# Strong convergence theorems for variational inequalities and fixed points of a countable family of nonexpansive mappings

## Abstract

A new general iterative method for finding a common element of the set of solutions of variational inequality and the set of common fixed points of a countable family of nonexpansive mappings is introduced and studied. A strong convergence theorem of the proposed iterative scheme to a common fixed point of a countable family of nonexpansive mappings and a solution of variational inequality of an inverse strongly monotone mapping are established. Moreover, we apply our main result to obtain strong convergence theorems for a countable family of nonexpansive mappings and a strictly pseudocontractive mapping, and a countable family of uniformly k-strictly pseudocontractive mappings and an inverse strongly monotone mapping. Our main results improve and extend the corresponding result obtained by Klin-eam and Suantai (J Inequal Appl 520301, 16 pp, 2009).

Mathematics Subject Classification (2000): 47H09, 47H10

## 1 Introduction

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. In this paper, we always assume that a bounded linear operator A on H is strongly positive, that is, there is a constant $\stackrel{̄}{\gamma }>0$ such that $⟨Ax,x⟩\ge \stackrel{̄}{\gamma }||x|{|}^{2}$ for all x H. Recall that a mapping T of H into itself is called nonexpansive if ||Tx - Ty|| ≤ ||x - y|| for all x, y H. The set of all fixed points of T is denoted by F(T), that is, F(T) = {x C : x = Tx}. A self-mapping f : HH is a contraction on H if there is a constant α [0, 1) such that ||f(x) - f(y) || ≤ α ||x - y|| for all x, y H.

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on H:

$\underset{x\in F}{min}\frac{1}{2}⟨Ax,x⟩-⟨x,b⟩,$
(1.1)

where F is the fixed point set of a nonexpansive mapping T on H and b is a given point in H. A mapping B of C into H is called monotone if 〈Bx - By, x - y〉 ≥ 0 for all x, y C. The variational inequality problem is to find x C such that 〈Bx, y - x〉 ≥ 0 for all y C. The set of solutions of the variational inequality is denoted by VI(C, B). A mapping B of C to H is called inverse strongly monotone if there exists a positive real number β such that 〈x - y, Bx - By〉 ≥ β ||Bx - By||2 for all x, y C.

Starting with an arbitrary initial x0 H, define a sequence {x n } recursively by

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

It is proved by Xu  that the sequence {x n } generated by (1.2) converges strongly to the unique solution of the minimization problem (1.1) provided the sequence {α n } satisfies certain conditions.

On the other hand, Moudafi  introduced the viscosity approximation method for nonexpansive mappings. Let f be a contraction on H. Starting with an arbitrary initial x0 H, define a sequence {x n } recursively by

${x}_{n+1}=\left(1-{\sigma }_{n}\right)T{x}_{n}+{\sigma }_{n}f\left({x}_{n}\right)\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}n\ge 0,$
(1.3)

where {σ n } is a sequence in (0, 1). It is proved by Moudafi  and Xu  that under certain appropriate conditions imposed on {σ n }, the sequence {x n } generated by (1.3) strongly converges to the unique solution x* in C of the variational inequality

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

Recently, Marino and Xu  combined the iterative method (1.2) with the viscosity approximation method (1.3) and considered the following general iteration process:

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

and 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

$⟨\left(A-\gamma f\right){x}^{*},x-{x}^{*}⟩\ge 0\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}x\in C$

which is the optimality condition for the minimization problem

$\underset{x\in C}{min}\frac{1}{2}⟨Ax,x⟩-h\left(x\right),$

where h is a potential function for γ f (i.e., h'(x) = γ f(x) for x H).

Chen, Zhang and Fan  introduced the following iterative process: x0 C,

${x}_{n+1}={\alpha }_{n}f\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)T{P}_{C}\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}n\ge 0,$
(1.5)

where {α n } (0, 1) and {λ n } [a, b] for some a, b with 0 < a < b < 2β.

They proved that under certain appropriate conditions imposed on {α n } and {λ n }, the sequence {x n } generated by (1.5) converges strongly to a common element of the set of fixed points of nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping (say $\stackrel{̄}{x}\in C$), which solves the variational inequality

$⟨\left(I-f\right)\stackrel{̄}{x},x-\stackrel{̄}{x}⟩\ge 0\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\forall x\in F\left(T\right)\cap VI\left(C,B\right).$

Klin-eam and Suantai  modify the iterative methods (1.4) and (1.5) by proposing the following general iterative method: x0 C,

${x}_{n+1}={P}_{C}\left({\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right)T{P}_{C}\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}n\ge 0,$
(1.6)

where P C is the projection of H onto C, f is a contraction, A is a strongly positive linear bounded operator, B is a β-inverse strongly monotone mapping, {α n } (0, 1) and {λ n } [a, b] for some a, b with 0 < a < b < 2β. They noted that when A = I and γ = 1, the iterative scheme (1.6) reduced to the iterative scheme (1.5).

Wangkeeree, Petrot and Wangkeeree  introduced the following iterative process:

$\left\{\begin{array}{c}{x}_{0}=x\in H,\\ {y}_{n}={\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){T}_{n}{x}_{n},\\ {x}_{n+1}={\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){y}_{n},n\ge 0\end{array}\right\$
(1.7)

where {α n }and{β n } (0, 1) and T n is a countable family of nonexpansive mappings, f is a contraction, and A is a strongly positive linear bounded operator. They proved that under certain appropriate conditions imposed on {α n }, {β n } and {T n }, the sequence {x n } converges strongly to $\stackrel{̃}{x}$, which solves the variational inequality:

$⟨\left(A-\gamma f\right)\stackrel{̃}{x},\stackrel{̃}{x}-z⟩\le 0\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}z\in F\left(T\right).$

In this paper, motivated and inspired by Klin-eam and Suantai , we introduced the following iteration to find some solutions of variational inequality and fixed points of countable family of nonexpansive mappings in a Hilbert spaces H: x0 C,

${x}_{n+1}={P}_{C}\left({\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{n}{P}_{C}\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)\right),\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}n\ge 0,$
(1.8)

where P C is the projection of H onto C, f is a contraction, A is a strongly positive linear bounded operator, T n is a countable family of nonexpansive mappings of C into itself, B is a β-inverse strongly monotone mapping, {α n } (0, 1), and {λ n } [a, b] for some a, b with 0 < a < b < 2β.

## 2 Preliminaries

Let H be a real Hilbert space with inner product 〈·,·〉 and norm || · ||, and let C be a closed convex subset of H. We write x n x to indicate that the sequence {x n } converges weakly to x, and x n x implies that {x n } converges strongly to x. For every point x H, there exists a unique nearest point in C, denoted by P C x, such that ||x - P C x|| ≤ ||x - y|| for all y C and P C x is called the metric projection of H onto C. We know that P C is a nonexpansive mapping of H onto C. It is also known that P C satisfies 〈x - y, P C x - P C y〉 ≥ ||P C x - P C y||2 for every x, y H. Moreover, P C x is characterized by the properties: P C x C and 〈x - P C x, P C x - y〉 ≥ 0 for all y C. In the context of the variational inequality problem, this implies that

$u\in VI\left(C,A\right)⇔u={P}_{C}\left(u-\lambda Au\right),\phantom{\rule{1em}{0ex}}\forall \lambda >0.$

A set-valued mapping T : H → 2H is called monotone if for all x, y H, f Tx and g Ty imply 〈x - y, f - g〉 ≥ 0. A monotone mapping T : H → 2H is maximal if the graph G(T) of T is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f ) H × H, 〈x - y, f - g〉 ≥ 0 for every (y, g) G(T) implies f Tx. Let A be an inverse strongly monotone mapping of C into H, and let N C v be the normal cone to C at v C, i.e., N C v = {w H : 〈v - u, w〉 ≥ 0, u C}, and define

$Tv=\left\{\begin{array}{cc}\hfill Av+{N}_{C}v,\hfill & \hfill v\in C,\hfill \\ \hfill \varnothing ,\hfill & \hfill v\notin C.\hfill \end{array}\right\$

Then, T is maximal monotone and 0 Tv if and only if v V I(C, A).

Lemma 2.1 Let C be a closed convex subset of a real Hilbert space H. Given x H and y C, then

(i) y = P C x if and only if the inequalityx - y, y - z〉 ≥ 0 for all z C,

(ii) P C is nonexpansive,

(iii)x - y, P C x - P C y〉 ≥ ||P C x - P C y||2 for all x, y H,

(iv)x - P C x, P C x - y〉 ≥ 0 for all x H and y C.

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  Assume {a n } is a sequence of nonnegative real numbers such that

${a}_{n+1}\le \left(1-{\gamma }_{n}\right){a}_{n}+{\delta }_{n},n\ge 0$

where {γ n } (0, 1) and {δ n } is a sequence in such that

(i) ${\sum }_{n=1}^{\infty }{\gamma }_{n}=\infty$,

(ii) lim supn→∞δ n n ≤ 0 or ${\sum }_{n=1}^{\infty }|{\delta }_{n}|<\infty$.

Then, lim n→∞a n = 0.

Lemma 2.4  Let C be a closed convex subset of a real Hilbert space H, and let T : CC be a nonexpansive mapping such that F(T) . If a sequence {x n } in C such that x n z and x n - Tx n → 0, then z = Tz.

To deal with a family of mappings, the following conditions are introduced: Let C be a subset of a real Banach space E, and let ${\left\{{T}_{n}\right\}}_{n=1}^{\infty }$ be a family of mappings of C such that ${\cap }_{n=1}^{\infty }F\left({T}_{n}\right)\ne \varnothing$. Then, {T n } is said to satisfy the AKTT-condition  if for each bounded subset B of C,

$\sum _{n=1}^{\infty }sup\left\{||{T}_{n+1}z-{T}_{n}z||:z\in B\right\}<\infty .$

Lemma 2.5  Let C be a nonempty and closed subset of a Banach space E and let {T n } be a family of mappings of C into itself which satisfies the AKTT-condition. Then, for each x C, {T n x} converges strongly to a point in C. Moreover, let the mapping T be defined by

$Tx=\underset{n\to \infty }{lim}{T}_{n}x\phantom{\rule{1em}{0ex}}\forall x\in C.$

Then, for each bounded subset B of C,

$\underset{n\to \infty }{\mathrm{lim}\mathrm{sup}}\left\{||Tz-{T}_{n}z||\text{‷}:z\in B\right\}=0.$

In the sequel, we will write ({T n }, T ) satisfies the AKTT-condition if {T n } satisfies the AKTT-condition, and T is defined by Lemma 2.5 with $F\left(T\right)={\cap }_{n=1}^{\infty }F\left({T}_{n}\right)$.

## 3 Main results

In this section, we prove a strong convergence theorem for a countable family of nonexpansive mappings.

Theorem 3.1 Let C be a closed convex subset of a real Hilbert space H, and let B : CH be a β-inverse strongly monotone mapping, also let A be a strongly positive linear bounded operator of H into itself with coefficient $\stackrel{̄}{\gamma }>0$ such that ||A|| = 1 and let f : CC be a contraction with coefficient α(0 < α < 1). Assume that $0<\gamma <\stackrel{̄}{\gamma }∕\alpha$. Let {T n } be a countable family of nonexpansive mappings from a subset C into itself with $F={\cap }_{n=1}^{\infty }F\left({T}_{n}\right)\cap VI\left(C,B\right)\ne \varnothing$. Suppose {x n } is the sequence generated by the following algorithm: x0 C,

${x}_{n+1}={P}_{C}\left({\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{n}{P}_{C}\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)\right)$

for all n = 0, 1, 2, ..., where {α n } (0, 1) and {λ n } (0, 2β ). If {α n } and {λ n } are chosen so that λ n [a, b] for some a, b with 0 < a < b < 2β ,

$\begin{array}{cc}\hfill \left(C1\right)\underset{n\to 0}{lim}{\alpha }_{n}=0;\hfill & \hfill \left(C2\right)\sum _{n=1}^{\infty }{\alpha }_{n}=\infty ;\hfill \\ \hfill \left(C3\right)\sum _{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_{n}\right|<\infty ;\hfill & \hfill \left(C4\right)\sum _{n=1}^{\infty }\left|{\lambda }_{n+1}-{\lambda }_{n}\right|<\infty .\hfill \end{array}$

Suppose that ({T n }, T ) satisfies the AKTT-condition. Then, {x n } converges strongly to q F, where q = P F (γ f + I - A)(q) which solves the following variational inequality:

$⟨\left(\gamma f-A\right)q,p-q⟩\le 0\phantom{\rule{1em}{0ex}}\forall p\in F.$

Proof. First, we show that the sequence {x n } is bounded. Consider the mapping I -λ n B. Since B is a β-inverse strongly monotone mapping, we have that for all x, y C,

For 0 < λ n < 2β, implies that ||k(I - λ n B)x - (I- λ n B)y||2 ≤ ||x - y||2.

So, the mapping I - λ n B is nonexpansive.

Put y n = P C (x n - λ n Bx n ) for all n ≥ 0. Let u F. Then u = P C (u - λ n Bu).

From P C is nonexpansive implies that

$\begin{array}{lll}\hfill ||{y}_{n}-u||& =\phantom{\rule{0.3em}{0ex}}||{P}_{C}\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)-{P}_{C}\left(u-{\lambda }_{n}Bu\right)||\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le ||\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)-\left(u-{\lambda }_{n}Bu\right)||\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =||\left(I-{\lambda }_{n}B\right){x}_{n}-\left(I-{\lambda }_{n}B\right)u||.\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$

Since I - λ n B is nonexpansive, we have that ||y n - u|| ≤ ||x n - u||. Then

$\begin{array}{lll}\hfill ||{x}_{n+1}-u||& =||{P}_{C}\left({\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{n}{y}_{n}\right)-u||\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le ||{\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{n}{y}_{n}-u||\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ =||{\alpha }_{n}\left(\gamma f\left({x}_{n}\right)-Au\right)+\left(I-{\alpha }_{n}A\right)\left({T}_{n}{y}_{n}-u\right)||.\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$

Since A is strongly positive linear bounded operator, we have

By contraction of f, we have

It follows from induction that $||{x}_{n}-u||\le max\left\{||{x}_{0}-u||,\frac{||\gamma f\left(u\right)-Au||}{\stackrel{̄}{\gamma }-\alpha \gamma }\right\}$, n ≥ 0.

Therefore, {x n } is bounded, so are {y n }, {T n y n }, {Bx n }, and {f (x n )}.

Next, we show that ||xn+1- x n || → 0 and ||y n - T n y n || → 0 as n → ∞.

Since P C is nonexpansive, we also have

Since I - λ n B is nonexpansive, we have

$||{y}_{n+1}-{y}_{n}||\le ||{x}_{n+1}-{x}_{n}||+|{\lambda }_{n}-{\lambda }_{n+1}|||B{x}_{n}||.$

So we obtain

$\begin{array}{l}||{x}_{n+1}-{x}_{n}||\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}||{P}_{C}\left({\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{n}{y}_{n}\right)-{P}_{C}\left({\alpha }_{n-1}\gamma f\left({x}_{n-1}\right)+\left(I-{\alpha }_{n-1}A\right){T}_{n-1}{y}_{n-1}\right)||\\ \le \phantom{\rule{0.1em}{0ex}}||{\alpha }_{n}\gamma \left(f\left({x}_{n}\right)-f\left({x}_{n-1}\right)\right)+\gamma \left({\alpha }_{n}-{\alpha }_{n-1}\right)f\left({x}_{n-1}\right)+\left(I-{\alpha }_{n}A\right)\left({T}_{n}{y}_{n}-{T}_{n-1}{y}_{n-1}\right)\\ \phantom{\rule{0.5em}{0ex}}+\left({\alpha }_{n}-{\alpha }_{n-1}\right)A{T}_{n-1}{y}_{n-1}||\\ \le {\alpha }_{n}\alpha \gamma ||{x}_{n}-{x}_{n-1}||+\gamma |{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||f\left({x}_{n-1}\right)||+\left(1-{\alpha }_{n}\overline{\gamma }\right)||{T}_{n}{y}_{n}-{T}_{n-1}{y}_{n-1}||\\ \phantom{\rule{0.5em}{0ex}}+|{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||A{T}_{n-1}{y}_{n-1}||\\ \le {\alpha }_{n}\alpha \gamma ||{x}_{n}-{x}_{n-1}||+\gamma |{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||f\left({x}_{n-1}\right)||+\left(1-{\alpha }_{n}\overline{\gamma }\right)\left(||{T}_{n}{y}_{n}-{T}_{n}{y}_{n-1}||\\ \phantom{\rule{0.5em}{0ex}}+||{T}_{n}{y}_{n-1}-{T}_{n-1}{y}_{n-1}||\right)+|{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||A{T}_{n-1}{y}_{n-1}||\\ \le {\alpha }_{n}\alpha \gamma ||{x}_{n}-{x}_{n-1}||+\gamma |{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||f\left({x}_{n-1}\right)||+\left(1-{\alpha }_{n}\overline{\gamma }\right)\left(||{y}_{n}-{y}_{n-1}||\\ \phantom{\rule{0.5em}{0ex}}+||{T}_{n}{y}_{n-1}-{T}_{n-1}{y}_{n-1}||\right)+|{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||A{T}_{n-1}{y}_{n-1}||\\ ={\alpha }_{n}\alpha \gamma ||{x}_{n}-{x}_{n-1}||+\gamma |{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||f\left({x}_{n-1}\right)||+\left(1-{\alpha }_{n}\overline{\gamma }\right)||{y}_{n}-{y}_{n-1}||\\ \phantom{\rule{0.5em}{0ex}}+\left(1-{\alpha }_{n}\overline{\gamma }\right)||{T}_{n}{y}_{n-1}-{T}_{n-1}{y}_{n-1}||+|{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||A{T}_{n-1}{y}_{n-1}||\\ \le {\alpha }_{n}\alpha \gamma ||{x}_{n}-{x}_{n-1}||+\gamma |{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||f\left({x}_{n-1}\right)||+\left(1-{\alpha }_{n}\overline{\gamma }\right)||{x}_{n}-{x}_{n-1}||\\ \phantom{\rule{0.5em}{0ex}}+\left(1-{\alpha }_{n}\overline{\gamma }\right)|{\lambda }_{n-1}-{\lambda }_{n}|\phantom{\rule{0.1em}{0ex}}||B{x}_{n-1}||+\left(1-{\alpha }_{n}\overline{\gamma }\right)||{T}_{n}{y}_{n-1}-{T}_{n-1}{y}_{n-1}||\\ \phantom{\rule{0.5em}{0ex}}+|{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||A{T}_{n-1}{y}_{n-1}||\\ =\left(1-\left(\overline{\gamma }-\alpha \gamma \right){\alpha }_{n}\right)||{x}_{n}-{x}_{n-1}||+\gamma |{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||f\left({x}_{n-1}\right)||\\ \phantom{\rule{0.5em}{0ex}}+\left(1-{\alpha }_{n}\overline{\gamma }\right)|{\lambda }_{n-1}-{\lambda }_{n}|\phantom{\rule{0.1em}{0ex}}||B{x}_{n-1}||+\left(1-{\alpha }_{n}\overline{\gamma }\right)||{T}_{n}{y}_{n-1}-{T}_{n-1}{y}_{n-1}||\\ \phantom{\rule{0.5em}{0ex}}+|{\alpha }_{n}-{\alpha }_{n-1}|\phantom{\rule{0.1em}{0ex}}||A{T}_{n-1}{y}_{n-1}||\\ \le \left(1-\left(\overline{\gamma }-\alpha \gamma \right){\alpha }_{n}\right)||{x}_{n}-{x}_{n-1}||+2L|{\alpha }_{n}-{\alpha }_{n-1}|+M|{\lambda }_{n-1}-{\lambda }_{n}|\\ \phantom{\rule{0.5em}{0ex}}+\underset{y\in \left\{{y}_{n}\right\}}{sup}||{T}_{n}y-{T}_{n-1}y||,\end{array}$

where L = max{supn||ATn- 1yn - 1||, supnγ ||f (xn - 1)||} and M = sup{||Bxn- 1|| : n}.

Since {T n } satisfies the AKTT-condition, we get that

$\sum _{n=1}^{\infty }\underset{y\in \left\{{y}_{n}\right\}}{sup}||{T}_{n}y-{T}_{n-1}y||<\infty .$

From condition (C3), (C4) and by Lemma 2.3, we have ||xn+1- x n || → 0.

For u F and u = P C (u - λ n Bu), we have

$\begin{array}{l}||{x}_{n+1}-u|{|}^{2}=\phantom{\rule{0.1em}{0ex}}||{P}_{C}\left({\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{n}{y}_{n}\right)-{P}_{C}\left(u\right)|{|}^{2}\\ \le \phantom{\rule{0.1em}{0ex}}||{\alpha }_{n}\left(\gamma f\left({x}_{n}\right)-Au\right)+\left(I-{\alpha }_{n}A\right)\left({T}_{n}{y}_{n}-u\right)|{|}^{2}\\ \le {\left({\alpha }_{n}||\gamma f\left({x}_{n}\right)-Au||+||I-{\alpha }_{n}A||\phantom{\rule{0.1em}{0ex}}||{T}_{n}{y}_{n}-u||\right)}^{2}\\ \le {\left({\alpha }_{n}||\gamma f\left({x}_{n}\right)-Au||+\left(1-{\alpha }_{n}\overline{\gamma }\right)||{y}_{n}-u||\right)}^{2}\\ \le {\alpha }_{n}||\gamma f\left({x}_{n}\right)-Au|{|}^{2}+\left(1-{\alpha }_{n}\overline{\gamma }\right)||{y}_{n}-u|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}+2{\alpha }_{n}\left(1-{\alpha }_{n}\overline{\gamma }\right)||\gamma f\left({x}_{n}\right)-Au||\phantom{\rule{0.1em}{0ex}}||{y}_{n}-u||\\ \le {\alpha }_{n}||\gamma f\left({x}_{n}\right)-Au|{|}^{2}+\left(1-{\alpha }_{n}\overline{\gamma }\right)||\left(I-{\lambda }_{n}B\right){x}_{n}-\left(I-{\lambda }_{n}B\right)u|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}+2{\alpha }_{n}\left(1-{\alpha }_{n}\overline{\gamma }\right)||\gamma f\left({x}_{n}\right)-Au||\phantom{\rule{0.1em}{0ex}}||{y}_{n}-u||\\ \le {\alpha }_{n}||\gamma f\left({x}_{n}\right)-Au|{|}^{2}+\left(1-{\alpha }_{n}\overline{\gamma }\right)\left(||{x}_{n}-u|{|}^{2}-2{\lambda }_{n}〈{x}_{n}-u,B{x}_{n}-Bu〉\\ \phantom{\rule{0.5em}{0ex}}+{\lambda }_{n}^{2}||B{x}_{n}-Bu|{|}^{2}\right)+2{\alpha }_{n}\left(1-{\alpha }_{n}\overline{\gamma }\right)||\gamma f\left({x}_{n}\right)-Au||\phantom{\rule{0.1em}{0ex}}||{y}_{n}-u||\\ \le {\alpha }_{n}||\gamma f\left({x}_{n}\right)-Au|{|}^{2}+\left(1-{\alpha }_{n}\overline{\gamma }\right)\left(||{x}_{n}-u|{|}^{2}-2{\lambda }_{n}\beta ||B{x}_{n}-Bu|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}+{\lambda }_{n}^{2}||B{x}_{n}-Bu|{|}^{2}\right)+2{\alpha }_{n}\left(1-{\alpha }_{n}\overline{\gamma }\right)||\gamma f\left({x}_{n}\right)-Au||\phantom{\rule{0.1em}{0ex}}||{y}_{n}-u||\\ ={\alpha }_{n}||\gamma f\left({x}_{n}\right)-Au|{|}^{2}+\left(1-{\alpha }_{n}\overline{\gamma }\right)\left(||{x}_{n}-u|{|}^{2}+{\lambda }_{n}\left({\lambda }_{n}-2\beta \right)||B{x}_{n}-Bu|{|}^{2}\right)\\ \phantom{\rule{0.5em}{0ex}}+2{\alpha }_{n}\left(1-{\alpha }_{n}\overline{\gamma }\right)||\gamma f\left({x}_{n}\right)-Au||\phantom{\rule{0.1em}{0ex}}||{y}_{n}-u||\\ \le {\alpha }_{n}||\gamma f\left({x}_{n}\right)-Au|{|}^{2}+||{x}_{n}-u|{|}^{2}+\left(1-{\alpha }_{n}\overline{\gamma }\right)b\left(b-2\beta \right)||B{x}_{n}-Bu|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}+2{\alpha }_{n}\left(1-{\alpha }_{n}\overline{\gamma }\right)||\gamma f\left({x}_{n}\right)-Au||\phantom{\rule{0.1em}{0ex}}||{y}_{n}-u||.\end{array}$

So, we obtain

$\begin{array}{c}-\left(1-{\alpha }_{n}\stackrel{̄}{\gamma }\right)b\left(b-2\beta \right)||B{x}_{n}-Bu|{|}^{2}\\ \phantom{\rule{1em}{0ex}}\le {\alpha }_{n}||\gamma f\left({x}_{n}\right)-Au|{|}^{2}+\left(||{x}_{n}-u||+||{x}_{n+1}-u||\right)\left(||{x}_{n}-u||-||{x}_{n+1}-u||\right)+{\epsilon }_{n}\\ \phantom{\rule{1em}{0ex}}\le {\alpha }_{n}||\gamma f\left({x}_{n}\right)-Au|{|}^{2}+{\epsilon }_{n}+||{x}_{n}-{x}_{n+1}||\left(||{x}_{n}-u||+||{x}_{n+1}-u||\right),\end{array}$

where ${\epsilon }_{n}=2{\alpha }_{n}\left(1-{\alpha }_{n}\stackrel{̄}{\gamma }\right)||\gamma f\left({x}_{n}\right)-Au||||{y}_{n}-u||$.

Since α n → 0 and ||xn+1- x n || → 0, we obtain ||Bx n - Bu|| → 0 as n → ∞.

Further, by Lemma 2.1, we have

$\begin{array}{l}||{y}_{n}-u|{|}^{2}=\phantom{\rule{0.1em}{0ex}}||{P}_{C}\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)-{P}_{C}\left(u-{\lambda }_{n}Bu\right)|{|}^{2}\\ \le 〈\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)-\left(u-{\lambda }_{n}Bu\right),{y}_{n}-u〉\\ =\frac{1}{2}\left(||\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)-\left(u-{\lambda }_{n}Bu\right)|{|}^{2}+||{y}_{n}-u|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}-||\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)-\left(u-{\lambda }_{n}Bu\right)-\left({y}_{n}-u\right)|{|}^{2}\right)\\ \le \frac{1}{2}\left(||{x}_{n}-u|{|}^{2}+||{y}_{n}-u|{|}^{2}-||\left({x}_{n}-{y}_{n}\right)-{\lambda }_{n}\left(B{x}_{n}-Bu\right)|{|}^{2}\right)\\ \le \phantom{\rule{0.1em}{0ex}}||{x}_{n}-u|{|}^{2}-||{x}_{n}-{y}_{n}|{|}^{2}+2{\lambda }_{n}〈{x}_{n}-{y}_{n},B{x}_{n}-Bu〉-{\lambda }_{n}^{2}||B{x}_{n}-Bu|{|}^{2}.\end{array}$

So, we have

which implies

Since α n → 0, ||xn+1- x n || → 0, and ||Bx n - Bu|| → 0, we obtain ||x n - y n || → 0 as n → ∞.

Next, we have

$\begin{array}{lll}\hfill ||{x}_{n+1}-{T}_{n}{y}_{n}||& =||{P}_{C}\left({\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{n}{y}_{n}\right)-{P}_{C}\left({T}_{n}{y}_{n}\right)||\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le ||{\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{n}{y}_{n}-{T}_{n}{y}_{n}||\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ ={\alpha }_{n}||\gamma f\left({x}_{n}\right)-A{T}_{n}{y}_{n}||.\phantom{\rule{2em}{0ex}}& \hfill \text{(3)}\\ \hfill \text{(4)}\end{array}$

Since α n → 0 and {f (x n )}, {AT n y n } are bounded, we have ||xn+1- T n y n || → 0 as n → ∞. Since

$||{x}_{n}-{T}_{n}{y}_{n}||\le ||{x}_{n}-{x}_{n+1}||+||{x}_{n+1}-{T}_{n}{y}_{n}||,$

it implies that ||x n - T n y n || → 0 as n → ∞. Since

$\begin{array}{lll}\hfill ||{x}_{n}-{T}_{n}{x}_{n}||& \le ||{x}_{n}-{T}_{n}{y}_{n}||+||{T}_{n}{y}_{n}-{T}_{n}{x}_{n}||\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le ||{x}_{n}-{T}_{n}{y}_{n}||+||{y}_{n}-{x}_{n}||,\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$

we obtain ||x n - T n x n || → 0 as n → ∞. Moreover, from

$||{y}_{n}-{T}_{n}{y}_{n}||\le ||{y}_{n}-{x}_{n}||+||{x}_{n}-{T}_{n}{y}_{n}||,$

it follows that ||y n - T n y n || → 0 as n → ∞.

By ||y n - x n || → 0, ||T n y n - x n || → 0 and Lemma 2.5, we have

$\begin{array}{lll}\hfill ||T{x}_{n}-{x}_{n}||& \le ||T{x}_{n}-T{y}_{n}||+||T{y}_{n}-{T}_{n}{y}_{n}||+||{T}_{n}{y}_{n}-{x}_{n}||\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \le ||{x}_{n}-{y}_{n}||+sup\left\{||{T}_{n}z-Tz||:z\in \left\{{y}_{n}\right\}\right\}+||{T}_{n}{y}_{n}-{x}_{n}||.\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$

Hence, limn→∞||Tx n - x n || = 0. Observe that P F (γ f +I - A) is a contraction.

By Lemma 2.2, we have that $||I-A||\le 1-\stackrel{̄}{\gamma }$, and since $0<\gamma <\stackrel{̄}{\gamma }∕\alpha$, we get

Then, Banach's contraction mapping principle guarantees that P F (γ f +I - A) has a unique fixed point, say q H. That is, q = P F (γ f + I - A)q. By Lemma 2.1, we obtain

$⟨\left(\gamma f-A\right)q,x-q⟩\le 0\mathsf{\text{for}}\mathsf{\text{all}}x\in F.$
(3.1)

Choose a subsequence $\left\{{y}_{{n}_{k}}\right\}$ of {y n } such that

$\underset{n\to \infty }{\text{lim}\text{‷}\text{sup}}〈\left(\gamma f-A\right)q,{T}_{n}{y}_{n}-q〉=\underset{k\to \infty }{\text{lim}}〈\left(\gamma f-A\right)q,{T}_{{n}_{k}}{y}_{{n}_{k}}-q〉.$

As $\left\{{y}_{{n}_{k}}\right\}$ is bounded, there exists a subsequence $\left\{{y}_{{n}_{{k}_{i}}}\right\}$ of $\left\{{y}_{{n}_{k}}\right\}$ which converges weakly to p. Without loss of generality, we may assume that ${y}_{{n}_{k}}⇀p$.

Since ||y n - T n y n || → 0, we obtain ${T}_{{n}_{k}}{y}_{{n}_{k}}⇀p$. Since ||x n - Tx n || → 0, ||x n - y n || → 0 and by Lemma 2.4-2.5, we have $p\in {\cap }_{n=1}^{\infty }F\left({T}_{n}\right)$. Let

$Sv=\left\{\begin{array}{cc}\hfill Bv+{N}_{C}v,\hfill & \hfill v\in C,\hfill \\ \hfill \varnothing ,\hfill & \hfill v\notin C.\hfill \end{array}\right\$

where N C v is normal cone to C at v C, that is N C v = {w H : 〈v - u, w〉 ≥ 0, u C}. Then S is a maximal monotone. Let (v, w) G(S). Since w - Bv N C v and y n C, we have 〈v - y n , w - Bv〉 ≥ 0. On the other hand, by Lemma 2.1 and from y n = P C (x n - λ n Bx n ), we have

$\begin{array}{lll}\hfill ⟨v-{y}_{n},{y}_{n}-\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)⟩& \ge 0\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ \hfill ⟨v-{y}_{n},\left({y}_{n}-{x}_{n}\right)∕{\lambda }_{n}+B{x}_{n}⟩& \ge 0.\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$

Hence,

This implies 〈v - p, w〉 ≥ 0. Since S is maximal monotone, we have p S -10 and hence p V I(C, B). We obtain that p F. By (3.1), we have 〈(γ f - A)q, p - q〉 ≤ 0. It follows that

$\underset{n\to \infty }{limsup}⟨\left(\gamma f-A\right)q,{T}_{n}{y}_{n}-q⟩=\underset{k\to \infty }{lim}⟨\left(\gamma f-A\right)q,{T}_{{n}_{k}}{y}_{{n}_{k}}-q⟩=⟨\left(\gamma f-A\right)q,p-q⟩\le 0.$

Finally, we prove x n q. By ||y n - u|| ≤ ||x n - u|| and Schwarz inequality, we have

$\begin{array}{l}||{x}_{n+1}-q|{|}^{2}=\phantom{\rule{0.1em}{0ex}}||{P}_{C}\left({\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{n}{y}_{n}\right)-{P}_{C}\left(q\right)|{|}^{2}\\ \le \phantom{\rule{0.1em}{0ex}}||{\alpha }_{n}\left(\gamma f\left({x}_{n}\right)-Aq\right)+\left(I-{\alpha }_{n}A\right)\left({T}_{n}{y}_{n}-q\right)|{|}^{2}\\ \le \phantom{\rule{0.1em}{0ex}}||\left(I-{\alpha }_{n}A\right)\left({T}_{n}{y}_{n}-q\right)|{|}^{2}+{\alpha }_{n}^{2}||\gamma f\left({x}_{n}\right)-Aq|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}+2{\alpha }_{n}〈\left(I-{\alpha }_{n}A\right)\left({T}_{n}{y}_{n}-q\right),\gamma f\left({x}_{n}\right)-Aq〉\\ \le {\left(1-{\alpha }_{n}\overline{\gamma }\right)}^{2}||{y}_{n}-q|{|}^{2}+{\alpha }_{n}^{2}||\gamma f\left({x}_{n}\right)-Aq|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}+2{\alpha }_{n}〈{T}_{n}{y}_{n}-q,\gamma f\left({x}_{n}\right)-Aq〉-2{\alpha }_{n}^{2}〈A\left({T}_{n}{y}_{n}-q\right),\gamma f\left({x}_{n}\right)-Aq〉\\ \le {\left(1-{\alpha }_{n}\overline{\gamma }\right)}^{2}||{x}_{n}-q|{|}^{2}+{\alpha }_{n}^{2}||\gamma f\left({x}_{n}\right)-Aq|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}+2{\alpha }_{n}〈{T}_{n}{y}_{n}-q,\gamma f\left({x}_{n}\right)-\gamma f\left(q\right)〉+2{\alpha }_{n}〈{T}_{n}{y}_{n}-q,\gamma f\left(q\right)-Aq〉\\ \phantom{\rule{0.5em}{0ex}}-2{\alpha }_{n}^{2}〈A\left({T}_{n}{y}_{n}-q\right),\gamma f\left({x}_{n}\right)-Aq〉\\ \le {\left(1-{\alpha }_{n}\overline{\gamma }\right)}^{2}||{x}_{n}-q|{|}^{2}+{\alpha }_{n}^{2}||\gamma f\left({x}_{n}\right)-Aq|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}+2{\alpha }_{n}||{T}_{n}{y}_{n}-q||\phantom{\rule{0.1em}{0ex}}||\gamma f\left({x}_{n}\right)-\gamma f\left(q\right)||+2{\alpha }_{n}〈{T}_{n}{y}_{n}-q,\gamma f\left(q\right)-Aq〉\\ \phantom{\rule{0.5em}{0ex}}-2{\alpha }_{n}^{2}〈A\left({T}_{n}{y}_{n}-q\right),\gamma f\left({x}_{n}\right)-Aq〉\\ \le {\left(1-{\alpha }_{n}\overline{\gamma }\right)}^{2}||{x}_{n}-q|{|}^{2}+{\alpha }_{n}^{2}||\gamma f\left({x}_{n}\right)-Aq|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}+2\gamma \alpha {\alpha }_{n}||{y}_{n}-q||\phantom{\rule{0.1em}{0ex}}||{x}_{n}-q||+2{\alpha }_{n}〈{T}_{n}{y}_{n}-q,\gamma f\left(q\right)-Aq〉\\ \phantom{\rule{0.5em}{0ex}}-2{\alpha }_{n}^{2}〈A\left({T}_{n}{y}_{n}-q\right),\gamma f\left({x}_{n}\right)-Aq〉\\ \le {\left(1-{\alpha }_{n}\overline{\gamma }\right)}^{2}||{x}_{n}-q|{|}^{2}+{\alpha }_{n}^{2}||\gamma f\left({x}_{n}\right)-Aq|{|}^{2}\\ \phantom{\rule{0.5em}{0ex}}+2\gamma \alpha {\alpha }_{n}||{x}_{n}-q|{|}^{2}+2{\alpha }_{n}〈{T}_{n}{y}_{n}-q,\gamma f\left(q\right)-Aq〉\\ \phantom{\rule{0.5em}{0ex}}-2{\alpha }_{n}^{2}〈A\left({T}_{n}{y}_{n}-q\right),\gamma f\left({x}_{n}\right)-Aq〉\\ \le \left({\left(1-{\alpha }_{n}\overline{\gamma }\right)}^{2}+2\gamma \alpha {\alpha }_{n}\right)||{x}_{n}-q|{|}^{2}+{\alpha }_{n}\left(2〈{T}_{n}{y}_{n}-q,\gamma f\left(q\right)-Aq〉\\ \phantom{\rule{0.5em}{0ex}}+{\alpha }_{n}||\gamma f\left({x}_{n}\right)-Aq|{|}^{2}+2{\alpha }_{n}||A\left({T}_{n}{y}_{n}-q\right)||\phantom{\rule{0.1em}{0ex}}||\gamma f\left({x}_{n}\right)-Aq||\right)\\ =\left(1-2\left(\overline{\gamma }-\gamma \alpha \right){\alpha }_{n}\right)||{x}_{n}-q|{|}^{2}+{\alpha }_{n}\left(2〈{T}_{n}{y}_{n}-q,\gamma f\left(q\right)-Aq〉\\ \phantom{\rule{0.5em}{0ex}}+{\alpha }_{n}||\gamma f\left({x}_{n}\right)-Aq|{|}^{2}+2{\alpha }_{n}||A\left({T}_{n}{y}_{n}-q\right)||\phantom{\rule{0.1em}{0ex}}||\gamma f\left({x}_{n}\right)-Aq||\\ \phantom{\rule{0.5em}{0ex}}+{\alpha }_{n}{\overline{\gamma }}^{2}||{x}_{n}-q|{|}^{2}\right).\end{array}$

Since {x n }, {f (x n )} and {T n y n } are bounded, we can take a constant η > 0 such that

$\eta \ge ||\gamma f\left({x}_{n}\right)-Aq|{|}^{2}+2||A\left({T}_{n}{y}_{n}-q\right)||||\gamma f\left({x}_{n}\right)-Aq||+{\stackrel{̄}{\gamma }}^{2}||{x}_{n}-q|{|}^{2}$

for all n ≥ 0. It follows that

$||{x}_{n+1}-q|{|}^{2}\le \left(1-2\left(\stackrel{̄}{\gamma }-\gamma \alpha \right){\alpha }_{n}\right)||{x}_{n}-q|{|}^{2}+{\alpha }_{n}{\beta }_{n},$
(3.2)

where β n = 2〈T n y n - q, γ f(q) - Aq〉 +ηα n . By lim supn→∞〈(γ f - A)q, T n y n - q〉 ≤ 0, we get lim sup n→∞β n ≤ 0. By Lemma 2.3 and (3.2), we can conclude that x n q. This completes the proof. ■

Corollary 3.2 Let C be a closed convex subset of a real Hilbert space H, and let B : CH be a β-inverse strongly monotone mapping, also let f : CC be a contraction with coefficient α(0 < α < 1). Let {T n } be a countable family of nonexpansive mappings from a subset C into itself with $F={\cap }_{n=1}^{\infty }F\left({T}_{n}\right)\cap VI\left(C,B\right)\ne \varnothing$. Suppose {x n } is the sequence generated by the following algorithm: x0 C,

${x}_{n+1}={\alpha }_{n}f\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right){T}_{n}{P}_{C}\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)$

for all n = 0, 1, 2, ..., where {α n } (0, 1) and {λ n } (0, 2β ). If {α n } and {λ n } are chosen so that λ n [a, b] for some a, b with 0 < a < b < 2β,

$\begin{array}{cc}\hfill \left(C1\right)\underset{n\to 0}{lim}{\alpha }_{n}=0;\hfill & \hfill \left(C2\right)\sum _{n=1}^{\infty }{\alpha }_{n}=\infty ;\hfill \\ \hfill \left(C3\right)\sum _{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_{n}\right|<\infty ;\hfill & \hfill \left(C4\right)\sum _{n=1}^{\infty }\left|{\lambda }_{n+1}-{\lambda }_{n}\right|<\infty .\hfill \end{array}$

Suppose that ({T n }, T ) satisfies the AKTT-condition. Then {x n } converges strongly to q F, where q = P F (γ f + I - A)(q) which solves the following variational inequality:

$⟨\left(\gamma f-I\right)q,p-q⟩\le 0\phantom{\rule{1em}{0ex}}\forall p\in F.$

Proof. Taking A = I and γ = 1 in Theorem 3.1, we get the results. ■

## 4 Applications

In this section, we apply the iterative scheme (1.8) and Theorem 3.1 for finding a common fixed point of countable family of nonexpansive mappings and strictly pseudocontractive mapping and inverse strongly monotone mapping.

A mapping T : CC is called strictly pseudocontractive if there exists k with 0 ≤ k < 1 such that

$||Tx-Ty|{|}^{2}\le ||x-y|{|}^{2}+k||\left(I-T\right)x-\left(I-T\right)y|{|}^{2}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\forall x,y\in C.$

If k = 0, then T is nonexpansive. Put B = I - T, where T : CC is a strictly pseudocontractive mapping with k. Then, B is ((1 - k)/2)-inverse strongly monotone and B -1 (0) = F(T). Hence, for all x, y C,

$||\left(I-B\right)x-\left(I-B\right)y|{|}^{2}\le ||x-y|{|}^{2}+k||Bx-By|{|}^{2}.$

Conversely, since H is a real Hilbert space, we have

$||\left(I-B\right)x-\left(I-B\right)y|{|}^{2}\le ||x-y|{|}^{2}+||Bx-By|{|}^{2}-2⟨x-y,Bx-By⟩.$

Thus, we have

$⟨x-y,Bx-By⟩\ge \frac{1-k}{2}||Bx-By|{|}^{2}.$

Theorem 4.1 Let C be a closed convex subset of a real Hilbert space H, and let A be a strongly positive linear bounded operator of H into itself with coefficient $\stackrel{̄}{\gamma }>0$ such that ||A|| = 1 and let f : CC be a contraction with coefficient α(0 < α < 1). Assume that $0<\gamma <\stackrel{̄}{\gamma }∕\alpha$. Let {T n } be a family of nonexpansive mappings of C into itself and let S be a strictly pseudocontractive mapping of C into itself with β such that $F={\cap }_{n=1}^{\infty }F\left({T}_{n}\right)\cap F\left(S\right)\ne \varnothing$. Suppose {x n } is a sequence generated by the following algorithm: x0 C,

${x}_{n+1}={P}_{C}\left({\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){T}_{n}\left(\left(1-{\lambda }_{n}\right){x}_{n}-{\lambda }_{n}S{x}_{n}\right)\right)$

for all n = 0, 1, 2, ..., where {α n } [0, 1) and {λ n } [0, 1 - β). If {α n } and {λ n } are chosen so that λ n [a, b] for some a, b with 0 < a < b < 1 - β,

$\begin{array}{cc}\hfill \left(C1\right)\underset{n\to 0}{lim}{\alpha }_{n}=0;\hfill & \hfill \left(C2\right)\sum _{n=1}^{\infty }{\alpha }_{n}=\infty ;\hfill \\ \hfill \left(C3\right)\sum _{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_{n}\right|<\infty ;\hfill & \hfill \left(C4\right)\sum _{n=1}^{\infty }\left|{\lambda }_{n+1}-{\lambda }_{n}\right|<\infty .\hfill \end{array}$

Suppose that ({T n }, T) satisfies the AKTT-condition. Then, {x n } converges strongly to q F, such that

$⟨\left(\gamma f-A\right)q,p-q⟩\le 0\phantom{\rule{1em}{0ex}}\forall p\in F.$

Proof. Put B = I - S, then B is ((1 - k)/2)-inverse strongly monotone and F(S) = V I(C, B) and P C (x n - λ n Bx n ) = (1 - λ n )x n +λ n Sx n . Therefore, by Theorem 3.1, the conclusion follows. ■

Lemma 4.2  Let T : CH be a k-strictly pseudocontractive, then

(i) the fixed point set F(T) of T is closed convex so that the projection PF(T)is well defined;

(ii) define a mapping S : CH by

$Sx=\mu x+\left(1-\mu \right)Tx,x\in C.$
(4.1)

If μ [k, 1), then S is a nonexpansive mapping such that F(T) = F(S).

A family of mappings ${\left\{{T}_{n}:C\to H\right\}}_{n=1}^{\infty }$ is called a family of uniformly k-strictly pseudocontractions, if there exists a constant k [0, 1) such that $||{T}_{n}x-{T}_{n}y|{|}^{2}\le ||x-y|{|}^{2}+k||\left(I-{T}_{n}\right)x-\left(I-{T}_{n}\right)y|{|}^{2}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\forall x,y\in C,\forall n\ge 1.$

Let {T n : CC} be a countable family of uniformly k-strictly pseudocontractions. Let ${\left\{{S}_{n}:C\to C\right\}}_{n=1}^{\infty }$ be the sequence of mappings defined by (4.1), i.e.,

${S}_{n}x=\mu x+\left(1-\mu \right){T}_{n}x,\phantom{\rule{1em}{0ex}}x\in C,\forall n\ge 1\mathsf{\text{with}}\mu \in \left[k,1\right).$

Corollary 4.3 Let C be a closed convex subset of a real Hilbert space H, and let B : CH be a β-inverse strongly monotone mapping, also let A be a strongly positive linear bounded operator of H into itself with coefficient $\stackrel{̄}{\gamma }>0$ such that ||A|| = 1 and let f : CC be a contraction with coefficient α(0 < α < 1). Assume that $0<\gamma <\stackrel{̄}{\gamma }∕\alpha$. Let {T n } be a countable family of uniformly k-strictly pseudocontractions from a subset C into itself with $F={\cap }_{n=1}^{\infty }F\left({T}_{n}\right)\cap VI\left(C,B\right)\ne \varnothing$. Suppose {x n } is the sequence generated by the following algorithm: x0 C,

${x}_{n+1}={P}_{C}\left({\alpha }_{n}\gamma f\left({x}_{n}\right)+\left(I-{\alpha }_{n}A\right){S}_{n}{P}_{C}\left({x}_{n}-{\lambda }_{n}B{x}_{n}\right)\right)$

for all n = 0, 1, 2, ..., where {α n } (0, 1) and {λ n } (0, 2β). If {α n } and {λ n } are chosen so that λ n [a, b] for some a, b with 0 < a < b < 2β,

$\begin{array}{cc}\hfill \left(C1\right)\underset{n\to 0}{lim}{\alpha }_{n}=0;\hfill & \hfill \left(C2\right)\sum _{n=1}^{\infty }{\alpha }_{n}=\infty ;\hfill \\ \hfill \left(C3\right)\sum _{n=1}^{\infty }\left|{\alpha }_{n+1}-{\alpha }_{n}\right|<\infty ;\hfill & \hfill \left(C4\right)\sum _{n=1}^{\infty }\left|{\lambda }_{n+1}-{\lambda }_{n}\right|<\infty .\hfill \end{array}$

Then, {x n } converges strongly to q F, where q = P F (γ f + I - A)(q) which solves the following variational inequality:

$⟨\left(\gamma f-A\right)q,p-q⟩\le 0\phantom{\rule{1em}{0ex}}\forall p\in F.$

Proof. Let {T n } be a countable family of uniformly k-strictly pseudo-contractions from a subset C into itself. Set S n = μI + (1 - μ)T n where μ [k, 1). By Lemma 4.2, we have S n is nonexpansive and F (S n ) = F (T n ). Therefore, by Theorem 3.1, the conclusion follows. ■

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## Acknowledgements

The authors would like to thank the Centre of Excellence in Mathematics for financial support under the project RG-1-53-02-2. The first author is also supported by the Graduate School, Chiang Mai University, Thailand.

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Authors

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Correspondence to Suthep Suantai.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

AB study and researched nonlinear analysis and also wrote this article. SS participated in the process of the study and helped to draft the manuscript. All authors read and approved the final manuscript.

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Bunyawat, A., Suantai, S. Strong convergence theorems for variational inequalities and fixed points of a countable family of nonexpansive mappings. Fixed Point Theory Appl 2011, 47 (2011). https://doi.org/10.1186/1687-1812-2011-47

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• DOI: https://doi.org/10.1186/1687-1812-2011-47

### Keywords

• countable family of nonexpansive mappings
• variational inequality
• inverse strongly monotone mapping
• strictly pseudocontractive mapping
• countable family of uniformly k-strictly pseudocontractive mappings 