Strong Convergence of the Iterative Methods for Hierarchical Fixed Point Problems of an Infinite Family of Strictly Nonself Pseudocontractions

Abstract

This paper deals with a new iterative algorithm for solving hierarchical fixed point problems of an infinite family of pseudocontractions in Hilbert spaces by 𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+(1−𝛽_𝑛)𝑥_𝑛, 𝑥_𝑛+1=𝑃_𝐶[𝛼_𝑛𝑓(𝑥_𝑛)+(1−𝛼_𝑛)∑^∞_𝑖=1𝜇_𝑖^(𝑛)𝑇_𝑖𝑦_𝑛], and ∀𝑛≥0, where 𝑇_𝑖∶𝐶↦𝐻 is a nonself 𝑘_𝑖-strictly pseudocontraction. Under certain approximate conditions, the sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈⋂^∞_𝑖=1𝐹(𝑇_𝑖), which solves some variational inequality. The results here improve and extend some recent results.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅⟩ and norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset of 𝐻. A mapping 𝑓∶𝐶↦𝐻 is called a contraction with coefficient 𝛾 if there exits a constant 𝛾∈[0,1) such that 𝑓(𝑥)−𝑓𝑦≤𝛾𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.1)

A mapping 𝑇∶𝐶↦𝐶 is called nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1.2)

A mapping 𝑇∶𝐶↦𝐻 is called 𝑘-strictly pseudocontraction if there exits a constant 𝑘∈[0,1) such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (1.3)

Write 𝐹(𝑇) as the set of fixed points of 𝑇, that is, 𝐹(𝑇)={𝑥∈𝐶,𝑇𝑥=𝑥}. In 2000, Moudafi [1] introduced an iterative scheme for nonexpansive mappings 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥0, (1.4) where 𝑓 be a contraction on 𝐻 and the sequence {𝑥_𝑛} started with arbitrary initial 𝑥₀∈𝐻. In 2004, Xu [2] proved that the sequence {𝑥_𝑛} generated by (1.4) converges strongly to a fixed point of 𝑇 under certain conditions on the parameters, which also solves the variational inequality 𝐼−𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐹(𝑇). (1.5)

Recently, some authors studied the problems of fixed points of nonexpansive mappings with strongly positive operators, Lipschitizian, strongly monotone operators, and extragradient methods, and many convergence results were obtained (such as, see [3–9]).

In 2008, Yao et al. [10] introduced the following iterative scheme: 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛,∀𝑛≥0, (1.6) where 𝑓 is a contraction on 𝐶 and 𝑇∶𝐶↦𝐶 is nonexpansive mapping. In 2012, Song et al. [11] analyzed the following iterative algorithm: 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝑃_𝐶𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛^∞_𝑖=1𝜇_𝑖^(𝑛)𝑇_𝑖𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝐼−𝛼_𝑛𝐹𝑦_𝑛,∀𝑛≥0, (1.7) where 𝑇_𝑖 is a 𝑘_𝑖-strictly pseudocontraction, 𝐹∶𝐶↦𝐶 is a lipschitzian and strongly monotone operator, 𝑓∶𝐶↦𝐶 is a contraction, and 𝑃_𝐶 is the metric projection from 𝐻 onto 𝐶. Under certain conditions on the parameters, the sequence {𝑥_𝑛} generated by (1.7) converges strongly to a fixed point of a countable family of 𝑘_𝑖-strictly pseudocontraction, which is the solution of some variational inequality.

On the other hand, in 2010, Yao et al. [12] introduced the iterative algorithm for solving hierarchical fixed point of nonexpansive mappings and gave the following theorem.

Theorem YCL
Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶↦𝐻 be a contraction with coefficient 𝛾∈[0,1). Suppose the following conditions are satisfied:(i) lim_𝑛→∞𝛼_𝑛=0 and Σ^∞_𝑛=0𝛼_𝑛=∞;(ii) lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0;(iii) lim_𝑛→∞(|𝛼_𝑛+1−𝛼_𝑛|/𝛼_𝑛)=0 and lim_𝑛→∞(|𝛽_𝑛+1−𝛽_𝑛|/𝛽_𝑛)=0.Then the sequence {𝑥_𝑛} generated by 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛,∀𝑛≥0, (1.8) converges strongly to a point of 𝑥^∗∈𝐻, which is the unique solution of the variational inequality 𝑥^∗∈𝐹(𝑇),⟨𝐼−𝑓𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐹(𝑇). (1.9)

Motivated and inspired by the iterative schemes (1.7) and (1.8), we introduce and study the hybrid iterative algorithm for solving some hierarchical fixed point problem of infinite family of strictly nonself pseudocontractions: 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛^∞_𝑖=1𝜇_𝑖^(𝑛)𝑇_𝑖𝑦_𝑛,∀𝑛≥0, (1.10) where 𝑆, 𝑓, and 𝑃_𝐶 are the same in (1.8), 𝑇_𝑖∶𝐶↦𝐻 is a nonself 𝑘_𝑖-strictly pseudocontraction. Under certain conditions on the parameters, the sequence {𝑥_𝑛} generated by (1.10) converges strongly to a common fixed point of infinite family of 𝑘_𝑖-strictly pseudocontractions, which solves the variational inequality 𝑥^∗∈^∞_𝑖=1𝐹𝑇_𝑖,⟨𝐼−𝑓𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈^∞_𝑖=1𝐹𝑇_𝑖. (1.11) So, our results extend and improve some results of other authors (such as [10–12]) from self-mappings to nonself-mappings, from nonexpansive mappings to 𝑘_𝑖-strictly pseudocontraction, and from one mapping to a infinite family mappings.

2. Preliminaries

In this section, we recall some basic facts that will be needed in the proof of the main results.

Lemma 2.1 (see [13] demiclosedness principle). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝑇∶𝐶↦𝐶 be a nonexpansive mapping with 𝐹(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦; in particular if 𝑦=0, then 𝑥∈𝐹(𝑇).

Lemma 2.2 (see [9]). Let 𝑥∈𝐻 and 𝑧∈𝐶 be any points. The following results hold:(1) that 𝑧=𝑃_𝐶𝑥 if and only if there holds the relation: 𝑥−𝑦,𝑦−𝑧≤0,∀𝑦∈𝐶; (2.1)(2) that 𝑧=𝑃_𝐶𝑥 if and only if there holds the relation: ‖𝑥−𝑧‖²≤𝑥−𝑦²−𝑦−𝑧²,∀𝑦∈𝐶; (2.2)(3) there holds the ration: 𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦≥𝑃_𝐶𝑥−𝑃_𝐶𝑦²,∀𝑥,𝑦∈𝐻. (2.3)

Lemma 2.3 (see [14]). For all 𝑥,𝑦∈𝐻, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑥+𝑦. (2.4)

Lemma 2.4 (see [3]). Let 𝑓∶𝐶↦𝐻 be a contraction with coefficient 𝛾∈[0,1) and let 𝑇∶𝐶↦𝐶 be a nonexpansive mapping. Then for all 𝑥,𝑦∈𝐶:(1) the mapping (𝐼−𝑓) is strongly monotone with coefficient (1−𝛾), that is, 𝑥−𝑦,𝐼−𝑓𝑥−𝐼−𝑓𝑦≥1−𝛾𝑥−𝑦², (2.5)(2) the mapping (𝐼−𝑇) is monotone: 𝑥−𝑦,(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦≥0. (2.6)

Lemma 2.5 (see [15]). Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty convex subset of 𝐻. Let 𝑇∶𝐶↦𝐻 be a 𝑘-strictly pseudocontractive mapping with 𝐹(𝑇)≠∅. Then 𝐹(𝑃_𝐶𝑇)=𝐹(𝑇).

Lemma 2.6 (see [16]). Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty convex subset of 𝐻. Let 𝑇∶𝐶↦𝐻 be a 𝑘-strictly pseudocontractive mapping. Define a mapping 𝐽𝑥=𝛿𝑥+(1−𝛿)𝑇𝑥 for all 𝑥∈𝐶. Then as 𝛿∈[𝑘,1), 𝐽 is a nonexpansive mapping such that 𝐹(𝐽)=𝐹(𝑇).

Lemma 2.7 (see [11]). Let 𝐻 be a Hilbert space and let 𝐶 be a nonempty convex subset of 𝐻. Assume that 𝑇_𝑖∶𝐶↦𝐻 is a countable family of 𝑘_𝑖-strictly pseudocontraction for some 0≤𝑘_𝑖<1 and sup{𝑘_𝑖∶𝑖∈𝑁}<1 such that ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Assume that {𝜇_𝑖} is a positive sequence such that ∑^∞_𝑖=1𝜇_𝑖=1. Then ∑^∞_𝑖=1𝜇_𝑖𝑇_𝑖∶𝐶↦𝐻 is a 𝑘-strictly pseudocontraction with coefficient 𝑘=sup{𝑘_𝑖∶𝑖∈𝑁} and 𝐹(∑^∞_𝑖=1𝜇_𝑖𝑇_𝑖)=⋂^∞_𝑖=1𝐹(𝑇_𝑖).

Lemma 2.8 (see [17]). Let {𝛼_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝛼_𝑛+1≤(1−𝛾_𝑛)𝛼_𝑛+𝛿_𝑛, where (1){𝛾_𝑛}⊂(0,1), Σ^∞_𝑛=1𝛾_𝑛=∞;(2)limsup_𝑛→∞(𝛿_𝑛/𝛾_𝑛)≤0 or Σ^∞_𝑛=1|𝛿_𝑛|<∞, then lim_𝑛→∞𝛼_𝑛=0.

3. Main Results

In this section, we prove some strong convergence results on the iterative algorithm for solving hierarchical fixed point problem.

Theorem 3.1. Let 𝐻 be a real Hilbert space and let 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑓∶𝐶↦𝐻 be a (possibly nonself) contraction with coefficient 𝛾∈[0,1), and let 𝑆∶𝐶↦𝐶 be a nonexpansive mapping. Let 𝑇_𝑖∶𝐶↦𝐻 be a countable family of 𝑘_𝑖-strictly (possibly nonself) pseudocontraction with 0≤𝑘_𝑖≤𝑘<1 such that ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Let the sequence {𝑥_𝑛} be generated by (1.10) with {𝛼_𝑛}, {𝛽_𝑛} in [0,1]. Suppose for each 𝑛, ∑^∞_𝑖=1𝜇_𝑖^(𝑛)=1, for all 𝑛 and 𝜇_𝑖^(𝑛)≥0, for all𝑖∈𝑁. Assume that the parameters satisfy the following conditions:(i) lim_𝑛→∞𝛼_𝑛=0 and Σ^∞_𝑛=0𝛼_𝑛=∞;(ii) lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0;(iii) lim_𝑛→∞(|𝛼_𝑛+1−𝛼_𝑛|/𝛼_𝑛)=0, lim_𝑛→∞(|𝛽_𝑛+1−𝛽_𝑛|/𝛼_𝑛)=0, and lim_𝑛→∞(∑^∞_𝑖=1|𝜇_𝑖^(𝑛)−𝜇_𝑖^(𝑛−1)|/𝛼_𝑛)=0;(iv) lim_𝑛→∞∑^∞_𝑖=1|𝜇_𝑖^(𝑛)−𝜇_𝑖|=0 and ∑^∞_𝑖=1𝜇_𝑖=1(𝜇_𝑖>0).
Then the sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈⋂^∞_𝑖=1𝐹(𝑇_𝑖), which solves the variational inequality 𝐼−𝑓𝑥^∗,𝑥−𝑥∗≥0,∀𝑥∈^∞_𝑖=1𝐹𝑇_𝑖. (3.1)

Proof. The proof is divided into four steps.
Step 1. We show that the sequences {𝑥_𝑛} and {𝑦_𝑛} are bounded.
For each 𝑛≥0, write 𝐵_𝑛=∑^∞_𝑖=1𝜇_𝑖^(𝑛)𝑇_𝑖 and by Lemma 2.7, we have 𝐵_𝑛 is a 𝑘-strictly pseudocontraction on 𝐶 and 𝐹(𝐵_𝑛)=⋂^∞_𝑖=1𝐹(𝑇_𝑖), for all 𝑛∈𝑁. Therefore, the iterative algorithm (1.10) can be written as 𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛,∀𝑛≥0. (3.2)
By condition (ii), without loss of generality, we may assume 𝛽_𝑛≤𝛼_𝑛, for all 𝑛≥0. Take 𝑝∈⋂^∞_𝑖=1𝐹(𝑇_𝑖) and we estimate ‖𝐵_𝑛𝑦_𝑛−𝑝‖. For fixed approximate 𝛿∈[𝑘,1), define a mapping 𝐽𝑥=𝛿𝑥+(1−𝛿)𝐵_𝑛𝑥 and by Lemma 2.6, 𝐽 is a nonexpansive mapping and Fix(𝐽)=Fix(𝐵_𝑛). So 𝐵_𝑛𝑦_𝑛=1−𝑝1−𝛿𝐽𝑦_𝑛−𝛿𝑦_𝑛≤1−𝑝1−𝛿𝐽𝑦_𝑛−𝐽𝑝+𝛿𝑦_𝑛≤−𝑝1+𝛿𝑦1−𝛿^𝑛−𝑝. (3.3)
Together with (3.2) and (3.3), we get 𝑥_𝑛+1−𝑝=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛−𝑃_𝐶𝑝≤𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑝+𝛼_𝑛𝑓𝑝−𝑝+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛−𝑝≤𝛼_𝑛𝛾𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝+1−𝛼_𝑛1+𝛿𝑦1−𝛿^𝑛−𝑝≤𝛼_𝑛𝛾𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝+1−𝛼_𝑛1+𝛿𝛽1−𝛿^𝑛𝑆𝑥_𝑛+−𝑝1−𝛼_𝑛1+𝛿1−𝛿1−𝛽_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝑥_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝑝+1−𝛼_𝑛1+𝛿𝛽1−𝛿^𝑛𝑆𝑥_𝑛+−𝑆𝑝1−𝛼_𝑛1+𝛿𝛽1−𝛿^𝑛𝑆𝑝−𝑝+1−𝛼_𝑛1+𝛿1−𝛿1−𝛽_𝑛𝑥_𝑛−𝑝≤𝛼_𝑛𝛾𝑥_𝑛−𝑝+1−𝛼_𝑛1+𝛿𝑥1−𝛿^𝑛−𝑝+1+𝛿𝛼1−𝛿^𝑛𝑆𝑝−𝑝+𝛼_𝑛𝑓𝑝=−𝑝1−𝛼_𝑛1+𝛿𝑥1−𝛿−𝛾_𝑛−𝑝+𝛼_𝑛1+𝛿11−𝛿−𝛾((1+𝛿)/(1−𝛿))−𝛾1+𝛿+𝑓𝑝.1−𝛿𝑆𝑝−𝑝−𝑝 (3.4)
Therefore, we obtain 𝑥_𝑛−𝑝≤max𝑥₀,1−𝑝((1+𝛿)/(1−𝛿))−𝛾1+𝛿+𝑓𝑝,1−𝛿𝑆𝑝−𝑝−𝑝 (3.5) which gives the results that the sequence {𝑥_𝑛} is bounded and so are {𝑓(𝑥_𝑛)}, {𝑦_𝑛}, {𝐵_𝑛𝑥_𝑛}, {𝐵_𝑛𝑦_𝑛}.
Step 2. Now we show that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞. Let 𝑣_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛. (3.6) Next we estimate ‖𝑥_𝑛+1−𝑥_𝑛‖. From (3.2),we have 𝑥_𝑛+1−𝑥_𝑛=𝑃_𝐶𝑣_𝑛−𝑃_𝐶𝑣_𝑛−1≤𝑣_𝑛−𝑣_𝑛−1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−1−𝛼_𝑛−1𝐵_𝑛−1𝑦_𝑛−1=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛−𝐵_𝑛𝑦_𝑛−1+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛−1+1−𝛼_𝑛−1𝐵_𝑛𝑦_𝑛−1−𝐵_𝑛−1𝑦_𝑛−1≤𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1−𝐵_𝑛𝑦_𝑛−1+1−𝛼_𝑛1+𝛿𝑦1−𝛿^𝑛−𝑦_𝑛−1+1−𝛼_𝑛−1^∞_𝑖=1𝜇_𝑖^(𝑛)−𝜇_𝑖^(𝑛−1)𝑇_𝑖𝑦_𝑛−1≤𝛼_𝑛𝛾𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛1+𝛿𝑦1−𝛿^𝑛−𝑦_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+^∞_𝑖=1𝜇_𝑖^(𝑛)−𝜇_𝑖^(𝑛−1), (3.7) where 𝑀 is a constant such that sup_𝑛∈𝑁𝑓𝑥_𝑛−1−𝐵_𝑛𝑦_𝑛−1+𝑆𝑥_𝑛−1−𝑥_𝑛−1+𝑇_𝑖𝑦_𝑛−1≤𝑀. (3.8)
From (3.2), we also obtain 𝑦_𝑛−𝑦_𝑛−1=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1−1−𝛽_𝑛−1𝑥_𝑛−1=𝛽_𝑛𝑆𝑥_𝑛−𝑆𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−1−𝛽_𝑛𝑥_𝑛−1=𝛽_𝑛𝑆𝑥_𝑛−𝑆𝑥_𝑛−1+1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1−𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑀. (3.9) Together with (3.7) and (3.9), we have 𝑥_𝑛+1−𝑥_𝑛≤𝑣_𝑛−𝑣_𝑛−1≤𝛼_𝑛𝛾𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛1+𝛿𝑥1−𝛿^𝑛−𝑥_𝑛−1+1−𝛼_𝑛𝛽_𝑛−𝛽_𝑛−11+𝛿𝑀+𝛼1−𝛿_𝑛−𝛼_𝑛−1𝑀+1−𝛼_𝑛−1^∞_𝑖=1𝜇_𝑖^(𝑛)−𝜇_𝑖^(𝑛−1)𝑀≤1−𝛼_𝑛1+𝛿𝑥1−𝛿−𝛾_𝑛−𝑥_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−11+𝛿+1−𝛿^∞_𝑖=1𝜇_𝑖^(𝑛)−𝜇_𝑖^(𝑛−1)=1−𝛼_𝑛1+𝛿𝑥1−𝛿−𝛾_𝑛−𝑥_𝑛−1+𝑀𝛼_𝑛1+𝛿×11−𝛿−𝛾𝛼_𝑛)/(×𝛼((1+𝛿1−𝛿))−𝛾_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−11+𝛿+1−𝛿^∞_𝑖=1𝜇_𝑖^(𝑛)−𝜇_𝑖^(𝑛−1). (3.10) By Lemma 2.8 and conditions (i)–(iii), we immediately get ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞.
Step 3. Next we prove that ‖𝑥_𝑛−𝑃_𝐶(∑^∞_𝑖=1𝜇_𝑖𝑇_𝑖)𝑥_𝑛‖→0 as 𝑛→∞.
Let 𝐵=∑^∞_𝑖=1𝜇_𝑖𝑇_𝑖. By Lemma 2.7 and condition (iv), we get the results that 𝐵∶𝐶↦𝐻 is a 𝑘-strictly pseudocontraction with 𝐹(𝐵)=⋂^∞_𝑖=1𝐹(𝑇_𝑖) and 𝐵_𝑛𝑥→𝐵𝑥 as 𝑛→∞, for any 𝑥∈𝐶, 𝑥_𝑛−𝑃_𝐶𝐵𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑃_𝐶𝐵𝑥_𝑛=𝑥_𝑛−𝑥_𝑛+1+𝑃_𝐶𝑣_𝑛−𝑃_𝐶𝐵𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛−𝐵_𝑛𝑥_𝑛+𝐵_𝑛𝑥_𝑛−𝐵𝑥_𝑛=𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑓𝑥_𝑛−𝐵_𝑛𝑥_𝑛+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛−𝐵_𝑛𝑥_𝑛+𝐵_𝑛𝑥_𝑛−𝐵𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑓𝑥_𝑛−𝐵_𝑛𝑥_𝑛+1−𝛼_𝑛1+𝛿𝑦1−𝛿^𝑛−𝑥_𝑛+𝐵_𝑛𝑥_𝑛−𝐵𝑥_𝑛=𝑥_𝑛−𝑥_𝑛+1+𝛼_𝑛𝑓𝑥_𝑛−𝐵_𝑛𝑥_𝑛+1−𝛼_𝑛1+𝛿𝛽1−𝛿^𝑛𝑆𝑥_𝑛−𝑥_𝑛+𝐵_𝑛𝑥_𝑛−𝐵𝑥_𝑛. (3.11) Because 𝛼_𝑛→0,𝛽_𝑛→0, ‖𝑥_𝑛+1−𝑥_𝑛‖→0, and 𝐵_𝑛𝑥→𝐵𝑥, so we obtain ‖𝑥_𝑛−∑^∞_𝑖=1𝜇_𝑖𝑇_𝑖𝑥_𝑛‖→0 as 𝑛→∞.
Step 4. Now we show that limsup_𝑛→∞⟨𝑓(𝑥^∗)−𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0, where 𝑥^∗=𝑃_𝐹(𝐵)𝑓(𝑥^∗).
Since the sequence {𝑥_𝑛} is bounded, we take 𝑠 subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞⟨𝑓(𝑥^∗)−𝑥^∗,𝑥_𝑛−𝑥^∗⟩=lim_𝑘→∞⟨𝑓(𝑥^∗)−𝑥^∗,𝑥_𝑛𝑘−𝑥^∗⟩ and 𝑥_𝑛𝑘0𝑥00086⇀𝑥^′. Notice that ‖𝑥_𝑛−𝑃_𝐶𝐵𝑥_𝑛‖→0 and by Lemmas 2.1 and 2.5, we have 𝑥^′∈Fix(𝑃_𝐶𝐵)=𝐹(𝐵)=⋂^∞_𝑖=1𝐹(𝑇_𝑖). Then lim_𝑘→∞𝑓𝑥^∗−𝑥^∗,𝑥_𝑛𝑘−𝑥^∗=⟨𝑓𝑥^∗−𝑥^∗,𝑥^′−𝑥^∗⟩≤0,𝑥^′∈𝐹(𝐵). (3.12)
Now, by Lemma 2.2, we get ⟨𝑃_𝐶[𝑣_𝑛]−𝑣_𝑛,𝑃_𝐶[𝑣_𝑛]−𝑥^∗⟩≤0. Therefore, we have 𝑥_𝑛+1−𝑥^∗2=𝑃_𝐶𝑣_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗=⟨𝑃_𝐶𝑣_𝑛−𝑣_𝑛,𝑥_𝑛+1−𝑥^∗⟩+𝑣_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗=𝑃_𝐶𝑣_𝑛−𝑣_𝑛,𝑃_𝐶𝑣_𝑛−𝑥^∗+𝑣_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝑣_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗=𝛼_𝑛𝑓𝑥_𝑛−𝑓𝑥^∗,𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗+1−𝛼_𝑛𝐵_𝑛𝑦_𝑛−𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝛾𝑥_𝑛−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗+1−𝛼_𝑛𝐵_𝑛𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝐵_𝑛𝑥^∗,𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝛾𝑥_𝑛−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑆𝑥_𝑛−𝑆𝑥^∗+𝛽_𝑛𝑆𝑥^∗−𝑥^∗+1−𝛽_𝑛𝑥_𝑛−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗≤𝛼_𝑛𝛾𝑥_𝑛−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗+1−𝛼_𝑛𝑥_𝑛−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑆𝑥^∗−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗=1−1−𝛾𝛼_𝑛𝑥_𝑛−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑆𝑥^∗−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗≤1−1−𝛾𝛼_𝑛2𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+𝛼_𝑛𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗+1−𝛼_𝑛𝛽_𝑛𝑆𝑥^∗−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗. (3.13)
Hence it follows that 𝑥_𝑛+1−𝑥^∗2≤2𝛼1−1−𝛾_𝑛1+1−𝛾𝛼_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛1+1−𝛾𝛼_𝑛⟨𝑓𝑥^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗⟩+21−𝛼_𝑛𝛽_𝑛1+1−𝛾𝛼_𝑛𝑆𝑥^∗−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗=2𝛼1−1−𝛾_𝑛1+1−𝛾𝛼_𝑛𝑥_𝑛−𝑥^∗2+21−𝛾𝛼_𝑛1+1−𝛾𝛼_𝑛1𝑓𝑥1−𝛾^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗+1−𝛼_𝑛𝛽_𝑛1−𝛾𝛼_𝑛𝑆𝑥^∗−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗. (3.14)
Now, by Lemma 2.8, conditions (i)–(iii), and limsup_𝑛→∞1𝑓𝑥1−𝛾^∗−𝑥^∗,𝑥_𝑛+1−𝑥^∗+1−𝛼_𝑛𝛽_𝑛1−𝛾𝛼_𝑛𝑆𝑥^∗−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗≤0, (3.15) we have 𝑥_𝑛→𝑥^∗∈⋂^∞_𝑖=1𝐹(𝑇_𝑖) as 𝑛→∞ and 𝑥^∗ also solves the variational inequality 𝐼−𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈^∞_𝑖=1𝐹𝑇_𝑖. (3.16) This completes the proof.

From Theorem 3.1, if we take 𝑆=𝐼 or 𝛽_𝑛=0, for all 𝑛∈𝑁, we get the following corollary.

Corollary 3.2. Let 𝐻 be a real Hilbert space and 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑓∶𝐶↦𝐻 be a (possibly nonself) contraction with coefficient 𝛾∈[0,1) and let 𝑇_𝑖∶𝐶↦𝐻 be a countable family of 𝑘_𝑖-strictly (possibly nonself) pseudocontraction with 0≤𝑘_𝑖<1 and sup{𝑘_𝑖∶𝑖∈𝑁}<1 such that ⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Let the sequence {𝑥_𝑛} be generated by 𝑥₀𝑥=𝑥∈𝐶,_𝑛+1=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛^∞_𝑖=1𝜇_𝑖^(𝑛)𝑇_𝑖𝑥_𝑛,∀𝑛≥0, (3.17) with {𝛼_𝑛} in [0,1]. Suppose for each 𝑛, ∑^∞_𝑖=1𝜇_𝑖^(𝑛)=1, for all 𝑛 and 𝜇_𝑖^(𝑛)≥0, for all 𝑖∈𝑁. Assume that the parameters satisfied the following conditions:(i) lim_𝑛→∞𝛼_𝑛=0 and Σ^∞_𝑛=0𝛼_𝑛=∞;(ii) lim_𝑛→∞(|𝛼_𝑛+1−𝛼_𝑛|/𝛼_𝑛)=0, and lim_𝑛→∞((∑^∞_𝑖=1|𝜇_𝑖^(𝑛)−𝜇_𝑖^(𝑛−1)|)/𝛼_𝑛)=0;(iii) lim_𝑛→∞∑^∞_𝑖=1|𝜇_𝑖^(𝑛)−𝜇_𝑖|=0 and ∑^∞_𝑖=1𝜇_𝑖=1(𝜇_𝑖>0).
Then the the sequence {𝑥_𝑛} converges strongly to 𝑥^∗∈⋂^∞_𝑖=1𝐹(𝑇_𝑖), which solves the variational inequality 𝐼−𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈^∞_𝑖=1𝐹𝑇_𝑖. (3.18)

Remark 3.3. Theorem 3.1 extends and improves Theorem YCL in the following way. The nonexpasnsive self-mapping 𝑇∶𝐶↦𝐶 is extended to a infinite family of nonself 𝑘_𝑖-strictly pseudocontraction 𝑇_𝑖∶𝐶↦𝐻. If we take 𝑘_𝑖=0,𝑖∈𝑁 in Theorem 3.1, then 𝐵=∑^∞_𝑖=1𝜇_𝑖𝑇_𝑖 reduces to a nonexpasnsive (possibly nonself) mapping, thus Theorem 3.1 reduces to Theorem YCL.

Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This paper was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and NSFC (11071169).