A Strong Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces

Abstract

Demiclosedness principle for total asymptotically pseudocontractive mappings in Hilbert spaces is established. The strong convergence to a fixed point of total asymptotically pseudocontraction in Hilbert spaces is obtained based on demiclosedness principle, metric projective operator, and hybrid iterative method. The main results presented in this paper extend and improve the corresponding results of Zhou (2009), Qin, Cho, and Kang (2011) and of many other authors.

1. Introduction

Throughout this article we assume that 𝐻 is a real Hilbert space, whose inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively; 𝐶 is a nonempty closed convex subset of 𝐻; ℕ and ℝ⁺ denote the natural number set and the set of nonnegative real numbers, respectively. Let 𝑇∶𝐶→𝐶 be a nonlinear mapping; 𝐹(𝑇) denotes the set of fixed points of mapping 𝑇. We use “→” to stand for strong convergence and “⇀” for weak convergence.

Recently, the iterative approximation of fixed points for asymptotically pseudocontractive mappings, total asymptotically pseudocontractive mappings in Hilbert, or Banach spaces has been studied extensively by many authors, see for example [1–5].

The asymptotically pseudocontractions and total asymptotically pseudo-contractions are defined as follows.

Definition 1.1 (see [3]). 𝑇∶𝐶→𝐶 is said to be asymptotically pseudocontraction if there exists a sequence {𝑘_𝑛} of positive real numbers with 𝑘_𝑛≥1, lim_𝑛→+∞𝑘_𝑛=1, such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑥−𝑦≤𝑘_𝑛𝑥−𝑦² (1.1) holds; in [4], 𝑇∶𝐶→𝐶 is said to be total asymptotically pseudo-contraction if there exists sequences {𝜇_𝑛},{𝜈_𝑛} with 𝜇_𝑛,𝜈_𝑛→0 as 𝑛→∞ and strictly increasing continuous functions 𝜓∶ℝ⁺→ℝ⁺ with 𝜓(0)=0, such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑥−𝑦≤𝑥−𝑦²+𝜇_𝑛𝜓𝑥−𝑦+𝜈_𝑛,𝑛∈ℕ (1.2) holds; Zhou [3, page 3144] have proved the following Theorem.

Theorem Zhou
Let 𝐶 be a bounded and closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian asymptotically pseudo-contraction with a fixed point. Suppose the control sequence 𝛼_𝑛 is chosen so that 𝛼_𝑛∈[𝑎,𝑏], for some 𝑎,𝑏∈(0,1/(1+𝐿)). Let 𝑥_𝑛 be a sequence generated in the following manner: 𝑥₀𝑦∈𝐶,chosenarbitrarily;𝑛≥0,_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝛼_𝑛1−(1+𝐿)𝛼_𝑛𝑥_𝑛−𝑇^𝑛𝑥_𝑛²≤𝑥_𝑛−𝑧,𝑦_𝑛−𝑇^𝑛𝑦_𝑛+𝑘_𝑛−1(diam𝐶)²,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛𝑥≥0,_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀. (1.3) Then the iterative sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥₀.

Qin et al. [5] introduced the class of total asymptotically pseudocontractive mappings in Hilbert spaces and established the following weak convergence theorem of fixed points.

Theorem Qin
Let 𝐶 be a bounded and closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian total asymptotically pseudo-contractive mapping. Assume that 𝐹(𝑇) is nonempty and there exist positive constants 𝑀 and 𝑀^∗ such that 𝜓(𝜆)≤𝑀^∗𝜆² for all 𝜆≥𝑀. Let 𝑥_𝑛 be a sequence generated in the following manner: 𝑥₁𝑦∈𝐶,_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑦_𝑛,∀𝑛≥1, (1.4) where 𝛼_𝑛 and 𝛽_𝑛 are sequences in (0,1). Assume that the following restrictions are satisfied:(a)Σ^∞_𝑛=1𝜇_𝑛<∞ and Σ^∞_𝑛=1𝜈_𝑛<∞.(b)𝑎<𝛼_𝑛<𝛽_𝑛<𝑏 for some 𝑎>0 and some 𝑏∈(0,𝐿⁻²[1+𝐿²−1]).Then the iterative sequence {𝑥_𝑛} converges weakly to fixed point of 𝑇.

The purpose of this article is to prove the strong convergence for total asymptotically pseudo-contraction in Hilbert spaces. The results presented in the article improve and extend the corresponding results of Zhou [3], Qin et al. [5], and many other authors.

2. Preliminaries

A mapping 𝑇∶𝐶→𝐶 is said to be uniformly 𝐿-Lipschitzian if there exists some 𝐿>0 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝐿𝑥−𝑦 (2.1) holds for all 𝑥,𝑦∈𝐶 and for all 𝑛∈ℕ. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. For every point 𝑥∈𝐻 there exists a unique nearest point in 𝐶, denoted by 𝑃_𝐶𝑥, such that ‖𝑥−𝑃_𝐶𝑥‖≤‖𝑥−𝑦‖ holds for all 𝑦∈𝐶, where 𝑃_𝐶 is called the metric projection of 𝐻 onto 𝐶.

In order to prove the results of this article, we will need the following lemmas.

Lemma 2.1 (see [6]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Given 𝑥∈𝐻 and 𝑧∈𝐶, then 𝑧=𝑃_𝐶𝑥 if and only if there holds the relation ⟨𝑥−𝑧,𝑦−𝑧⟩≤0,∀𝑦∈𝐶. (2.2)

Lemma 2.2 (see [6]). Let C be a nonempty closed convex subset of a real Hilbert space H and 𝑃_𝐶∶𝐻→𝐶 the metric projection from H onto C. Then the following inequality holds: 𝑦−𝑃_𝐶𝑥²+𝑥−𝑃_𝐶𝑥²≤𝑥−𝑦²,∀𝑥∈𝐻,𝑦∈𝐶. (2.3)

3. Main Results

Theorem 3.1 (demiclosedness principle). Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian and total asymptotically pseudo-contraction. Suppose there exists 𝑀^∗>0, such that 𝜓(𝜉_𝑛)≤𝑀^∗𝜉_𝑛, then 𝐼−𝑇 is demiclosed at zero, where 𝐼 is the identical mapping.

Proof. Assume that {𝑥_𝑛}⊂𝐶, with 𝑥_𝑛⇀𝑥 and 𝑥_𝑛−𝑇𝑥_𝑛→0 as 𝑛→∞. We want to prove 𝑥∈𝐶 and 𝑥=𝑇𝑥. Since 𝐶 is a closed convex subset of 𝐻, so 𝑥∈𝐶. In the following we prove 𝑥=𝑇𝑥.
Now we choose 𝛼∈(0,1/(1+𝐿)) and let 𝑦_𝛼,𝑚=(1−𝛼)𝑥+𝛼𝑇^𝑚𝑥 for arbitrary fixed 𝑚≥1. Because 𝑇 is uniformly 𝐿-Lipschitzian, we have 𝑥_𝑛−𝑇^𝑚𝑥_𝑛≤𝑥_𝑛−𝑇𝑥_𝑛+𝑇𝑥_𝑛−𝑇²(𝑡)𝑥_𝑛+⋯+𝑇^𝑚−1𝑥_𝑛−𝑇^𝑚𝑥_𝑛≤𝑚𝐿𝑥_𝑛−𝑇𝑥_𝑛→0,as𝑛→∞. (3.1) Since 𝑇 is total asymptotically pseudo-contraction, we have 𝑥−𝑦_𝛼,𝑚,𝐼−𝑇^𝑚𝑦_𝛼,𝑚=𝑥−𝑥_𝑛,𝐼−𝑇^𝑚𝑦_𝛼,𝑚+𝑥_𝑛−𝑦_𝛼,𝑚,𝐼−𝑇^𝑚𝑦_𝛼,𝑚=⟨𝑥−𝑥_𝑛,𝐼−𝑇^𝑚𝑦_𝛼,𝑚⟩+𝑥_𝑛−𝑦_𝛼,𝑚,𝐼−𝑇^𝑚𝑥_𝑛+⟨𝑥_𝑛−𝑦_𝛼,𝑚,𝐼−𝑇^𝑚𝑦_𝛼,𝑚−𝐼−𝑇^𝑚𝑥_𝑛⟩≤⟨𝑥−𝑥_𝑛,𝐼−𝑇^𝑚𝑦_𝛼,𝑚⟩+⟨𝑥_𝑛−𝑦_𝛼,𝑚,𝐼−𝑇^𝑚𝑥_𝑛⟩+𝜇_𝑚𝜓𝑥_𝑛−𝑦_𝛼,𝑚+𝜈_𝑚. (3.2) By assumption 𝑥_𝑛⇀𝑥, 𝑥_𝑛−𝑇𝑥_𝑛→0 and ‖𝑥_𝑛−𝑇^𝑚𝑥_𝑛‖→0 as 𝑛→∞, we have ⟨𝑥−𝑦_𝛼,𝑚,𝐼−𝑇^𝑚𝑦_𝛼,𝑚⟩≤𝜇_𝑚𝜓𝑥_𝑛−𝑦_𝛼,𝑚+𝜈_𝑚≤𝜇_𝑚𝑀^∗𝑥_𝑛−𝑦_𝛼,𝑚+𝜈_𝑚≤𝜇_𝑚𝑀^∗(diam𝐶)+𝜈_𝑚. (3.3) By the 𝐿-Lipschitz of 𝑇 and the definition of 𝑦_𝛼,𝑚, we have ⟨𝑥−𝑦_𝛼,𝑚,𝐼−𝑇^𝑚𝑥−𝐼−𝑇^𝑚𝑦_𝛼,𝑚⟩≤(1+𝐿)𝑥−𝑦_𝛼,𝑚²≤(1+𝐿)𝛼²𝑥−𝑇^𝑚𝑥². (3.4) Thus we have 𝑥−𝑇^𝑚𝑥²=𝑥−𝑇^𝑚𝑥,𝑥−𝑇^𝑚𝑥=1𝛼𝑥−𝑦_𝛼,𝑚,𝑥−𝑇^𝑚𝑥=1𝛼𝑥−𝑦_𝛼,𝑚,𝑥−𝑇^𝑚𝑥−𝑦_𝛼,𝑚−𝑇^𝑚𝑦_𝛼,𝑚+1𝛼𝑥−𝑦_𝛼,𝑚,𝑦_𝛼,𝑚−𝑇^𝑚𝑦_𝛼,𝑚≤𝛼(1+𝐿)𝑥−𝑇^𝑚𝑥²+1𝛼𝑥−𝑦_𝛼,𝑚,𝑦_𝛼,𝑚−𝑇^𝑚𝑦_𝛼,𝑚≤𝛼(1+𝐿)𝑥−𝑇^𝑚𝑥²+1𝛼𝜇_𝑚𝑀^∗(diam𝐶)+𝜈_𝑚, (3.5) which implies that 𝛼[1−𝛼(1+𝐿)]𝑥−𝑇^𝑚𝑥²≤𝜇_𝑚𝑀^∗(diam𝐶)+𝜈_𝑚,∀𝑚∈ℕ. (3.6) When 𝑚→∞,𝜇_𝑚,𝜈_𝑚→0, so we have ‖𝑥−𝑇^𝑚𝑥‖→0,𝑚→∞, that is, 𝑇^𝑚𝑥→𝑥,𝑚→∞, so 𝑇^𝑚+1𝑥→𝑇𝑥,𝑚→∞. By the continuity of 𝑇, we have 𝑇𝑥=𝑥.

Theorem 3.2. Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian and total asymptotically pseudo-contraction. Suppose there exists 𝑀^∗>0, such that 𝜓(𝜉_𝑛)≤𝑀^∗𝜉_𝑛, then 𝐹(𝑇) is a closed convex subset of C.

Proof. Since 𝑇 is uniformly 𝐿-Lipschitzian continuous, 𝐹(𝑇) is closed. We need to show that 𝐹(𝑇) is convex. We let 𝑝₁,𝑝₂∈𝐹(𝑇), and 𝑝=𝑡𝑝₁+(1−𝑡)𝑝₂ for 𝑡∈(0,1). We take 𝛼∈(0,1/(1+𝐿)) and let 𝑦_𝛼,𝑛=(1−𝛼)𝑝+𝛼𝑇^𝑛𝑝, 𝑛∈ℕ. Then for any 𝑧∈𝐹(𝑇), we have 𝑝−𝑇^𝑛𝑝²=⟨𝑝−𝑇^𝑛𝑝,𝑝−𝑇^𝑛1𝑝⟩=𝛼⟨𝑝−𝑦_𝛼,𝑛,𝑝−𝑇^𝑛=1𝑝⟩𝛼𝑝−𝑦_𝛼,𝑛,𝑝−𝑇^𝑛𝑝−𝑦_𝛼,𝑛−𝑇^𝑛𝑦_𝛼,𝑛+1𝛼𝑝−𝑦_𝛼,𝑛,𝑦_𝛼,𝑛−𝑇^𝑛𝑦_𝛼,𝑛≤1+𝐿𝛼𝑝−𝑦_𝛼,𝑛²+1𝛼𝑝−𝑧,𝑦_𝛼,𝑛−𝑇^𝑛𝑦_𝛼,𝑛+1𝛼𝑧−𝑦_𝛼,𝑛,𝑦_𝛼,𝑛−𝑇^𝑛𝑦_𝛼,𝑛≤1+𝐿𝛼𝑝−𝑦_𝛼,𝑛²+1𝛼⟨𝑝−𝑧,𝑦_𝛼,𝑛−𝑇^𝑛𝑦_𝛼,𝑛1⟩+𝛼𝜇_𝑛𝑀^∗(diam𝐶)+𝜈_𝑛≤𝛼(1+𝐿)𝑝−𝑇^𝑛𝑝²+1𝛼𝑝−𝑧,𝑦_𝛼,𝑛−𝑇^𝑛𝑦_𝛼,𝑛+1𝛼𝜇_𝑚𝑀^∗(diam𝐶)+𝜈_𝑚. (3.7) This implies that 𝛼[1−𝛼(1+𝐿)]𝑝−𝑇^𝑛𝑝²≤⟨𝑝−𝑧,𝑦_𝛼,𝑛−𝑇^𝑛𝑦_𝛼,𝑛⟩+𝜇_𝑚𝑀^∗(diam𝐶)+𝜈_𝑚. (3.8) Now we take 𝑧=𝑝₁,𝑝₂, multiplying 𝑡 and 1−𝑡 on both sides of above inequality, respectively, and adding up, and we can get 𝛼[1−𝛼(1+𝐿)]𝑝−𝑇^𝑛𝑝²≤𝜇_𝑚𝑀^∗(diam𝐶)+𝜈_𝑚. (3.9) By 𝑛→∞, we get 𝑇^𝑛𝑝→𝑝. Since 𝑇 is continuous, we have 𝑇^𝑛+1𝑝→𝑇𝑝 as 𝑛→∞, so that 𝑝=𝑇𝑝.

Theorem 3.3. Let 𝐶 be a bounded and closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian and total asymptotically pseudo-contraction. Suppose there exists 𝑀^∗>0, such that 𝜓(𝜉_𝑛)≤𝑀^∗𝜉_𝑛, 𝐹(𝑇)≠∅, 𝛼_𝑛 is a sequence in [𝑎,𝑏], where 𝑎,𝑏∈(0,1/(1+𝐿)). Let 𝑥_𝑛 be a sequence generated by 𝑥₁𝑦=𝑥∈𝐶,∀𝑛∈ℕ,_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝐻_𝑛=𝑧∈𝐶∶𝛼_𝑛1−(1+𝐿)𝛼_𝑛𝑥_𝑛−𝑇^𝑛𝑥_𝑛²≤𝑥_𝑛−𝑧,𝑦_𝑛−𝑇^𝑛𝑦_𝑛+𝜁_𝑛,𝑊_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥−𝑥_𝑛𝑥≥0,_𝑛+1=𝑃_{H𝑛∩𝑊𝑛}𝑥, (3.10) where 𝜁_𝑛=𝜇_𝑚𝑀^∗(diam𝐶)+𝜈_𝑚, then the iterative sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)𝑥 in 𝐶.

Proof. We divide the proof into seven steps.
(I) 𝑃_𝐹(𝑇) is well defined for every 𝑥∈𝐶.
By Theorem 3.2, we know 𝐹(𝑇) is closed and convex subset of 𝐶. Moreover, by our assumption that 𝐹(𝑇) is nonempty, therefore, 𝑃_𝐹(𝑇)𝑥 is well defined for every 𝑥∈𝐶.
(II) 𝐻_𝑛 and 𝑊_𝑛 are closed and convex for all 𝑛∈ℕ.
From the definitions of 𝑊_𝑛 and 𝐻_𝑛, it is obvious that 𝐻_𝑛 and 𝑊_𝑛 are closed and convex for each 𝑛∈ℕ. We omit the details.
(III) We prove 𝐹(𝑇)⊂𝐻_𝑛⋂𝑊_𝑛 for each 𝑛∈ℕ.
We first show 𝐹(𝑇)⊂𝐻_𝑛. Let 𝑧∈𝐹(𝑇), by (3.10), and the uniform 𝐿-Lipschitz continuity of 𝑇 and the total asymptotical pseudo-contractiveness of 𝑇, we have 𝑥_𝑛−𝑇^𝑛𝑥_𝑛²=𝑥_𝑛−𝑇^𝑛𝑥_𝑛,𝑥_𝑛−𝑇^𝑛𝑥_𝑛=1𝛼_𝑛𝑥_𝑛−𝑦_𝑛,𝑥_𝑛−𝑇^𝑛𝑥_𝑛=1𝛼_𝑛⟨𝑥_𝑛−𝑦_𝑛,𝑥_𝑛−𝑇^𝑛𝑥_𝑛−𝑦_𝑛−𝑇^𝑛𝑦_𝑛1⟩+𝛼_𝑛𝑥_𝑛−𝑦_𝑛,𝑦_𝑛−𝑇^𝑛𝑦_𝑛≤1+𝐿𝛼_𝑛𝑥_𝑛−𝑦_𝑛²+1𝛼_𝑛⟨𝑥_𝑛−𝑧,𝑦_𝑛−𝑇^𝑛𝑦_𝑛1⟩+𝛼_𝑛𝜇_𝑛𝑀^∗(diam𝐶)+𝜈_𝑛=(1+𝐿)𝛼_𝑛𝑥_𝑛−𝑇^𝑛𝑥_𝑛²+1𝛼_𝑛⟨𝑥_𝑛−𝑧,𝑦_𝑛−𝑇^𝑛𝑦_𝑛⟩+1𝛼_𝑛𝜇_𝑛𝑀^∗(diam𝐶)+𝜈_𝑛. (3.11) This implies that 𝛼_𝑛1−𝛼_𝑛(1+𝐿)𝑥_𝑛−𝑇^𝑛𝑥_𝑛²≤𝑥_𝑛−𝑧,𝑦_𝑛−𝑇^𝑛𝑦_𝑛+𝜇_𝑛𝑀^∗(diam𝐶)+𝜈_𝑛. (3.12) This shows that 𝑧∈𝐻_𝑛 for all 𝑛∈ℕ. So 𝐹(𝑇)⊂𝐻_𝑛 for all 𝑛∈ℕ. Next we prove 𝐹(𝑇)⊂𝑊_𝑛 for all 𝑛∈ℕ. By induction, for 𝑛=1, we have 𝐹(𝑇)⊂𝐶=𝑊₁. Assume that 𝐹(𝑇)⊂𝑊_𝑛. Since 𝑥_𝑛+1 is the projection of 𝑥 onto 𝐻_𝑛⋂𝑊_𝑛, by Lemma 2.1, we have 𝑥_𝑛+1−𝑧,𝑥−𝑥_𝑛+1≥0, (3.13) for any 𝑧∈𝐻_𝑛⋂𝑊_𝑛, by the definition of 𝑊_𝑛+1, this shows that 𝑧∈𝑊_𝑛+1. So 𝐹(𝑇)⊂𝐻_𝑛⋂𝑊_𝑛, for all 𝑛∈ℕ.
(IV) We prove that lim_𝑛→∞‖𝑥_𝑛−𝑥‖ exists.
From (3.10) and Lemma 2.1, we have 𝑥_𝑛=𝑃_𝑤𝑛𝑥, this with 𝑥_𝑛+1∈𝑊_𝑛+1 show ‖𝑥_𝑛−𝑥‖≤‖𝑥_𝑛+1−𝑥‖, for all 𝑛∈ℕ. As 𝑧∈𝐹(𝑇)⊂𝑊_𝑛, we also have ‖𝑥_𝑛−𝑥‖≤‖𝑧−𝑥‖, for all 𝑛∈ℕ. Consequently, lim_𝑛→∞‖𝑥_𝑛−𝑥‖ exists and {𝑥_𝑛} is bounded.
(V) We prove that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞.
By Lemma 2.2, we have 𝑥_𝑛+1−𝑥_𝑛²≤𝑥_𝑛+1−𝑥²−𝑥_𝑛−𝑥²→0, (3.14) as 𝑛→∞.
(VI) Now we prove ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0 as 𝑛→∞.
It follows from ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞, 𝑥_𝑛+1⊂𝐻_𝑛, {𝑦_𝑛} is bounded, {𝑇^𝑛𝑦_𝑛} is bounded, and 𝛼_𝑛∈(𝑎,𝑏) that 𝛼_𝑛1−𝛼_𝑛(1+𝐿)𝑥_𝑛−𝑇^𝑛𝑥_𝑛²≤𝑥_𝑛−𝑧,𝑦_𝑛−𝑇^𝑛𝑦_𝑛+𝜇_𝑛𝑀^∗(diam𝐶)+𝜈_𝑛≤𝑥_𝑛−𝑧𝑦_𝑛−𝑇^𝑛𝑦_𝑛+𝜇_𝑛𝑀^∗(diam𝐶)+𝜈_𝑛→0,𝑛→∞. (3.15) So ‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖→0 as 𝑛→∞. Additional 𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑇^𝑛+1𝑥_𝑛+1−𝑥_𝑛+1+𝑇^𝑛+1𝑥_𝑛−𝑇^𝑛+1𝑥_𝑛+1+𝑇𝑥_𝑛−𝑇^𝑛+1𝑥_𝑛≤(𝐿+1)𝑥_𝑛+1−𝑥_𝑛+𝑇^𝑛+1𝑥_𝑛+1−𝑥_𝑛+1+𝐿𝑥_𝑛−𝑇^𝑛𝑥_𝑛. (3.16) So ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0 as 𝑛→∞.
(VII) Finally, we prove 𝑥_𝑛→𝑃_𝐹(𝑇)𝑥 as 𝑛→∞.
Let 𝑥_𝑛𝑘 be a subsequence of 𝑥_𝑛𝑘 such that 𝑥_𝑛𝑘⇀𝑥∈𝐶, then by Theorem 3.1, we have 𝑥∈𝐹(𝑇). We let 𝜔∈𝑃_𝐹(𝑇)𝑥. For any 𝑛∈ℕ, 𝑥_𝑛+1=𝑃_{𝐻𝑛⋂𝑊𝑛}𝑥 and 𝜔∈𝑃_𝐹(𝑇)𝑥⊂𝐻_𝑛⋂𝑊_𝑛, so we get ‖𝑥_𝑛+1−𝑥‖≤‖𝜔−𝑥‖.
On the other hand, from the weak lower semicontinuity of the norm, we have ‖𝑥−𝑥‖²=‖𝑥‖²−2⟨𝑥,𝑥⟩+‖𝑥‖²≤liminf_𝑛→∞𝑥_𝑛𝑘²−2⟨𝑥_𝑛𝑘²,𝑥⟩+‖𝑥‖²=liminf_𝑛→∞𝑥_𝑛𝑘−𝑥²≤limsup_𝑛→∞𝑥_𝑛𝑘−𝑥²≤‖𝜔−𝑥‖². (3.17) From the definition of 𝑃_𝐹(𝑇)𝑥, we obtain 𝑥=𝜔 and hence limsup_𝑛→∞‖𝑥_𝑛𝑘−𝑥‖²=‖𝜔−𝑥‖². So we have limsup_𝑘→∞‖𝑥_𝑛𝑘‖=‖𝜔‖. Thus we obtain that 𝑥_𝑛𝑘 converges strongly to 𝑃_𝐹(𝑇)𝑥. Since 𝑥_𝑛𝑘 is an arbitrary weakly convergent sequence of 𝑥_𝑛, we can conclude that 𝑥_𝑛 converges strongly to 𝑃_𝐹(𝑇)𝑥. This completes the proof of Theorem 3.3.

Remark 3.4. Theorem 3.3 extends the main results of Zhou [3] and improves the main results of Qin et al. [5] and of many other authors.

Acknowledgments

The authors are grateful to the referees for their helpful and useful comments. This research is partially supported by the Fundamental Research Funds for the Central Universities (JBK120117), by the National Research Foundation of Yibin University (No. 2011B07) and by Scientific Research Fund Project of SiChuan Provincial Education Department (No. 12ZB345).