[Retracted] The Split Common Fixed Point Problem for Quasi-Total Asymptotically Nonexpansive Uniformly Lipschitzian Mappings

Abstract

We introduce an algorithm for solving the split common fixed point problem for quasi-total asymptotically nonexpansive uniformly Lipschitzian mapping in Hilbert spaces. The results presented in this paper improve and extend some recent corresponding results.

1. Introduction and Preliminaries

Let 𝐻₁ and 𝐻₂ be real Hilbert spaces with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Let 𝐶 and 𝑄 be nonempty closed convex subsets of 𝐻₁ and 𝐻₂, respectively. The split feasibility problem (SFP) is formulated as finding a point 𝑥^∗ with the property𝑥^∗∈𝐶,𝐴𝑥^∗∈𝑄, (1.1) where 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator.

The SFP in finite-dimensional Hilbert 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 [1, 3–7]. The SFP in infinite-dimensional Hilbert spaces can be found in [8–16].

The split common fixed point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP), see [4]. Let 𝑆∶𝐻₁→𝐻₁ and 𝑇∶𝐻₂→𝐻₂ be two mappings satisfying 𝐹(𝑆)={𝑥∈𝐻₁∶𝑆𝑥=𝑥}≠𝜙 and 𝐹(𝑇)={𝑥∈𝐻₂∶𝑇𝑥=𝑥}≠𝜙, respectively. The split common fixed point problem for mappings 𝑆 and 𝑇 is to find a point 𝑞∈𝐻₁ with the property𝑞∈𝐹(𝑆),𝐴𝑞∈𝐹(𝑇), (1.2) where 𝐴 is a bounded linear operator from 𝐻₁ to 𝐻₂. We use Γ to denote the set of solutions of SCFP (1.2).

We first recall some definitions, notations, and conclusions which will be used 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^∗.

A Banach space 𝐸 is said to satisfy Opial's condition, if for any sequence {𝑥_𝑛} in 𝐸, 𝑥_𝑛⇀𝑥^∗ implies thatliminf_𝑛→∞𝑥_𝑛−𝑥^∗<liminf_𝑛→∞𝑥_𝑛−𝑦,∀𝑦∈𝐸with𝑦≠𝑥^∗. (1.3) It is well known that every Hilbert space satisfies Opial's condition.

Definition 1.1. Let 𝐻 be a real Hilbert space, 𝑇 a mapping from 𝐻 into itself, and the fixed point set 𝐹(𝑇) of 𝑇 nonempty.(1)𝑇 is said to be quasi-nonexpansive if for all (𝑥,𝑞)∈𝐻×𝐹(𝑇)𝑇𝑥−𝑞≤𝑥−𝑞. (1.4)(2)𝑇 is said to be quasi-asymptotically nonexpansive if there exists a sequence {𝑘_𝑛}⊂[1,∞) with 𝑘_𝑛→1 as 𝑛→∞ such that for all (𝑥,𝑞)∈𝐻×𝐹(𝑇)𝑇^𝑛𝑥−𝑞≤𝑘_𝑛𝑥−𝑞,∀𝑛≥1, (1.5)(3)𝑇 is said to be ({𝜇_𝑛},{𝜉_𝑛},𝜙)-quasi-total asymptotically nonexpansive if for all (𝑥,𝑞)∈𝐻×𝐹(𝑇)𝑇^𝑛𝑥−𝑞²≤𝑥−𝑞²+𝜇_𝑛𝜙𝑥−𝑞+𝜉_𝑛,∀𝑛≥1, (1.6) where 𝜙∶[0,∞)→[0,∞) is a continuous and strictly increasing function with 𝜙(0)=0 and {𝜇_𝑛} and {𝜉_𝑛} are two nonnegative real sequences satisfying 𝜇_𝑛→0 and 𝜉_𝑛→0 as 𝑛→∞. The class of mappings was introduced by Alber et al. [17] in 2006.(4)𝑇 is said to be uniformly L-Lipschitzian if there exists a constant 𝐿>0 such that for all (𝑥,𝑦)∈𝐻×𝐻𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝐿𝑥−𝑦. (1.7)(5)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 1.2. From Definition 1.1 we can see that a quasi-nonexpansive mapping or an asymptotically quasi-nonexpansive mapping is a ({𝜇_𝑛},{𝜉_𝑛},𝜙)-quasi-total asymptotically nonexpansive mapping. But the converse does not hold.
In [9], Moudafi proposed the following iterative algorithm for solving split common fixed problem of quasi-nonexpansive 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, (1.8) 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 quasi-nonexpansive mappings and 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator.
Inspired by the work of Moudafi [9, 10], very recently, Qin et al. [12] introduced the following iterative algorithm to study multiples-sets feasibility problem of a finite family of asymptotically quasi-nonexpansive mappings. For arbitrarily chosen 𝑥₁∈𝐻₁, {𝑥_𝑛} is defined as follows: 𝑦_𝑛=𝑥_𝑛+𝛾𝛽𝐴^∗𝑇^𝑛_𝑛−𝐼𝐴𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑦_𝑛+𝛼_𝑛𝑆^𝑛_𝑛𝑦_𝑛,𝑛≥1, (1.9) where 𝑆^𝑛_𝑛=𝑆^𝑛_{𝑛(mod𝑁)}, 𝑇^𝑛_𝑛=𝑇^𝑛_{𝑛(mod𝑁)}, 𝑛≥1, 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator. They also proved that {𝑥_𝑛} converges strongly or weakly to a multiple-sets split common fixed point of a finite family of asymptotically quasi-nonexpansive mappings in Hilbert spaces under some suitable conditions.
Motivated and inspired by the work of Moudafi [9, 10] and Qin et al. [12], in this paper, we study the SCFP (1.2) of ({𝜇_𝑛},{𝜉_𝑛},𝜙)-quasi-total asymptotically nonexpansive mappings in Hilbert spaces and obtain some of the strong and weak convergence of the presented algorithm to some 𝑞∈Γ. The results obtained in this paper improve and extend the result of Qin et al. [12], Xu [14], and Yang [16] and others.
By using the well-known equality ⟨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. The proof is omitted.

Proposition 1.3. Let 𝑇∶𝐻→𝐻 be a ({𝜇_𝑛},{𝜉_𝑛},𝜙)-quasi-total asymptotically nonexpansive mapping. Then for each 𝑞∈𝐹(𝑇) and 𝑥∈𝐻, the following inequalities hold: 𝑥−𝑇^𝑛≥1𝑥,𝑥−𝑞2𝑥−𝑇^𝑛𝑥²−𝜇_𝑛2𝜙−𝜉𝑥−𝑞^𝑛2, (1.10)𝑥−𝑇^𝑛𝑥,𝑞−𝑇^𝑛𝑥≤12𝑥−𝑇^𝑛𝑥²+𝜇_𝑛2𝜙+𝜉𝑥−𝑞^𝑛2. (1.11)

Lemma 1.4 (18). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝛿_𝑛} be sequences of nonnegative real numbers satisfying 𝑎_𝑛+1≤1+𝛿_𝑛𝑎_𝑛+𝑏_𝑛. (1.12) If ∑^∞_𝑛=1𝛿_𝑛<∞ and ∑^∞_𝑛=1𝑏_𝑛<∞, then the lim_𝑛→∞𝑎_𝑛 exists.

2. Main Results

For solving split common fixed point problem (1.2), we assume that the following conditions are satisfied:(i)𝐻₁ and 𝐻₂ are two real Hilbert spaces, and 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator;(ii)𝑆∶𝐻₁→𝐻₁ is uniformly 𝐿₁-Lipschitzian and ({𝜇_𝑛⁽¹⁾},{𝜉_𝑛⁽¹⁾},𝜙₁)-quasi-total asymptotically nonexpansive mapping and 𝑇∶𝐻₂→𝐻₂ is uniformly 𝐿₂-Lipschitzian and ({𝜇_𝑛⁽²⁾},{𝜉_𝑛⁽²⁾},𝜙₂)-quasi-total asymptotically nonexpansive mapping satisfying the following conditions:   (a) 𝐶∶=𝐹(𝑆)≠∅, 𝑄∶=𝐹(𝑇)≠∅;   (b) 𝜇_𝑛=max{𝜇_𝑛⁽¹⁾,𝜇_𝑛⁽²⁾}, 𝜉_𝑛=max{𝜉_𝑛⁽¹⁾,𝜉_𝑛⁽²⁾}, 𝑛≥1, and ∑^∞_𝑛=1𝜇_𝑛<∞,∑^∞_𝑛=1𝜉_𝑛<∞;  (c) 𝜙=max{𝜙₁,𝜙₂} and there exists two positive constant 𝑀 and 𝑀^∗ such that   𝜙(𝜆)≤𝑀^∗𝜆² for all 𝜆≥𝑀.

Theorem 2.1. Let 𝐻₁,𝐻₂,𝐴,𝑆,𝑇,𝐶,𝑄,𝐿₁,𝐿₂,{𝜇_𝑛},{𝜉_𝑛}, and 𝜙 be the same as above. Let {𝑥_𝑛} be the sequence generated by: 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, (2.1) where {𝛼_𝑛} is a sequence in [𝛼,1−𝛼] for some 𝛼∈(0,1) and 𝛾>0 is a constant satisfying the following condition:
  (d)𝛾∈(0,1/‖𝐴‖²).
If 𝑇 and 𝑆 both are demi-closed at origin and Γ≠∅, then(I)the sequence {𝑥_𝑛} converges weakly to a split common fixed point 𝑥^∗∈Γ;(II)in addition, if 𝑆 is also semicompact, then {𝑥_𝑛} and {𝑢_𝑛} both converge strongly to a 𝑥^∗∈Γ.

Proof. (I) The proof will be divided into 4 steps.Step 1. We prove that lim_𝑛→∞‖𝑥_𝑛−𝑞‖ and lim_𝑛→∞‖𝑢_𝑛−𝑞‖ exist, and lim_𝑛→∞‖𝑥_𝑛−𝑞‖=lim_𝑛→∞‖𝑢_𝑛−𝑞‖for each 𝑞∈Γ. Since 𝜙 is a continuous and increasing function, as 𝜆≥0, we can obtain that 𝜙(𝜆)≤𝜙(𝑀)+𝑀^∗𝜆². (2.2) Taking 𝑞∈Γ, that is 𝑞∈𝐹(𝑆)=𝐶 and 𝐴𝑞∈𝐹(𝑇)=𝑄, and using (1.10), (2.1), and (2.2), we have 𝑥_𝑛+1−𝑞²≤𝑢_𝑛−𝑞−𝛼_𝑛𝑢_𝑛−𝑆^𝑛𝑢_𝑛²=𝑢_𝑛−𝑞²−2𝛼_𝑛𝑢_𝑛−𝑞,𝑢_𝑛−𝑆^𝑛𝑢_𝑛+𝛼²_𝑛𝑢_𝑛−𝑆^𝑛𝑢_𝑛²≤𝑢_𝑛−𝑞²−𝛼_𝑛1−𝛼_𝑛𝑢_𝑛−𝑆^𝑛𝑢_𝑛²+𝛼_𝑛𝜇_𝑛𝜙(𝑀)+𝑀^∗𝑢_𝑛−𝑞²+𝛼_𝑛𝜉_𝑛=1+𝛼_𝑛𝜇_𝑛𝑀^∗𝑢_𝑛−𝑞²−𝛼_𝑛1−𝛼_𝑛𝑢_𝑛−𝑆^𝑛𝑢_𝑛²+𝛼_𝑛𝜇_𝑛𝜙(𝑀)+𝛼_𝑛𝜉_𝑛,𝑢_𝑛−𝑞²=𝑥_𝑛+𝛾𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛−𝑞²=𝑥_𝑛−𝑞²+𝛾²𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛²+2𝛾𝑥_𝑛−𝑞,𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛=𝑥_𝑛−𝑞²+𝛾²𝑇^𝑛−𝐼𝐴𝑥_𝑛,𝐴𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛+2𝛾𝑥_𝑛−𝑞,𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛, (2.3)(2.4) where 𝛾²𝑇^𝑛−𝐼𝐴𝑥_𝑛,𝐴𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛≤‖𝐴‖²𝛾²𝑇^𝑛−𝐼𝐴𝑥_𝑛,𝑇^𝑛−𝐼𝐴𝑥_𝑛≤‖𝐴‖²𝛾²(𝑇^𝑛−𝐼)𝐴𝑥_𝑛²,2𝛾𝑥_𝑛−𝑞,𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛=2𝛾𝐴𝑥_𝑛−𝑞,𝑇^𝑛−𝐼𝐴𝑥_𝑛=2𝛾𝐴𝑥_𝑛−𝑞+𝑇^𝑛−𝐼𝐴𝑥_𝑛−𝑇^𝑛−𝐼𝐴𝑥_𝑛,𝑇^𝑛−𝐼𝐴𝑥_𝑛=2𝛾𝐴𝑥_𝑛−𝑞+𝑇^𝑛−𝐼𝐴𝑥_𝑛−𝑇^𝑛−𝐼𝐴𝑥_𝑛,𝑇^𝑛−𝐼𝐴𝑥_𝑛=2𝛾𝑇^𝑛𝐴𝑥_𝑛−𝐴𝑞,𝑇^𝑛−𝐼𝐴𝑥_𝑛−𝑇^𝑛−𝐼𝐴𝑥_𝑛². (2.5)(2.6) Using (1.11), we have 2𝛾𝑇^𝑛𝐴𝑥_𝑛−𝐴𝑞,𝑇^𝑛−𝐼𝐴𝑥_𝑛−𝑇^𝑛−𝐼𝐴𝑥_𝑛²≤𝛾𝑇^𝑛−𝐼𝐴𝑥_𝑛²+𝛾𝜇_𝑛𝜙𝐴𝑥_𝑛−𝐴𝑞+𝛾𝜉_𝑛−2𝛾𝑇^𝑛−𝐼𝐴𝑥_𝑛²≤−𝛾𝑇^𝑛−𝐼𝐴𝑥_𝑛²+𝛾𝜇_𝑛𝑀^∗‖𝐴‖²𝑥_𝑛−𝑞+𝛾𝜇_𝑛𝜙(𝑀)+𝛾𝜉_𝑛. (2.7) Substituting (2.5) and (2.7) into (2.4) and simplifying, we obtain 𝑢_𝑛−𝑞²≤𝑥_𝑛−𝑞²+𝛾²‖𝐴‖²𝑇^𝑛−𝐼𝐴𝑥_𝑛²−𝛾𝑇^𝑛−𝐼𝐴𝑥_𝑛²+𝛾𝜇_𝑛𝑀^∗‖𝐴‖²𝑥_𝑛−𝑞²+𝛾𝜇_𝑛𝜙(𝑀)+𝛾𝜉_𝑛≤1+𝛾𝜇_𝑛𝑀^∗‖𝐴‖²𝑥_𝑛−𝑞²−𝛾1−𝛾‖𝐴‖²𝑇^𝑛−𝐼𝐴𝑥_𝑛²+𝛾𝜇_𝑛𝜙(𝑀)+𝛾𝜉_𝑛. (2.8) Substituting (2.8) into (2.3) and simplifying, we have 𝑥_𝑛+1−𝑞²≤1+𝛼_𝑛𝜇_𝑛𝑀^∗1+𝛾𝜇_𝑛𝑀^∗‖𝐴‖²𝑥_𝑛−𝑞²−𝛾1−𝛾‖𝐴‖²𝑇^𝑛−𝐼𝐴𝑥_𝑛²+𝛾𝜇_𝑛𝜙(𝑀)+𝛾𝜉_𝑛−𝛼_𝑛1−𝛼_𝑛𝑢_𝑛−𝑆^𝑛𝑢_𝑛²+𝛼_𝑛𝜇_𝑛𝜙(𝑀)+𝛼_𝑛𝜉_𝑛=1+𝑏_𝑛𝑥_𝑛−𝑞²−1+𝛼_𝑛𝜇_𝑛𝑀^∗𝛾1−𝛾‖𝐴‖²𝑇^𝑛−𝐼𝐴𝑥_𝑛²−𝛼_𝑛1−𝛼_𝑛𝑢_𝑛−𝑆^𝑛𝑢_𝑛²+𝑐_𝑛, (2.9) where 𝑏_𝑛=𝛼_𝑛𝜇_𝑛𝑀^∗+𝛾𝜇_𝑛𝑀^∗‖𝐴‖²+𝛾‖𝐴‖²𝛼_𝑛𝜇²_𝑛𝑀^∗2,𝑐_𝑛=1+𝛼_𝑛𝜇_𝑛𝑀^∗𝛾𝜇_𝑛𝜙(𝑀)+𝛼_𝑛𝜉_𝑛+𝛼_𝑛𝜇_𝑛𝜙(𝑀)+𝛼_𝑛𝜉_𝑛. (2.10) Since ∑^∞_𝑛=1𝜇_𝑛<∞ and ∑^∞_𝑛=1𝜉_𝑛<∞, so ∑^∞_𝑛=1𝑏_𝑛<∞ and ∑^∞_𝑛=1𝑐_𝑛<∞. By Condition (4), we have 𝑥_𝑛+1−𝑞²≤1+𝑏_𝑛𝑥_𝑛−𝑞²+𝑐_𝑛. (2.11) Therefore, it follows from Lemma 1.4 that lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists.We now prove that lim_𝑛→∞‖𝑢_𝑛−𝑞‖ exists for each 𝑞∈Γ. Since lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists, from (2.9), we have 1+𝛼_𝑛𝜇_𝑛𝑀^∗𝛾1−𝛾‖𝐴‖²𝑇^𝑛−𝐼𝐴𝑥_𝑛²+𝛼_𝑛1−𝛼_𝑛𝑢_𝑛−𝑆^𝑛𝑢_𝑛²≤𝑥_𝑛−𝑞²−𝑥_𝑛+1−𝑞²−𝑏_𝑛𝑥_𝑛−𝑞²⟶0,as𝑛⟶∞. (2.12) This together with Condition (4) implies that lim_𝑛→∞𝑢−𝑆^𝑛𝑢_𝑛=0,lim_𝑛→∞𝑇^𝑛−𝐼𝐴𝑥_𝑛=0. (2.13)(2.14) Thus, since lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists, it follows from (2.4) and (2.14) that lim_𝑛→∞‖𝑢_𝑛−𝑞‖ exists and lim_𝑛→∞‖𝑥_𝑛−𝑞‖=lim_𝑛→∞‖𝑢_𝑛−𝑞‖.Step 2. Now we 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 (2.1) that 𝑥_𝑛+1−𝑥_𝑛=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑆^𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛾𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^𝑛𝑢_𝑛−𝑢_𝑛+𝛼_𝑛𝑢_𝑛−𝑥_𝑛=1−𝛼_𝑛𝛾𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^𝑛𝑢_𝑛−𝑢_𝑛+𝛼_𝑛𝛾𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛=𝛾𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛+𝛼_𝑛𝑆^𝑛𝑢_𝑛−𝑢_𝑛. (2.15)
In view of (2.13) and (2.14) we have that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (2.16) Similarly, it follows from (2.1), (2.14), and (2.16) that 𝑢_𝑛+1−𝑢_𝑛=𝑥_𝑛+1+𝛾𝐴^∗𝑇^𝑛+1−𝐼𝐴𝑥_𝑛+1−𝑥_𝑛+𝛾𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛾𝐴^∗𝑇^𝑛+1−𝐼𝐴𝑥_𝑛+1+𝛾𝐴^∗𝑇^𝑛−𝐼𝐴𝑥_𝑛⟶0,(𝑛⟶∞). (2.17)Step 3. Next, we prove that ‖𝑢_𝑛−𝑆𝑢_𝑛‖→0 and ‖𝐴𝑥_𝑛−𝑇𝐴𝑥_𝑛‖→0 as 𝑛→∞.
Setting 𝜂_𝑛∶=‖𝑢_𝑛−𝑆^𝑛𝑢_𝑛‖, since 𝑆 is uniformly L-Lipschitzian continuous, it follows from (2.13), (2.16), and (2.17) that 𝑢_𝑛−𝑆𝑢_𝑛≤𝑢_𝑛−𝑆^𝑛𝑢_𝑛+𝑆^𝑛𝑢_𝑛−𝑆𝑢_𝑛≤𝜂_𝑛+𝐿𝑆^𝑛−1𝑢_𝑛−𝑢_𝑛≤𝜂_𝑛+𝐿𝑆^𝑛−1𝑢_𝑛−𝑆^𝑛−1𝑢_𝑛−1+𝑆^𝑛−1𝑢_𝑛−1−𝑢_𝑛≤𝜂_𝑛+𝐿²𝑢_𝑛−𝑢_𝑛−1+𝐿𝑆^𝑛−1𝑢_𝑛−1−𝑢_𝑛−1+𝑢_𝑛−1−𝑢_𝑛≤𝜂_𝑛+𝐿(1+𝐿)𝑢_𝑛−𝑢_𝑛−1+𝐿𝜂_𝑛−1⟶0,(𝑛⟶∞). (2.18) Similarly, we have 𝐴𝑥_𝑛−𝑇𝐴𝑥_𝑛⟶0,(𝑛⟶∞). (2.19)Step 4. Finally, we prove that 𝑥_𝑛⇀𝑥^∗ and 𝑢_𝑛⇀𝑥^∗, where 𝑥^∗∈Γ. Since {𝑢_𝑛} is bounded, there exists a subsequence {𝑢_𝑛𝑖} of {𝑢_𝑛} such that 𝑢_𝑛𝑖⇀𝑥^∗(some point in 𝐻₁). From (2.18) we have lim_𝑖→∞‖𝑢_𝑛𝑖−𝑆𝑢_𝑛𝑖‖=0. Since 𝑆 is demi-closed at zero, we know that 𝑥^∗∈𝐹(𝑆). Moreover, it follows from (2.1) and (2.14) that 𝑥_𝑛𝑖=𝑢_𝑛𝑖−𝛾𝐴^∗𝑇^𝑛𝑖−𝐼𝐴𝑥_𝑛𝑖⇀𝑥^∗. (2.20) Since 𝐴 is a linear bounded operator, it gets 𝐴𝑥_𝑛𝑖⇀𝐴𝑥^∗. In view of (2.19) we have lim_𝑖→∞‖𝐴𝑥_𝑛𝑖−𝑇𝐴𝑥_𝑛𝑖‖=0.
Again since 𝑇 is demi-closed at zero, we know that 𝐴𝑥^∗∈𝐹(𝑇). This implies that 𝑥^∗∈Γ.
Assume that there exists another subsequence {𝑢_𝑛𝑘} of {𝑢_𝑛} such that {𝑢_𝑛𝑘} converges weakly to a point 𝑦^∗∈𝐻 with 𝑦^∗≠𝑥^∗. Using the same argument above, we know that 𝑦^∗∈Γ. Since each Hilbert space possesses Opial’s property, we have liminf_𝑖→∞𝑢_𝑛𝑖−𝑥^∗<liminf_𝑖→∞𝑢_𝑛𝑖−𝑦^∗=lim_𝑛→∞𝑢_𝑛−𝑦^∗=liminf_𝑘→∞𝑢_𝑛𝑘−𝑦^∗<liminf_𝑘→∞𝑢_𝑛𝑘−𝑥^∗=lim_𝑛→∞𝑢_𝑛−𝑥^∗=liminf_𝑖→∞𝑢_𝑛𝑖−𝑥^∗, (2.21) which is a contradiction. This implies that {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 of conclusion (I) is completed.

Proof of Conclusion (II). Since 𝑆 is semi-compact, it follows from Step 4 that there exists a subsequence of {𝑢_𝑛𝑖} (without loss of generality, we still denote it by {𝑢_𝑛𝑖}) such that {𝑢_𝑛𝑖}→𝑢^∗∈𝐻 (some point in 𝐻). Since {𝑢_𝑛𝑖}⇀x^∗, this implies that 𝑥^∗=𝑢^∗. And so {𝑢_𝑛𝑖}→𝑥^∗∈Γ as 𝑖→∞. Since lim_𝑛→∞‖𝑥_𝑛−𝑞‖ and lim_𝑛→∞‖𝑢_𝑛−𝑞‖ exist, and lim_𝑛→∞‖𝑥_𝑛−𝑞‖=lim_𝑛→∞‖𝑢_𝑛−𝑞‖ for each 𝑞∈Γ, we know that lim_𝑛→∞‖𝑢_𝑛−𝑥^∗‖=lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖=0. This implies that {𝑥_𝑛} and {𝑢_𝑛} both converge strongly to a 𝑥^∗∈Γ. The proof is completed.

Acknowldgments

The authors would like to express their thanks to the referees for helpful suggestions and comments. This work was supported by the Natural Science Foundation of Yunnan Province (Grant no. 2011FB074), P.R. China.