that 𝑥_𝑛+1−𝑥_𝑛=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑆^𝑛_𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛾𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^𝑛_𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^𝑛_𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^𝑛_𝑛𝑢_𝑛−𝑢_𝑛+𝛼_𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^𝑛_𝑛𝑢_𝑛−𝑢_𝑛+𝛼_𝑛𝛾A^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛=𝛾𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^𝑛_𝑛𝑢_𝑛−𝑢_𝑛. (3.19) In view of (3.16) and (3.17), we have lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.20)
Similarly, it follows from (3.1), (3.17), and (3.20) that 𝑢_𝑛+1−𝑢_𝑛=𝑥_𝑛+1+𝛾𝐴^∗𝑇^𝑛+1_𝑛+1−𝐼𝐴𝑥_𝑛+1−𝑥_𝑛+𝛾𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛾𝐴^∗𝑇^𝑛+1_𝑛+1−𝐼𝐴𝑥_𝑛+1+𝛾𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛⟶0(𝑛⟶∞). (3.21) The conclusion (3.18) is proved.
Step 4. Next, we prove that, for each 𝑗=1,2,…,𝑀, 𝑢_{𝑖𝑀+𝑗}−𝑆_𝑗𝑢_{𝑖𝑀+𝑗}⟶0,𝐴𝑥_{𝑖𝑀+𝑗}−𝑇_𝑗𝐴𝑥_{𝑖𝑀+𝑗}⟶0(𝑖⟶∞). (3.22) In fact, from (3.16), it follows that 𝜁_{𝑖𝑀+𝑗}∶=𝑢_{𝑖𝑀+𝑗}−𝑆_𝑗^{𝑖𝑀+𝑗}𝑢_{𝑖𝑀+𝑗}⟶0(𝑖⟶∞). (3.23) Since 𝑆_𝑗 is uniformly 𝐿_𝑗-Lipschitzian continuous, it follows from (3.18) and (3.23) that 𝑢_{𝑖𝑀+𝑗}−𝑆_𝑗𝑢_{𝑖𝑀+𝑗}≤𝑢_{𝑖𝑀+𝑗}−𝑆_𝑗^{𝑖𝑀+𝑗}𝑢_{𝑖𝑀+𝑗}+𝑆_𝑗^{𝑖𝑀+𝑗}𝑢_{𝑖𝑀+𝑗}−𝑆_𝑗𝑢_{𝑖𝑀+𝑗}≤𝜁_{𝑖𝑀+𝑗}+𝐿_𝑗𝑆_𝑗^{𝑖𝑀+𝑗−1}𝑢_{𝑖𝑀+𝑗}−𝑢_{𝑖𝑀+𝑗}≤𝜁_{𝑖𝑀+𝑗}+𝐿_𝑗𝑆_𝑗^{𝑖𝑀+𝑗−1}𝑢_{𝑖𝑀+𝑗}−𝑆_𝑗^{𝑖𝑀+𝑗−1}𝑢_{𝑖𝑀+𝑗−1}+𝑆_𝑗^{𝑖𝑀+𝑗−1}𝑢_{𝑖𝑀+𝑗−1}−𝑢_{𝑖𝑀+𝑗}≤𝜁_{𝑖𝑀+𝑗}+𝐿²_𝑗𝑢_{𝑖𝑀+𝑗}−𝑢_{𝑖𝑀+𝑗−1}+𝐿_𝑗𝑆_𝑗^{𝑖𝑀+𝑗−1}𝑢_{𝑖𝑀+𝑗−1}−𝑢_{𝑖𝑀+𝑗−1}+𝑢_{𝑖𝑀+𝑗−1}−𝑢_{𝑖𝑀+𝑗}≤𝜁_{𝑖𝑀+𝑗}+𝐿_𝑗1+𝐿_𝑗𝑢_{𝑖𝑀+𝑗}−𝑢_{𝑖𝑀+𝑗−1}+𝐿_𝑗𝜁_{𝑖𝑀+𝑗−1}⟶0(𝑖⟶∞). (3.24)
Similarly, for each 𝑗=1,2,…,𝑀, it follows from (3.17) that 𝜉_{𝑖𝑀+𝑗}∶=𝐴𝑥_{𝑖𝑀+𝑗}−𝑇_𝑗^{𝑖𝑀+𝑗}𝐴𝑥_{𝑖𝑀+𝑗}⟶0(𝑖⟶∞). (3.25) Since 𝑇_𝑗 is uniformly 𝐿_𝑗-Lipschitzian continuous, by the same way as above, from (3.18) and (3.25), we can also prove that 𝐴𝑥_{𝑖𝑀+𝑗}−𝑇_𝑗𝐴𝑥_{𝑖𝑀+𝑗}⟶0(𝑖⟶∞). (3.26)
Step 5. Finally, we prove that 𝑥_𝑛⇀𝑥^∗ and 𝑢_𝑛⇀𝑥^∗, which is a solution of the problem (MSSFP). In fact, since {𝑢_𝑛} is bounded, there exists a subsequence {𝑢_𝑛𝑖}⊂{𝑢_𝑛} such that 𝑢_𝑛𝑖⇀𝑥^∗∈𝐻₁. Hence, for any positive integer 𝑗=1,2,…,𝑀, there exists a subsequence {𝑛_𝑖(𝑗)}⊂{𝑛_𝑖} with 𝑛_𝑖(𝑗)(mod𝑀)=𝑗 such that 𝑢_{𝑛𝑖(𝑗)}⇀𝑥^∗. Again, from (3.22), it follows that 𝑢_{𝑛𝑖(𝑗)}−𝑆_𝑗𝑢_{𝑛𝑖(𝑗)}⟶0𝑛_𝑖𝑗⟶∞. (3.27) Since 𝑆_𝑗 is demiclosed at zero (see Remark 2.4), it follows that 𝑥^∗∈𝐹(𝑆_𝑗). By the arbitrariness of 𝑗=1,2,…,𝑀, we have 𝑥^∗∈𝐶∶=⋂^𝑀_𝑗=1𝐹(𝑆_𝑗).
Moreover, it follows from (3.1) and (3.17) that 𝑥_𝑛𝑖=𝑢_𝑛𝑖−𝛾𝐴^∗𝑇^𝑛𝑖_𝑛𝑖−𝐼𝐴𝑥_𝑛𝑖⇀𝑥^∗. (3.28) Since 𝐴 is a linear bounded operator, it follows that 𝐴𝑥_𝑛𝑖⇀𝐴𝑥^∗. For any positive integer 𝑘=1,2,…,𝑀, there exists a subsequence {𝑛_𝑖(𝑘)}⊂{𝑛_𝑖} with 𝑛_𝑖(𝑘)(mod𝑀)=𝑘 such that 𝐴𝑥_{𝑛𝑖(𝑘)}⇀𝐴𝑥^∗. In view of (3.22), we have 𝐴𝑥_{𝑛𝑖(𝑘)}−𝑇_𝑘𝐴𝑥_𝑛i(𝑘)⟶0𝑛_𝑖(𝑘)⟶∞. (3.29) Since 𝑇_𝑘 is demiclosed at zero, we have 𝐴𝑥^∗∈𝐹(𝑇_𝑘). By the arbitrariness of 𝑘=1,2,…,𝑀, it follows that 𝐴𝑥^∗∈𝑄∶=⋂^𝑀_𝑘=1𝐹(𝑇_𝑘). This together with 𝑥^∗∈𝐶 shows that 𝑥^∗∈Γ, that is, 𝑥^∗ is a solution to the problem (MSSFP).
Now, we prove that 𝑥_𝑛⇀𝑥^∗ and 𝑢_𝑛⇀𝑥^∗. In fact, assume that there exists another subsequence {𝑢_𝑛𝑗}⊂{𝑢_𝑛} such that 𝑢_𝑛𝑗⇀𝑦^∗∈Γ with 𝑦^∗≠𝑥^∗. Consequently, by virtue of (3.2) and Opial’s property of Hilbert space, we have liminf_{𝑛𝑖→∞}𝑢_𝑛𝑖−𝑥^∗<liminf_{𝑛𝑖→∞}𝑢_𝑛𝑖−𝑦^∗=lim_𝑛→∞𝑢_𝑛−𝑦^∗=lim_{𝑛𝑗→∞}𝑢_𝑛𝑗−𝑦^∗<liminf_{𝑛𝑗→∞}𝑢_𝑛𝑗−𝑥^∗=lim_𝑛→∞𝑢_𝑛−𝑥^∗=liminf_{𝑛𝑖→∞}𝑢_𝑛𝑖−𝑥^∗. (3.30) This is a contradiction. Therefore, 𝑢_𝑛⇀𝑥^∗. By using (3.1) and (3.17), we have 𝑥_𝑛=𝑢_𝑛−𝛾𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛⇀𝑥^∗. (3.31) Therefore, the conclusion (I) follows.
(2) Without loss of generality, we can assume that 𝑆₁ is semicompact. It follows from (3.27) that 𝑢_𝑛𝑖(1)−𝑆₁𝑢_𝑛𝑖(1)⟶0𝑛_𝑖(1)⟶∞. (3.32) Therefore, there exists a subsequence of {𝑢_𝑛𝑖(1)} (for the sake of convenience, we still denote it by {𝑢_𝑛𝑖(1)}) such that 𝑢_𝑛𝑖(1)→𝑢^∗∈𝐻. Since 𝑢_𝑛𝑖(1)⇀𝑥^∗, 𝑥^∗=𝑢^∗ and so 𝑢_𝑛𝑖(1)→𝑥^∗∈Γ. By virtue of (3.2), we know that lim_𝑛→∞𝑢_𝑛−𝑥^∗=0,lim_𝑛→∞𝑥_𝑛−𝑥^∗=0, (3.33) that is, {𝑢_𝑛} and {𝑥_𝑛} both converge strongly to the point 𝑥^∗∈Γ. This completes the proof.

If we put 𝛾=0 in Theorem 3.1, we can get the following.

Corollary 3.2. Let 𝐻, 𝐶, 𝐿 and {𝑘_𝑛} be the same as above and {𝑆_𝑖} a family of asymptotically nonexpansive mappings. Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝐻₁𝑥chosenarbitrarily,_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑆^𝑛_𝑛𝑥_𝑛,∀𝑛≥1, (3.34) where 𝑆^𝑛_𝑛=𝑆^𝑛_{𝑛(mod𝑀)} for all 𝑛≥1 and {𝛼_𝑛} is a sequence in [0,1] satisfying the following conditions.(e)𝛼_𝑛∈(𝛿,1−𝛽) for all 𝑛≥1, where 𝛿∈(0,1−𝛽) is a positive constant.(1) If Γ≠∅, then the sequence {𝑥_𝑛} converges weakly to a point 𝑥^∗∈Γ.(2) In addition, if there exists a positive integer 𝑗 such that 𝑆_𝑗 is semicompact, then the sequence {𝑥_𝑛} converges strongly to a point 𝑥^∗∈Γ.

The following theorem can be obtained from Theorem 3.1 immediately.

Theorem 3.3. Let 𝐻₁ and 𝐻₂ be two real Hilbert spaces, 𝐴∶𝐻₁→𝐻₂ a bounded linear operator, 𝑆_𝑖∶𝐻₁→𝐻₁, 𝑖=1,2,…,𝑀, a uniformly 𝐿_𝑖-Lipschitzian and 𝛽_𝑖-strict pseudocontraction, and 𝑇_𝑖∶𝐻₂→𝐻₂,𝑖=1,2…,𝑀, a uniformly 𝐿_𝑖-Lipschitzian and 𝜇_𝑖-strict pseudocontraction satisfying the following conditions: (a)𝐶∶=⋂^𝑀_𝑖=1𝐹(𝑆_𝑖)≠∅ and 𝑄∶=⋂^𝑀_𝑖=1𝐹(𝑇_𝑖)≠∅,(b)𝛽=max_{1≤𝑖≤𝑀}𝛽_𝑖<1 and 𝜇=sup_{1≤𝑖≤𝑀}𝜇_𝑖<1.Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝐻₁𝑥chosenarbitrarily,_𝑛+1=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑆_𝑛𝑢_𝑛,𝑢_𝑛=𝑥_𝑛+𝛾𝐴^∗𝑇_𝑛−𝐼𝐴𝑥_𝑛,∀𝑛≥1, (3.35) where 𝑆_𝑛=𝑆_{𝑛(mod𝑀)}, 𝑇_𝑛=𝑇_{𝑛(mod𝑀)}, {𝛼_𝑛} is a sequence in [0,1], and 0<𝛾<1 is a constant. If Γ≠∅ and the following condition is satisfied:(c)𝛼_𝑛∈(𝛿,1−𝛽) for all 𝑛≥1 and 𝛾∈(0,(1−𝜇)/‖𝐴‖²), where 𝛿∈(0,1−𝛽) is a constant,then the sequence {𝑥_𝑛} converges weakly to a point 𝑥^∗∈Γ. In addition, if there exists a positive integer 𝑗 such that 𝑆_𝑗 is semicompact, then the sequences {𝑥_𝑛} and {𝑢_𝑛} both converge strongly to the point 𝑥^∗.

Proof. By the same way as given in the proof of Theorem 3.1 and using the case of strict pseudocontraction with the sequence {𝑘_𝑛=1}, we can prove that, for each 𝑝∈Γ, the limits lim_𝑛→∞‖𝑥_𝑛−𝑝‖ and lim_𝑛→∞‖𝑢_𝑛−𝑝‖ exist, 𝑢_𝑛−𝑆_𝑛𝑢_𝑛→0,𝐴𝑥_𝑛−𝑇_𝑛𝐴𝑥_𝑛→0,𝑢_𝑛−𝑢_𝑛+1→0,𝑥_𝑛−𝑥_𝑛+1𝑥→0,_𝑛⇀𝑥^∗,𝑢_𝑛⇀𝑥^∗∈Γ. (3.36)
In addition, if there exists a positive integer 𝑗 such that 𝑆_𝑗 is semicompact, we can also prove that {𝑥_𝑛} and {𝑢_𝑛} both converge strongly to the point 𝑥^∗. This completes the proof.

If you put 𝑆_𝑖=𝑇_𝑖 or 𝑇_𝑖=𝐼 (: the identity mapping) for each 𝑖=1,2…,𝑀 in Theorem 3.3, then we have the following.

Corollary 3.4. Let 𝐻 be a real Hilbert space and 𝑆_𝑖∶𝐻→𝐻, 𝑖=1,2,…,𝑀, a uniformly 𝐿_𝑖-Lipschitzian and 𝛽_𝑖-strict pseudocontraction satisfying the following conditions: (a)𝐶∶=⋂^𝑀_𝑖=1𝐹(𝑆_𝑖)≠∅,(b)𝛽=max_{1≤𝑖≤𝑀}𝛽_𝑖<1. Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝐻₁𝑥chosenarbitrarily,_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑆_𝑛𝑥_𝑛,∀𝑛≥1, (3.37) where 𝑆_𝑛=𝑆_{𝑛(mod𝑀)} and {𝛼_𝑛} is a sequence in [0,1]. If Γ≠∅ and the following condition is satisfied:(c)𝛼_𝑛∈(𝛿,1−𝛽) for all 𝑛≥1, where 𝛿∈(0,1−𝛽) is a constant,then the sequence {𝑥_𝑛} converges weakly to a point 𝑥^∗∈Γ. In addition, if there exists a positive integer 𝑗 such that 𝑆_𝑗 is semicompact, then the sequences {𝑥_𝑛} converges strongly to the point 𝑥^∗.

Remark 3.5. Theorems 3.1 and 3.3 improve and extend the corresponding results of Censor et al. [1, 4, 5], Byrne [2], Yang [7], Moudafi [12], Xu [13], Censor and Segal [14], Masad and Reich [15], and others in the following aspects:(1)for the framework of spaces, we extend the space from finite dimension Hilbert space to infinite dimension Hilbert space;(2)for the mappings, we extend the mappings from nonexpansive mappings, quasi-nonexpansive mapping or demicontractive mappings to finite families of asymptotically strictly pseudocontractions;(3)for the algorithms, we propose some new hybrid iterative algorithms which are different from ones given in [1, 2, 4, 5, 7, 14, 15]. And, under suitable conditions, some weak and strong convergences for the algorithms are proved.

Acknowledgments

The authors would like to express their thanks to the referees for their helpful suggestions and comments. This work was supported by the Natural Science Foundation of Yunnan University of Economics and Finance and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011–0021821).