General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities

Abstract

This paper deals with new methods for approximating a solution to the fixed point problem; find 𝑥̃∈𝐹(𝑇), where 𝐻 is a Hilbert space, 𝐶 is a closed convex subset of 𝐻, 𝑓 is a 𝜌-contraction from 𝐶 into 𝐻, 0<𝜌<1, 𝐴 is a strongly positive linear-bounded operator with coefficient 𝛾>0, 0<𝛾<𝛾/𝜌, 𝑇 is a nonexpansive mapping on 𝐶, and 𝑃_𝐹(𝑇) denotes the metric projection on the set of fixed point of 𝑇. Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality ⟨(𝐴−𝛾𝑓)𝑥̃+𝜏(𝐼−𝑆)𝑥̃,𝑥−𝑥̃⟩≥0 for 𝑥∈𝐹(𝑇), where 𝜏∈[0,∞). Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part.

1. Introduction

Throughout this paper, we assume that 𝐻 is a real Hilbert space where inner product and norm are denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively, and let 𝐶 be a nonempty closed convex subset of 𝐻. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.1)

We use 𝐹(𝑇) to denote the set of fixed points of 𝑇, that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. It is assumed throughout the paper that 𝑇 is a nonexpansive mapping such that 𝐹(𝑇)≠∅.

Recall that a mapping 𝑓∶𝐶→𝐻 is a contraction on 𝐶 if there exists a constant 𝜌∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝜌‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(1.2)

A mapping 𝐴∶𝐻→𝐻 is called a strongly positive linear bounded operator on 𝐻 if there exists a constant 𝛾>0 with property ⟨𝐴𝑥,𝑥⟩≥𝛾‖𝑥‖²,∀𝑥∈𝐻.(1.3)

A mapping 𝑀∶𝐻→𝐻 is called a strongly monotone operator with 𝛼 if ⟨𝑥−𝑦,𝑀𝑥−𝑀𝑦⟩≥𝛼‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐻,(1.4) and 𝑀 is called a monotone operator if ⟨𝑥−𝑦,𝑀𝑥−𝑀𝑦⟩≥0,∀𝑥,𝑦∈𝐻.(1.5) We easily prove that the mapping (𝐼−𝑇) is monotone operator, if 𝑇 is nonexpansive mapping.

The metric (or nearest point) projection from 𝐻 onto 𝐶 is mapping 𝑃_𝐶[⋅]∶𝐻→𝐶 which assigns to each point 𝑥∈𝐶 the unique point 𝑃_𝐶[𝑥]∈𝐶 satisfying the property ‖‖𝑥−𝑃_𝐶[𝑥]‖‖=inf_𝑦∈𝐶‖𝑥−𝑦‖=∶𝑑(𝑥,𝐶).(1.6)

The variational inequality for a monotone operator, 𝑀∶𝐻→𝐻 over 𝐶, is to find a point in VI(𝐶,𝑀)∶={𝑥̃∈𝐶∶⟨𝑥−𝑥̃,𝑀𝑥̃⟩≥0,∀𝑥∈𝐶}.(1.7)

A hierarchical fixed point problem is equivalent to the variational inequality for a monotone operator over the fixed point set. Moreover, to find a hierarchically fixed point of a nonexpansive mapping 𝑇 with respect to another nonexpansive mapping 𝑆, namely, we find 𝑥̃∈𝐹(𝑇) such that ⟨𝑥−𝑥̃,(𝐼−𝑆)𝑥̃⟩≥0,∀𝑥∈𝐹(𝑇).(1.8)

Iterative methods for nonexpansive mappings have recently been applied to solve a convex minimization problem; see, for example, [1–5] 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⟩,(1.9) where 𝑏 is a given point in 𝐻. In [5], it is proved that the sequence {𝑥_𝑛} defined by the iterative method below, with the initial guess 𝑥₀∈𝐻 chosen arbitrarily, 𝑥_𝑛+1=𝐼−𝛼_𝑛𝐴𝑇𝑥_𝑛+𝛼_𝑛𝑏,𝑛≥0,(1.10) converges strongly to the unique solution of the minimization problem (1.9) provided the sequence {𝛼_𝑛} of parameters satisfies certain appropriate 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). Starting with an arbitrary initial 𝑥₀∈𝐻, define a sequence {𝑥_𝑛} recursively by 𝑥_𝑛+1=𝜎_𝑛𝑓𝑥_𝑛+1−𝜎_𝑛𝑇𝑥_𝑛,𝑛≥0,(1.11) where {𝜎_𝑛} is a sequence in (0,1). It is proved in [6, 7] that under certain appropriate conditions imposed on {𝜎_𝑛}, the sequence {𝑥_𝑛} generated by (1.11) strongly converges to the unique solution 𝑥^∗ in 𝐶 of the variational inequality ⟨(𝐼−𝑓)𝑥^∗,𝑥−𝑥^∗⟩≥0,𝑥∈𝐶.(1.12)

In 2006, Marino and Xu [8] introduced a general iterative method for nonexpansive mapping. Starting with an arbitrary initial 𝑥₀∈𝐻, define a sequence {𝑥_𝑛} recursively by 𝑥_𝑛+1=𝜖_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝜖_𝑛𝐴𝑇𝑥_𝑛,𝑛≥0.(1.13) They proved that if the sequence {𝜖_𝑛} of parameters satisfies appropriate conditions, then the sequence {𝑥_𝑛} generated by (1.13) strongly converges to the unique solution 𝑥̃=𝑃_𝐹(𝑇)(𝐼−𝐴+𝛾𝑓)𝑥̃ of the variational inequality ⟨(𝐴−𝛾𝑓)𝑥̃,𝑥−𝑥̃⟩≥0,∀𝑥∈𝐹(𝑇),(1.14) which is the optimality condition for the minimization problem min_{𝑥∈𝐹(𝑇)}12⟨𝐴𝑥,𝑥⟩−ℎ(𝑥),(1.15) where ℎ is a potential function for 𝛾𝑓 (i.e., ℎ′(𝑥)=𝛾𝑓(𝑥) for 𝑥∈𝐻).

In 2010, Yao et al. [9] introduced an iterative algorithm for solving some hierarchical fixed point problem, starting with an arbitrary initial guess 𝑥₀∈𝐶, define a sequence {𝑥_𝑛} iteratively by 𝑥_𝑛+1𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛,∀𝑛≥1.(1.16) They proved that if the sequences {𝛼_𝑛} and {𝛽_𝑛} of parameters satisfies appropriate conditions, then the sequence {𝑥_𝑛} generated by (1.16) strongly converges to the unique solution 𝑧 in 𝐻 of the variational inequality 𝑧∈𝐹(𝑇),⟨(𝐼−𝑓)𝑧,𝑥−𝑧⟩≥0,∀𝑥∈𝐹(𝑇).(1.17)

In this paper we will combine the general iterative method (1.13) with the iterative algorithm (1.16) and consider the following iterative algorithm: 𝑥_𝑛+1𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑦_𝑛,∀𝑛≥1.(1.18) We will prove in Section 3 that if the sequences {𝛼_𝑛} and {𝛽_𝑛} of parameters satisfy appropriate conditions and lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=𝜏∈(0,∞) then the sequence {𝑥_𝑛} generated by (1.18) converges strongly to the unique solution 𝑥̃ in 𝐻 of the following variational inequality 𝑥̃∈𝐹(𝑇),1𝜏(𝐴−𝛾𝑓)𝑥̃+(𝐼−𝑆)𝑥̃,𝑥−𝑥̃≥0,∀𝑥∈𝐹(𝑇).(1.19) In particular, if we take 𝜏=0, under suitable difference assumptions on parameter, then the sequence {𝑥_𝑛} generated by (1.18) converges strongly to the unique solution 𝑥̃ in 𝐻 of the following variational inequality 𝑥̃∈𝐹(𝑇),⟨(𝐴−𝛾𝑓)𝑥̃,𝑥−𝑥̃⟩≥0,∀𝑥∈𝐹(𝑇).(1.20) Our results improve and extend the recent results of Yao et al. [9] and some authors. Furthermore, we give an example which supports our main theorem in the last part.

2. Preliminaries

This section collects some lemma which can 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 (Browder [10]). Let 𝐻 be a Hilbert space, 𝐶 be a closed convex subset of 𝐻, and 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. Let 𝑥∈𝐻 and 𝑧∈𝐶 be any points. Then one has the following:
(1) That 𝑧=𝑃_𝐶[𝑥] if and only if there holds the relation: ⟨𝑥−𝑧,𝑦−𝑧⟩≤0,∀𝑦∈𝐶.(2.1)(2) That 𝑧=𝑃_𝐶[𝑥] if and only if there holds the relation: ‖𝑥−𝑧‖²≤‖𝑥−𝑦‖²−‖𝑦−𝑧‖²,∀𝑦∈𝐶.(2.2)(3) There holds the relation: ⟨𝑃_𝐶[𝑥]−𝑃_𝐶[𝑦],𝑥−𝑦⟩≥‖‖𝑃_𝐶[𝑥]−𝑃_𝐶[𝑦]‖‖²,∀𝑥,𝑦∈𝐻.(2.3) Consequently, 𝑃_𝐶 is nonexpansive and monotone.

Lemma 2.3 (Marino and Xu [8]). Let 𝐻 be a Hilbert space, 𝐶 be a closed convex subset of 𝐻, 𝑓∶𝐶→𝐻 be a contraction with coefficient 0<𝜌<1, and 𝑇∶𝐶→𝐶 be nonexpansive mapping. Let 𝐴 be a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾⁻>0. Then, for 0<𝛾<𝛾⁻/𝜌, for 𝑥,𝑦∈𝐶, (1)the mapping (𝐼−𝑓) is strongly monotone with coefficient (1−𝜌), that is, ⟨𝑥−𝑦,(𝐼−𝑓)𝑥−(𝐼−𝑓)𝑦⟩≥(1−𝜌)‖𝑥−𝑦‖²,(2.4)(2)the mapping (𝐴−𝛾𝑓) is strongly monotone with coefficient 𝛾⁻−𝛾𝜌 that is ⟨𝑥−𝑦,(𝐴−𝛾𝑓)𝑥−(𝐴−𝛾𝑓)𝑦⟩≥𝛾⁻−𝛾𝜌‖𝑥−𝑦‖².(2.5)

Lemma 2.4 (Xu [4]). Assume that {𝑎_𝑛} is a sequence of nonnegative numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛,∀𝑛≥0,(2.6) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in ℝ such that (1)∑^∞𝑛=1𝛾_𝑛=∞, (2)limsup_𝑛→∞(𝛿_𝑛/𝛾_𝑛)≤0 or ∑^∞𝑛=1|𝛿_𝑛|<∞. Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.5 (Marino and Xu [8]). Assume 𝐴 is a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾⁻>0 and 0<𝛼≤‖𝐴‖⁻¹. Then ‖𝐼−𝛼𝐴‖≤1−𝛼𝛾⁻.

Lemma 2.6 (Acedo and Xu [11]). Let 𝐶 be a closed convex subset of 𝐻. Let {𝑥_𝑛} be a bounded sequence in 𝐻. Assume that (1)The weak 𝜔-limit set 𝜔_𝑤(𝑥_𝑛)⊂𝐶,(2)For each 𝑧∈𝐶, lim_𝑛→∞‖𝑥_𝑛−𝑧‖ exists. Then {𝑥_𝑛} is weakly convergent to a point in 𝐶.

NotationWe use → for strong convergence and ⇀ for weak convergence.

3. Main Results

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝐻 be a 𝜌-contraction with 𝜌∈(0,1). Let 𝑆,𝑇∶𝐶→𝐶 be two nonexpansive mappings with 𝐹(𝑇)≠∅. Let 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient 𝛾⁻>0. {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) and 0<𝛾<𝛾⁻/𝜌. Starting with an arbitrary initial guess 𝑥₀∈𝐶 and {𝑥_𝑛} is a sequence generated by 𝑥_𝑛+1𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑦_𝑛,∀𝑛≥1.(3.1)
Suppose that the following conditions are satisfied: (C1)  lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞, (C2)  lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=𝜏=0, (C3)  lim_𝑛→∞(|𝛼_𝑛−𝛼_𝑛−1|/𝛼_𝑛)=0 and lim_𝑛→∞(|𝛽_𝑛−𝛽_𝑛−1|/𝛽_𝑛)=0, or (C4)  ∑^∞𝑛=1|𝛼_𝑛−𝛼_𝑛−1|<∞ and ∑^∞𝑛=1|𝛽_𝑛−𝛽_𝑛−1|<∞. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑥̃∈𝐻, which is the unique solution of the variational inequality: 𝑥̃∈𝐹(𝑇),⟨(𝐴−𝛾𝑓)𝑥̃,𝑥−𝑥̃⟩≥0,∀𝑥∈𝐹(𝑇).(3.2) Equivalently, one has 𝑃_𝐹(𝑇)(𝐼−𝐴+𝛾𝑓)𝑥̃=𝑥̃.

Proof. We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of 𝐴−𝛾𝑓. Suppose 𝑥⁻∈𝐹(𝑇) and 𝑥̃∈𝐹(𝑇) both are solutions to (3.2), then ⟨(𝐴−𝛾𝑓)𝑥⁻,𝑥⁻−𝑥̃⟩≤0 and ⟨(𝐴−𝛾𝑓)𝑥̃,𝑥̃−𝑥⁻⟩≤0. It follows that (𝐴−𝛾𝑓)𝑥⁻,𝑥⁻−𝑥̃+(𝐴−𝛾𝑓)𝑥̃,𝑥̃−𝑥⁻==⟨(𝐴(𝐴−−𝛾𝛾𝑓𝑓))₋𝑥⁻,𝑥⁻−𝑥̃−(𝐴−𝛾𝑓)𝑥̃,𝑥⁻−𝑥̃𝑥−(𝐴−𝛾𝑓)𝑥̃,𝑥⁻−𝑥̃⟩≤0.(3.3) The strong monotonicity of 𝐴−𝛾𝑓 (Lemma 2.3) implies that 𝑥⁻=𝑥̃ and the uniqueness is proved.
Next, we prove that the sequence {𝑥_𝑛} is bounded. Since 𝛼_𝑛→0 and lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0 by condition (C1) and (C2), respectively, we can assume, without loss of generality, that 𝛼_𝑛<‖𝐴‖⁻¹ and 𝛽_𝑛<𝛼_𝑛 for all 𝑛≥1. Take 𝑢∈𝐹(𝑇) and from (3.1), we have ‖‖𝑥_𝑛+1−𝑢‖‖≤=‖‖‖‖𝛼𝑃_𝐶𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑦_𝑛−𝑃_𝐶[𝑢]‖‖≤𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑦_𝑛−𝑢‖‖𝛾‖‖𝑓𝑥_𝑛−𝑓(𝑢)‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖+‖‖𝐼−𝛼_𝑛𝐴𝑇𝑦_𝑛−𝑢‖‖.(3.4) Since ‖𝐼−𝛼_𝑛𝐴‖≤1−𝛼𝛾_𝑛− and by Lemma 2.5, we note that ‖‖𝑥_𝑛+1−𝑢‖‖≤≤𝛼𝛼_𝑛𝑛𝛾‖‖𝑓𝑥_𝑛−𝑓(𝑢)‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖+1−𝛼𝛾_𝑛−‖‖𝑇𝑦_𝑛−𝑢‖‖≤𝛼_𝑛𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖+1−𝛼𝛾_𝑛−‖‖𝑇𝑦_𝑛−𝑇𝑢‖‖≤𝛼_𝑛𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖+1−𝛼𝛾_𝑛−‖‖𝑦_𝑛−𝑢‖‖+𝛾𝜌1‖‖−𝑥_𝑛𝛼−_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖≤𝛼_𝑛𝛾𝛽_𝑛‖‖𝑆𝑥_𝑛−𝑆𝑢‖‖+𝛽_𝑛‖𝑆𝑢−𝑢‖+1−𝛽_𝑛‖‖𝑥_𝑛−𝑢‖‖+𝛾𝜌1‖‖−𝑥_𝑛𝛼−_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖=1−𝛼_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑢‖‖+𝛽_𝑛‖𝑆𝑢−𝑢‖+1−𝛽_𝑛‖‖𝑥_𝑛−𝑢‖‖≤1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖+1−𝛼𝛾_𝑛−𝛽_𝑛‖𝑆𝑢−𝑢‖≤1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖+𝛽_𝑛‖𝑆𝑢−𝑢‖=1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖+𝛼_𝑛‖𝑆𝑢−𝑢‖=1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛[‖𝛾𝑓(𝑢)−𝐴𝑢‖+‖𝑆𝑢−𝑢‖]𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛𝛾⁻−𝛾𝜌‖𝛾𝑓(𝑢)−𝐴𝑢‖+‖𝑆𝑢−𝑢‖𝛾⁻−𝛾𝜌.(3.5) By induction, we can obtain ‖‖𝑥_𝑛+1−𝑢‖‖≤max⎪⎪⎧⎨⎩‖‖𝑥₀−𝑢‖‖,‖𝛾𝑓(𝑢)−𝐴𝑢‖+‖𝑆𝑢−𝑢‖𝛾⁻−𝛾𝜌⎫⎪⎬⎪⎭,(3.6) which implies that the sequence {𝑥_𝑛} is bounded and so are the sequences {𝑓(𝑥_𝑛)}, {𝑆𝑥_𝑛}, and {𝐴𝑇𝑦_𝑛}.
Set 𝑤_𝑛∶=𝛼_𝑛𝛾𝑓(𝑥_𝑛)+(𝐼−𝛼_𝑛𝐴)𝑇𝑦_𝑛,𝑛≥1. We get ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤=‖‖‖‖𝑤𝑃_𝐶_𝑛+1𝑤_𝑛+1−𝑃_𝐶𝑤_𝑛‖‖−𝑤_𝑛‖‖.(3.7) It follows that ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤≤𝛼‖‖_𝑛𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑦_𝑛−𝛼_𝑛−1𝛾𝑓𝑥_𝑛−1+𝐼−𝛼_𝑛−1𝐴𝑇𝑦_𝑛−1‖‖+𝛾‖‖1𝑓−𝛼𝑥_𝑛_𝑛−−𝑓𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||‖‖𝛾𝑓𝑥_𝑛−1−𝐴𝑇𝑦_𝑛−1‖‖≤𝛼_𝑛𝛾‖‖𝑇𝑦_𝑛−𝑇𝑦_𝑛−1‖‖+𝛾𝜌1‖‖−𝑥_𝑛𝛼−_𝑛−𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||‖‖𝛾𝑓𝑥_𝑛−1−𝐴𝑇𝑦_𝑛−1‖‖𝛾‖‖𝑦_𝑛−𝑦_𝑛−1‖‖.(3.8) By (3.7) and (3.8), we get ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤𝛼+_𝑛𝛾𝜌1‖‖−𝑤𝛼_𝑛−_𝑛−𝑤_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||‖‖𝛾𝑓𝑥_𝑛−1−𝐴𝑇𝑦_𝑛−1‖‖𝛾‖‖𝑦_𝑛−𝑦_𝑛−1‖‖.(3.9) From (3.1), we obtain ‖‖𝑦_𝑛−𝑦_𝑛−1‖‖==‖‖‖‖𝛽_𝑛𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1+1−𝛽_𝑛−1𝑥_𝑛−1‖‖≤‖‖𝑥_𝑛𝑆𝑥_𝑛−𝑆𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑆𝑥_𝑛−1−𝑥_𝑛−1+1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1‖‖≤‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+||𝛽_𝑛−𝛽_𝑛−1||‖‖𝑆𝑥_𝑛−1−𝑥_𝑛−1‖‖−𝑥_𝑛−1‖‖+||𝛽_𝑛−𝛽_𝑛−1||𝑀,(3.10) where 𝑀 is a constant such that sup_𝑛∈ℕ‖‖𝛾𝑓𝑥_𝑛−1−𝐴𝑇𝑦_𝑛−1‖‖+‖‖𝑆𝑥_𝑛−1−𝑥_𝑛−1‖‖≤𝑀.(3.11) Substituting (3.10) into (3.8) to obtain ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤𝛼+_𝑛𝛾𝜌1‖‖−𝑥_𝑛𝛼−_𝑛−𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||‖‖𝛾𝑓𝑥_𝑛−1−𝐴𝑇𝑦_𝑛−1‖‖≤𝛼_𝑛𝛾‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+||𝛽_𝑛−𝛽_𝑛−1||𝑀+𝛾𝜌1‖‖−𝑥_𝑛𝛼−_𝑛−𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||𝑀=1−𝛼_𝑛𝛾‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+||𝛽_𝑛−𝛽_𝑛−1||𝑀𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+𝑀||𝛼_𝑛−𝛼_𝑛−1≤1−𝛼_𝑛||+||𝛽_𝑛−𝛽_𝑛−1||𝛾⁻−𝛾𝜌‖‖𝑤_𝑛−𝑤_𝑛−1‖‖+𝑀||𝛼_𝑛−𝛼_𝑛−1||+||𝛽_𝑛−𝛽_𝑛−1||.(3.12) At the same time, we can write (3.12) as ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑤_𝑛−𝑤_𝑛−1‖‖+𝑀𝛼_𝑛||𝛼^𝑛−𝛼_𝑛−1||𝛼_𝑛+||𝛽^𝑛−𝛽_𝑛−1||≤1−𝛼_𝑛𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑤_𝑛−𝑤_𝑛−1‖‖+𝑀𝛼_𝑛||𝛼^𝑛−𝛼_𝑛−1||𝛽_𝑛+||𝛽^𝑛−𝛽_𝑛−1||𝛽_𝑛.(3.13) From (3.12), (C4), and Lemma 2.5 or from (3.13), (C3), and Lemma 2.5, we can deduce that ‖𝑥_𝑛+1−𝑥_𝑛‖→0, respectively.
From (3.1), we have ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖≤=‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑇𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑃_𝐶𝑤_𝑛−𝑃_𝐶𝑇𝑥_𝑛‖‖=‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑤_𝑛−𝑇𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑦_𝑛−𝑇𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑇𝑥_𝑛‖‖+1−𝛼𝛾_𝑛−‖‖𝑇𝑦_𝑛−𝑇𝑥_𝑛‖‖=‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑇𝑥_𝑛‖‖+1−𝛼𝛾_𝑛−‖‖𝑦_𝑛−𝑥_𝑛‖‖−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝛾𝑓𝑥_𝑛−𝐴𝑇𝑥_𝑛‖‖+1−𝛼𝛾_𝑛−𝛽_𝑛‖‖𝑆𝑥_𝑛−𝑥_𝑛‖‖.(3.14) Notice that 𝛼_𝑛→0, 𝛽_𝑛→0, and ‖𝑥_𝑛+1−𝑥_𝑛‖→0, so we obtain ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖⟶0.(3.15) Next, we prove limsup_𝑛→∞⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛−𝑧⟩≤0,(3.16) where 𝑧=𝑃_𝐹(𝑇)(𝐼−𝐴+𝛾𝑓)𝑧. Since the sequence {𝑥_𝑛} is bounded we can take a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that limsup_𝑛→∞⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛−𝑧⟩=lim_𝑘→∞𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛𝑘−𝑧,(3.17) and 𝑥_𝑛𝑘⇀0𝑥00086𝑥̃. From (3.15) and by Lemma 2.1, it follows that 𝑥̃∈𝐹(𝑇). Hence, by Lemma 2.2(1) that limsup_𝑛→∞⟨𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛−𝑧⟩≤===0⟨⟨li.𝛾(m𝐼𝑓_𝑘→∞−(𝑧𝛾𝐴)𝑓−+(𝐴𝑧𝛾)𝑧𝑓−,)𝑥𝑧𝐴̃−−𝑧,𝑧𝑧𝑥⟩,𝑥_𝑛𝑘̃−−𝑧𝑧⟩(3.18) Now, by Lemma 2.2(1), we observe that 𝑃_𝐶𝑤_𝑛−𝑤_𝑛,𝑃_𝐶𝑤_𝑛−𝑧≤0,(3.19) and so ‖‖𝑥_𝑛+1−𝑧‖‖²==𝑃𝑃_𝐶𝑤_𝑛−𝑧,𝑃_𝐶𝑤_𝑛−𝑧_𝐶≤𝑤𝑤_𝑛−𝑤_𝑛,𝑃_𝐶𝑤_𝑛−𝑧+𝑤_𝑛−𝑧,𝑃_𝐶𝑤_𝑛−𝑧_𝑛=𝛼−𝑧,𝑃_𝐶𝑤_𝑛−𝑧_𝑛≤𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑦_𝑛−𝑧,𝑥_𝑛+1−𝑧+𝛾‖‖1𝑓−𝛼𝑥_𝑛_𝑛−−𝑓(𝑧)‖‖‖‖𝑥_𝑛+1−𝑧‖‖+𝛼_𝑛𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛+1−𝑧≤𝛼_𝑛𝛾‖‖𝑇𝑦_𝑛−𝑧‖‖‖‖𝑥_𝑛+1−𝑧‖‖+𝛾𝜌1‖‖−𝑥_𝑛𝛼−_𝑛−𝑧‖‖‖‖𝑥_𝑛+1−𝑧‖‖+𝛼_𝑛𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛+1−𝑧=𝛼_𝑛𝛾‖‖𝑦_𝑛−𝑧‖‖‖‖𝑥_𝑛+1−𝑧‖‖+𝛾𝜌1‖‖−𝑥_𝑛𝛼−_𝑛−𝑧‖‖‖‖𝑥_𝑛+1−𝑧‖‖+𝛼_𝑛𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛+1−𝑧≤𝛼_𝑛𝛾‖‖𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝑧‖‖‖‖𝑥_𝑛+1−𝑧‖‖+𝛾𝜌1‖‖−𝑥_𝑛𝛼−_𝑛−𝑧‖‖‖‖𝑥_𝑛+1−𝑧‖‖+𝛼_𝑛𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛+1−𝑧≤𝛼_𝑛𝛾𝛽_𝑛‖‖𝑆𝑥_𝑛−𝑆𝑧‖‖+𝛽_𝑛‖𝑆𝑧−𝑧‖+1−𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖‖𝑥_𝑛+1−𝑧‖+𝛾𝜌1‖‖−𝑥_𝑛𝛼−_𝑛−𝑧‖‖‖‖𝑥_𝑛+1−𝑧‖‖+𝛼_𝑛𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛+1−𝑧=1−𝛼_𝑛𝛾𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖+𝛽_𝑛‖𝑆𝑧−𝑧‖+1−𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖‖‖𝑥_𝑛+1−𝑧‖‖+1−𝛼𝛾⁻_𝑛−−𝛾𝜌‖‖𝑥_𝑛−𝑧‖‖‖‖𝑥_𝑛+1−𝑧‖‖+𝛼_𝑛𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛+1−𝑧≤1−𝛼_𝑛𝛾𝛽_𝑛‖𝑆𝑧−𝑧‖‖‖𝑥_𝑛+1−𝑧‖‖𝛾⁻−𝛾𝜌2‖‖𝑥^𝑛+1−𝛼_𝑛−−𝑧‖‖²+‖‖𝑥_𝑛+1−𝑧‖‖²+𝛼^𝑛𝛾𝑓(𝑧)−𝐴𝑧,𝑥_𝑛+1−𝑧𝛾𝛽_𝑛‖𝑆𝑧−𝑧‖‖‖𝑥_𝑛+1−𝑧‖‖.(3.20) Hence, it follows that ‖‖𝑥_𝑛+1−𝑧‖‖²≤1−𝛼^𝑛𝛾⁻−𝛾𝜌1+𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥^𝑛−𝑧‖‖²+2𝛼^𝑛1+𝛼_𝑛𝛾⁻−𝛾𝜌𝛾𝑓(𝑧)−𝐴𝑧,𝑥^𝑛+1+21−𝛼_𝑛−−𝑧𝛾𝛽_𝑛1+𝛼_𝑛𝛾⁻−𝛾𝜌‖𝑆𝑧−𝑧‖‖‖𝑥^𝑛+1=⎡⎢⎢⎣2𝛼_𝑛−𝑧‖‖𝛾⁻−𝛾𝜌1+𝛼_𝑛𝛾⁻−𝛾𝜌⎤⎥⎥⎦⎡⎢⎢⎣𝛼_𝑛𝛾⁻1−𝛾𝜌𝛾𝑓(𝑧)−𝐴𝑧,𝑥^𝑛+1+𝛽−𝑧_𝑛1−𝛼𝛾_𝑛−𝛼_𝑛𝛾⁻−𝛾𝜌‖𝑆𝑧−𝑧‖‖‖𝑥^𝑛+1×⎡⎢⎢⎣1−2𝛼_𝑛−𝑧‖‖⎤⎥⎥⎦𝛾⁻−𝛾𝜌1+𝛼_𝑛𝛾⁻−𝛾𝜌⎤⎥⎥⎦‖‖𝑥^𝑛−𝑧‖‖².(3.21) We observe that limsup_𝑛→∞⎡⎢⎢⎣1𝛼_𝑛𝛾⁻−𝛾𝜌𝛾𝑓(𝑧)−𝐴𝑧,𝑥^𝑛+1−𝑧+𝛽^𝑛1−𝛼𝛾_𝑛−𝛼_𝑛𝛾⁻−𝛾𝜌‖𝑆𝑧−𝑧‖‖‖𝑥^𝑛+1−𝑧‖‖⎤⎥⎥⎦≤0.(3.22) Thus, by Lemma 2.4, 𝑥_𝑛→𝑧 as 𝑛→∞. This is completes.

From Theorem 3.1, we can deduce the following interesting corollary.

Corollary 3.2 (Yao et al. [9]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝐻 be a 𝜌-contraction (possibly nonself) with 𝜌∈(0,1). Let 𝑆,𝑇∶𝐶→𝐶 be two nonexpansive mappings with 𝐹(𝑇)≠∅.  {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1). Starting with an arbitrary initial guess 𝑥₀∈𝐶 and {𝑥_𝑛} is a sequence generated by 𝑥_𝑛+1𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛,∀𝑛≥1.(3.23) Suppose that the following conditions are satisfied: (C1)  lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞, (C2)  lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0, (C3)  lim_𝑛→∞(|𝛼_𝑛−𝛼_𝑛−1|/𝛼_𝑛)=0 and lim_𝑛→∞(|𝛽_𝑛−𝛽_𝑛−1|/𝛽_𝑛)=0, or (C4)  ∑^∞𝑛=1|𝛼_𝑛−𝛼_𝑛−1|<∞ and ∑^∞𝑛=1|𝛽_𝑛−𝛽_𝑛−1|<∞. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑥̃∈𝐻, which is the unique solution of the variational inequality: 𝑥̃∈𝐹(𝑇),⟨(𝐼−𝑓)𝑥̃,𝑥−𝑥̃⟩≥0,∀𝑥∈𝐹(𝑇).(3.24) Equivalently, one has 𝑃_𝐹(𝑇)(𝑓)𝑥̃=𝑥̃. In particular, if one takes 𝑓=0, then the sequence {𝑥_𝑛} converges in norm to the Minimum norm fixed point 𝑥̃ of 𝑇, namely, the point 𝑥̃ is the unique solution to the quadratic minimization problem: 𝑧=argmin_{𝑥∈𝐹(𝑇)}‖𝑥‖².(3.25)

Proof. As a matter of fact, if we take 𝐴=𝐼 and 𝛾=1 in Theorem 3.1. This completes the proof.

Under different conditions on data we obtain the following result.

Theorem 3.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝐻 be a 𝜌-contraction (possibly nonself) with 𝜌∈(0,1). Let 𝑆,𝑇∶𝐶→𝐶 be two nonexpansive mappings with 𝐹(𝑇)≠∅. Let 𝐴 be a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient 𝛾⁻>0 and 0<𝛾<𝛾⁻/𝜌. {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1). Starting with an arbitrary initial guess 𝑥₀∈𝐶 and {𝑥_𝑛} is a sequence generated by 𝑥_𝑛+1𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,=𝑃_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝐼−𝛼_𝑛𝐴𝑇𝑦_𝑛,∀𝑛≥1.(3.26) Suppose that the following conditions are satisfied: (C1)  lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞, (C2)  lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=𝜏∈(0,∞), (C5)  lim_𝑛→∞((|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|)/𝛼_𝑛𝛽_𝑛)=0, (C6)  there exists a constant 𝐾>0 such that (1/𝛼_𝑛)|1/𝛽_𝑛−1/𝛽_𝑛−1|≤𝐾. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑥̃∈𝐻, which is the unique solution of the variational inequality: 𝑥̃∈𝐹(𝑇),1𝜏(𝐴−𝛾𝑓)𝑥̃+(𝐼−𝑆)𝑥̃,𝑥−𝑥̃≥0,∀𝑥∈𝐹(𝑇).(3.27)

Proof. First of all, we show that (3.27) has the unique solution. Indeed, let 𝑥⁻ and 𝑥̃ be two solutions. Then (𝐴−𝛾𝑓)𝑥̃,𝑥̃−𝑥⁻≤𝜏(𝐼−𝑆)𝑥̃,𝑥⁻−𝑥̃.(3.28) Analogously, we have (𝐴−𝛾𝑓)𝑥⁻,𝑥⁻−𝑥̃≤𝜏(𝐼−𝑆)𝑥⁻,𝑥̃−𝑥⁻.(3.29) Adding (3.28) and (3.29), by Lemma 2.3, we obtain 𝛾⁻−𝛾𝜌‖‖𝑥̃−𝑥⁻‖‖²≤≤−(𝜏𝐴−(𝐼𝛾−𝑓)𝑆𝑥)̃−𝑥̃(−𝐴(−𝐼−𝛾𝑓𝑆))𝑥⁻₋,𝑥̃−𝑥⁻≤0,𝑥,𝑥̃−𝑥⁻(3.30) and so 𝑥̃=𝑥⁻. From (C2), we can assume, without loss of generality, that 𝛽_𝑛≤(𝜏+1)𝛼_𝑛 for all 𝑛≥1. By a similar argument in Theorem 3.1, we have ‖‖𝑥_𝑛+1−𝑢‖‖≤𝛼+_𝑛𝛾𝜌1‖‖−𝑥_𝑛𝛼−_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖=1−𝛼_𝑛𝛾‖‖𝑥_𝑛−𝑢‖‖+𝛽_𝑛‖𝑆𝑢−𝑢‖+1−𝛽_𝑛‖‖𝑥_𝑛−𝑢‖‖≤1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖+1−𝛼𝛾_𝑛−𝛽_𝑛‖𝑆𝑢−𝑢‖≤1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖+𝛽_𝑛‖𝑆𝑢−𝑢‖=1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛‖𝛾𝑓(𝑢)−𝐴𝑢‖+(𝜏+1)𝛼_𝑛‖𝑆𝑢−𝑢‖=1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛[‖𝛾𝑓(𝑢)−𝐴𝑢‖+(𝜏+1)‖𝑆𝑢−𝑢‖]𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑢‖‖+𝛼_𝑛𝛾⁻−𝛾𝜌‖𝛾𝑓(𝑢)−𝐴𝑢‖+(𝜏+1)‖𝑆𝑢−𝑢‖𝛾⁻−𝛾𝜌.(3.31) By induction, we obtain ‖‖𝑥_𝑛−𝑢‖‖≤max‖‖𝑥₀−𝑢‖‖,₋1𝛾−𝛾𝜌[‖𝛾𝑓(𝑢)−𝐴𝑢‖+(𝜏+1)‖𝑆𝑢−𝑢‖],(3.32) which implies that the sequence {𝑥_𝑛} is bounded. Since (C5) implies (C4) then, from Theorem 3.1, we can deduce ‖𝑥_𝑛+1−𝑥_𝑛‖→0.
From (3.1), we note that 𝑥_𝑛+1=𝑃_𝐶𝑤_𝑛−𝑤_𝑛+𝑤_𝑛+𝑦_𝑛−𝑦_𝑛=𝑃_𝐶𝑤_𝑛−𝑤_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛+𝑇𝑦_𝑛−𝑦_𝑛+𝑦_𝑛−𝛼_𝑛𝐴𝑇𝑦_𝑛.(3.33) Hence, it follows that 𝑥_𝑛−𝑥_𝑛+1==𝑤𝑤_𝑛𝑛−𝑃_𝐶𝑤_𝑛+𝛼_𝑛𝐴𝑥_𝑛−𝛾𝑓𝑥_𝑛+𝑦_𝑛−𝑇𝑦_𝑛+𝑥_𝑛−𝑦_𝑛+𝛼_𝑛𝐴𝑇𝑦_𝑛−𝐴𝑥_𝑛−𝑃_𝐶𝑤_𝑛+𝛼_𝑛(𝐴−𝛾𝑓)𝑥_𝑛+(𝐼−𝑇)𝑦_𝑛+𝛽_𝑛(𝐼−𝑆)𝑥_𝑛+𝛼_𝑛𝐴𝑇𝑦_𝑛−𝑥_𝑛,(3.34) and so 𝑥_𝑛−𝑥_𝑛+11−𝛼_𝑛𝛽_𝑛=11−𝛼_𝑛𝛽_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛+𝛼^𝑛1−𝛼_𝑛𝛽_𝑛(𝐴−𝛾𝑓)𝑥_𝑛+11−𝛼_𝑛𝛽_𝑛(𝐼−𝑇)𝑦_𝑛+11−𝛼_𝑛(𝐼−𝑆)𝑥^𝑛+𝛼^𝑛1−𝛼_𝑛𝛽_𝑛𝐴𝑇𝑦_𝑛−𝑥_𝑛.(3.35) Set 𝑣_𝑛∶=(𝑥_𝑛−𝑥_𝑛+1)/(1−𝛼_𝑛)𝛽_𝑛. Then, we have 𝑣_𝑛=11−𝛼_𝑛𝛽_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛+𝛼^𝑛1−𝛼_𝑛𝛽_𝑛(𝐴−𝛾𝑓)𝑥_𝑛+11−𝛼_𝑛𝛽_𝑛(𝐼−𝑇)𝑦_𝑛+11−𝛼_𝑛(𝐼−𝑆)𝑥^𝑛+𝛼^𝑛1−𝛼_𝑛𝛽_𝑛𝐴𝑇𝑦_𝑛−𝑥_𝑛.(3.36) From (3.12) in Theorem 3.1 and (C6), we obtain ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖𝛽_𝑛≤1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛽_𝑛+𝑀||𝛼^𝑛−𝛼_𝑛−1||𝛽_𝑛+||𝛽^𝑛−𝛽_𝑛−1||=1−𝛼_𝑛𝛽_𝑛𝛾⁻−𝛾𝜌‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛽_𝑛+1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛽_𝑛−1−1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛽_𝑛−1+𝑀||𝛼^𝑛−𝛼_𝑛−1||𝛽_𝑛+||𝛽^𝑛−𝛽_𝑛−1||=1−𝛼_𝑛𝛽_𝑛𝛾⁻−𝛾𝜌‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛽_𝑛−1+1−𝛼_𝑛𝛾⁻−𝛾𝜌‖‖𝑥_𝑛−𝑥_𝑛−1‖‖1𝛽_𝑛−1+𝑀||𝛼_𝑛𝛽_𝑛−1−𝛼_𝑛−1||𝛽_𝑛+||𝛽^𝑛−𝛽_𝑛−1||≤1−𝛼_𝑛𝛽_𝑛𝛾⁻−𝛾𝜌‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛽_𝑛−1+‖‖𝑥_𝑛−𝑥_𝑛−1‖‖||||1𝛽_𝑛−1𝛽_𝑛−1+𝑀||𝛼_𝑛||||−𝛼_𝑛−1||𝛽_𝑛+||𝛽^𝑛−𝛽_𝑛−1||≤1−𝛼_𝑛𝛽_𝑛𝛾⁻−𝛾𝜌‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛽_𝑛−1+𝑀||𝛼_𝑛+𝛼_𝑛𝐾‖‖𝑥_𝑛−𝑥_𝑛−1‖‖−𝛼_𝑛−1||𝛽_𝑛+||𝛽^𝑛−𝛽_𝑛−1||≤1−𝛼_𝑛𝛽_𝑛𝛾⁻−𝛾𝜌‖‖𝑤^𝑛−𝑤_𝑛−1‖‖𝛽_𝑛−1+𝑀||𝛼_𝑛+𝛼_𝑛𝐾‖‖𝑥_𝑛−𝑥_𝑛−1‖‖−𝛼_𝑛−1||𝛽_𝑛+||𝛽^𝑛−𝛽_𝑛−1||𝛽_𝑛.(3.37) This together with Lemma 2.4 and (C2) imply that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖𝛽_𝑛=lim_𝑛→∞‖‖𝑤_𝑛+1−𝑤_𝑛‖‖𝛽_𝑛=lim_𝑛→∞‖‖𝑤_𝑛+1−𝑤_𝑛‖‖𝛼_𝑛=0.(3.38) From (3.36), for 𝑧∈𝐹(𝑇), we have ⟨𝑣_𝑛,𝑥_𝑛−𝑧⟩=11−𝛼_𝑛𝛽_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1−𝑧+𝛼^𝑛1−𝛼_𝑛𝛽_𝑛+1⟨(𝐴−𝛾𝑓)𝑥_𝑛,𝑥_𝑛−𝑧⟩1−𝛼_𝑛𝛽_𝑛⟨(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛−𝑧⟩+11−𝛼_𝑛⟨(𝐼−𝑆)𝑥^𝑛+𝛼,𝑥_𝑛−𝑧⟩_𝑛1−𝛼_𝑛𝛽_𝑛=1𝐴𝑇𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧1−𝛼_𝑛𝛽_𝑛+1𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−𝑧1−𝛼_𝑛𝛽_𝑛+𝛼𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1−𝑃_𝐶𝑤_𝑛_𝑛1−𝛼_𝑛𝛽_𝑛⟨(𝐴−𝛾𝑓)𝑥_𝑛−(𝐴−𝛾𝑓)𝑧,𝑥_𝑛−𝑧⟩+𝛼^𝑛1−𝛼_𝑛𝛽_𝑛+1⟨(𝐴−𝛾𝑓)𝑧,𝑥_𝑛−𝑧⟩1−𝛼_𝑛⟨(𝐼−𝑆)𝑥^𝑛−(𝐼−𝑆)𝑧,𝑥_𝑛−𝑧⟩+11−𝛼_𝑛⟨(𝐼−𝑆)𝑧,𝑥^𝑛+1−𝑧⟩1−𝛼_𝑛𝛽_𝑛⟨(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛−𝑧⟩+𝛼^𝑛1−𝛼_𝑛𝛽_𝑛𝐴𝑇𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧.(3.39) By Lemmas 2.2 and 2.3, we obtain ⟨𝑣_𝑛,𝑥_𝑛−𝑧⟩≥11−𝛼_𝑛𝛽_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1−𝑃_𝐶𝑤_𝑛+⁻𝛾−𝛾𝜌𝛼_𝑛1−𝛼_𝑛𝛽_𝑛‖‖𝑥_𝑛−𝑧‖‖²+𝛼^𝑛1−𝛼_𝑛𝛽_𝑛⟨(𝐴−𝛾𝑓)𝑧,𝑥_𝑛−𝑧⟩+11−𝛼_𝑛⟨(𝐼−𝑆)𝑧,𝑥^𝑛+1−𝑧⟩1−𝛼_𝑛𝛽_𝑛⟨(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛−𝑧⟩+𝛼^𝑛1−𝛼_𝑛𝛽_𝑛𝐴𝑇𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧.(3.40) Now, we observe that ‖‖𝑥_𝑛−𝑧‖‖²≤1−𝛼^𝑛𝛽_𝑛𝛾⁻−𝛾𝜌𝛼_𝑛⟨𝑣,𝑥_𝑛−𝑧⟩−𝛽^𝑛𝛾⁻−𝛾𝜌𝛼_𝑛−1⟨(𝐼−𝑆)𝑧,𝑥_𝑛−𝑧⟩𝛾⁻−𝛾𝜌⟨(𝐴−𝛾𝑓)𝑧,𝑥^𝑛−𝑧⟩−1𝛾⁻−𝛾𝜌𝛼_𝑛−1⟨(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛−𝑧⟩𝛾⁻−𝛾𝜌𝐴𝑇𝑦^𝑛−1−𝑥_𝑛,𝑥_𝑛−𝑧𝛾⁻−𝛾𝜌𝛼_𝑛≤1−𝛼_𝑛𝑤_𝑛−𝑃_𝐶𝑤_𝑛,𝑃_𝐶𝑤_𝑛−1−𝑃_𝐶𝑤_𝑛𝛽_𝑛𝛾⁻−𝛾𝜌𝛼_𝑛⟨𝑣,𝑥_𝑛−𝑧⟩−𝛽^𝑛𝛾⁻−𝛾𝜌𝛼_𝑛−1⟨(𝐼−𝑆)𝑧,𝑥_𝑛−𝑧⟩𝛾⁻−𝛾𝜌⟨(𝐴−𝛾𝑓)𝑧,𝑥^𝑛−𝑧⟩−1𝛾⁻−𝛾𝜌𝛼_𝑛−1⟨(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛−𝑧⟩𝛾⁻−𝛾𝜌𝐴𝑇𝑦^𝑛−𝑥_𝑛,𝑥_𝑛−𝑧+‖‖𝑤^𝑛−𝑤_𝑛−1‖‖𝛾⁻−𝛾𝜌‖‖𝑤^𝑛−𝑃_𝐶𝑤_𝑛‖‖.(3.41) From (C1) and (C2), we have 𝛽_𝑛→0. Hence, from (3.1), we deduce ‖𝑦_𝑛−𝑥_𝑛‖→0 and ‖𝑥_𝑛+1−𝑇𝑦_𝑛‖→0. Therefore, ‖‖𝑦_𝑛−𝑇𝑦_𝑛‖‖≤‖‖𝑦_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑇𝑦_𝑛‖‖→0.(3.42)
Since 𝑣_𝑛→0, (𝐼−𝑇)𝑦_𝑛→0, 𝐴(𝑇𝑦_𝑛−𝑥_𝑛)→0, and ‖𝑤_𝑛−𝑤_𝑛−1‖/(𝛾⁻−𝛾𝜌)→0, every weak cluster point of {𝑥_𝑛} is also a strong cluster point. Note that the sequence {𝑥_𝑛} is bounded, thus there exists a subsequence {𝑥_𝑛𝑘} converging to a point 𝑥̃∈𝐻. For all 𝑧∈𝐹(𝑇), it follows from (3.39) that =1(𝐴−−𝛼_𝑛𝑘𝛾𝑓)𝑥_𝑛𝑘,𝑥_𝑛𝑘−𝑧𝛽_𝑛𝑘𝛼_𝑛𝑘𝑣_𝑛𝑘,𝑥_𝑛𝑘−𝑧−1𝛼_𝑛𝑘(𝐼−𝑇)𝑦_𝑛𝑘,𝑥_𝑛𝑘−𝑧−𝛽^𝑛𝑘𝛼_𝑛𝑘(𝐼−𝑆)𝑥_𝑛𝑘−𝐴𝑇𝑦_𝑛𝑘,𝑥_𝑛𝑘−𝑧−𝑥_𝑛𝑘,𝑥_𝑛𝑘−𝑧−1𝛼_𝑛𝑘𝑤_𝑛𝑘≤1−𝛼_𝑛𝑘−𝑃_𝐶𝑤_𝑛𝑘,𝑃_𝐶𝑤_𝑛𝑘−1−𝑧𝛽_𝑛𝑘𝛼_𝑛𝑘𝑣_𝑛𝑘,𝑥_𝑛𝑘−𝑧−1𝛼_𝑛𝑘(𝐼−𝑇)𝑦_𝑛𝑘,𝑥_𝑛𝑘−𝑧−𝛽^𝑛𝑘𝛼_𝑛𝑘(𝐼−𝑆)𝑥_𝑛𝑘−𝐴𝑇𝑦_𝑛𝑘,𝑥_𝑛𝑘−𝑧−𝑥_𝑛𝑘,𝑥_𝑛𝑘−𝑧−1𝛼_𝑛𝑘𝑤_𝑛𝑘−𝐴𝑇𝑦_𝑛𝑘−𝑃_𝐶𝑤_𝑛𝑘,𝑃_𝐶𝑤_𝑛𝑘−1−𝑃_𝐶𝑤_𝑛𝑘−𝑥_𝑛𝑘,𝑥_𝑛𝑘−𝑧−1𝛼_𝑛𝑘𝑤_𝑛𝑘≤1−𝛼_𝑛𝑘−𝑃_𝐶𝑤_𝑛𝑘,𝑃_𝐶𝑤_𝑛𝑘−1−𝑧𝛽_𝑛𝑘𝛼_𝑛𝑘𝑣_𝑛𝑘,𝑥_𝑛𝑘−𝑧−1𝛼_𝑛𝑘(𝐼−𝑇)𝑦_𝑛𝑘,𝑥_𝑛𝑘−𝑧−𝛽^𝑛𝑘𝛼_𝑛𝑘(𝐼−𝑆)𝑥_𝑛𝑘−𝐴𝑇𝑦_𝑛𝑘,𝑥_𝑛𝑘−𝑧−𝑥_𝑛𝑘,𝑥_𝑛𝑘−𝑧+‖‖𝑤^𝑛𝑘−𝑤_𝑛𝑘−1‖‖𝛼_𝑛𝑘‖‖𝑤_𝑛𝑘−𝑃_𝐶𝑤_𝑛𝑘‖‖.(3.43) Letting 𝑘→∞, we obtain ⟨(𝐴−𝛾𝑓)𝑥̃,𝑥̃−𝑧⟩≤−𝜏⟨(𝐼−𝑆)𝑥̃,𝑥̃−𝑧⟩,∀𝑧∈𝐹(𝑇).(3.44) By Lemma 2.6 and (3.27) having the unique solution, it follows that 𝜔_𝑤(𝑥_𝑛)={𝑥̃}. Therefore, 𝑥_𝑛→𝑥̃ as 𝑛→∞. This completes the proof.

From Theorem 3.3, we can deduce the following interesting corollary.

Corollary 3.4 (Yao et al. [9]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝐻 be a 𝜌-contraction (possibly nonself) with 𝜌∈(0,1). Let 𝑆,𝑇∶𝐶→𝐶 be two nonexpansive mappings with 𝐹(𝑇)≠∅. {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) Starting with an arbitrary initial guess 𝑥₀∈𝐶 and {𝑥_𝑛} is a sequence generated by 𝑥_𝑛+1𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,=𝑃_𝐶𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛,∀𝑛≥1.(3.45) Suppose that the following conditions are satisfied: (C1)  lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞, (C2)  lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=𝜏∈(0,∞), (C5)  lim_𝑛→∞((|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|)/𝛼_𝑛𝛽_𝑛)=0, (C6)  there exists a constant 𝐾>0 such that (1/𝛼_𝑛)|1/𝛽_𝑛−1/𝛽_𝑛−1|≤𝐾. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑥̃∈𝐻, which is the unique solution of the variational inequality: 𝑥̃∈𝐹(𝑇),1𝜏(𝐼−𝑓)𝑥̃+(𝐼−𝑆)𝑥̃,𝑥−𝑥̃≥0,∀𝑥∈𝐹(𝑇).(3.46)

Proof. As a matter of fact, if we take 𝐴=𝐼 and 𝛾=1 in Theorem 3.3 then this completes the proof.

Corollary 3.5 (Yao et al. [9]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑆,𝑇∶𝐶→𝐶 be two nonexpansive mappings with 𝐹(𝑇)≠∅. {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1). Starting with an arbitrary initial guess 𝑥₀∈𝐶 and suppose {𝑥_𝑛} is a sequence generated by 𝑥_𝑛+1𝑦_𝑛=𝛽_𝑛𝑆𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,=𝑃_𝐶1−𝛼_𝑛𝑇𝑦_𝑛,∀𝑛≥1.(3.47) Suppose that the following conditions are satisfied: (C1)  lim_𝑛→∞𝛼_𝑛=0 and ∑^∞𝑛=1𝛼_𝑛=∞, (C2)  lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=1, (C5)  lim_𝑛→∞((|𝛼_𝑛−𝛼_𝑛−1|+|𝛽_𝑛−𝛽_𝑛−1|)/𝛼_𝑛𝛽_𝑛)=0, (C6)  there exists a constant 𝐾>0 such that (1/𝛼_𝑛)|1/𝛽_𝑛−1/𝛽_𝑛−1|≤𝐾. Then the sequence {𝑥_𝑛} converges strongly to a point 𝑥̃∈𝐻, which is the unique solution of the variational inequality: 𝑥̃∈𝐹(𝑇),𝐼−𝑆2𝑥̃,𝑥−𝑥̃≥0,∀𝑥∈𝐹(𝑇).(3.48)

Proof. As a matter of fact, if we take 𝐴=𝐼, 𝑓=0, and 𝛾=1 in Theorem 3.3 then this is completes the proof.

Remark 3.6. Prototypes for the iterative parameters are, for example, 𝛼_𝑛=𝑛^−𝜃 and 𝛽_𝑛=𝑛^−𝜔 (with 𝜃,𝜔>0). Since |𝛼_𝑛−𝛼_𝑛−1|≈𝑛^−𝜃 and |𝛽_𝑛−𝛽_𝑛−1|≈𝑛^−𝜔, it is not difficult to prove that (C5) is satisfied for 0<𝜃,𝜔<1 and (C6) is satisfied if 𝜃+𝜔≤1.

Remark 3.7. Our results improve and extend the results of Yao et al. [9] by taking 𝐴=𝐼 and 𝛾=1 in Theorems 3.1 and 3.3.

The following is an example to support Theorem 3.3.

Example 3.8. Let 𝐻=ℝ,𝐶=[−1/4,1/4],𝑇=𝐼,𝑆=−𝐼,𝐴=𝐼, 𝑓(𝑥)=𝑥², 𝑃_𝐶=𝐼,𝛽_𝑛=1/√𝑛, 𝛼_𝑛=1/√𝑛 for every 𝑛∈ℕ, we have 𝜏=1 and choose 𝛾⁻=1/2, 𝜌=1/3 and