The Split Feasibility Problems for Countable Families of Asymptotically Strict Pseudocontractions

Abstract

An up-to-date algorithm for solving the split feasibility problem for countable families of asymptotically strict pseudocontractions is introduced in the framework of Hilbert spaces. Our results greatly improve and extend those of other authors whose related research studies are restricted to the situation of at most finitely many such mappings.

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].

Let 𝐻₁ and 𝐻₂ be two real Hilbert spaces with inner product ⟨⋅,⋅⟩ and the corresponding norm ‖⋅‖. Let 𝐶 and 𝑄 be nonempty closed convex subsets of 𝐻₁ and 𝐻₂, respectively. The purpose of this paper is to introduce and study the following multiple-set split feasibility problem for an infinite family of asymptotically strict pseudocontractions (MSSFP) in the framework of infinite-dimensional Hilbert spaces. Find 𝑥^∗∈𝐶 such that 𝐴𝑥^∗∈𝑄, (1) where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator.

In the sequel, we use Γ to denote the set of solutions of the problem (MSSFP), that is, Γ={𝑥∈𝐶,𝐴𝑥∈𝑄}. (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, then 𝑥^∗=𝑇𝑥^∗, where 𝑥_𝑛⇀𝑥^∗ denotes that {𝑥_𝑛} converges weakly to 𝑥^∗.

A Banach space 𝐸 is said to satisfy Opial’s condition, if for any sequence {𝑥_𝑛} in 𝐸, 𝑥_𝑛⇀𝑥^∗ implies that liminf_𝑛→∞𝑥_𝑛−𝑥^∗<liminf_𝑛→∞𝑥_𝑛−𝑦,∀𝑦∈𝐸with𝑦≠𝑥^∗. (3) It is well known that every Hilbert space satisfies Opial's condition.

Definition 1. Let 𝐻 be a real Hilbert space, 𝑇 be a mapping from 𝐻 into itself and the fixed point set 𝐹(𝑇) of 𝑇 be nonempty.(1)𝑇is 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 𝑇^𝑛𝑥−𝑇^𝑛𝑦²≤𝑘_𝑛𝑥−𝑦²+𝛾𝐼−𝑇^𝑛𝑥−𝐼−𝑇^𝑛𝑦²,∀𝑥,𝑦∈𝐻. (4) Especially, if 𝑘_𝑛=1 for each 𝑛≥1 in (4) and there exists a 𝛾∈[0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝛾(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐻, (5) then 𝑇∶𝐻→𝐻 is called a 𝛾-strict pseudocontraction.(2)𝑇 is said to be uniformly 𝐿-Lipschitzian if there exists a constant 𝐿>0 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦²≤𝐿𝑥−𝑦,∀𝑥,𝑦∈𝐻,𝑛≥1. (6)(3)𝑇 is said to be semicompact if for any bounded sequence {𝑥_𝑛}⊂𝐻 with lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0, there exists a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that {𝑥_𝑛𝑖} converges strongly to a point 𝑥^∗∈𝐻.

Remark 2. (1) If we put 𝛾=0 in (4), then the mapping 𝑇∶𝐻→𝐻 is asymptotically nonexpansive.
(2) If we put 𝛾=0 in (5), then the mapping 𝑇∶𝐻→𝐻 is nonexpansive.
(3) Each (𝛾,{𝑘_𝑛})-asymptotically strict pseudocontraction and each 𝛾-strict pseudocontraction both are demiclosed at origin [9].

In 2011, Moudafi [10] proposed the following iterative algorithm for solving split common fixed problem of quasinonexpansive mappings: for arbitrarily chosen 𝑥₁∈𝐻₁, 𝑢_𝑛=𝑥_𝑛+𝛾𝛽𝐴^∗(𝑇−𝐼)𝐴𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑈𝑢_𝑛,𝑛∈ℕ, (7) and proved that {𝑥_𝑛} converges weakly to a split common fixed point 𝑥^∗∈Γ, where 𝑈∶𝐻₁→𝐻₁ and 𝑇∶𝐻₂→𝐻₂ are two quasinonexpansive mappings, 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator, and 𝐴^∗ denotes the adjoint of 𝐴.

Motivated and inspired by the studies of Moudafi [10, 11] and Chang et al. [12], in this paper, we introduce an algorithm for solving the split feasibility problems for countable families of asymptotically strict pseudocontractions and prove some strong and weak convergence theorems for such mappings in Hilbert spaces. The results extend those of the authors [12] whose related research studies are restricted to the situation of at most finite families of such mappings.

By using the well-known inequality ⟨𝑥,𝑦⟩=(1/2)‖𝑥‖²+(1/2)‖𝑦‖²−(1/2)‖𝑥−𝑦‖² in Hilbert spaces, we can easily show the following proposition, whose proof is omitted.

Proposition 3 (see [12]). Let 𝑇∶𝐻→𝐻 be a (𝛾,{𝑘_𝑛})-asymptotically strict pseudocontraction. If Γ≠∅, then for each 𝑝∈𝐹(𝑇) and 𝑥∈𝐻, the following inequalities hold and they are equivalent: 𝑇^𝑛𝑥−𝑝²≤𝑘_𝑛𝑥−𝑝²+𝛾𝑥−𝑇^𝑛𝑥²;𝑥−𝑇^𝑛𝑥,𝑥−𝑝≥1−𝛾2𝑥−𝑇^𝑛𝑥²−𝑘_𝑛−12𝑥−𝑝²;𝑥−𝑇^𝑛𝑥,𝑝−𝑇^𝑛𝑥≤1+𝛾2𝑥−𝑇^𝑛𝑥²+𝑘_𝑛−12𝑥−𝑝². (8)(9)(10)

Lemma 4 (see [13]). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝛿_𝑛} be the sequences of nonnegative real numbers satisfying 𝑎_𝑛+1≤1+𝛿_𝑛𝑎_𝑛+𝑏_𝑛. (11) If ∑^∞_𝑛=1𝛿_𝑛<∞ and ∑^∞_𝑛=1𝑏_𝑛<∞, then the lim_𝑛→∞𝑎_𝑛 exists.

Lemma 5 (see [14]). Let 𝐾 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑇 a nonexpansive mapping from 𝐾 into itself. If 𝑇 has a fixed point, then 𝐼−𝑇 is demiclosed at zero, where 𝐼 is the identity mapping of 𝐻.

Lemma 6 (see [15]). The unique solutions to the positive integer equation ()𝑚𝑛=𝑖+𝑚−12,𝑚≥𝑖,𝑛=1,2,3,… (12) are ()𝑚𝑖=𝑛−𝑚−121,𝑚=−2−12𝑛+4,𝑛=1,2,3,…, (13) where [𝑥] denotes the maximal integer that is not larger than 𝑥.

3. Main Results

In the sequel, we assume that the following conditions are satisfied:(a)𝐻₁ and 𝐻₂ are two real Hilbert spaces, 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator, and 𝐴^∗ denotes the adjoint of 𝐴;(b){𝑆_𝑖}∶𝐻₁→𝐻₁ is a sequence of uniformly 𝐿₁-Lipschitzian and (𝛽_𝑖,{𝑘^(𝑖)_1,𝑛})-asymptotically strict pseudocontractions and {𝑇_𝑖}∶𝐻₂→𝐻₂ is a sequence of uniformly 𝐿₂-Lipschitzian and (𝜇_𝑖,{𝑘^(𝑖)_2,𝑛})-asymptotically strict pseudocontractions satisfying the following conditions: (1)   𝐶∶=∩^∞_𝑖=1𝐹(𝑆_𝑖)≠∅, 𝑄∶=∩^∞_𝑖=1𝐹(𝑇_𝑖)≠∅; (2)𝛽∶=sup_𝑖≥1{𝛽_𝑖}<1 and 𝜇∶=sup_𝑖≥1{𝜇_𝑖}<1; (3)for each 𝑖≥1, 𝑘_𝑛^(𝑖)∶=max{𝑘^(𝑖)_1,𝑛,𝑘^(𝑖)_2,𝑛}, and ∑^∞_𝑖=1∑^∞_𝑛=1(𝑘_𝑛^(𝑖)−1)<∞.

The multiple-set split feasibility problem for infinite families of nonlinear mappings {𝑆_𝑖} and {𝑇_𝑖} is to find a point 𝑞∈𝐶suchthat𝐴𝑞∈𝑄, (14) whose set of solutions is denoted by Γ.

Lemma 7. Let 𝐻₁,𝐻₂,𝐴,{𝑆_𝑖},{𝑇_𝑖},𝐶,𝑄,𝛽,𝜇,𝐿₁,𝐿₂ and {𝑘_𝑛^(𝑖)} be the same as those mentioned above. Let {𝑥_𝑛} be the following sequence generated by an arbitrarily chosen 𝑥₁∈𝐻₁𝑢_𝑛=𝑥_𝑛+𝛾𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛,𝑛∈ℕ, (15) where 𝑇^∗_𝑛=𝑇_𝑖𝑛, 𝑆^∗_𝑛=𝑆_𝑖𝑛 with 𝑖_𝑛 and 𝑚_𝑛 being the solutions to the positive integer equation: 𝑛=𝑖+(𝑚−1)𝑚/2(𝑚≥𝑖,𝑛=1,2,…); that is, for each 𝑛≥1, there exist unique 𝑖_𝑛 and 𝑚_𝑛 such that 𝑖₁=1,𝑖₂=1,𝑖₃=2,𝑖₄𝑖=1,₅=2,𝑖₆=3,𝑖₇=1,𝑖₈𝑚=2,…;₁=1,𝑚₂=2,𝑚₃=2,𝑚₄=3,𝑚₅𝑚=3,₆=3,𝑚₇=4,𝑚₈=4,…; (16){𝛼_𝑛} is a sequence in [0,1], and 𝛾 is a constant satisfying the following condition: (4)∶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. If Γ≠∅, for any 𝑞∈Γ, then(I)lim_𝑛→∞‖𝑥_𝑛−𝑞‖ and lim_𝑛→∞‖𝑢_𝑛−𝑞‖ exist and have the same values;(II) for each 𝑖≥1, there exists a corresponding subsequence {𝑥_𝑛}_{𝑛∈ℕ𝑖} of {𝑥_𝑛} such that lim_{ℕ𝑖∋𝑛→∞}𝑢_𝑛−𝑆_𝑖𝑢_𝑛=𝐴𝑥_𝑛−𝑇_𝑖𝐴𝑥_𝑛=0, (17)where ℕ_𝑖∶={𝑛∈ℕ∶𝑛=𝑖+(𝑚−1)𝑚/2,𝑚≥𝑖,𝑚∈ℕ}.

Proof. (I) Taking 𝑞∈Γ, that is, 𝑞∈𝐶 and 𝐴𝑞∈𝑄, and using (15) and (9), we have 𝑥_𝑛+1−𝑞²≤𝑢_𝑛−𝑞−𝛼_𝑛𝑢_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛²=𝑢_𝑛−𝑞²−2𝛼_𝑛𝑢_𝑛−𝑞,𝑢_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛+𝛼²_𝑛𝑢_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛²≤𝑢_𝑛−𝑞²−𝛼_𝑛1−𝛽𝑢_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛²−𝑘^(𝑖𝑛)_𝑚𝑛−1𝑢_𝑛−𝑞²+𝛼²_𝑛𝑢_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛²=1+𝛼_𝑛𝑘^(𝑖𝑛)_𝑚𝑛−1𝑢_𝑛−𝑞²−𝛼_𝑛1−𝛽−𝛼_𝑛𝑢_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛²,𝑢_𝑛−𝑞²=𝑥_𝑛+𝛾𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛−𝑞²=𝑥_𝑛−𝑞²+𝛾²𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²+2𝛾𝑥_𝑛−𝑞,𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛=𝑥_𝑛−𝑞²+𝛾²𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛,𝐴𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛+2𝛾𝑥_𝑛−𝑞,𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛, (18)(19) where 𝛾²𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛,𝐴𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛≤‖𝐴‖²𝛾²𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛,𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛≤‖𝐴‖²𝛾²𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²,2𝛾𝑥_𝑛−𝑞,𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛=2𝛾𝐴𝑥_𝑛−𝑞,𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛=2𝛾𝐴𝑥_𝑛−𝑞+𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛−𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛,𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛=2𝛾𝑇^∗_𝑛^𝑚𝑛𝐴𝑥_𝑛−𝐴𝑞,𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛−𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛². (20)(21) Further, letting 𝑥=𝐴𝑥_𝑛, 𝑇^𝑛=(𝑇^∗_𝑛)^𝑚𝑛, 𝑝=𝐴𝑞, 𝛾=𝜇 in (10) and noting 𝐴𝑞∈𝐹(𝑇^∗_𝑛), we have 𝑇^∗_𝑛^𝑚𝑛𝐴𝑥_𝑛−𝐴𝑞,𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛≤1+𝜇2𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²+𝑘^(𝑖𝑛)_𝑚𝑛−12𝐴𝑥_𝑛−𝐴𝑞²≤1+𝜇2𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²+𝑘^(𝑖𝑛)_𝑚𝑛−1‖𝐴‖²2𝑥_𝑛−𝑞². (22) Substituting (22) into (21) and simplifying it, we have 2𝛾𝑥_𝑛−𝑞,𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛≤𝛾𝜇−1𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²+𝑘^(𝑖𝑛)_𝑚𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑞². (23) Substituting (20) and (23) into (19) and simplifying it, we have 𝑢_𝑛−𝑞²≤𝑥_𝑛−𝑞²+𝛾²‖𝐴‖²𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²+𝛾𝜇−1𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²+𝑘^(𝑖𝑛)_𝑚𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑞²=𝑥_𝑛−𝑞²−𝛾1−𝜇−𝛾‖𝐴‖²×𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²+𝑘^(𝑖𝑛)_𝑚𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑞². (24) Again, substituting (24) into (18) and simplifying it, we have 𝑥_𝑛+1−𝑞²≤1+𝛼_𝑛𝑘^(𝑖𝑛)_𝑚𝑛×𝑥−1_𝑛−𝑞²−𝛾1−𝜇−𝛾‖𝐴‖²×𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²+𝑘^(𝑖𝑛)_𝑚𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑞²−𝛼_𝑛1−𝛽−𝛼_𝑛𝑢_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛²≤1+𝛼_𝑛𝑘^(𝑖𝑛)_𝑚𝑛−1𝑥_𝑛−𝑞²−𝛾1−𝜇−𝛾‖𝐴‖²𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²+1+𝛼_𝑛𝑘^(𝑖𝑛)_𝑚𝑛−1𝑘^(𝑖𝑛)_𝑚𝑛−1𝛾‖𝐴‖²𝑥_𝑛−𝑞²−𝛼_𝑛1−𝛽−𝛼_𝑛𝑢_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛². (25)
By condition (4), we have 𝑥_𝑛+1−𝑞²≤1+𝛼_𝑛𝑘^(𝑖𝑛)_𝑚𝑛−1𝑥_𝑛−𝑞²+1+𝛼_𝑛𝑘^(𝑖𝑛)_𝑚𝑛−1𝑘^(𝑖𝑛)_𝑚𝑛‖𝐴‖−1×𝛾²𝑥_𝑛−𝑞²≤1+𝐾𝑘^(𝑖𝑛)_𝑚𝑛−1𝑥_𝑛−𝑞², (26) where 𝐾=sup_𝑛≥1𝛼_𝑛+1+𝛼_𝑛𝑘^(𝑖𝑛)_𝑚𝑛−1𝛾‖𝐴‖²<∞. (27) Note that ∑^∞_𝑛=1(𝑘^(𝑖𝑛)_𝑚𝑛−1)=∑^∞_𝑖=1∑^∞_𝑛=𝑖(𝑘_𝑛^(𝑖)−1)≤∑^∞_𝑖=1∑^∞_𝑛=1(𝑘_𝑛^(𝑖)−1)<∞. Hence, from Lemma 4, we know that the following limit exists: lim_𝑛→∞𝑥_𝑛−𝑞. (28) We now prove that for each 𝑞∈Γ, the limit lim_𝑛→∞𝑢_𝑛−𝑞 (29) exists. In fact, from (25) and (28), it follows that 𝛾1−𝜇−𝛾‖𝐴‖²𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛²+𝛼_𝑛1−𝛽−𝛼_𝑛𝑢_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛²≤𝑥_𝑛−𝑞²−𝑥_𝑛+1−𝑞²+𝐾𝑘^(𝑖𝑛)_𝑚𝑛−1𝑥_𝑛−𝑞²⟶0(𝑛⟶∞). (30) This, combined with condition (4), implies that lim_𝑛→∞𝑢_𝑛−𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛=0,lim_𝑛→∞𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛=0. (31)(32) Therefore, it follows from (19), (28), and (32) that lim_𝑛→∞‖𝑢_𝑛−𝑞‖ exists.
(II) We firstly prove that lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑢_𝑛+1−𝑢_𝑛‖=0. As a matter of fact, it follows from (15) that 𝑥_𝑛+1−𝑥_𝑛=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛾𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛−𝑢_𝑛+𝛼_𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛−𝑢_𝑛+𝛼_𝑛𝛾𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛=𝛾𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^∗_𝑛^𝑚𝑛𝑢_𝑛−𝑢_𝑛. (33) In view of (31) and (32), we have that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (34) Similarly, it follows from (15), (32), and (34) that 𝑢_𝑛+1−𝑢_𝑛=𝑥_𝑛+1+𝛾𝐴^∗𝑇^∗_𝑛+1^𝑚𝑛+1−𝐼𝐴𝑥_𝑛+1−𝑥_𝑛+𝛾𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛+𝛾𝐴^∗𝑇^∗_𝑛+1^𝑚𝑛+1−𝐼𝐴𝑥_𝑛+1+𝛾𝐴^∗𝑇^∗_𝑛^𝑚𝑛−𝐼𝐴𝑥_𝑛⟶0(𝑛⟶∞). (35) Next, for each 𝑖∈ℕ, we consider the corresponding subsequence {𝑢_𝑛}_{𝑛∈ℕ𝑖} of {𝑢_𝑛}. For example, by Lemma 6 and the definition of ℕ₁, we have ℕ₁={1,2,4,7,11,16,…} and 𝑖₁=𝑖₂=𝑖₄=𝑖₇=𝑖₁₁=𝑖₁₆=⋯=1. Note that {𝑚_𝑛}_{𝑛∈ℕ𝑖}={𝑖,𝑖+1,𝑖+2,…}, that is, 𝑚_𝑛−1=𝑚_𝑛−1, and 𝑆^∗_𝑛=𝑆_𝑖𝑛=𝑆_𝑖 whenever 𝑛∈ℕ_𝑖. Set 𝜂_𝑛∶=‖𝑢_𝑛−𝑆^𝑚𝑛_𝑖𝑢_𝑛‖. Since {𝑆_𝑖} are uniformly 𝐿₁-Lipschitzian and 𝑚_𝑛≥1 for 𝑛≥1, we have, for each 𝑛∈ℕ_𝑖 and 𝑛≥2, 𝑢_𝑛−𝑆_𝑖𝑢_𝑛≤𝑢_𝑛−𝑆^𝑚𝑛_𝑖𝑢_𝑛+𝑆^𝑚𝑛_𝑖𝑢_𝑛−𝑆_𝑖𝑢_𝑛≤𝜂_𝑛+𝐿₁𝑆^𝑚𝑛_𝑖⁻¹𝑢_𝑛−𝑢_𝑛≤𝜂_𝑛+𝐿₁𝑆^𝑚𝑛_𝑖⁻¹𝑢_𝑛−𝑆^𝑚𝑛_𝑖⁻¹𝑢_𝑛−1+𝑆^𝑚𝑛_𝑖⁻¹𝑢_𝑛−1−𝑢_𝑛≤𝜂_𝑛+𝐿₁𝑆^𝑚𝑛_𝑖⁻¹𝑢_𝑛−𝑆^𝑚𝑛_𝑖⁻¹𝑢_𝑛−1+𝐿₁𝑆^𝑚𝑛_𝑖⁻¹𝑢_𝑛−1−𝑢_𝑛−1+𝑢_𝑛−𝑢_𝑛−1≤𝜂_𝑛+𝐿²₁𝑢_𝑛−𝑢_𝑛−1+𝐿₁𝑆^𝑚𝑛_𝑖⁻¹𝑢_𝑛−1−𝑢_𝑛−1+𝑢_𝑛−1−𝑢_𝑛≤𝜂_𝑛+𝐿₁1+𝐿₁𝑢_𝑛−𝑢_𝑛−1+𝐿₁𝜂_𝑛−1. (36) Thus, it follows from (31) and (35) that, for each 𝑖≥1, lim_{ℕ𝑖∋𝑛→∞}𝑢_𝑛−𝑆_𝑖𝑢_𝑛=0. (37) Similarly, we have, for each 𝑖≥1, lim_{ℕ𝑖∋𝑛→∞}𝐴𝑥_𝑛−𝑇_𝑖𝐴𝑥_𝑛=0. (38) This completes the proof.

Theorem 8. Let 𝐻₁,𝐻₂,𝐴,{𝑆_𝑖},{𝑇_𝑖},𝐶,𝑄,𝛽,𝜇,𝐿₁,𝐿₂ and {𝑘_𝑛^(𝑖)} be the same as those in Lemma 7. Suppose that {𝑥_𝑛} is a sequence defined by (15). If Γ≠∅ and there exist mappings 𝑆_𝑖0∈{𝑆_𝑖}^∞_𝑖=1 and 𝑇_𝑖0∈{𝑇_𝑖}^∞_𝑖=1 and nondecreasing functions 𝑓,ℎ∶[0,∞)→[0,∞) with 𝑓(0)=ℎ(0)=0 and 𝑓(𝑟),ℎ(𝑟)>0 for all 𝑟∈(0,∞) such that 𝑓(𝑑(𝑥_𝑛,Γ))≤‖𝑢_𝑛−𝑆_𝑖0𝑢_𝑛‖ and ℎ(𝑑(𝐴𝑥_𝑛,𝑄))≤‖𝐴𝑥_𝑛−𝑇_𝑖0𝐴𝑥_𝑛‖ for all 𝑛≥1, then {𝑥_𝑛} converges strongly to some member of Γ.

Proof. By Lemma 7, there exists a subsequence {𝑢_𝑛}_{𝑛∈ℕ𝑖0} of {𝑢_𝑛} such that lim_{ℕ𝑖0∋𝑛→∞}𝑢_𝑛−𝑆_𝑖0𝑢_𝑛=0. (39) Since for all 𝑛∈ℕ_𝑖0, 𝑓𝑑𝑥_𝑛,Γ≤𝑢_𝑛−𝑆_𝑖0𝑢_𝑛, (40) by taking limsup as ℕ_𝑖0∋𝑛→∞ on both sides in the inequality above, we have lim_{ℕ𝑖0∋𝑛→∞}𝑓𝑑𝑥_𝑛,Γ=0, (41) which implies lim_{ℕ𝑖0∋𝑛→∞}𝑑(𝑥_𝑛,Γ)=0 by the definition of the function 𝑓.
Now we will show that {𝑥_𝑛}_{𝑛∈ℕ𝑖0} is a Cauchy sequence. By Lemma 7, there exists a constant 𝑀>0 such that ‖𝑥_𝑛−𝑞‖²≤𝑀‖𝑥_𝑚−𝑞‖² for any 𝑞∈Γ and all 𝑛>𝑚. And for any 𝜖>0, there exists a positive integer 𝑁 such that 𝑑²(𝑥_𝑛,Γ)<𝜖/4𝑀 for all 𝑛≥𝑁 and 𝑛∈ℕ_𝑖0. Then, for any 𝑞∈Γ and 𝑛,𝑚≥𝑁 and 𝑛,𝑚∈ℕ_𝑖0, we have 𝑥_𝑛−𝑥_𝑚²≤2𝑥_𝑛−𝑞²+𝑥_𝑚−𝑞²≤4𝑀𝑥_𝑁−𝑞². (42) Taking the infimum in the above inequalities for all 𝑞∈Γ yields that 𝑥_𝑛−𝑥_𝑚²≤4𝑀𝑑²𝑥_𝑁,Γ<𝜖, (43) which implies that {𝑥_𝑛}_{𝑛∈ℕ𝑖0} is a Cauchy sequence. Therefore, there exists a 𝑝∈𝐻₁ such that 𝑥_𝑛→𝑝 as ℕ_𝑖0∋𝑛→∞ since 𝐻₁ is complete. Firstly, we show that 𝑝∈𝐶. lim_{ℕ𝑖0∋𝑛→∞}𝑑(𝑥_𝑛,Γ)=0 shows that 𝑑(𝑝,Γ)=0, which implies that 𝑝∈𝐶 since Γ⊂𝐶. Secondly, we show that 𝐴𝑝∈𝑄. Since {𝑥_𝑛}_{𝑛∈ℕ𝑖0} converges to 𝑝 and ℎ(𝑑(𝐴𝑥_𝑛,𝑄))≤‖𝐴𝑥_𝑛−𝑇_𝑖0𝐴𝑥_𝑛‖ for all 𝑛∈ℕ_𝑖0, then 𝑑(𝐴𝑝,𝑄)=0. This implies that 𝐴𝑝∈𝑄 because of the closedness of 𝑄, and so 𝑝∈Γ. It finally follows from the existence of lim_𝑛→∞‖𝑥_𝑛−𝑝‖ that 𝑥_𝑛→𝑝 as 𝑛→∞. This completes the proof.

Example 9. Let 𝐻₁=𝐻₂=ℝ¹ with the standard norm ‖⋅‖=|⋅| and 𝐶=[−1,1]. Let {𝑆_𝑖}={𝑇_𝑖}∶𝐶→𝐶 be two sequences of mappings defined by 𝑆_𝑖𝑥𝑥=(],[].𝑖+1,𝑥∈0,1𝑥,𝑥∈−1,0 (44) It is easily shown that {𝑆_𝑖} is uniformly 𝐿-Lipschitzian and a sequence of (0,{𝑘_𝑛=1})-asymptotically strict pseudocontractions. We now prove that the sequence {𝑥_𝑛} defined by (15) converges strongly to some member of Γ. Let 𝐴𝑥=𝐴^∗𝑥=𝑥/2 for all 𝑥∈𝐶 with ‖𝐴‖=1/2 and 𝛾=3∈(0,1/‖𝐴‖²). If {𝑥_𝑛}⊂(0,1], we then have 𝑢_𝑛−𝑆₁𝑢_𝑛=12𝑢_𝑛=12𝑥_𝑛+321𝑖_𝑛+1^𝑚𝑛𝑥_𝑛2−𝑥_𝑛2=𝜅_𝑛𝑥_𝑛, (45) where 𝜅_𝑛∶=(1/2)|1+(3/4)[(1/(𝑖_𝑛+1))^𝑚𝑛−1]| with 𝜅∶=inf_𝑛≥1𝜅_𝑛>0. Define a nondecreasing function 𝑓∶[0,∞)→[0,∞) by 𝑓(𝑥)=𝜅𝑥. Since Γ=[−1,0], we then have 𝑓𝑑𝑥_𝑛,Γ=𝑓𝑥_𝑛=𝜅𝑥_𝑛≤𝑢_𝑛−𝑆₁𝑢_𝑛. (46) Similarly, we also can define a nondecreasing function ℎ∶[0,∞)→[0,∞) with ℎ(0)=0 such that ℎ𝑑𝐴𝑥_𝑛,𝑄≤𝐴𝑥_𝑛−𝑇_𝑖0𝐴𝑥_𝑛 (47) for some 𝑖₀≥1, which implies that, by Lemma 7 and Theorem 8, 𝑥_𝑛→𝑥^∗∈Γ as 𝑛→∞.

Theorem 10. Let 𝐻₁,𝐻₂,𝐴,𝐶 and 𝑄 be the same as those in Lemma 7. Let {𝑆_𝑖}∶𝐻₁→𝐻₁ and {𝑇_𝑖}∶𝐻₂→𝐻₂ be two sequences of nonexpansive mappings. Let {𝑥_𝑛} be the following sequence generated by an arbitrarily chosen 𝑥₁∈𝐻₁𝑢_𝑛=𝑥_𝑛+𝛾𝐴^∗𝑇^∗_𝑛−𝐼𝐴𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑆^∗_𝑛𝑢_𝑛,𝑛≥1, (48) where {𝛼_𝑛} is a sequence in [𝛼,1−𝛼] for some 𝛼∈(0,1); 𝛾∈(0,1/‖𝐴‖²); 𝑇^∗_𝑛=𝑇_𝑖𝑛,𝑆^∗_𝑛=𝑆_𝑖𝑛 with 𝑖_𝑛 satisfying the positive integer equation: 𝑛=𝑖+(𝑚−1)𝑚/2(𝑚≥𝑖,𝑛=1,2,…). Then {𝑥_𝑛} converges weakly to an 𝑥^∗∈Γ.

Proof. It is clear that both {𝑆_𝑖} and {𝑇_𝑖} are asymptotically strict pseudocontractions. Then, by the proof of Lemma 7, we have lim_𝑛→∞𝑢_𝑛−𝑆^∗_𝑛𝑢_𝑛=0,lim_𝑛→∞𝑇^∗_𝑛−𝐼𝐴𝑥_𝑛=0. (49)(50) In addition, we also have lim_𝑛→∞𝑢_𝑛+1−𝑢_𝑛=𝑥_𝑛+1−𝑥_𝑛=0, (51) which implies that, by induction, for any nonnegative integer 𝑘, lim_𝑛→∞𝑢_𝑛+𝑘−𝑢_𝑛=lim_𝑛→∞𝑥_𝑛+𝑘−𝑥_𝑛=0. (52) For each 𝑘≥1, since 𝑢_𝑛−𝑆^∗_𝑛+𝑘𝑢_𝑛≤𝑢_𝑛−𝑢_𝑛+𝑘+𝑢_𝑛+𝑘−𝑆^∗_𝑛+𝑘𝑢_𝑛≤𝑢_𝑛−𝑢_𝑛+𝑘+𝑢_𝑛+𝑘−𝑆^∗_𝑛+𝑘𝑢_𝑛+𝑘+𝑆^∗_𝑛+𝑘𝑢_𝑛+𝑘−𝑆^∗_𝑛+𝑘𝑢_𝑛≤2𝑢_𝑛−𝑢_𝑛+𝑘+𝑢_𝑛+𝑘−𝑆^∗_𝑛+𝑘𝑢_𝑛+𝑘, (53) it follows from (49) and (52) that lim_𝑛→∞𝑢_𝑛−𝑆^∗_𝑛+𝑘𝑢_𝑛=0. (54) Now, for each 𝑖≥1, we claim that lim_𝑛→∞𝑢_𝑛−𝑆_𝑖𝑢_𝑛=0. (55) As a matter of fact, setting 𝑛=𝑁_𝑚+𝑖, (56) where 𝑁_𝑚=(𝑚−1)𝑚/2, 𝑚≥𝑖, we obtain that 𝑢_𝑛−𝑆_𝑖𝑢_𝑛≤𝑢_𝑛−𝑢_𝑁𝑚+𝑢_𝑁𝑚−𝑆_𝑖𝑢_𝑛≤𝑢_𝑛−𝑢_𝑁𝑚+𝑢_𝑁𝑚−𝑆^∗_{𝑁𝑚+𝑖}𝑢_𝑁𝑚+𝑆^∗_{𝑁𝑚+𝑖}𝑢_𝑁𝑚−𝑆_𝑖𝑢_𝑛=𝑢_𝑛−𝑢_𝑁𝑚+𝑢_𝑁𝑚−𝑆^∗_{𝑁𝑚+𝑖}𝑢_𝑁𝑚+𝑆_𝑖𝑢_𝑁𝑚−𝑆_𝑖𝑢_𝑛≤2𝑢_𝑛−𝑢_𝑁𝑚+𝑢_𝑁𝑚−𝑆^∗_{𝑁𝑚+𝑖}𝑢_𝑁𝑚=2𝑢_𝑛−𝑢_𝑛−𝑖+𝑢_𝑁𝑚−𝑆^∗_{𝑁𝑚+𝑖}𝑢_𝑁𝑚. (57) Then, since 𝑁_𝑚→∞ as 𝑛→∞, it follows from (52) and (54) that (55) holds obviously. Similarly, we have, for each 𝑖≥1, lim_𝑛→∞𝐴𝑥_𝑛−𝑇_𝑖𝐴𝑥_𝑛=0. (58) Next, since {𝑢_𝑛} is bounded, there exists a subsequence {𝑢_𝑛𝑖}⊂{𝑢_𝑛} such that 𝑢_𝑛𝑖⇀𝑥^∗ (some point in 𝐻₁). From (55) we have lim_𝑖→∞‖𝑢_𝑛𝑖−𝑆_𝑗𝑢_𝑛𝑖‖=0 for each 𝑗≥1. By Lemma 5, each 𝑆_𝑗 is demiclosed at zero, so we know that 𝑥^∗∈∩^∞_𝑗=1𝐹(𝑆_𝑗). Moreover, it follows from (48) and (50) that 𝑥_𝑛𝑖=𝑢_𝑛𝑖−𝛾𝐴^∗𝑇^∗_𝑛𝑖−𝐼𝐴𝑥_𝑛𝑖⇀𝑥^∗(𝑖⟶∞). (59) Since 𝐴 is a linear bounded operator, it yields that 𝐴𝑥_𝑛𝑖⇀𝐴𝑥^∗. In view of (58) we have lim_𝑖→∞𝐴𝑥_𝑛𝑖−𝑇_𝑗𝐴𝑥_𝑛𝑖=0,∀𝑗≥1. (60) Again since each 𝑇_𝑗 is demiclosed at zero, we know that 𝐴𝑥^∗∈∩^∞_𝑗=1𝐹(𝑇_𝑗). This implies that 𝑥^∗∈Γ.
Note that each Hilbert space possesses Opial property, which guarantees that the weakly subsequential limit of {𝑢_𝑛} is unique. Consequently, {𝑢_𝑛} converges weakly to the point 𝑥^∗∈Γ. Since 𝑥_𝑛=𝑢_𝑛−𝛾𝐴^∗(𝑇^∗_𝑛−𝐼)𝐴𝑥_𝑛, we know that {𝑥_𝑛} converges weakly to 𝑥^∗∈Γ. The proof is completed.

Remark 11. Note that, from Remark 2(3), the class of (𝛾,{𝑘_𝑛})-asymptotically strict pseudocontractions is demiclosed at zero. Then, together with nonexpansiveness replaced by Lipschitz continuity, the two sequences of nonexpansive mappings {𝑆_𝑖} and {𝑇_𝑖} in Theorem 10 can be extended to (𝛾,{𝑘_𝑛})-asymptotically strict pseudocontractions as in Lemma 7.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is very grateful to the referees for their useful suggestions, by which the contents of this paper has been improved. This work is supported by the General Project of Scientific Research Foundation of Yunnan University of Finance and Economics (YC2013A02).