# A general iterative method for quasi-nonexpansive mappings in Hilbert space

## Abstract

Iterative algorithms have been extensively studied over the class of nonexpansive mappings in Hilbert spaces. Recall that nonexpansive mappings belong to quasi-nonexpansive mappings. The aim of this article is expanding the general approximation method proposed by Marino and Xu to quasi-nonexpansive mappings in Hilbert spaces.

## 1. Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉, and induced norm || · ||. A mapping T: HH is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y H. The set of the fixed points of T is denoted by Fix(T): = {x H: Tx = x}.

Iterative theory and methods for nonlinear mappings and variational inequalities have recently been applied to solve convex minimization problems, zero point problems and many others; see, e.g.,  and references therein.

The viscosity approximation method was first introduced by Moudafi . Starting with an arbitrary initial x0 H, define a sequence {x n } generated by:

${x}_{n+1}=\frac{{\epsilon }_{n}}{1+{\epsilon }_{n}}f\left({x}_{n}\right)+\frac{1}{1+{\epsilon }_{n}}T{x}_{n},\phantom{\rule{2.77695pt}{0ex}}\forall n\ge 0,$
(1.1)

where f is a contraction with a coefficient α [0,1) on H, i.e., ||f(x) - f(y)|| ≤ α||x - y|| for all x, y H, and {ε n } is a sequence in (0,1) satisfying the following given conditions:

1. (1)

limn→∞ε n = 0;

2. (2)

${\sum }_{n=0}^{\infty }{\epsilon }_{n}=\infty$;

3. (3)

${\mathrm{lim}}_{n\to \infty }\left(\frac{1}{{\epsilon }_{n}}-\frac{1}{{\epsilon }_{n+1}}\right)=0$.

It is proved that the sequence {x n } generated by (1.1) converges strongly to the unique solution x* C(C: = Fix(T)) of the variational inequality:

$⟨\left(I-f\right){x}^{*},x-{x}^{*}⟩\ge 0,\phantom{\rule{1em}{0ex}}\forall x\in Fix\left(T\right).$

In , Xu proved that the sequence {x n } defined by the below process started with an arbitrary initial x0 H:

${x}_{n+1}={\alpha }_{n}b+\left(I-{\alpha }_{n}A\right)T{x}_{n},\phantom{\rule{1em}{0ex}}\forall n\ge 0,$
(1.2)

converges strongly to the unique solution of the minimization problem (1.3) provided the the sequence {α n } satisfies certain conditions:

$\underset{x\in C}{\text{min}}\frac{1}{2}⟨Ax,x⟩-⟨x,b⟩,$
(1.3)

where C is the set of fixed points set of T on H and b is a given point in H.

In , Marino and Xu combined the iterative method (1.2) with the viscosity approximation method (1.1) and considered the following general iterative method:

${x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right)T{x}_{n},\phantom{\rule{2.77695pt}{0ex}}\forall n\ge 0.$
(1.4)

It is proved that if the sequence {α n } satisfies appropriate conditions, the sequence {x n } generated by (1.4) converges strongly to the unique solution of the variational inequality:

$⟨\left(\gamma f-A\right)\stackrel{̃}{x},x-\stackrel{̃}{x}⟩\le 0,\phantom{\rule{2.77695pt}{0ex}}\forall x\in C,$
(1.5)

or equivalently $\stackrel{̃}{x}={P}_{Fix\left(T\right)}\left(I-A+\gamma f\right)\stackrel{̃}{x}$, where C is the fixed point set of a nonexpansive mapping T.

In , Maingé considered the viscosity approximation method (1.1), and expanded the strong convergence to quasi-nonexpansive mappings in Hilbert space. Motivated by Marino and Xu  and Maingé , we consider the following iterative process:

$\left\{\begin{array}{c}{x}_{0}=x\in H\phantom{\rule{1em}{0ex}}arbitrarily\phantom{\rule{2.77695pt}{0ex}}chosen,\hfill \\ {x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{\omega }{x}_{n},\phantom{\rule{1em}{0ex}}\forall n\ge 0,\hfill \end{array}\right\$
(1.6)

where T ω = (1 - ω)I + ωT, and T is a quasi-nonexpansive mapping. Under some appropriate conditions on ω and {α n }, we obtain strong convergence over the class of quasi-nonexpansive mappings in Hilbert spaces. Our result is more general than Maingé's  conclusion, and also extends the iterative method (1.4) to quasi-nonexpansive mappings.

## 2. Preliminaries

Throughout this article, we write x n x to indicate that the sequence {x n } converges weakly to x. x n x implies that the sequence {x n } converges strongly to x. The following lemmas are useful for our article.

The following identities are valid in a Hilbert space H: for each x,y H, t [0, 1]

1. (i)

||x + y||2 ≤ ||x||2 + 2〈y, x + y〉;

2. (ii)

||(1 - t)x + ty||2 = (1 - t)||x||2 + t||y||2 - (1 - t) t||x - y||2;

3. (iii)

$⟨x,y⟩=-\frac{1}{2}{∥x-y∥}^{2}+\frac{1}{2}{∥x∥}^{2}+\frac{1}{2}{∥y∥}^{2}$.

Lemma 2.1.  Let H be a Hilbert space H. Given x H, C is a closed convex subset of H, f : HH is a contraction with coefficient 0 < α < 1, and A is a strongly positive linear bounded operator with coefficient $\stackrel{̄}{\gamma }$. Then for $0<\gamma <\stackrel{̄}{\gamma }/\alpha$,

$⟨x-y,\left(A-\gamma f\right)x-\left(A-\gamma f\right)y⟩\ge \left(\stackrel{̄}{\gamma }-\gamma \alpha \right){∥x-y∥}^{2},\phantom{\rule{1em}{0ex}}\forall x,y\in H.$

That is, A - γ f is strongly monotone with coefficient $\stackrel{̄}{\gamma }-\gamma \alpha$.

Lemma 2.2.  Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient $\stackrel{̄}{\gamma }>0$ and 0 < ρ ≤ ||A||-1. Then $∥I-\rho A∥\le 1-\rho \stackrel{̄}{\gamma }$.

Lemma 2.3.  Let T ω : = (1 - ω)I + ωT, with T being a quasi-nonexpansive mapping on H, $Fix\left(T\right)\ne 0̸$, and ω (0, 1]. Then the following statements are reached:

(a1) Fix(T) = Fix(T ω );

(a2) T ω is quasi-nonexpansive;

(a3) ||T ω x - q||2 ≤ ||x - q||2 - ω(1 - ω)||Tx - x||2 for all x H and q Fix(T);

(a4) $⟨x-{T}_{\omega }x,x-q⟩\ge \frac{\omega }{2}{∥x-Tx∥}^{2}$ for all x H and q Fix(T).

Remark 2.4. (a4) was revised by Wongchan and Saejung  (Proposition 2).

Lemma 2.5.  Let n } be a sequence of real numbers that does not decrease at infinity, in the sense that there exist a subsequence ${\left\{{\Gamma }_{{n}_{j}}\right\}}_{j\ge 0}$ of n } which satisfies ${\Gamma }_{{n}_{j}}<{\Gamma }_{{n}_{j}+1}$ for all j ≥ 0. Also consider the sequence of integers ${\left\{\tau \left(n\right)\right\}}_{n\ge {n}_{0}}$ defined by

$\tau \left(n\right)=\text{max}\left\{k\le n|{\Gamma }_{k}<{\Gamma }_{k+1}\right\}.$

Then ${\left\{\tau \left(n\right)\right\}}_{n\ge {n}_{0}}$ is a nondecreasing sequence verifying limn→∞τ(n) = ∞ and for all nn0, it holds that Γτ(n)< Γτ(n)+1and we have

${\Gamma }_{n}\le {\Gamma }_{\tau \left(n\right)+1}.$

Recall the metric projection P K form a Hilbert space H to a closed convex subset K of H is defined: for each x H, there exists a unique element P K x K such that

$∥x-{P}_{K}x∥:=\text{inf}\left\{∥x-y∥:y\in K\right\}.$

Lemma 2.6. Let K be a closed convex subset of H. Given x H, and z K, z = P K x, if and only if there holds the inequality:

$⟨x-z,y-z⟩\le 0,\phantom{\rule{2.77695pt}{0ex}}\forall y\in K.$

Lemma 2.7. If x* is the solution of the variational inequality (1.5) with demi-closedness of T and {y n } H is a bounded sequence such that ||Ty n - y n || → 0, then

(2.1)

Proof. We assume that there exists a subsequence $\left\{{y}_{{n}_{j}}\right\}$ of {y n } such that ${y}_{{n}_{j}}⇀ỹ$. From the given conditions ||Ty n - y n || → 0 and T: HH demi-closed, we have that any weak cluster point of {y n } belongs to the fixed point set Fix(T). Hence, we conclude that $ỹ\in Fix\left(T\right)$, and also have that

Recalling the (1.5), we immediately obtain

This completes the proof.

## 3. Main results

Let H be a real Hilbert space, let A be a bounded linear operator on H, and let T be a quasi-nonexpansive mapping on H, and f is a contraction with coefficient α; that is ||f (x) - f(y)|| ≤ α||x - y|| for all x, y H. Assume the set Fix(T) of fixed points of T is nonempty and we note that Fix(T) is closed and convex (see  for more general results).

Throughout this article, we assume that A is strongly positive; that is, there exist a constant $\stackrel{̄}{\gamma }>0$ such that $⟨Ax,x⟩\ge \stackrel{̄}{\gamma }{∥x∥}^{2}$, for all x H. Let $0<\gamma <\stackrel{̄}{\gamma }/\alpha$.

Theorem 3.1. Starting with an arbitrary chosen x0 H, let the sequence {x n } be generated by

${x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{\omega }{x}_{n},$
(3.1)

where the sequence {α n } (0,1) satisfies limn→∞α n = 0, and ${\sum }_{n=0}^{\infty }{\alpha }_{n}=\infty$. Also $\omega \in \left(0,\frac{1}{2}\right)$, T ω : = (1 - ω)I + ωT with two conditions on T:

(C1) ||Tx - q|| ≤ ||x - q|| for any x H, and q Fix(T); this means that T is a quasi-nonexpansive mapping;

(C2) T is demiclosed on H; that is: if {y k } H, y k z, and (I - T)y k → 0, then z Fix(T).

Then {x n } converges strongly to the x* Fix(T) which is the unique solution of the VIP:

$⟨\left(\gamma f-A\right){x}^{*},x-{x}^{*}⟩\le 0,\phantom{\rule{2.77695pt}{0ex}}\forall x\in Fix\left(T\right).$
(3.2)

Remark 3.2. Equivalently, from the VIP (3.2), we have

${x}^{*}={P}_{Fix\left(T\right)}\circ \left(I-A+\gamma f\right){x}^{*}.$
(3.3)

Proof. First we show that {x n } is bounded.

Take any p Fix(T), from Lemma 2.3 (a3), we have

$\begin{array}{ll}\hfill ∥{x}_{n+1}-p∥& =∥{\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{\omega }{x}_{n}-p∥\phantom{\rule{2em}{0ex}}\\ =∥{\alpha }_{n}\gamma \left(f\left({x}_{n}\right)-f\left(p\right)\right)+{\alpha }_{n}\left(\gamma f\left(p\right)-Ap\right)+\left(I-{\alpha }_{n}A\right)\left({T}_{\omega }{x}_{n}-p\right)∥\phantom{\rule{2em}{0ex}}\\ \le {\alpha }_{n}\gamma \alpha ∥f\left({x}_{n}\right)-f\left(p\right)∥+{\alpha }_{n}∥\gamma f\left(p\right)-Ap∥+\left(1-{\alpha }_{n}\stackrel{̄}{\gamma }\right)∥{x}_{n}-p∥\phantom{\rule{2em}{0ex}}\\ \le \left(1-{\alpha }_{n}\left(\stackrel{̄}{\gamma }-\gamma \alpha \right)\right)∥{x}_{n}-p∥+{\alpha }_{n}∥\gamma f\left(p\right)-Ap∥.\phantom{\rule{2em}{0ex}}\end{array}$
(3.4)

By induction

$∥{x}_{n}-p∥\le \text{max}\left\{∥{x}_{0}-p∥,\frac{∥\gamma f\left(p\right)-Ap∥}{\stackrel{̄}{\gamma }-\gamma \alpha }\right\},\phantom{\rule{1em}{0ex}}\forall n\ge 0.$

Hence {x n } is bounded, so are the {f(x n )} and {A(x n )}.

Let x* = PFix(T)o(I - A + γf)x* From (3.1), we have

${x}_{n+1}-{x}_{n}+{\alpha }_{n}\left(A{x}_{n}-\gamma f\left({x}_{n}\right)\right)=\left(I-{\alpha }_{n}A\right)\left({T}_{\omega }{x}_{n}-{x}_{n}\right).$
(3.5)

Since x* Fix(T), from (a4), and together with (3.5), we obtain

$\begin{array}{c}⟨{x}_{n+1}-{x}_{n}+{\alpha }_{n}\left(A{x}_{n}-\gamma f\left({x}_{n}\right)\right),{x}_{n}-{x}^{*}⟩\\ \phantom{\rule{1em}{0ex}}=⟨\left(I-{\alpha }_{n}A\right)\left({T}_{\omega }{x}_{n}-{x}_{n}\right),{x}_{n}-{x}^{*}⟩\\ \phantom{\rule{1em}{0ex}}=\left(1-{\alpha }_{n}\right)⟨{T}_{\omega }{x}_{n}-{x}_{n},{x}_{n}-{x}^{*}⟩+{\alpha }_{n}⟨\left(I-A\right)\left({T}_{\omega }{x}_{n}-{x}_{n}\right),{x}_{n}-{x}^{*}⟩\\ \phantom{\rule{1em}{0ex}}\le -\frac{\omega }{2}\left(1-{\alpha }_{n}\right){∥{x}_{n}-T{x}_{n}∥}^{2}+\omega {\alpha }_{n}⟨\left(I-A\right)\left(T-I\right){x}_{n},{x}_{n}-{x}^{*}⟩,\end{array}$

it follows from the previous inequality that

$\begin{array}{ll}\hfill -⟨{x}_{n}-{x}_{n+1},{x}_{n}-{x}^{*}⟩& \le -{\alpha }_{n}⟨\left(A-\gamma f\right){x}_{n},{x}_{n}-{x}^{*}⟩-\frac{\omega }{2}\left(1-{\alpha }_{n}\right){∥{x}_{n}-T{x}_{n}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\omega {\alpha }_{n}⟨\left(I-A\right)\left(T-I\right){x}_{n},{x}_{n}-{x}^{*}⟩.\phantom{\rule{2em}{0ex}}\end{array}$
(3.6)

From (iii), we obviously have

$⟨{x}_{n}-{x}_{n+1},{x}_{n}-{x}^{*}⟩=-\frac{1}{2}{∥{x}_{n+1}-{x}^{*}∥}^{2}+\frac{1}{2}{∥{x}_{n}-{x}^{*}∥}^{2}+\frac{1}{2}{∥{x}_{n+1}-{x}_{n}∥}^{2}.$
(3.7)

Set ${\Gamma }_{n}:=\frac{1}{2}{∥{x}_{n}-{x}^{*}∥}^{2}$, and combine with (3.6), it follows that

$\begin{array}{ll}\hfill {\Gamma }_{n+1}-{\Gamma }_{n}-\frac{1}{2}{∥{x}_{n+1}-{x}_{n}∥}^{2}& \le -{\alpha }_{n}⟨\left(A-\gamma f\right){x}_{n},{x}_{n}-{x}^{*}⟩-\frac{\omega }{2}\left(1-{\alpha }_{n}\right){∥{x}_{n}-T{x}_{n}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+\omega {\alpha }_{n}⟨\left(I-A\right)\left(T-I\right){x}_{n},{x}_{n}-{x}^{*}⟩.\phantom{\rule{2em}{0ex}}\end{array}$
(3.8)

Now, we calculate ||xn+1- x n ||.

From the given condition: T ω : = (1 - ω)I + ωT, it is easy to deduce that ||T ω x n - x n || = ω||x n - Txn||. Thus, it follows from (3.5) that

$\begin{array}{ll}\hfill {∥{x}_{n+1}-{x}_{n}∥}^{2}& ={∥{\alpha }_{n}\left(\gamma f\left({x}_{n}\right)-A{x}_{n}\right)+\left(I-{\alpha }_{n}A\right)\left({T}_{\omega }{x}_{n}-{x}_{n}\right)∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le 2{\alpha }_{n}^{2}{∥\gamma f\left({x}_{n}\right)-A{x}_{n}∥}^{2}+2{\left(1-{\alpha }_{n}\stackrel{̄}{\gamma }\right)}^{2}{∥{T}_{\omega }{x}_{n}-{x}_{n}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le 2{\alpha }_{n}^{2}{∥\gamma f\left({x}_{n}\right)-A{x}_{n}∥}^{2}+2\left(1-{\alpha }_{n}\stackrel{̄}{\gamma }\right){∥{T}_{\omega }{x}_{n}-{x}_{n}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \le 2{\alpha }_{n}^{2}{∥\gamma f\left({x}_{n}\right)-A{x}_{n}∥}^{2}+2{\omega }^{2}\left(1-{\alpha }_{n}\stackrel{̄}{\gamma }\right){∥T{x}_{n}-{x}_{n}∥}^{2}.\phantom{\rule{2em}{0ex}}\end{array}$
(3.9)

Then from (3.8) and (3.9), we have

(3.10)

Finally, we prove x n x*. To this end, we consider two cases.

Case 1: Suppose that there exists n0 such that ${\left\{{\Gamma }_{n}\right\}}_{n\ge {n}_{0}}$ is nonincreasing, it is equal to Γ n+1≤ Γ n for all nn0. It follows that limn→∞Γ n exists, so we conclude that

$\underset{n\to \infty }{\text{lim}}\left({\Gamma }_{n+1}-{\Gamma }_{n}\right)=0.$
(3.11)

It follows from (3.10), (3.11) and the fact that limn→∞α n = 0, we have limn→∞||x n -Tx n || = 0. Again, from (3.10), we have

$\begin{array}{c}-{\alpha }_{n}\left[{\alpha }_{n}{∥\gamma f\left({x}_{n}\right)-A{x}_{n}∥}^{2}-⟨\left(A-\gamma f\right){x}_{n},{x}_{n}-{x}^{*}⟩+\omega ⟨\left(I-A\right)\left(T-I\right){x}_{n},{x}_{n}-{x}^{*}⟩\right]\\ \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\le {\Gamma }_{n}-{\Gamma }_{n+1}.\end{array}$
(3.12)

Then, by ${\sum }_{n=0}^{\infty }{\alpha }_{n}=\infty$, we conclude that

(3.13)

Since {f(x n )} and {x n } are both bounded, as well as α n → 0, and limn→∞||x n - Tx n || = 0, it follows from (3.13) that

(3.14)

From Lemma 2.1, it is obvious that

$⟨\left(A-\gamma f\right){x}_{n},{x}_{n}-{x}^{*}⟩\ge ⟨\left(A-\gamma f\right){x}^{*},{x}_{n}-{x}^{*}⟩+2\left(\stackrel{̄}{\gamma }-\gamma \alpha \right){\Gamma }_{n}.$
(3.15)

Thus, from (3.14), (3.15) and the fact that limn→∞Γ n exists, we immediately obtain

(3.16)

or equivalently

(3.17)

Finally, by Lemma 2.7, we have

$2\left(\stackrel{̄}{\gamma }-\gamma \alpha \right)\underset{n\to \infty }{\text{lim}}{\Gamma }_{n}\le 0,$
(3.18)

so we conclude that limn→∞Γ n = 0, which equivalently means that {x n } converges strongly to x*.

Case 2: Assume that there exists a subsequence${\left\{{\Gamma }_{{n}_{j}}\right\}}_{j\ge 0}$ of {Γ n }n≥0such that ${\Gamma }_{{n}_{j}}<{\Gamma }_{{n}_{j}+1}$ for all j . In this case, it follows from Lemma 2.5 that there exists a subsequence {Γτ(n)} of {Γ n } such that Γτ(n)+1> Γτ(n), and {τ(n)} is defined as in Lemma 2.5.

Invoking the (3.10) again, it follows that

$\begin{array}{l}{\Gamma }_{\tau \left(n\right)+1}-{\Gamma }_{\tau \left(n\right)}+\left[\frac{\omega }{2}\left(1-{\alpha }_{\tau \left(n\right)}\right)-{\omega }^{2}\left(1-{\alpha }_{\tau \left(n\right)}\overline{\gamma }\right)\right]{‖{x}_{\tau \left(n\right)}-T{x}_{\tau \left(n\right)}‖}^{2}\\ \le {\alpha }_{\tau \left(n\right)}\left[{\alpha }_{\tau \left(n\right)}{‖\gamma f\left({x}_{\tau \left(n\right)}\right)-A{x}_{\tau \left(n\right)}‖}^{2}-〈\left(A-\gamma f\right){x}_{\tau \left(n\right)},{x}_{\tau \left(n\right)}-{x}^{*}〉\\ +\omega 〈\left(I-A\right)\left(T-I\right){x}_{\tau \left(n\right)},{x}_{\tau \left(n\right)}-{x}^{*}〉\right].\end{array}$

Recalling the fact that Γτ(n)+1> Γτ(n), we have

$\begin{array}{l}\left[\frac{\omega }{2}\left(1-{\alpha }_{\tau \left(n\right)}\right)-{\omega }^{2}\left(1-{\alpha }_{\tau \left(n\right)}\overline{\gamma }\right)\right]{‖{x}_{\tau \left(n\right)}-T{x}_{\tau \left(n\right)}‖}^{2}\\ \le {\alpha }_{\tau \left(n\right)}\left[{\alpha }_{\tau \left(n\right)}{‖\gamma f\left({x}_{\tau \left(n\right)}\right)-A{x}_{\tau \left(n\right)}‖}^{2}-〈\left(A-\gamma f\right){x}_{\tau \left(n\right)},{x}_{\tau \left(n\right)}-{x}^{*}〉\\ +\omega 〈\left(I-A\right)\left(T-I\right){x}_{\tau \left(n\right)},{x}_{\tau \left(n\right)}-{x}^{*}〉\right].\end{array}$
(3.19)

From the preceding results, we get the boundedness of {x n } and α n → 0, which obviously lead to

$\underset{n\to \infty }{\text{lim}}∥{x}_{\tau \left(n\right)}-T{x}_{\tau \left(n\right)}∥=0.$
(3.20)

Hence, combining (3.19) with (3.20), we immediately deduce that

$\begin{array}{ll}\hfill ⟨\left(A-\gamma f\right){x}_{\tau \left(n\right)},{x}_{\tau \left(n\right)}-{x}^{*}⟩& \le {\alpha }_{\tau \left(n\right)}{∥\gamma f\left({x}_{\tau \left(n\right)}\right)-A{x}_{\tau \left(n\right)}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\omega ⟨\left(I-A\right)\left(T-I\right){x}_{\tau \left(n\right)},{x}_{\tau \left(n\right)}-{x}^{*}⟩.\phantom{\rule{2em}{0ex}}\end{array}$
(3.21)

Again, (3.15) and (3.21) yield

$\begin{array}{ll}\hfill ⟨\left(A-\gamma f\right){x}^{*},{x}_{\tau \left(n\right)}-{x}^{*}⟩+2\left(\stackrel{̄}{\gamma }-\gamma \alpha \right){\Gamma }_{\tau \left(n\right)}& \le {\alpha }_{\tau \left(n\right)}{∥\gamma f\left({x}_{\tau \left(n\right)}\right)-A{x}_{\tau \left(n\right)}∥}^{2}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\omega ⟨\left(I-A\right)\left(T-I\right){x}_{\tau \left(n\right)},{x}_{\tau \left(n\right)}-{x}^{*}⟩.\phantom{\rule{2em}{0ex}}\end{array}$
(3.22)

Recall that limn→∞α τ(n)= 0 and (3.20), we immediately have

$2\left(\stackrel{̄}{\gamma }-\gamma \alpha \right)\underset{n\to \infty }{\text{lim sup}}{\Gamma }_{\tau \left(n\right)}\le -\underset{n\to \infty }{\text{lim inf}}⟨\left(A-\gamma f\right){x}^{*},{x}_{\tau \left(n\right)}-{x}^{*}⟩$
(3.23)

By Lemma 2.7, we have

$\underset{n\to \infty }{\text{lim inf}}⟨\left(A-\gamma f\right){x}^{*},{x}_{\tau \left(n\right)}-{x}^{*}⟩\ge 0.$
(3.24)

Consider (3.23) again, we conclude that

$\underset{n\to \infty }{\text{lim sup}}{\Gamma }_{\tau \left(n\right)}=0,$
(3.25)

which means that limn→∞Γτ(n)= 0. By Lemma 2.5, it follows that Γ n ≤ Γτ(n), thus, we get limn→∞Γ n = 0, which is equivalent to x n x*.

Corollary 3.3.  Let the sequence {x n } be generated by

${x}_{n+1}={\alpha }_{n}f\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right){T}_{\omega }{x}_{n},$
(3.26)

where the sequence {α n } (0,1) satisfies limn→∞α n = 0, and ${\sum }_{n=0}^{\infty }{\alpha }_{n}=\infty$. Also $\omega \in \left(0,\frac{1}{2}\right)$, and T ω : = (1 - ω)I + ωT with two conditions on T:

(C1) ||Tx - q|| ≤ ||x - q|| for any x H, and q Fix(T); this means that T is a quasi-nonexpansive mapping;

(C2) T is demiclosed on H; that is: if{yk} H, y k z, and (I - T)y k → 0, z Fix(T).

Then {x n } converges strongly to the x* Fix(T) which is the unique solution of the VIP (3.27):

$⟨\left(I-f\right){x}^{*},x-{x}^{*}⟩\ge 0,\phantom{\rule{2.77695pt}{0ex}}\forall x\in Fix\left(T\right).$
(3.27)

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## Acknowledgements

M. Tian was supported by the Fundamental Research Funds for the Central Universities (No. ZXH2011C002).

## Author information

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Correspondence to M Tian.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors read and approved the final manuscript.

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Tian, M., Jin, X. A general iterative method for quasi-nonexpansive mappings in Hilbert space. J Inequal Appl 2012, 38 (2012). https://doi.org/10.1186/1029-242X-2012-38

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• DOI: https://doi.org/10.1186/1029-242X-2012-38

### Keywords

• quasi-nonexpansive mapping
• iterative analysis
• variational inequality
• fixed point
• viscosity method 