A General Iterative Algorithm with Strongly Positive Operators for Strict Pseudo-Contractions

Abstract

This paper deals with a new iterative algorithm {𝑥_𝑛} with a strongly positive operator A for a k-strict pseudo-contraction T and a non-self-Lipschitzian mapping S in Hilbert spaces. Under certain appropriate conditions, the sequence {𝑥_𝑛} converges strongly to a fixed point of T, which solves some variational inequality. The results here improve and extend some recent related results.

1. Introduction

Let 𝐶 be a closed convex subset of Hilbert space 𝐻 with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖, 𝑇∶𝐶→𝐻 be a nonlinear mapping. The fixed point set of 𝑇 is denoted by Fix(𝑇); that is, Fix(𝑇)={𝑥∈𝐶,𝑇𝑥=𝑥}. Fixed point problem is very general in the sense that it includes, as spacial cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and others.

Recall that a mapping 𝑇∶𝐶↦𝐶 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. A mapping 𝑇∶𝐻↦𝐻 is said to be strongly positive, if there exists a constant 𝛾>0 such that ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖² for all 𝑥∈𝐻. In 2000, Moudafi [1] investigated the fixed point problem of nonexpansive mapping with viscosity approximation method. Let 𝑓 be a contraction on 𝐻; that is, there exists a constant 𝛼∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶; define a sequence {𝑥_𝑛} by 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛≥0, (1) where 𝑥₀ is an arbitrary starting point in 𝐻 and {𝛼_𝑛} is a sequence in (0,1). In 2004 Xu [2] proved that if the parameter {𝛼_𝑛} satisfies some approximate conditions, the sequence {𝑥_𝑛} generated by (1) converges strongly to not only a fixed point of 𝑇 but also the unique solution 𝑥^∗ of the variational inequality 𝐼−𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈Fix(𝑇). (2)

In 2010, Tian [3] considered a general hybrid steepest-descent method: 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛,∀𝑛≥0, (3) where 𝐹 is a Lipschitzian and strongly monotone operator. Under certain conditions, he proved that the sequence {𝑥_𝑛} generated by (3) converges strongly to the unique solution 𝑥^∗ of the variational inequality 𝛾𝑓−𝜇𝐹𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈Fix(𝑇). (4) On the other hand, Marino and Xu [4] introduced the following iterative scheme: 𝑥_𝑛+1=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛,∀𝑛≥0, (5) where 𝐴 is a strongly positive bounded linear operator. It was proven that under certain conditions on the parameters, the sequence {𝑥_𝑛} generated by (5) converges strongly to the unique solution 𝑥^∗ of the variational inequality 𝛾𝑓−𝐴𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈Fix(𝑇). (6)

It is well known that a typical convex minimization is that of minimizing a quadratic function on the sets of the fixed points of a nonexpansive mapping: Min_{(𝑇)𝑥∈Fix}12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩, (7) where 𝑏 is a given point of 𝐻. The solution 𝑥^∗ is also the optimality condition for the minimization problem Min_{(𝑇)𝑥∈Fix}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥), (8) where ℎ(𝑥) is a potential function for 𝛾𝑓; that is, ℎ^′(𝑥)=𝛾𝑓(𝑥),∀𝑥∈𝐻. Some authors investigated each iterative method for nonexpansive mappings for solving convex minimization problems and got some convergence results; see, for example [5–7].

In 2011, Ceng et al. [8] introduced a general iterative algorithm with strongly positive operators for nonexpansive mappings: 𝑦_𝑛=𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛,𝑥_𝑛+1=𝐼−𝛽_𝑛𝐴𝑇𝑥_𝑛+𝛽_𝑛𝑦_𝑛,∀𝑛≥0 (9) and proved that under certain conditions on the parameters the sequence {𝑥_𝑛} generated by (9) converges strongly to a fixed point 𝑥^∗ of 𝑇, which also solves the variational inequality (𝐼−𝐴)𝑥^∗,𝑥−𝑥^∗≤0,∀𝑥∈Fix(𝑇). (10)

Recently the problems of the approximation of the common fixed points of nonexpansive mappings were extended to the case of a family of finite or infinite pseudo-contractions; see, for example, [9–11].

Motivated and inspired by the above research works, we consider some fixed point problems with non-self mappings and introduce a new general iterative algorithm with strongly positive operators for 𝑘-strict pseudo-contractions which is a wider map class then the nonexpansive map class 𝑦_𝑛=𝑃_𝐶𝛼_𝑛𝜏𝑆𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛,𝑥_𝑛+1=𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑦_𝑛+𝛾_𝑛𝑥_𝑛+𝛿_𝑛𝑇𝑥_𝑛,∀𝑛≥0, (11) where 𝑃_𝐶∶𝐻→𝐶 is the metric projection, 𝑆∶𝐶↦𝐻 is a non-self-Lipschitzian mapping, 𝑇∶𝐶↦𝐶 is a 𝑘-strict pseudo-contraction, and 𝐴∶𝐶↦𝐶 is a strongly positive bounded linear operator. Under certain conditions on the parameters, we prove that the sequence {𝑥_𝑛} generated by (11) converges strongly to a fixed point 𝑥^∗ of 𝑇, which solves the variational inequality (𝐼−𝐴)𝑥^∗,𝑧−𝑥^∗≤0,∀𝑧∈Fix(𝑇). (12)

2. Preliminaries

In this section, we recall some useful definitions and lemmas for the proof of the main results.

Definition 1. A mapping 𝑇∶𝐶↦𝐶 is said to be 𝐿-Lipschitzian, if there exists a constant 𝐿>0 such that 𝑓(𝑥)−𝑓𝑦≤𝐿𝑥−𝑦∀𝑥,𝑦∈𝐶. (13) A mapping 𝑇∶𝐶↦𝐶 is said to be 𝑘-strict pseudo-contraction, if there exists a constant 𝑘∈[0,1)  such that 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦²,∀𝑥,𝑦∈𝐶. (14)
It is clear that a Lipschitzian map is a contractive map when 0<𝐿<1 and is a nonexpansive map when 𝐿=1. If 𝑘=0; then a 𝑘-strict pseudo-contraction map is a nonexpansive map.

Definition 2. A mapping 𝑃_𝐶∶𝐻↦𝐶 is said to be the metric projection, if for any 𝑥∈𝐻, there exists a unique nearest point in C denoted by 𝑃_𝐶𝑥 such that 𝑥−𝑃_𝐶𝑥≤𝑥−𝑦,∀𝑦∈𝐶. (15) And it is well known that if 𝐶 is a nonempty closed convex subset of 𝐻, then the 𝑃_𝐶 exists (e.g., see [12]).

Lemma 3 (see [13]). Let 𝑥∈𝐻 and 𝑧∈𝐶 be any points. There holds 𝑃_𝐶𝑥−𝑃_𝐶𝑦,𝑥−𝑦≥𝑃_𝐶𝑥−𝑃_𝐶𝑦²,∀𝑥,𝑦∈𝐻. (16) And 𝑧=𝑃_𝐶𝑥 if and only if there holds 𝑥−𝑧,𝑦−𝑧≤0,∀𝑦∈𝐶, (17) and if and only if there holds the relation ‖𝑥−𝑧‖²≤𝑥−𝑦²−𝑦−𝑧²,∀𝑦∈𝐶. (18)

Lemma 4 (see [9], Demiclosedness princple). 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 5 (see [14]). Let 𝜆 be a number in [0,1] and 𝜇≥0. Let 𝐹∶𝐻↦𝐻 be a t-Lipschitzian and 𝜂-strongly monotone operator on a Hilbert space. Associate with a nonexpansive mapping 𝑇∶𝐻↦𝐻 and define the mapping 𝑇^𝜆∶𝐻↦𝐻 by 𝑇^𝜆𝑥∶=𝑇𝑥−𝜆𝜇𝐹(𝑇𝑥),∀𝑥∈𝐻. (19) Then 𝑇^𝜆 is a contraction provided 𝜇≤2𝜂/𝑡²; that is, 𝑇^𝜆𝑥−𝑇^𝜆𝑦≤1−𝜆𝜇𝜂−𝜇𝑡²2𝑥−𝑦,∀𝑥,𝑦∈𝐻. (20)

Lemma 6 (see [4]). Assume that 𝐴 is a strongly positive bounded linear operator on a Hilbert space 𝐻 with coefficient 𝛾>0 and 0<𝜌≤‖𝐴‖⁻¹; then ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Lemma 7 (see [15]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶↦𝐶 be a 𝑘-strict pseudo-contractive mapping. Let 𝛾 and 𝛿 be two nonnegative real numbers such that (𝛾+𝛿)𝑘≤𝛾; then 𝛾𝑥−𝑦+𝛿𝑇𝑥−𝑇𝑦≤𝛾+𝛿𝑥−𝑦,∀𝑥,𝑦∈𝐶. (21)

Lemma 8 (see [16]). Let 𝐻 be a Hilbert space and 𝐶 a nonempty convex subset of 𝐻. Let 𝑇∶𝐶↦𝐻 be a 𝑘-strict pseudo-contractive 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 9 (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 (i) {𝛾_𝑛}⊂(0,1), ∑^∞_𝑛=1𝛾_𝑛=∞; (ii) limsup_𝑛→∞(𝛿_𝑛/𝛾_𝑛)=0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞; then lim_𝑛→∞𝛼_𝑛=0.

3. Main Results

In this section, we prove the strong convergence results on the iterative algorithm for 𝑘-strict pseudo-contractions.

Theorem 10. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, 𝑆∶𝐶↦𝐻 a non-self-L-Lipschitzian mapping, and 𝑇∶𝐶↦𝐶 a 𝑘-strict pseudo-contractive mapping such that Fix(𝑇)≠∅. Let 𝐹∶𝐶↦𝐻 be a t-Lipschitzian and 𝜂-strongly monotone mapping and 𝐴∶𝐶↦𝐶 a 𝛾-strongly positive bounded linear operator. For a given 𝑥₀∈𝐶, let the sequences {𝑥_𝑛} and {𝑦_𝑛} generated by (11), where {𝛼_𝑛}, {𝛾_𝑛}, {𝛿_𝑛}∈[0,1], satisfy the following conditions:(i)[1−𝜇(𝜂−𝜇𝑡²/2)]((1+𝑘)/(1−𝑘))≤1, 𝜇(𝜂−𝜇𝑡²/2)−𝜏𝐿>0, 𝛾∈(1,2); (ii)lim_𝑛→∞𝛾_𝑛=0, lim_𝑛→∞𝛿_𝑛=0, ∑^∞_𝑛=0𝛾_𝑛=∞, ∑^∞_𝑛=0𝛿_𝑛=∞, (𝛾_𝑛+𝛿_𝑛)𝑘≤𝛾_𝑛;(iii)lim_𝑛→∞(𝛼_𝑛/(𝛾_𝑛+𝛿_𝑛))=0, ∑^∞_𝑛=1|𝛼_𝑛−𝛼_𝑛−1|<∞, ∑^∞_𝑛=1|𝛾_𝑛−𝛾_𝑛−1|<∞, ∑^∞_𝑛=1|𝛿_𝑛−𝛿_𝑛−1|<∞.Then the sequence {𝑥_𝑛} converges strongly to a fixed point 𝑥^∗ of 𝑇, which solves the variational inequality (𝐼−𝐴)𝑥^∗,𝑧−𝑥^∗≤0,∀𝑧∈Fix(𝑇). (22)

Proof. The proof is divided into five steps.
Step  1. We first show that the sequences {𝑥_𝑛}, {𝑦_𝑛} are bounded. Take 𝑝∈Fix(𝑇), own to 𝑇∶𝐶↦𝐶 be a 𝑘-strict pseudo-contractive mapping, and define 𝐽𝑥=𝑘𝑥+(1−𝑘)𝑇𝑥. By Lemma 8  𝐽 is nonexpansive and Fix(𝐽)=Fix(𝑇); therefore 𝑇𝑥=(1/(1−𝑘))(𝐽𝑥−𝑘𝑥): 𝑇𝑥_𝑛=1−𝑇𝑝1−𝑘𝐽𝑥_𝑛−𝑘𝑥_𝑛−1=11−𝑘𝐽𝑝−𝑘𝑝1−𝑘𝐽𝑥_𝑛−𝐽𝑝−𝑘𝑥_𝑛≤−𝑝1+𝑘𝑥1−𝑘^𝑛−𝑝. (23)
Thus we immediately get that 𝑇 is a (1+𝑘)/(1−𝑘)-Lipschitzian mapping. Then we estimate ‖𝑦_𝑛−𝑝‖: 𝑦_𝑛=𝑃−𝑝_𝐶𝛼_𝑛𝜏𝑆𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛−𝑃_𝐶𝑝≤𝛼_𝑛𝜏𝑆𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛=𝛼−𝑝_𝑛𝜏𝑆𝑥_𝑛−𝜇𝐹𝑝+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛−𝐼−𝜇𝛼_𝑛𝐹≤𝑇𝑝1−𝜇𝜂−𝜇𝑡²21+𝑘𝑥1−𝑘^𝑛−𝑝+𝛼_𝑛𝜏𝑆𝑥_𝑛≤𝑥−𝜇𝐹𝑝_𝑛−𝑝+𝛼_𝑛𝜏𝑆𝑥_𝑛−𝜇𝐹𝑝. (24)
On the other hand, notice that lim_𝑛→∞𝛾_𝑛=0, lim_𝑛→∞𝛿_𝑛=0; without loss of generality, we may assume that 𝛾_𝑛+𝛿_𝑛≤‖𝐴‖⁻¹; thus 𝑥_𝑛+1=𝛾−𝑝𝐼−_𝑛+𝛿_𝑛𝐴𝑦_𝑛+𝛾_𝑛𝑥_𝑛+𝛿_𝑛𝑇𝑥_𝑛≤𝛾−𝑝𝐼−_𝑛+𝛿_𝑛𝐴𝑦_𝑛−𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑝+𝛾_𝑛𝑥_𝑛−𝑝+𝛿_𝑛𝑇𝑥_𝑛+𝛾−𝑇𝑝_𝑛+𝛿_𝑛)𝑝≤𝛾(𝐼−𝐴1−_𝑛+𝛿_𝑛𝛾𝑦_𝑛+𝛾−𝑝_𝑛+𝛿_𝑛𝑥_𝑛+𝛾−𝑝_𝑛+𝛿_𝑛‖(𝐼−𝐴)‖⋅𝑝. (25) Together with (24), we have 𝑥_𝑛+1≤𝛾−𝑝1−_𝑛+𝛿_𝑛𝛾−1𝑥_𝑛−𝑝+𝛼_𝑛𝜏𝑆𝑥_𝑛−𝜇𝐹𝑝+𝛾_𝑛+𝛿_𝑛⋅𝑝=𝛾‖(𝐼−𝐴)‖1−_𝑛+𝛿_𝑛𝛾−1𝑥_𝑛+𝛾−𝑝_𝑛+𝛿_𝑛1𝛾−1𝛾_𝑛+𝛿_𝑛×𝛼𝛾−1_𝑛𝜏𝑆𝑥_𝑛+𝛾−𝜇𝐹𝑝_𝑛+𝛿_𝑛⋅𝑝𝑥‖(𝐼−𝐴)‖≤max_𝑛,1−𝑝𝛼𝛾−1𝛾_𝑛+^𝑛𝛿_𝑛𝜏𝑆𝑥_𝑛+⋅𝑝.−𝜇𝐹𝑝‖(𝐼−𝐴)‖ (26) By conditions (ii) and (iii), we get that {𝑥_𝑛} is bounded, and so are {𝑦_𝑛},{𝑆𝑥_𝑛},{𝑇𝑥_𝑛},{𝐹𝑇𝑥_𝑛}.
Step  2. Now we prove that ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞. Denote 𝜉=𝜇(𝜂−(𝜇𝑡²/2)): 𝑦_𝑛−𝑦_𝑛−1=𝑃_𝐶𝛼_𝑛𝜏𝑆𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛−𝑃_𝐶𝛼_𝑛−1𝜏𝑆𝑥_𝑛−1+𝐼−𝜇𝛼_𝑛−1𝐹𝑇𝑥_𝑛−1≤𝛼_𝑛𝜏𝑆𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛−𝛼_𝑛−1𝜏𝑆𝑥_𝑛−1+𝐼−𝜇𝛼_𝑛−1𝐹𝑇𝑥_𝑛−1=𝛼_𝑛𝜏𝑆𝑥_𝑛−𝑆𝑥_𝑛−1+𝜏𝛼_𝑛−𝛼_𝑛−1𝑆𝑥_𝑛−1+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛−𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛−1−𝜇𝛼_𝑛−𝛼_𝑛−1𝐹𝑇𝑥_𝑛−1≤𝛼_𝑛𝜏𝐿𝑥_𝑛−𝑥_𝑛−1+1−𝛼_𝑛𝜉𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝜏𝑆𝑥_𝑛−1+𝜇𝐹𝑇𝑥_𝑛−1=1−𝛼_𝑛(𝜉−𝜏𝐿)𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑀≤𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑀, (27) where 𝑀 is a constant such that Sup𝜏𝑆𝑥_𝑛−1+𝜇𝐹𝑇𝑥_𝑛−1+𝑥_𝑛−1+𝑇𝑥_𝑛−1+2𝐴𝑦_𝑛−1𝑥≤𝑀,_𝑛+1−𝑥_𝑛=𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑦_𝑛+𝛾_𝑛𝑥_𝑛+𝛿_𝑛𝑇𝑥_𝑛−𝐼−𝛾_𝑛−1+𝛿_𝑛−1𝐴𝑦_𝑛−1+𝛾_𝑛−1𝑥_𝑛−1+𝛿_𝑛−1𝑇𝑥_𝑛−1=𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑦_𝑛−𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑦_𝑛−1+𝛾_𝑛−1−𝛾_𝑛+𝛿_𝑛−1−𝛿_𝑛𝐴𝑦_𝑛−1+𝛾_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛿_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑇𝑥_𝑛−1≤1−𝛾_𝑛+𝛿_𝑛𝛾𝑦_𝑛−𝑦_𝑛−1+𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝐴𝑦_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑥_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑇𝑥_𝑛−1≤1−𝛾_𝑛+𝛿_𝑛𝛾𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑀+𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1𝑥_𝑛−1+𝐴𝑦_𝑛−1+𝛿_𝑛−𝛿_𝑛−1𝑇𝑥_𝑛−1+𝐴𝑦_𝑛−1≤1−𝛾_𝑛+𝛿_𝑛𝛾−1𝑥_𝑛−𝑥_𝑛−1+𝑀𝛼_𝑛−𝛼_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝛿_𝑛−𝛿_𝑛−1. (28)(29) By the conditions (i), (ii), and (iii) and Lemma 9, we get ‖𝑥_𝑛+1−𝑥_𝑛‖→0 as 𝑛→∞.
Step  3. Now we prove that ‖𝑥_𝑛+1−𝑇𝑥_𝑛‖→0 as 𝑛→∞: 𝑥_𝑛+1−𝑇𝑥_𝑛=𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑦_𝑛+𝛾_𝑛𝑥_𝑛+𝛿_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛=𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑦_𝑛−𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑇𝑥_𝑛+𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑇𝑥_𝑛−1−𝛾_𝑛+𝛿_𝑛𝑇𝑥_𝑛+𝛾_𝑛𝑥_𝑛−𝑇𝑥_𝑛+𝛿_𝑛𝑇𝑥_𝑛−𝑇𝑥_𝑛≤1−𝛾_𝑛+𝛿_𝑛𝛾𝑦_𝑛−𝑇𝑥_𝑛+𝛾_𝑛+𝛿_𝑛𝑇𝑥_𝑛−𝐴𝑇𝑥_𝑛+𝛾_𝑛𝑥_𝑛−𝑇𝑥_𝑛. (30) On the other hand, 𝑦_𝑛−𝑇𝑥_𝑛=𝑃_𝐶𝛼_𝑛𝜏𝑆𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛−𝑃_𝐶𝑇𝑥_𝑛≤𝛼_𝑛𝜏𝑆𝑥_𝑛+𝐼−𝜇𝛼_𝑛𝐹𝑇𝑥_𝑛−𝑇𝑥_𝑛=𝛼_𝑛𝜏𝑆𝑥_𝑛−𝜇𝐹𝑇𝑥_𝑛. (31) Thus we have ‖𝑥_𝑛+1−𝑇𝑥_𝑛‖→0 as 𝑛→∞. Observe that 𝑥_𝑛−𝑇𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇𝑥_𝑛; (32) we immediately get ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0 as 𝑛→∞.
Step  4. Now we show that limsup_𝑛→∞⟨(𝐼−𝐴)𝑥^∗,𝑥_𝑛−𝑥^∗⟩≤0, where 𝑥^∗∈Fix(𝑇) is the unique solution of the variational inequality. Take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞(𝐼−𝐴)𝑥^∗,𝑥_𝑛−𝑥^∗=lim_𝑘→∞(𝐼−𝐴)𝑥^∗,𝑥_𝑛𝑘−𝑥^∗. (33)
Observe that the sequence {𝑥_𝑛} is bounded; without loss of generality we may assume that 𝑥_𝑛𝑘⇀𝑥^′. By Lemma 4, we get 𝑥^′∈Fix(𝑇). Therefore by Lemma 3, we have limsup_𝑛→∞(𝐼−𝐴)𝑥^∗,𝑥_𝑛−𝑥^∗=(𝐼−𝐴)𝑥^∗,𝑥^′−𝑥^∗≤0. (34)
Step  5. Next we prove that ‖𝑥_𝑛+1−𝑥^∗‖→0 as 𝑛→∞: 𝑥_𝑛+1−𝑥^∗2=𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑦_𝑛−𝐼−𝛾_𝑛+𝛿_𝑛𝐴𝑥^∗+𝛾_𝑛𝑥_𝑛−𝑥^∗+𝛿_𝑛𝑇𝑥_𝑛−𝑇𝑥^∗+𝛾_𝑛+𝛿_𝑛(𝐼−𝐴)𝑥^∗2≤(𝐼−(𝛾_𝑛+𝛿_𝑛)𝐴)(𝑦_𝑛−𝑥^∗)²+2𝛾_𝑛𝑥_𝑛−𝑥^∗+𝛿_𝑛𝑇𝑥_𝑛−𝑇𝑥^∗,𝑥_𝑛+1−𝑥^∗+2𝛾_𝑛+𝛿_𝑛(𝐼−𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛾_𝑛+𝛿_𝑛𝛾²𝑦_𝑛−𝑥^∗2+2𝛾_𝑛𝑥_𝑛−𝑥^∗+𝛿_𝑛𝑇𝑥_𝑛−𝑇𝑥^∗⋅𝑥_𝑛+1−𝑥^∗+2𝛾_𝑛+𝛿_𝑛(𝐼−𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗≤1−𝛾_𝑛+𝛿_𝑛𝛾²𝑦_𝑛−𝑥^∗2+2𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑥^∗⋅𝑥_𝑛+1−𝑥^∗+2𝛾_𝑛+𝛿_𝑛(𝐼−𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗. (35)
Notice that 𝑦_𝑛−𝑥^∗2≤𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑥_𝑛−𝑥^∗⋅𝜏𝑆𝑥_𝑛−𝜇𝐹𝑥^∗+𝛼²_𝑛𝜏𝑆𝑥_𝑛−𝜇𝐹𝑥^∗2; (36) thus 𝑥_𝑛+1−𝑥^∗2≤1−𝛾_𝑛+𝛿_𝑛𝛾²×𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑥_𝑛−𝑥^∗⋅𝜏𝑆𝑥_𝑛−𝜇𝐹𝑥^∗+𝛼²_𝑛𝜏𝑆𝑥_𝑛−𝜇𝐹𝑥^∗2×𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑥^∗2+𝑥_𝑛+1−𝑥^∗2+2𝛾_𝑛+𝛿_𝑛(𝐼−𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗.1−𝛾_𝑛+𝛿_𝑛𝑥_𝑛+1−𝑥^∗2≤1−𝛾_𝑛+𝛿_𝑛𝛾²𝑥_𝑛−𝑥^∗2+𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑥_𝑛−𝑥^∗⋅𝜏𝑆𝑥_𝑛−𝜇𝐹𝑥^∗+𝛼²_𝑛𝜏𝑆𝑥_𝑛−𝜇𝐹𝑥^∗2+2𝛾_𝑛+𝛿_𝑛(𝐼−𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗,𝑥_𝑛+1−𝑥^∗2=2𝛾1−^𝑛+𝛿_𝑛𝛾−11−𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑥^∗2+11−𝛾_𝑛+𝛿_𝑛×𝛾_𝑛+𝛿_𝑛²𝛾²𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑥_𝑛−𝑥^∗⋅𝜏𝑆𝑥_𝑛−𝜇𝐹𝑥^∗+𝛼²_𝑛𝜏𝑆𝑥_𝑛−𝜇𝐹𝑥^∗2+2𝛾_𝑛+𝛿_𝑛(𝐼−𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗=2𝛾1−^𝑛+𝛿_𝑛𝛾−11−𝛾_𝑛+𝛿_𝑛𝑥_𝑛−𝑥^∗2+2𝛾_𝑛+𝛿_𝑛𝛾−11−𝛾_𝑛+𝛿_𝑛×12𝛾_𝑛+𝛿_𝑛×𝛾𝛾−1_𝑛+𝛿_𝑛²𝛾²𝑥_𝑛−𝑥^∗2+2𝛼_𝑛𝑥_𝑛−𝑥^∗⋅𝜏𝑆𝑥_𝑛−𝜇𝐹𝑥^∗+𝛼²_𝑛𝜏𝑆𝑥_𝑛−𝜇𝐹𝑥^∗2+2𝛾_𝑛+𝛿_𝑛×(𝐼−𝐴)𝑥^∗,𝑥_𝑛+1−𝑥^∗. (37) By the conditions (ii), (iii) and Lemma 9, we conclude that ‖𝑥_𝑛−𝑥^∗‖→0 as 𝑛→∞, which solves the variational inequality ⟨(𝐼−𝐴)𝑥^∗,𝑧−𝑥^∗⟩≤0,forall𝑧∈Fix(𝑇). This completes the proof.

Remark 11. The iterative algorithm in Theorem 10 here is a new approximating method, and Lemma 7 plays a key role in the proof of the main results which makes the proof simple.

Remark 12. The results in this paper improve and extend some recent related results. For example, Theorem 10 here improves and extends Theorem  3.2 in [8] in the following ways:(i)the nonexpansive mapping 𝑇∶𝐶↦𝐶 in [8] is extended to the case of 𝑘-strict pseudo-contractions 𝑇∶𝐶↦𝐶;(ii)the self-contraction 𝑓∶𝐶↦𝐶 in [8] is extended to the case of a (possiblly non-self) Lipschitzian mapping 𝑆∶𝐶↦𝐻.

Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This work is partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696, Y6110270) and the National Natural Science Foundation (11071169, 11271330).