Approximate Iteration Algorithm with Error Estimate for Fixed Point of Nonexpansive Mappings

Abstract

The purpose of this article is to present a general viscosity iteration process {𝑥_𝑛} which defined by 𝑥_𝑛+1=(𝐼−𝛼_𝑛𝐴)𝑇𝑥_𝑛+𝛽_𝑛𝛾𝑓(𝑥_𝑛)+(𝛼_𝑛−𝛽_𝑛)𝑥_𝑛 and to study the convergence of {𝑥_𝑛}, where T is a nonexpansive mapping and A is a strongly positive linear operator, if {𝛼_𝑛}, {𝛽_𝑛} satisfy appropriate conditions, then iteration sequence {𝑥_𝑛} converges strongly to the unique solution 𝑥^∗∈𝑓(𝑇) of variational inequality ⟨(𝐴−𝛾𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0, for all 𝑥∈𝑓(𝑇). Meanwhile, a approximate iteration algorithm is presented which is used to calculate the fixed point of nonexpansive mapping and solution of variational inequality, the error estimate is also given. The results presented in this paper extend, generalize, and improve the results of Xu, G. Marino and Xu and some others.

1. Introduction

Iteration methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [1–4] and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space 𝐻: min_{𝑥∈𝐹(𝑇)}12⟨𝐴𝑥,𝑥⟩−⟨𝑥,𝑏⟩,(1.1) where 𝐹(𝑇) is the fixed points set of a nonexpansive mapping 𝑇 on 𝐻, 𝑏 is a given point in 𝐻, and 𝐴∶𝐻→𝐻 is strongly positive operator, that is, there exists a constant 𝛿>0 with the property ⟨𝐴𝑥,𝑥⟩≥𝛿‖𝑥‖²,∀𝑥∈𝐻.(1.2)

Recall that 𝑇∶𝐻→𝐻 is nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻. Throughout the rest of this paper, we denote by 𝐹(𝑇) the fixed points set of 𝑇 and assume that 𝐹(𝑇) is nonempty. It is well known that 𝐹(𝑇) is closed convex (cf. [5]). In [4] (see also [2]), it is proved that the sequence {𝑥_𝑛} defined by the iteration method below, with the initial guess 𝑥₀ chosen arbitrarily, 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝑏,𝑛≥0,(1.3) converges strongly to the unique solution of minimization problem (1.1) provided the sequence {𝛼_𝑛} satisfies certain conditions.

On the other hand, Moudafi [6] introduced the viscosity approximation method for nonexpansive mappings (see [7] for further developments in both Hilbert and Banach spaces). Let 𝑓 be a contraction on 𝐻. Starting with an arbitrary initial guess 𝑥₀∈𝐻, define a sequence {𝑥_𝑛} recursively by 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝑓𝑥_𝑛,𝑛≥0,(1.4) where {𝛼_𝑛} is a sequence in (0,1). It is proved [6, 7] that under certain appropriate conditions imposed on {𝛼_𝑛}, the sequence {𝑥_𝑛} generated by (1.4) converges strongly to the unique solution 𝑥^∗ in 𝐹(𝑇) of the variational inequality ⟨(𝐼−𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,𝑥∈𝐹(𝑇).(1.5)

Recently (2006), Marino and Xu [2] combine the iteration method (1.3) with the viscosity approximation method (1.4) and consider the following general iteration method: 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛,𝑛≥0,(1.6) they have proved that if the sequence {𝛼_𝑛} of parameters satisfies appropriate conditions, then the sequence {𝑥_𝑛} generated by (1.6) converges strongly to the unique solution of the variational inequality ⟨(𝐴−𝛾𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,𝑥∈𝐹(𝑇),(1.7) which is the optimality condition for the minimization problem min_{𝑥∈𝐹(𝑇)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥),(1.8) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ^(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻).

The purpose of this paper is to present a general viscosity iteration process {𝑥_𝑛} which is defined by 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛽_𝑛𝛾𝑓𝑥_𝑛+𝛼_𝑛−𝛽_𝑛𝑥_𝑛(1.9) and to study the convergence of {𝑥_𝑛}, where 𝑇 is a nonexpansive mapping and 𝐴 is a strongly positive linear operator, if {𝛼_𝑛}, {𝛽_𝑛} satisfy appropriate conditions, then iteration sequence {𝑥_𝑛} converges strongly to the unique solution 𝑥^∗∈𝐹(𝑇) of variational inequality (1.7). Meanwhile, an approximate iteration algorithm 𝑥_𝑛+1=(𝐼−𝑠𝐴)𝑇𝑥_𝑛+𝑡𝛾𝑓𝑥_𝑛+(𝑠−𝑡)𝑥_𝑛(1.10) is presented which is used to calculate the fixed point of nonexpansive mapping and solution of variational inequality; the convergence rate estimate is also given. The results presented in this paper extend, generalize and improve the results of Xu [7], Marino and Xu [2], and some others.

2. Preliminaries

This section collects some lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.

Lemma 2.1 (see [3]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝜆_𝑛𝑎_𝑛+𝛿_𝑛,(2.1) where {𝜆_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in (−∞,+∞) such that(i)∑^∞𝑛=1𝜆_𝑛=∞; (ii)limsup_𝑛→∞𝛿_𝑛/𝑟_𝑛≤0, or ∑^∞𝑛=1|𝛿_𝑛|<∞. Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.2 (see [5]). Let 𝐻 be a Hilbert space, 𝐾 a closed convex subset of 𝐻, and 𝑇∶𝐾→𝐾 a nonexpansive mapping with nonempty fixed points set 𝐹(𝑇). If {𝑥_𝑛} is a sequence in 𝐾 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦.

The following lemma is not hard to prove.

Lemma 2.3. Let 𝐻 be a Hilbert space, 𝐾 a closed convex subset of 𝐻, 𝑓∶𝐻→𝐻 a contraction with coefficient 0<ℎ<1, and 𝐴 a strongly positive linear bounded operator with coefficient 𝛿>0. Then, for 0<𝛾<(𝛿/ℎ), ⟨𝑥−𝑦,(𝐴−𝛾𝑓)𝑥−(𝐴−𝛾𝑓)𝑦⟩≥(𝛿−𝛾ℎ)‖𝑥−𝑦‖²,𝑥,𝑦∈𝐻.(2.2) That is, 𝐴−𝛾𝑓 is strongly monotone with coefficient 𝛿−𝛾ℎ.

Recall the metric (nearest point) projection 𝑃_𝐾 from a real Hilbert space 𝐻 to a closed convex subset 𝐾 of 𝐻 is defined as follows: given 𝑥∈𝐻, 𝑃_𝐾𝑥 is the only point in 𝐾 with the property ‖‖𝑥−𝑃_𝐾𝑥‖‖=min_𝑦∈𝐾‖𝑥−𝑦‖.(2.3)𝑃_𝐾 is characterized as follows.

Lemma 2.4. Let 𝐾 be a closed convex subset of a real Hilbert space 𝐻. Given that 𝑥∈𝐻 and 𝑦∈𝐾. Then 𝑦=𝑃_𝐾𝑥 if and only if there holds the inequality ⟨𝑥−𝑦,𝑦−𝑧⟩≥0,∀𝑧∈𝐾.(2.4)

Lemma 2.5. Assume that 𝐴 is a strongly positive linear-bounded operator on a Hilbert space 𝐻 with coefficient 𝛿>0 and 0<𝜌≤‖𝐴‖⁻¹. Then ‖𝐼−𝜌𝐴‖≤(1−𝜌𝛿).

Proof. Recall that a standard result in functional analysis is that if 𝑉 is linear bounded self-adjoint operator on 𝐻, then ‖𝑉‖=sup||⟨𝑉𝑥,𝑥⟩||∶𝑥∈𝐻,‖𝑥‖=1.(2.5) Now for 𝑥∈𝐻 with ‖𝑥‖=1, we see that ⟨(𝐼−𝜌𝐴)𝑥,𝑥⟩=1−𝜌⟨𝐴𝑥,𝑥⟩≥1−𝜌‖𝐴‖≥0(2.6) (i.e., 𝐼−𝜌𝐴 is positive). It follows that ‖𝐼−𝜌𝐴‖==≤s1suu−pp𝜌{{𝛿1⟨.(−𝐼−𝜌⟨𝜌𝐴𝐴𝑥),𝑥𝑥,⟩𝑥⟩∶𝑥∶𝑥∈∈𝐻𝐻,‖,𝑥‖‖𝑥‖==1}1}(2.7)

The following lemma is also not hard to prove by induction.

Lemma 2.6. Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝜆_𝑛𝑎_𝑛+𝜆_𝑛+𝜇_𝑛𝑀,(2.8) where 𝑀 is a nonnegative constant and {𝜆_𝑛},{𝜇_𝑛} are sequences in [0,+∞) such that(i)∑^∞𝑛=0𝜆_𝑛=∞; (ii)∑^∞𝑛=0𝜇_𝑛<∞. Then {𝑎_𝑛} is bounded.

Notation. We use → for strong convergence and ⇀ for weak convergence.

3. A general Iteration Algorithm with Bounded Linear Operator

Let 𝐻 be a real Hilbert space, 𝐴 be a bounded linear operator on 𝐻, and 𝑇 be a nonexpansive mapping on 𝐻. Assume that the fixed point set 𝐹(𝑇)={𝑥∈𝐻∶𝑇𝑥=𝑥} of 𝑇 is nonempty. Since 𝐹(𝑇) is closed convex, the nearest point projection from 𝐻 onto 𝐹(𝑇) is well defined.

Throughout the rest of this paper, we always assume that 𝐴 is strongly positive, that is, there exists a constant 𝛿>0 such that ⟨𝐴𝑥,𝑥⟩≥𝛿‖𝑥‖²,∀𝑥∈𝐻.(3.1) (Note: 𝛿>0 is throughout reserved to be the constant such that (3.1) holds.)

Recall also that a contraction on 𝐻 is a self-mapping 𝑓 of 𝐻 such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤ℎ‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐻,(3.2) where ℎ∈(0,1) is a constant which is called contractive coefficient of 𝑓.

For given contraction 𝑓 with contractive coefficient 0<ℎ<1, and 𝑡∈[0,1),𝑠∈(0,1),𝑠≥𝑡 such that 0≤𝑡≤𝑠<‖𝐴‖⁻¹ and 0<𝛾<𝛿/ℎ, consider a mapping 𝑆_𝑡,𝑠 on 𝐻 defined by 𝑆_𝑡,𝑠𝑥=(𝐼−𝑠𝐴)𝑇𝑥+𝑡𝛾𝑓(𝑥)+(𝑠−𝑡)𝑥,𝑥∈𝐻.(3.3) Assume that 𝑠−𝑡𝑠→0,(3.4) it is not hard to see that 𝑆_𝑡,𝑠 is a contraction for sufficiently small 𝑠, indeed, by Lemma 2.5 we have ‖‖𝑆_𝑡,𝑠𝑥−𝑆_𝑡,𝑠𝑦‖‖=≤≤𝑡((𝛾1𝑡𝛾‖+ℎ𝑓𝑡+((𝑥𝛾1)ℎ−−−𝑠𝑓1𝛿)(+𝑦−)‖𝑠𝑠−(+𝛿𝑡‖)−‖(1𝐼𝑥)−)−‖𝑠𝑦𝑥𝐴‖−)(𝑦𝑇‖𝑥.−𝑇𝑦)‖+‖(𝑠−𝑡)(𝑥−𝑦)‖(3.5) Hence, 𝑆_𝑡,𝑠 has a unique fixed point, denoted by 𝑥_𝑡,𝑠, which uniquely solves the fixed point equation: 𝑥_𝑡,𝑠=(𝐼−𝑠𝐴)𝑇𝑥_𝑡,𝑠+𝑡𝛾𝑓𝑥_𝑡,𝑠+(𝑠−𝑡)𝑥_𝑡,𝑠.(3.6) Note that 𝑥_𝑡,𝑠 indeed depends on 𝑓 as well, but we will suppress this dependence of 𝑥_𝑡,𝑠 on 𝑓 for simplicity of notation throughout the rest of this paper. We will also always use 𝛾 to mean a number in (0,𝛿/ℎ).

The next proposition summarizes the basic properties of 𝑥_𝑡,𝑠,(𝑡≤𝑠).

Proposition 3.1. Let 𝑥_𝑡,𝑠 be defined via (3.6).(i){𝑥_𝑡,𝑠} is bounded for 𝑡∈[0,‖𝐴‖⁻¹),𝑠∈(0,‖𝐴‖⁻¹).(ii)lim_𝑠→0‖𝑥_𝑡,𝑠−𝑇𝑥_𝑡,𝑠‖=0.(iii){𝑥_𝑡,𝑠} defines a continuous surface for (𝑡,𝑠)∈[0,‖𝐴‖⁻¹)×(0,‖𝐴‖⁻¹),𝑡≤𝑠 into 𝐻.

Proof. Observe, for 𝑠∈(0,‖𝐴‖⁻¹), that ‖𝐼−𝑠𝐴‖≤1−𝑠𝛿 by Lemma 2.5.
To show (i) pick 𝑝∈𝐹(𝑇). We then have ‖‖𝑥_𝑡,𝑠−𝑝‖‖=≤‖‖(1(𝐼−−𝑠𝛿𝑠)𝐴‖‖)𝑥𝑇_𝑡,𝑠𝑥_𝑡,𝑠−𝑝+𝑡𝛾𝑓𝑥_𝑡,𝑠−𝐴𝑝−𝑠𝐴𝑝+𝑡𝐴𝑝‖‖≤(1−𝑠𝛿)‖‖𝑥_𝑡,𝑠−𝑝‖‖+𝑡‖‖𝛾𝑓𝑥_𝑡,𝑠−𝐴𝑝‖‖+(𝑠−𝑡)‖𝐴𝑝‖≤[1−𝑠(𝛿−𝛾ℎ−)]𝑝‖‖‖‖𝑥+_𝑡,𝑠𝑠𝛾ℎ‖‖𝑥_𝑡,𝑠−𝑝‖‖+‖𝛾𝑓(𝑝)−𝐴𝑝‖+(𝑠−𝑡)‖𝐴𝑝‖−𝑝‖‖+𝑠‖𝛾𝑓(𝑝)−𝐴𝑝‖+(𝑠−𝑡)‖𝐴𝑝‖.(3.7) It follows that ‖‖𝑥_𝑡,𝑠−𝑝‖‖≤‖𝛾𝑓(𝑝)−𝐴𝑝‖𝛿−𝛾ℎ+𝑠−𝑠𝑡𝛿‖𝐴−𝑝𝛾‖ℎ<+∞.(3.8) Hence {𝑥_𝑡,𝑠} is bounded.
(ii) Since the boundedness of {𝑥_𝑡,𝑠} implies that of {𝑓(𝑥_𝑡,𝑠)} and {𝐴𝑇𝑥_𝑡,𝑠}, and observe that ‖‖𝑥_𝑡,𝑠−𝑇𝑥_𝑡,𝑠‖‖=‖‖𝑡𝑓𝑥_𝑡,𝑠−𝑠𝐴𝑇𝑥_𝑡,𝑠+(𝑠−𝑡)𝑥_𝑡,𝑠‖‖,(3.9) we have lim_𝑠→0‖‖𝑥_𝑡,𝑠−𝑇𝑥_𝑡,𝑠‖‖=0.(3.10)
To prove (iii) take 𝑡,𝑡₀∈[0,‖𝐴‖⁻¹),𝑠,𝑠₀∈(0,‖𝐴‖⁻¹),𝑠≥𝑡,𝑠₀≥𝑡₀ and calculate ‖‖𝑥_𝑡,𝑠−𝑥_𝑡0,𝑠0‖‖=‖‖𝑡−𝑡₀𝛾𝑓𝑥_𝑡,𝑠+𝑡₀𝛾𝑓𝑥_𝑡,𝑠−𝑓𝑥_𝑡0,𝑠0−𝑠−𝑠₀𝐴𝑇𝑥_𝑡,𝑠≤||𝑡+−𝑡𝐼₀−𝑠₀𝐴𝑇𝑥_𝑡,𝑠−𝑇𝑥_𝑡0,𝑠0+(𝑠−𝑡)𝑥_𝑡,𝑠−𝑥_𝑡0,𝑠0+𝑠−𝑠₀+𝑡₀−𝑡𝑥_𝑡0,𝑠0‖‖+1−||𝑠𝛾₀‖‖𝑓𝑥_𝑡,𝑠‖‖+𝑡₀𝛾ℎ‖‖𝑥_𝑡,𝑠−𝑥_𝑡0,𝑠0‖‖+||𝑠−𝑠₀||‖‖𝐴𝑇𝑥_𝑡,𝑠‖‖+||𝑠−𝑠₀𝛿‖‖𝑥_𝑡,𝑠−𝑥_𝑡0,𝑠0‖‖+(𝑠−𝑡)‖‖𝑥_𝑡,𝑠−𝑥_𝑡0,𝑠0‖‖||+||𝑡−𝑡₀||‖‖𝑥_𝑡0,𝑠0‖‖,(3.11) which implies that 𝑠₀𝛿−𝑡₀𝛾ℎ+𝑡−𝑠‖‖𝑥_𝑡,𝑠−𝑥_𝑡0,𝑠0‖‖≤+||𝑡−||𝑠𝑡−₀||𝑠𝛾₀‖‖𝑓𝑥_𝑡,𝑠‖‖+||𝑠−𝑠₀||‖‖𝐴𝑇𝑥_𝑡,𝑠‖‖||+||𝑡−𝑡₀||‖‖𝑥_𝑡0,𝑠0‖‖→0(3.12) as 𝑡→𝑡₀,𝑠→𝑠₀. Note that lim_{𝑡→𝑡0,𝑠→𝑠0}𝑠₀𝛿−𝑡₀𝛾ℎ+𝑡−𝑠=𝑠₀(𝛿−1)−𝑡₀(𝛾ℎ−1)>0,(3.13) it is obvious that lim_{𝑡→𝑡0,𝑠→𝑠0}‖‖𝑥_𝑡,𝑠−𝑥_𝑡0,𝑠0‖‖=0.(3.14) This completes the proof of Proposition 3.1.

Our first main result below shows that 𝑥_𝑡,𝑠 converges strongly as 𝑠→0 to a fixed point of 𝑇 which solves some variational inequality.

Theorem 3.2. One has that 𝑥_𝑡,𝑠 converges strongly as 𝑠→0(𝑡≤𝑠) to a fixed point 𝑥̃ of 𝑇 which solves the variational inequality: ⟨(𝐴−𝛾𝑓)𝑥̃,𝑥̃−𝑧⟩≤0,𝑧∈𝐹(𝑇).(3.15) Equivalently, One has 𝑃_𝐹(𝑇)(𝐼−𝐴+𝛾𝑓)𝑥̃=𝑥̃, where 𝑃_𝐹(𝑇)(⋅) is the nearest point projection from 𝐻 onto 𝐹(𝑇).

Proof. We first shows the uniqueness of a solution of the variational inequality (3.15), which is indeed a consequence of the strong monotonicity of 𝐴−𝛾𝑓. Suppose 𝑥̃∈𝐹(𝑇) and 𝑥̂∈𝐹(𝑇) both are solutions to (3.15), then ⟨(⟨𝐴(𝐴−−𝛾𝑓𝛾𝑓)𝑥)̂,𝑥𝑥̃,̂𝑥−̃−𝑥̃⟩𝑥̂⟩≤≤0.0,(3.16) Adding up (3.16) gets ⟨(𝐴−𝛾𝑓)𝑥̃−(𝐴−𝛾𝑓)𝑥̂,𝑥̃−𝑥̂⟩≤0.(3.17) The strong monotonicity of 𝐴−𝛾𝑓 implies that 𝑥̃=𝑥̂ and the uniqueness is proved. Below we use 𝑥̃∈𝐹(𝑇) to denote the unique solution of (3.15).
To prove that 𝑥_𝑡,𝑠 converges strongly to 𝑥̃, we write, for a given 𝑧∈𝐹(𝑇), 𝑥_𝑡,𝑠−𝑧=𝑡𝛾𝑓𝑥_𝑡,𝑠−𝑠𝑡𝐴𝑧+(𝐼−𝑠𝐴)𝑇𝑥^𝑡,𝑠−𝑧+(𝑠−𝑡)𝑥_𝑡,𝑠(3.18) to derive that ‖‖𝑥_𝑡,𝑠−𝑧‖‖²=𝑡𝛾𝑓𝑥_𝑡,𝑠−𝑠𝑡𝐴𝑧+(𝐼−𝑠𝐴)𝑇𝑥^𝑡,𝑠=𝑡𝛾𝑓𝑥_𝑡,𝑠−𝑧+(𝑠−𝑡)𝑥_𝑡,𝑠,𝑥_𝑡,𝑠−𝑧+(𝑠−𝑡)𝑥−_𝑡,𝑠𝐴𝑧,𝑥_𝑡,𝑠−𝑧+(𝐼−𝑠𝐴)T𝑥_𝑡,𝑠−𝑧,𝑥_𝑡,𝑠−𝑧≤(1−𝑠𝛿)‖‖𝑥_𝑡,𝑠−𝐴𝑧,𝑥_𝑡,𝑠−𝑧−𝑧‖‖²+𝑡𝛾𝑓𝑥_𝑡,𝑠−𝐴𝑧,𝑥_𝑡,𝑠−𝑧+(𝑠−𝑡)𝑥_𝑡,𝑠−𝐴𝑧,𝑥_𝑡,𝑠−𝑧.(3.19) It follows that ‖‖𝑥_𝑡,𝑠−𝑧‖‖²≤𝑡𝑠𝛿𝛾𝑓𝑥^𝑡,𝑠−𝐴𝑧,𝑥_𝑡,𝑠−𝑧+𝑠−𝑡𝑠𝛿𝑥^𝑡,𝑠=𝑡−𝐴𝑧,𝑥_𝑡,𝑠−𝑧𝑠𝛿𝛾𝑓𝑥^𝑡,𝑠−𝑓(𝑧),𝑥_𝑡,𝑠+𝑠−𝑡−𝑧+𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑡,𝑠−𝑧𝑠𝛿𝑥^𝑡,𝑠≤𝑡−𝐴𝑧,𝑥_𝑡,𝑠−𝑧𝑠𝛿𝛾ℎ‖‖𝑥^𝑡,𝑠+𝑠−𝑡−𝑧‖‖²+𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑡,𝑠−𝑧𝑠𝛿𝑥^𝑡,𝑠−𝐴𝑧,𝑥_𝑡,𝑠−𝑧,(3.20) which leads to ‖‖𝑥_𝑡,𝑠−𝑧‖‖²≤𝑡𝑠𝛿−𝑡𝛾ℎ𝛾𝑓(𝑧)−𝐴𝑧,𝑥^𝑡,𝑠+𝑠−𝑡−𝑧𝑠𝛿−𝑡𝛾ℎ𝑥^𝑡,𝑠−𝐴𝑧,𝑥_𝑡,𝑠−𝑧.(3.21) Observe that condition (3.4) implies 𝑠−𝑡𝑠𝛿−𝑡𝛾ℎ→0,(3.22) as 𝑠→0. Since 𝑥_𝑡,𝑠 is bounded as 𝑠→0,𝑠≥𝑡, then there exists real sequences {𝑠_𝑛},{𝑡_𝑛} in [0,1] such that 𝑠_𝑛→0,𝑠_𝑛≥𝑡_𝑛 and {𝑥_{𝑡𝑛,𝑠𝑛}} converges weakly to a point 𝑥^∗∈𝐻. Using Proposition 3.1 and Lemma 2.2, we see that 𝑥^∗∈𝐹(𝑇), therefore by (3.21), we see 𝑥_{𝑡𝑛,𝑠𝑛}→𝑥^∗. We next prove that 𝑥^∗ solves the variational inequality (3.15). Since 𝑥_𝑡,𝑠=(𝐼−𝑠𝐴)𝑇𝑥_𝑡,𝑠+𝑡𝛾𝑓𝑥_𝑡,𝑠+(𝑠−𝑡)𝑥_𝑡,𝑠,(3.23) we derive that 𝑠(𝐴−𝛾𝑓)𝑥_𝑡,𝑠=𝑠𝐴𝑥_𝑡,𝑠−𝑥_𝑡,𝑠+𝑡𝛾𝑓𝑥_𝑡,𝑠−𝑠𝛾𝑓𝑥_𝑡,𝑠+(𝐼−𝑠𝐴)𝑇𝑥_𝑡,𝑠+(𝑠−𝑡)𝑥_𝑡,𝑠=(𝐼−𝑠𝐴)𝑇𝑥_𝑡,𝑠−𝑥_𝑡,𝑠+(𝑠−𝑡)𝑥_𝑡,𝑠−𝛾𝑓𝑥_𝑡,𝑠,(3.24) so that (𝐴−𝛾𝑓)𝑥_𝑡,𝑠=1𝑠(𝐼−𝑠𝐴)𝑇𝑥^𝑡,𝑠−𝑥_𝑡,𝑠+𝑠−𝑡𝑠𝑥^𝑡,𝑠−𝛾𝑓𝑥_𝑡,𝑠.(3.25) It follows that, for 𝑧∈𝐹(𝑇), (𝐴−𝛾𝑓)𝑥_𝑡,𝑠,𝑥_𝑡,𝑠−𝑧=+1𝑠𝑠−(𝐼𝑡−𝑠𝐴)𝑇𝑥^𝑡,𝑠−𝑥_𝑡,𝑠,𝑥_𝑡,𝑠−𝑧𝑠𝑥^𝑡,𝑠=−1−𝛾𝑓𝑥_𝑡,𝑠,𝑥_𝑡,𝑠−𝑧𝑠(𝐼−𝑇)𝑥^𝑡,𝑠+𝐴(𝐼−𝑇)𝑥_𝑡,𝑠−(𝐼−𝑇)𝑧,𝑥_𝑡,𝑠−𝑧+𝑠−𝑡,𝑥_𝑡,𝑠−𝑧𝑠𝑥^𝑡,𝑠≤𝐴(𝐼−𝑇)𝑥−_𝑡,𝑠𝛾𝑓𝑥_𝑡,𝑠,𝑥_𝑡,𝑠−𝑧+𝑠−𝑡,𝑥,𝑠_𝑡−𝑧𝑠𝑥^𝑡,𝑠−𝛾𝑓𝑥_𝑡,𝑠,𝑥_𝑡,𝑠−𝑧,(3.26) since 𝐼−𝑇 is monotone (i.e., ⟨𝑥−𝑦,(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦⟩≥0 for 𝑥,𝑦∈𝐻). This is due to the nonexpansivity of 𝑇. Now replacing 𝑡,𝑠 in (3.26) with 𝑡_𝑛,𝑠_𝑛 and letting 𝑛→∞, we, noticing that (𝐼−𝑇)𝑥_{𝑡𝑛,𝑠𝑛}→(𝐼−𝑇)𝑥^∗=0 for 𝑥^∗∈𝐹(𝑇), obtain ⟨(𝐴−𝛾𝑓)𝑥^∗,𝑥^∗−𝑧⟩≤0.(3.27) That is, 𝑥^∗∈𝐹(𝑇) is a solution of (3.15), hence 𝑥^∗=𝑥̃ by uniqueness. In a summary, we have shown that each cluster point of 𝑥_𝑡,𝑠 equals 𝑥̃. Therefore, 𝑥_𝑡,𝑠→𝑥̃ as 𝑠→0.
The variational inequality (3.15) can be rewritten as ⟨(𝐼−𝐴+𝛾𝑓)𝑥̃−𝑥̃,𝑥̃−𝑧⟩≥0,𝑧∈𝐹(𝑇).(3.28) This, by Lemma 2.4, is equivalent to the fixed point equation 𝑃_𝐹(𝑇)(𝐼−𝐴+𝛾𝑓)𝑥̃=𝑥̃.(3.29) This complete the proof.

Taking 𝑡=𝑠 in Theorem 3.2, we get

Corollary 3.3 (see [7]). One has that 𝑥_𝑡=𝑥_𝑡,𝑡 converges strongly as 𝑡→0 to a fixed point 𝑥̃ of 𝑇 which solves the variational inequality: ⟨(𝐴−𝛾𝑓)𝑥̃,𝑥̃−𝑧⟩≤0,𝑧∈𝐹(𝑇).(3.30) Equivalently, One has 𝑃_𝐹(𝑇)(𝐼−𝐴+𝛾𝑓)𝑥̃=𝑥̃, where 𝑃_𝐹(𝑇)(⋅) is the nearest point projection from 𝐻 onto 𝐹(𝑇).

Next we study a general iteration method as follows. The initial guess 𝑥₀ is selected in 𝐻 arbitrarily, and the (𝑛+1)th iterate 𝑥_𝑛+1 is recursively defined by 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛽_𝑛𝛾𝑓𝑥_𝑛+𝛼_𝑛−𝛽_𝑛𝑥_𝑛,(3.31) where {𝛼_𝑛}⊂(0,1),𝛽_𝑛∈[0,1),𝛽_𝑛≤𝛼_𝑛 are sequences satisfying following conditions: (C₁)𝛼_𝑛→0; (C₂)∑^∞𝑛=0𝛼_𝑛=∞; (C₃) either ∑^∞𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞ or lim_𝑛→∞(𝛼_𝑛+1/𝛼_𝑛)=1; (C₄)∑^∞𝑛=0(𝛼_𝑛−𝛽_𝑛)<∞.Below is the second main result of this paper.

Theorem 3.4. Let {𝑥_𝑛} be general by Algorithm (3.31) with the sequences {𝛼_𝑛},{𝛽_𝑛} of parameters satisfying conditions (C₁)–(C₄). Then {𝑥_𝑛} converges strongly to 𝑥̃ that is obtained in Theorem 3.2.

Proof. Since 𝛼_𝑛→0 by condition (C₁), we may assume, without loss of generality, that 𝛼_𝑛<‖𝐴‖⁻¹ for all 𝑛.
We now observe that {𝑥_𝑛} is bounded. Indeed, pick any 𝑝∈𝐹(𝑇) to obtain ‖‖𝑥_𝑛+1−𝑝‖‖=≤‖‖‖‖𝐼𝐼−−𝛼𝛼_𝑛𝑛𝐴𝑇𝑥_𝑛−𝑝+𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑝+𝛼_𝑛−𝛽_𝑛𝑥_𝑛−𝐴𝑝‖‖≤1−𝛼_𝑛𝐴‖‖‖‖𝑇𝑥_𝑛−𝑝‖‖+𝛽_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑝‖‖+𝛼_𝑛−𝛽_𝑛‖‖𝑥_𝑛−𝐴𝑝‖‖+𝛼𝛿‖‖𝑥_𝑛−𝑝‖‖+𝛽_𝑛𝛾‖‖𝑓𝑥_𝑛−𝑓(𝑝)‖‖+‖𝛾𝑓(𝑝)−𝐴𝑝_𝑛‖≤1−𝛿−𝛼𝛽_𝑛𝑛‖‖𝑥_𝑛−𝐴𝑝‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛𝛾ℎ‖‖𝑥_𝑛−𝑝‖‖−𝛼_𝑛𝛾ℎ‖‖𝑥_𝑛−𝑝‖‖=1−𝛾(ℎ𝛿‖‖−𝑥_𝑛𝛾ℎ−)𝑝𝛼‖‖_𝑛+𝛽_𝑛‖𝛾𝑓(𝑝)−𝐴𝑝‖+𝛼_𝑛−𝛽_𝑛‖‖𝑥_𝑛−𝐴𝑝‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖−𝛼_𝑛−𝛽_𝑛𝛾ℎ‖‖𝑥_𝑛−𝑝‖‖≤1−‖(𝛾𝛿𝑓−(𝑝𝛾)ℎ−)𝛼𝐴_𝑛𝑝‖+𝛼_𝑛−𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖+‖𝑝−𝐴𝑝‖+𝛽_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛−𝛽_𝑛(1−𝛾ℎ)‖‖𝑥_𝑛−𝑝‖‖≤1−‖𝛾(𝑓𝛿(−𝑝)𝛾−ℎ)𝐴𝛼𝑝_𝑛‖+𝛼_𝑛−𝛽_𝑛‖𝑝−𝐴𝑝‖+𝛼−𝛼_𝑛−𝛽_𝑛||1−𝛾ℎ||‖‖𝑥_𝑛−𝑝‖‖+𝛽_𝑛‖𝛾𝑓(𝑝)−𝐴𝑝_𝑛‖≤1−−(𝛿𝛽−_𝑛𝛾‖ℎ𝑝)−𝛼_𝑛𝐴𝑝‖−𝛼_𝑛−𝛽_𝑛||1−𝛾ℎ||‖‖𝑥_𝑛−𝑝‖‖+(𝛿−𝛾ℎ)𝛼_𝑛𝑀𝛿−𝛾ℎ,(3.32) where 𝑀≥‖𝛾𝑓(𝑝)−𝐴𝑝‖+‖𝑝−𝐴𝑝‖ is a constant. By Lemma 2.6 we see that {𝑥_𝑛} is bounded.
As a result, noticing 𝑥_𝑛+1−𝑇𝑥_𝑛=𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛽_𝑛𝛾𝑓𝑥_𝑛+𝛼_𝑛−𝛽_𝑛𝑥_𝑛(3.33) and 𝛼_𝑛→0, we obtain 𝑥_𝑛+1−𝑇𝑥_𝑛→0.(3.34) But the key is to prove that 𝑥_𝑛+1−𝑥_𝑛→0.(3.35) To see this, we calculate ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=‖‖𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑇𝑥_𝑛−1−𝛼_𝑛−𝛼_𝑛−1𝐴𝑇𝑥_𝑛−1≤+𝛾1−𝛼_𝑛(𝛿𝑓−𝛾𝑥ℎ_𝑛)𝛼−_𝑛𝑓𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑓𝑥_𝑛−1+𝛼_𝑛−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1‖‖+𝛼_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||‖‖𝐴𝑇𝑥_𝑛−1−𝑓𝑥_𝑛−1‖‖=1+𝛼−_𝑛𝛽_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛽_𝑛−(𝛿−𝛾ℎ)𝛼_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖=1−−(𝛿𝛼−_𝑛−1𝛾||ℎ‖‖𝐴−𝑇1𝑥)𝛼_𝑛−1𝑛−𝑓𝑥_𝑛−1‖‖+||𝛼_𝑛+𝛽_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖−𝛼_𝑛−1||‖‖𝐴𝑇𝑥_𝑛−1−𝑓𝑥_𝑛−1‖‖.(3.36) Since ^∞_𝑛=0(𝛿−𝛾ℎ−1)𝛼_𝑛+𝛽_𝑛=^∞_𝑛=0(𝛿−𝛾ℎ)𝛼_𝑛−𝛼_𝑛−𝛽_𝑛=∞(3.37) and condition (C₃) holds, an application of Lemma 2.1 to (3.36) implies (3.35) which combined with (3.34), in turns, implies 𝑥_𝑛−𝑇𝑥_𝑛→0.(3.38) Next we show that limsup_𝑛→∞⟨𝑇𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩≤0,(3.39) where 𝑥̃ is obtained in Theorem 3.2.
To see this, we take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞⟨𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩=lim_𝑘→∞𝑥_𝑛𝑘−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃.(3.40) We may also assume that 𝑥_𝑛𝑘⇀𝑧. Note that 𝑧∈𝐹(𝑇) in virtue of Lemma 2.2 and (3.38). It follows from the variational inequality (3.15) that limsup_𝑛→∞⟨𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩=⟨𝑧−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩≤0.(3.41) So (3.39) holds, thanks to (3.38).
Finally, we prove 𝑥_𝑛→𝑥̃. To this end, we calculate ‖‖𝑥_𝑛+1−𝑥̃‖‖²=‖‖(𝐼−𝛼_𝑛𝐴)(𝑇𝑥_𝑛−𝑥̃)+𝛼_𝑛(𝛾𝑓(𝑥_𝑛)−𝐴𝑥̃)−𝛼_𝑛𝛾𝑓(𝑥_𝑛)+𝛽_𝑛𝛾𝑓(𝑥_𝑛)+(𝛼_𝑛−𝛽_𝑛)𝑥_𝑛‖‖²=‖‖𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃+𝛼_𝑛−𝛽_𝑛𝑥_𝑛−𝛾𝑓(𝑥_𝑛)‖‖²=‖‖(𝐼−𝛼_𝑛𝐴)(𝑇𝑥_𝑛−𝑥̃)‖‖²+‖‖𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃‖‖²++‖‖2𝛼_𝑛𝐼−−𝛽𝛼_𝑛𝑛𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖²+2𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃+2𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝛼_𝑛−𝛽_𝑛𝑥_𝑛−𝛾𝑓𝑥_𝑛≤1−𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃,𝛼_𝑛−𝛽_𝑛𝑥_𝑛−𝛾𝑓𝑥_𝑛𝛿²‖‖𝑥_𝑛−𝑥̃‖‖²+𝛼^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥̃‖‖²+𝛼_𝑛−𝛽_𝑛²‖‖𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖²+−22𝛼𝛼_𝑛2𝑛𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝐴𝑥̃+2𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝐴𝑥̃−2𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛+2𝛼_𝑛𝛼_𝑛−𝛽_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛≤1−𝛼_𝑛𝛼_𝑛−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛𝛿²‖‖𝑥_𝑛−𝑥̃‖‖²+𝛼^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥̃‖‖²+𝛼_𝑛−𝛽_𝑛²‖‖𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖²+−22𝛼𝛼_𝑛2𝑛𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑥̃)+𝛾𝑓(𝑥̃)−𝐴𝑥̃−2𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝐴𝑥̃+2𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛+2𝛼_𝑛𝛼_𝑛−𝛽_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛≤1−𝛼_𝑛𝛼_𝑛−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛𝛿²‖‖𝑥_𝑛−𝑥̃‖‖²+𝛼^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥̃‖‖²+𝛼_𝑛−𝛽_𝑛²‖‖𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖²+−22𝛼𝛼_𝑛2𝑛𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝛾𝑓(𝑥̃)+2𝛼_𝑛⟨𝑇𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩−2𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝐴𝑥̃+2𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛+2𝛼_𝑛𝛼_𝑛−𝛽_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛≤1−𝛼_𝑛𝛼_𝑛−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛𝛿²‖‖𝑥_𝑛−𝑥̃‖‖²+𝛼^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥̃‖‖²+𝛼_𝑛−𝛽_𝑛²‖‖𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖²+2𝛼_𝑛𝛾ℎ‖‖𝑥_𝑛−𝑥̃‖‖²++22𝛼_𝑛𝛼⟨_𝑛𝑇𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩−2𝛼^2𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝐴𝑥̃−2𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛+2𝛼_𝑛𝛼_𝑛−𝛽_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛≤1−𝛼𝛼_𝑛𝑛−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛𝛿²+2𝛼_𝑛𝛾ℎ‖‖𝑥_𝑛−𝑥̃‖‖²+𝛼^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥̃‖‖²+−2𝛼𝛼_𝑛2𝑛−𝛽_𝑛²‖‖𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖²+2𝛼_𝑛⟨𝑇𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩−2𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝐴𝑥̃+2𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛≤1−2(𝛼𝛿_𝑛−−𝛾𝛽ℎ_𝑛)𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛+2𝛼_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛼^2𝑛𝛿²‖‖𝑥_𝑛−𝑥̃‖‖+𝛼^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥̃‖‖²+−2𝛼𝛼_𝑛2𝑛−𝛽_𝑛²‖‖𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖²+2𝛼_𝑛⟨𝑇𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩−2𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝐴𝑥̃+2𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛≤1−2𝛼(𝛿_𝑛−−𝛽𝛾_𝑛ℎ)𝛼𝐴_𝑛𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛+2𝛼_𝑛𝛼_𝑛−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛼^2𝑛𝛿²‖‖𝑥_𝑛−𝑥̃‖‖+𝛼^2𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥̃‖‖²+−2𝛼𝛼_𝑛2𝑛−𝛽_𝑛²‖‖𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖²+2𝛼_𝑛⟨𝑇𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩−2𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝐴𝑥̃+2𝛼_𝑛−𝛽_𝑛𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛+2𝛼_𝑛𝛼_𝑛−𝛽_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛≤1−2𝛼(𝛿_𝑛−−𝛽𝛾_𝑛ℎ)𝛼𝛾_𝑛𝑓𝑥_𝑛−𝐴𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+2𝛼_𝑛⟨𝑇𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩+𝛼_𝑛𝑀_𝑛,(3.42) where 𝑀_𝑛=𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑥̃‖‖²+𝛼^𝑛−𝛽_𝑛²𝛼_𝑛‖‖𝑥_𝑛−𝛾𝑓𝑥_𝑛‖‖²−2𝛼_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝛾𝑓𝑥_𝑛−𝐴𝑥̃+2𝛼^𝑛−𝛽_𝑛𝛼_𝑛𝑇𝑥_𝑛−2𝛼_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛+2𝛼_𝑛−𝛽_𝑛𝐴𝑇𝑥_𝑛−𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛−𝛽_𝑛𝛾𝑓𝑥_𝑛−𝐴𝑥̃,𝑥_𝑛−𝛾𝑓𝑥_𝑛.(3.43) That is, ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤1−2(𝛿−𝛾ℎ)𝛼_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛼_𝑛2⟨𝑇𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩+𝑀_𝑛.(3.44) Since {𝑥_𝑛} is bounded, by the conditions of Theorem 3.4, we get lim_𝑛→∞𝑀_𝑛=0 and ∑^∞𝑛=0(𝛿−𝛾ℎ)𝛼_𝑛=∞, this together with (3.39) implies that limsup_𝑛→∞2⟨𝑇𝑥_𝑛−𝑥̃,𝛾𝑓(𝑥̃)−𝐴𝑥̃⟩+𝑀_𝑛≤0.(3.45) Now applying Lemma 2.1 to (3.44) concludes that 𝑥_𝑛→𝑥̃. This complete the proof of Theorem 3.4.

If pick 𝛼_𝑛=𝛽_𝑛, we obtain the result of Marino and Xu [2].

4. Approximate Iteration Algorithm and Error Estimate

In this section, we use the following approximate iteration algorithm: 𝑦_𝑛+1=(𝐼−𝑠𝐴)𝑇𝑦_𝑛+𝑡𝛾𝑓𝑦_𝑛+(𝑠−𝑡)𝑦_𝑛,(4.1) for an arbitrary initial 𝑦₀∈𝐻, to calculate the fixed point of nonexpansive mapping and solution of variational inequality with bounded linear operator 𝐴, where 𝐴,𝑇,𝛾,𝑠,𝑡 and others 𝛿,ℎ as in the Section 3.

Meanwhile, the 𝑥̃∈𝐹(𝑇) is obtained in Theorem 3.2 which is unique solution of variational inequality (3.15) and {𝑥_𝑛}→𝑥̃, as 𝑛→∞,𝑥_𝑡,𝑠→𝑥̃ as 𝑠→0, where {𝑥_𝑛} and 𝑥_𝑡,𝑠 are respectively defined by (3.31) and (3.6).

The following lemma will be useful for the establish of formula of convergence rate estimate.

Lemma 4.1 (Banach's contractive mapping principle). Let 𝐻 be a Banach space and 𝑆 be a contraction from 𝐻 into self, that is, ‖𝑆𝑥−𝑆𝑦‖≤𝜃‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐻,(4.2) where 0<𝜃<1 is a constant. Then the Picard iterative sequence 𝑥_𝑛+1=𝑆𝑥_𝑛, for arbitrary initial 𝑥₀∈𝐻, converges strongly to a unique fixed point 𝑥^∗ of 𝑆 and ‖‖𝑥_𝑛−𝑥^∗‖‖≤𝜃^𝑛1−𝜃‖‖𝑥⁰−𝑆𝑥₀‖‖.(4.3)

For above 𝑇,𝐴,𝑓,𝛾,𝑠,𝑡,𝛿, we define the following contractive mapping: 𝑆_𝑡,𝑠𝑦=(𝐼−𝑠𝐴)𝑇𝑦+𝛾𝑓(𝑦)+(𝑠−𝑡)𝑦(4.4) from 𝐻 into self. In fact, it is not hard to see that 𝑆_𝑡,𝑠 is a contraction for sufficiently small 𝑠, indeed, by Lemma 2.5 we have, for any 𝑥,𝑦∈𝐻, that ‖‖𝑆_𝑡,𝑠𝑥−𝑆_𝑡,𝑠𝑦‖‖=≤≤𝑡((𝛾1𝑡𝛾‖+ℎ𝑓𝑡+((𝑥𝛾1)ℎ−−−𝑠𝑓1𝛿)(+𝑦−)‖𝑠𝑠−(+𝛿𝑡‖)−‖(1𝐼𝑥)−)−‖𝑠𝑦𝑥𝐴‖−)(𝑦𝑇‖𝑥.−𝑇𝑦)‖+‖(𝑠−𝑡)(𝑥−𝑦)‖(4.5) By using Lemma 4.1, then there exists unique fixed point 𝑥_𝑡,𝑠∈𝐻 of 𝑆_𝑡,𝑠 and the iterative sequence 𝑦_𝑛+1=𝑆_𝑡,𝑠𝑦_𝑛=(𝐼−𝑠𝐴)𝑇𝑦_𝑛+𝛾𝑓𝑦_𝑛+(𝑠−𝑡)𝑦_𝑛,𝑦₀∈𝐻,(4.6) converges strongly to this fixed point 𝑥_𝑡,𝑠. Meanwhile, from (4.3) and (4.5) we obtain ‖‖𝑦_𝑛−𝑥_𝑡,𝑠‖‖≤(1+𝑡(𝛾ℎ−1)−𝑠(𝛿−1))^𝑛𝑠(𝛿−1)−𝑡(𝛾ℎ−1)‖‖𝑦⁰−𝑆_𝑡,𝑠𝑦₀‖‖.(4.7) On the other hand, from (3.21) we have ‖‖𝑥_𝑡,𝑠−𝑥̃‖‖²≤𝑡𝑠𝛿−𝑡𝛾ℎ𝛾𝑓(𝑥̃)−𝐴𝑥̃,𝑥^𝑡,𝑠+𝑠−𝑡−𝑥̃𝑠𝛿−𝑡𝛾ℎ𝑥^𝑡,𝑠−𝑥̃,𝑥_𝑡,𝑠−𝑥̃+𝑥̃−𝐴𝑥̃,𝑥_𝑡,𝑠−𝑥̃,(4.8) which leads to 1−𝑠𝛿𝑠−−𝑡𝑡𝛾ℎ‖‖𝑥^𝑡,𝑠−𝑥̃‖‖²≤𝑡𝑠𝛿−𝑡𝛾ℎ𝛾𝑓(𝑥̃)−𝐴𝑥̃,𝑥^𝑡,𝑠+𝑠−𝑡−𝑥̃𝑠𝛿−𝑡𝛾ℎ𝑥̃−𝐴𝑥̃,𝑥^𝑡,𝑠−𝑥̃.(4.9) Therefore, 1−𝑠𝛿𝑠−−𝑡𝑡𝛾ℎ‖‖𝑥^𝑡,𝑠−𝑥̃‖‖≤𝑡‖‖𝑥𝑠𝛿−𝑡𝛾ℎ‖𝛾𝑓(𝑥̃)−𝐴𝑥̃‖+𝑠𝛿𝑠−−𝑡𝑡𝛾ℎ‖𝑥̃−𝐴𝑥̃‖,_𝑡,𝑠−𝑥̃‖‖≤𝑡𝑠𝛿−𝑡𝛾ℎ+𝑡−𝑠‖𝛾𝑓(𝑥̃)−𝐴𝑥̃‖+𝑠𝛿−𝑡𝑠𝛾ℎ−+𝑡𝑡−𝑠‖𝑥̃−𝐴𝑥̃‖.(4.10) Letting 𝐷₁=‖𝛾𝑓(𝑥̃)−𝐴𝑥̃‖,𝐷₂=‖𝑥̃−𝐴𝑥̃‖, it follows that ‖‖𝑥_𝑡,𝑠−𝑥̃‖‖≤𝑡𝑠𝛿−𝑡𝛾ℎ+𝑡−𝑠𝐷¹+𝑠−𝑡𝑠𝛿−𝑡𝛾ℎ+𝑡−𝑠𝐷².(4.11) From inequality (4.11) together with (4.7), and letting 𝐷₃=‖𝑦₀−𝑆_𝑡,𝑠𝑦₀‖, we get ‖‖𝑦_𝑛−𝑥̃‖‖≤𝑡𝑠𝛿−𝑡𝛾ℎ+𝑡−𝑠𝐷¹+𝑠−𝑡𝑠𝛿−𝑡𝛾ℎ+𝑡−𝑠𝐷²+(1+𝑡(𝛾ℎ−1)−𝑠(𝛿−1))^𝑛𝑠(𝛿−1)−𝑡(𝛾ℎ−1)𝐷³.(4.12) Inequality (4.12) is, namely, the error estimate for approximate fixed point 𝑦_𝑛. Now, we give several special cases of inequality (4.12).

Error Estimate 1
Consider limsup_𝑛→∞‖‖𝑦_𝑛−𝑥̃‖‖≤𝑡𝑠𝛿−𝑡𝛾ℎ+𝑡−𝑠𝐷¹+𝑠−𝑡𝑠𝛿−𝑡𝛾ℎ+𝑡−𝑠𝐷².(4.13)

Error Estimate 2
If 𝑡=𝑠, then ‖‖𝑦_𝑛−𝑥̃‖‖≤1(𝛿−𝛾ℎ)𝐷¹+(1−𝑠(𝛿−𝛾ℎ))^𝑛𝑠(𝛿−𝛾ℎ)𝐷³,(4.14) which can be used to estimate error for iterative scheme 𝑦_𝑛+1=(𝐼−𝑠𝐴)𝑇𝑦_𝑛+𝑠𝛾𝑓𝑦_𝑛,𝑦₀∈𝐻.(4.15)

Error Estimate 3
If 𝐴=𝐼, then ‖‖𝑦_𝑛−𝑥̃‖‖≤𝑡𝑠𝛿−𝑡𝛾ℎ+𝑡−𝑠𝐷¹+(1+𝑡(𝛾ℎ−1)−𝑠(𝛿−1))^𝑛𝑠(𝛿−1)−𝑡(𝛾ℎ−1)𝐷³,(4.16) which can be used to estimate error for iterative scheme 𝑦_𝑛+1=(1−𝑠)𝑇𝑦_𝑛+𝑡𝛾𝑓𝑦_𝑛+(𝑠−𝑡)𝑥_𝑛,𝑦₀∈𝐻.(4.17)

Acknowledgments

This paper is supported by the National Natural Science Foundation of China under Grant (11071279).