A new convergence theorem for families of asymptotically nonexpansive maps and solution of variational inequality problem

Abstract

A new strong convergence theorem for approximation of common fixed points of family of uniformly asymptotically regular asymptotically nonexpansive mappings, which is also a unique solution of some variational inequality problem is proved in the framework of a real Banach space. The Theorem presented here extend, generalize and unify many recently announced results.

1 Introduction

Let E be a real Banach space and $E^{*}$ be the dual space of E. A mapping $\varphi :[0,\infty )\rightarrow [0,\infty )$ is called a guage function if it is strictly increasing, continuous and $\varphi (0)=0.$ Let $\varphi$ be a gauge function, a generalized duality mapping with respect to $\varphi ,$ $J_{\varphi }:E\rightarrow 2^{E^{*}}$ is defined by, $x\in E,$

$$\begin{aligned} J_{\varphi }x=\{x^{*}\in E^{*}:\langle x,x^{*} \rangle =\Vert x \Vert \varphi (\Vert x\Vert ),\Vert x^{*}\Vert =\varphi (\Vert x\Vert )\}, \end{aligned}$$

where $\langle \cdot ,\cdot \rangle$ denotes the duality pairing between element of E and that of $E^{*}$. If $\varphi (t)=t,$ then $J_{\varphi }$ is simply called the normalized duality mapping and is denoted by J. For any $x\in E,$ an element of $J_{\varphi }x$ is denoted by $j_{\varphi }(x)$.

Let $S(E) : = \{x \in E : \Vert x\Vert = 1\}$ be the unit sphere of E. The space E is said to have G $\hat{a}$ teaux differentiable norm if for any $x \in S(E)$ the limit

$$\begin{aligned} \lim _{\lambda \rightarrow 0}\frac{\Vert x + \lambda y\Vert - \Vert x\Vert }{\lambda } \end{aligned}$$

(1.1)

exists $\forall y \in S(E)$. The norm of E is said to be uniformly G$\hat{a}$teaux differentiable if for each $y\in S(E),$ the limit (1.1) is attained uniformly for $x \in S(E).$

If E has a uniformly G$\hat{a}$teaux differentiable, then $J_{\varphi }:E\rightarrow 2^{E^*}$ is uniformly continuous on bounded subsets of E from the strong topology of E to the weak$^*$ topology of $E^*$. All $L_{p},\ell _{p}(1<p<\infty )$ spaces has uniformly G$\hat{a}$teaux differentiable.

A mapping $T : E \rightarrow E$ is said to be L-Lipschitz if there exists a constant $L>0$ such that

$$\begin{aligned} \Vert Tx - Ty\Vert \le L\Vert x - y\Vert \quad \mathrm{for all}\; x,y \in E. \end{aligned}$$

(1.2)

If in this case, (1.2) is satisfied with $L\in [0, 1),$ respectively $L=1,$ then the mapping T is called a contraction, respectively nonexpansive. A mapping $T :E \rightarrow E$ is called asymptotically nonexpansive if there exists a sequence $\{\rho _{n}\} \subset [1,\infty ), {\lim \nolimits _{n\rightarrow \infty }} \rho _{n} = 1$ such that for all $x, y\in E$

$$\begin{aligned} ||T^{n}x- T^{n}y||\le \rho _{n}||x- y||\quad \text { for all}\quad n\in N. \end{aligned}$$

(1.3)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [13] as an important generalization of the class of nonexpansive mappings. A point $x \in E$ is called a fixed point of T provided $Tx = x$. We denote by F(T) the set of all fixed point of T (i.e., $F(T) = \{x \in E : Tx = x\}$ ).

Goebel and Kirk [13] proved that if C is a nonempty, bounded, closed and convex subset of a real uniformly convex Banach space and T is a self asymptotically nonexpansive mapping of C, then T has a fixed point in C.

The mapping T is said to be asymptotically regular if

$$\begin{aligned} \underset{n\rightarrow \infty }{lim}\Vert T^{n+1}x-T^{n}x\Vert =0 \end{aligned}$$

for all $x\in C.$ It is said to be uniformly asymptotically regular if for any bounded subset K of C, 

$$\begin{aligned} \underset{n\rightarrow \infty }{lim}\underset{x\in K}{sup}\Vert T^{n+1}x-T^{n}x\Vert =0. \end{aligned}$$

A mapping $G : E \rightarrow E$ is said to be accretive if for all $x,y\in E,$ there exists $j(x - y) \in J(x -y)$ such that

$$\begin{aligned} \langle Gx - Gy, j(x - y)\rangle \ge 0. \end{aligned}$$

For some positive real numbers $\eta$ and $\mu$ the mapping G is called $\eta$-strongly accretive if

$$\begin{aligned} \langle Gx - Gy, j(x - y)\rangle \ge \eta \Vert x - y\Vert ^{2} \end{aligned}$$

holds $\forall x,y\in E$ and $\mu$-strictly pseudocontractive if

$$\begin{aligned} \langle Gx - Gy, j(x - y)\rangle \le \Vert x - y\Vert ^{2} - \mu \Vert (I - G)x - (I - G)y\Vert ^{2} \end{aligned}$$

holds $\forall x,y\in E$. It is known that if G is $\mu$-strictly pseudocontractive then it is$(1+\frac{1}{\mu })-$Lipschitzian.

Let C be a nonempty closed convex subset of E, a variational inequality problem with respect to C and G, is to find $\bar{x}\in C$ such that

$$\begin{aligned} \langle G(\bar{x}),j(y - \bar{x})\rangle \ge 0 \quad \forall y \in E. \end{aligned}$$

(1.4)

The problem of solving variational inequality of the form (1.4) has been intensively studied by numerous authors due to its various applications in several physical problems, such as in operational research, economics, engineering, e.t.c.

A typical problem is to minimize a quadratic function over the set of fixed points of some nonexpansive mapping in a real Hilbert space H:

$$\begin{aligned} \min _{x\in F}\frac{1}{2}\langle Ax,x\rangle -\langle x,b\rangle . \end{aligned}$$

(1.5)

Here F is a fixed point set of some nonexpansive mapping T of H,  b is a point in H, and A is some bounded, linear and strongly positive operator on H,  where a map $A:H\rightarrow H$ is said to be strongly positive if there exist a constant $\overline{\gamma }>0$ such that

$$\begin{aligned} \langle Ax,x\rangle \ge \overline{\gamma }\Vert x\Vert ^{2},\quad \forall x\in H. \end{aligned}$$

Iterative methods for approximating fixed points of nonexpansive mappings and theirgeneralizations which solves some variational inequalities problems have been studied by a number of authors, see for examples [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23, 26, 30, 31] and the references contained in them.

In 2000, Moudafi [17] introduced viscosity approximation method for nonexpansivemappings. He proved that if a sequence $\{x_{n}\}$ is defined by

$$\begin{aligned} x_{n+1} = (1 - \alpha _{n})Tx_{n} + \alpha _{n}f(x_{n}),\quad n\ge 0 \end{aligned}$$

(1.6)

then $\{x_{n}\}$ converges strongly to the unique solution $x^{*}\in F$ of the variational inequality

$$\begin{aligned} \langle (I - f)x^{*}, x - x^{*} \rangle \ge 0,\quad \forall x \in F \end{aligned}$$

(1.7)

where $\{\alpha _{n}\} \subseteq (0,1)$ is a real sequence satisfying some conditions and $f : H \rightarrow H$ is a contraction map.

In 2003, Xu [29] proved that for a strongly positive linear bounded operator A on H a sequence $\{x_{n}\}$ defined by $x_{0} \in H$

$$\begin{aligned} x_{n+1} = (I - \alpha _{n}A)Tx_{n} + \alpha _{n}b,\quad n\ge 0, \end{aligned}$$

(1.8)

converges strongly to the unique solution of the minimization problem (1.5) provided the sequence $\{\alpha _{n}\}$ satisfies some control conditions.

In 2006, Marino and Xu [16] combined the iterative methods of Xu [29] and that of Moudafi [17] and studied the following general iterative method:

$$\begin{aligned} x_{n+1} = (I - \alpha _{n}A)Tx_{n} + \alpha _{n}\gamma f(x_{n}),\quad n\ge 0. \end{aligned}$$

(1.9)

They proved that if the sequence $\{\alpha _{n}\}$ satisfies appropriate conditions, then $\{x_{n}\}$ converges strongly to the unique solution of the variational inequality

$$\begin{aligned} \langle (A - \gamma f)x^{*}, x - x^{*} \rangle \ge 0,\quad \forall x \in F . \end{aligned}$$

(1.10)

Let $T_{k} : E \rightarrow E,$ $k = 1,2,3,\ldots N$ be a finite family of nonexpansive maps. For $n\in \mathbb {N},$ define a map $W_{n}:E\rightarrow E$ by

$$\begin{aligned} U_{n, 1}= & {} \gamma _{n, 1} T_{1} + (1 - \gamma _{n, 1})I, \\ U_{n, 2}= & {} \gamma _{n, 2} T_{2}U_{n, 1} + (1 - \gamma _{n, 2})I\\&\vdots \\ W_{n}= & {} U_{n, N} = \gamma _{n, N} T_{N}U_{n, N-1} + (1 - \gamma _{n, N})I, \end{aligned}$$

where $I = U_{n,0}$ and $\{\gamma _{n,k}\}^{N}_{k} \subseteq [0, 1]$. The mapping $W_{n}$ here is called the W mapping generated by $T_{1},T_{2},\ldots ,T_{N}$ and $\{\gamma _{n,k}\}_{n\ge 1},$ $k\in \{1,2,\ldots ,N\}$.

In 2007, Shang et al. [22] introduced a composite iterative scheme as follows: given$x_{0} = x \in C$ arbitrarily chosen,

$$\begin{aligned} y_{n}= & {} \beta _{n}x_{n} + (1 - \beta _{n})W_{n}x_{n}, \\ x_{n+1}= & {} \alpha _{n}\gamma f(x_{n}) + (I - \alpha _{n}A)y_{n}, \end{aligned}$$

where f is a contraction, and A is a strongly positive bounded linear operator on H.

In 2009, Kangtunyakarn and Suantai [15] introduced and studied the following scheme for approximation of common fixed point of a finite family of nonexpansive mappings $\{T_{k}\}^{N}_{k = 1},$ for $n\in \mathbb {N};$

$$\begin{aligned} U_{n, 1}= & {} \gamma _{n, 1} T_{1} + (1 - \gamma _{n, 1})I, \nonumber \\ U_{n, 2}= & {} \gamma _{n, 2} T_{2}U_{n, 1} + (1 - \gamma _{n, 2})U_{n, 1},\nonumber \\&\vdots \nonumber \\ K_{n}= & {} U_{n, N} = \gamma _{n, N} T_{N}U_{n, N-1} + (1 - \gamma _{n, N})U_{n, N-1}. \end{aligned}$$

(1.11)

The mapping $K_{n}$ here is called the K mapping generated by $T_{1},T_{2},\ldots ,T_{N}$ and $\{\gamma _{n,k}\}_{n\ge 1},$ $k\in \{1,2,\ldots ,N\}$.

Recently, Singthong and Suantai [24] studied the convergence of the following composite scheme $x_{0}\in C,$

$$\begin{aligned} y_{n}= & {} \beta _{n}x_{n} + (1 - \beta _{n})K_{n}x_{n}, \nonumber \\ x_{n+1}= & {} P_{C} (\alpha _{n}\gamma f(x_{n}) + (I - \alpha _{n}A)y_{n}), \end{aligned}$$

(1.12)

where C is a nonempty, closed convex subset of Hilbert space H,  $f:C\rightarrow C$ is a contraction, and A is a strongly positive bounded linear operator on H.

More recently, Ali et al. [2] introduce a modified iterative scheme for approximation of common fixed point of a finite family of nonexpansive mappings $\{T_{k}\}_{k=1}^{N},$ for $n\in \mathbb {N}$ and a sequence $\{\gamma _{n,k}\},\,k\in \{1,2,\ldots ,N\}$,

$$\begin{aligned} U_{n, 1}= & {} \gamma _{n, 1} T_{1} + (1 - \gamma _{n, 1})I, \nonumber \\ U_{n, 2}= & {} \gamma _{n, 2} T_{2}U_{n, 1} + (1 - \gamma _{n, 2})U_{n, 1},\nonumber \\&\vdots \nonumber \\ U_{n, N-1}= & {} \gamma _{n, N-1} T_{N-1}U_{n, N-2} + (1 - \gamma _{n, N-1})U_{n,N-2}\nonumber \\ K_{n}= & {} U_{n, N} = \gamma _{n, N} T_{N}U_{n, N-1} + (1 - \gamma _{n, N})I. \end{aligned}$$

(1.13)

They proved strong convergence of an iterative scheme to a common fixed point of a finite family of nonexpansive mappings which is also a unique solution of some variationalinequality problem in a framework of a Banach space much more general than Hilbert space. They actually proved the following theorems:

Theorem 1.1

(Ali et al. [2]) Let E be a real reflexive and strictly convex Banach space with a uniformly G$\hat{a}$teaux differentiable norm. Let $\{T_{i}\}^{N}_{i=1}$ be a finite family of nonexpansive mappings of E into itself and $F = \bigcap ^{N}_{i=1}F(T_{i}) \ne \emptyset .$ Let $f:E\rightarrow E$ be a contraction with constant $\alpha \in (0,1).$ Let $G:E\rightarrow E$ be an $\eta -$ strongly accretive and $\mu -$ strictly pseudocontractive with $\eta +\mu >1$ and let $\tau = 1-\sqrt{\frac{1-\eta }{\mu }}.$ Let $\gamma$ be a real number satisfying $0<\gamma <\frac{\tau }{\alpha }$ and let $K:E\rightarrow E$ be as in (1.13). Given $\beta \in (0,1),$ then for any $t\in (0,1)$. Let $\{z_{t}\}_{t\in (0,1)}$ be a path defined by

$$\begin{aligned} z_{t} = t\gamma f(z_{t}) + (I - tG)[\beta z_{t} + (1 - \beta )K z_{t}]. \end{aligned}$$

(1.14)

Then $\{z_{t}\}$ converges strongly to a common fixed point of the family say p which is a unique solution of the variational inequality

$$\begin{aligned} \langle (G-\gamma f)p,j(q-p)\rangle \ge 0,\,\,\forall q\in F. \end{aligned}$$

(1.15)

Theorem 1.2

(Ali et al. [2]) Let E be a real, reflexive and strictly convex Banach space with a uniformly G$\hat{a}$teaux differentiable norm, C a nonempty closed convex subset of E. Let $G : E \rightarrow E$ be an $\eta$-strongly accretive and $\mu$-strictly pseudocontractive with $\eta + \mu > 1$ and let $f : E \rightarrow E$ be a contraction with coefficient $\alpha \in (0,1)$. Let $\{T_{k}\}^{N}_{k=1}$ be a finite family of nonexpansive mappings of E into itself and $F = \bigcap ^{N}_{k=1}F(T_{k}) \ne \emptyset .$ Let $K_{n}$ be as in (1.13). Assume that $0< \gamma < \frac{\tau }{2\alpha }$, where $\tau := (1 - \sqrt{\frac{1 - \eta }{\mu }})$ and let $x_{0} \in C$. Let $\{\alpha _{n}\}^{\infty }_{n = 1}$ and $\{\beta _{n}\}^{\infty }_{n = 1}$ be sequences in (0, 1), and suppose that the following conditions are satisfied:

• (C1) $\alpha _{n} \rightarrow 0 \; as \; n \rightarrow \infty ;$

• (C2) $\Sigma ^{\infty }_{n=0}\alpha _{n} = \infty$

• (C3) $0< \liminf \nolimits _{n\rightarrow \infty }\beta _{n} \le \limsup \nolimits _{n\rightarrow \infty }\beta _{n} < 1;$

• (C4) $\Sigma ^{\infty }_{n=1}|\gamma _{n,k} - \gamma _{n-1,k}|< \infty , for \; all \; k = 1,2,3,\ldots ,N \;and\; \{\gamma _{n,k}\}^{N}_{k=1} \subset [a, b], \quad where \; 0<a\le b <1;$

• (C5) $\Sigma ^{\infty }_{n=1}|\alpha _{n+1} - \alpha _{n}| < \infty ;$

• (C6) $\Sigma ^{\infty }_{n=1}|\beta _{n+1} - \beta _{n}| < \infty .$

If $\{x_{n}\}^{\infty }_{n = 1}$ is a sequence defined by,

$$\begin{aligned} y_{n}= & {} \beta _{n}x_{n} + (1 - \beta _{n})K_{n}x_{n}, \nonumber \\ x_{n+1}= & {} \alpha _{n}\gamma f(x_{n}) + (I - \alpha _{n}G)y_{n},\qquad n \ge 0, \end{aligned}$$

(1.16)

then $\{x_{n}\}^{\infty }_{n = 1}$ converges strongly to $p \in F$, which also solves the following variational inequality problem,

$$\begin{aligned} \langle (G - \gamma f)x^{*}, x - x^{*} \rangle \ge 0,\quad \forall x \in F . \end{aligned}$$

(1.17)

It is our purpose in this paper to continue the study of the above problem and prove a new convergence theorems for approximation of common fixed point of finite family $\{T_k\}_{k=1}^{N}$ of asymptotically nonexpansive mappings which is also a unique solution of some variational inequality problem. The result presented here generalize and improve those recent ones such as in [2, 24]. In particular our Theorem extend the result in [24] to more general Banach space setting than Hilbert and generalizes it to family of asymptotically nonexpansive mappings. On the other hand our result also not only generalizes Theorems and 1.1 to the family of asymptotically nonexpansive mappings but also conditions C5 and C6 imposed in both Theorems 1.1 above and Theorem 2.1 of [24] are dispensed with.

2 Preliminaries

The following lemmas will be use for the main result.

Lemma 2.1

Let E be a real normed linear space. Then the following inequality holds:

$$\begin{aligned} \Vert x + y\Vert ^{2} \le \Vert x\Vert ^{2} + 2\langle y, j(x + y)\rangle ,\quad \forall x,y \in E, j(x + y) \in J(x + y). \end{aligned}$$

Lemma 2.2

(Suzuki [25]) Let $\{x_{n}\}$ and $\{y_{n}\}$ be bounded sequences in a Banach space E and let $\{\beta _{n}\}$ be a sequence in [0, 1] with $0<\liminf \beta _{n}\le \limsup \beta _{n}<1.$ Suppose that $x_{n+1}=\beta _{n}y_{n}+(1-\beta _{n})x_{n}$ for all integer $n\ge 1$ and

$$\begin{aligned} \underset{n\rightarrow \infty }{\limsup }(||y_{n+1}-y_{n}||-||x_{n+1}-x_{n}||)\le 0. \end{aligned}$$

Then, ${\lim \nolimits _{n\rightarrow \infty }}||y_{n}-x_{n}||=0.$

Lemma 2.3

(Xu [27]) Let E be a uniformly convex real Banach space. For arbitrary $r>0$, let $B_r(0):=\{x \in E:||x|| \le r\}$ and $\lambda \in [0,1]$. Then, there exists a continuous strictly increasing convex function

$$\begin{aligned} g:[0,2r]\rightarrow \mathbb {R},\,\,g(0)=0 \end{aligned}$$

such that for every $x,y \in B_r(0),$ the following inequality holds:

$||\lambda x +(1-\lambda ) y||^2 \le \lambda ||x||^2+(1-\lambda )||y||^2-\lambda (1-\lambda )g(||x-y||)$.

Lemma 2.4

(Xu [28]) Let $\{a_{n}\}$ be a sequence of nonegative real numbers satisfying the following relation:

$$\begin{aligned} a_{n+1}\le (1-\alpha _{n})a_{n}+\alpha _{n}\sigma _{n}+\gamma _{n},\,n\ge 0 \end{aligned}$$

where, (i) $\{\alpha _{n}\}\subset [0,1], \,\sum \alpha _{n}=\infty ;$ (ii) $\limsup \sigma _{n}\le 0;$ (iii) $\gamma _{n}\ge 0;~(n\ge 0),\sum \gamma _{n}<\infty .$ Then, $a_{n}\rightarrow 0$ as $n\rightarrow \infty .$

Lemma 2.5

(Chang et al. [9]) Let E be a uniformily convex Banach space, K be a nonempty closed convex subset of E and $T:K\rightarrow K$ be an asymptotically nonexpansive mapping, then $I-T$ is demiclosed at zero.

Lemma 2.6

(Piri and Vaezi [19] see also [1]) Let E be a real Banach space and $G : E \rightarrow E$ be a mapping.

(i):

If G is $\eta$-strongly accretive and $\mu$-strictly pseudo-contractive with $\eta + \mu > 1,$ then $I - G$ is contractive with constant $\sqrt{\frac{1-\eta }{\mu }}$.

(ii):

If G is $\eta$-strongly accretive and $\mu$-strictly pseudo-contractive with $\eta + \mu > 1,$ then for any fixed number $\kappa \in (0,1)$, $I - \kappa G$ is contractive with constant $1 - \kappa \Big (1 - \sqrt{\frac{1-\eta }{\mu }} \Big ).$

3 Main results

Lemma 3.1

Let C be a nonempty closed convex subset of a uniformly convex real Banach space E. Let $\{T_{k}\}^{N}_{k=1}$ be finite family of uniformly asymtotically regular asymptotically nonexpansive mappings of C into itself with sequences $\{\rho _{n,k}\}\subset [1,\infty )$, let $\{\gamma _{n,k}\}^{N}_{k=1}$ be a sequence in (0, 1) such that $0<\liminf \nolimits _{n\rightarrow \infty }\gamma _{n,k}\le \limsup \nolimits _{n\rightarrow \infty }\gamma _{n,k}<1$ and $\lim \nolimits _{n\rightarrow \infty }|\gamma _{n,k}-\gamma _{n-1,k}|=0\,\forall k\in \{1,2,3,\ldots ,N\}.$ Let $K_n$ be a mapping generated by $T_1,T_2,T_3,\ldots ,T_N$ and $\gamma _{n,1},\gamma _{n,2},\gamma _{n,3},\ldots ,\gamma _{n,N}$ as follows;

$$\begin{aligned} U_{n, 1}= & {} \gamma _{n, 1} T^{n}_{1} + (1 - \gamma _{n, 1})I, \nonumber \\ U_{n, 2}= & {} \gamma _{n, 2} T^{n}_{2}U_{n, 1} + (1 - \gamma _{n, 2})U_{n, 1},\nonumber \\&\vdots \nonumber \\ U_{n, N-1}= & {} \gamma _{n,N-1} T^{n}_{N-1}U_{n, N-2} + (1 - \gamma _{n,N-1})U_{n, N-2},\nonumber \\ K_{n} = U_{n, N}= & {} \gamma _{n, N} T^{n}_{N}U_{n, N-1} + (1 - \gamma _{n, N})I. \end{aligned}$$

(3.1)

Then, the following holds:

• (i) $\Vert K_{n}x - K_{n}y\Vert \le (1+v_{n})\Vert x - y\Vert$, where $v_n=\rho _{n,N}(1+\lambda _{n,N-1})-1,$ and $\{\lambda _{n,N}\}$ is some sequence in $[0,\infty )$, with $\lambda _{n,N}\rightarrow 0$ as $n\rightarrow \infty .$

• (ii) If ${\lim \nolimits _{n \rightarrow \infty }} \Vert T^{n+1}_{k}U_{n,k-1}z_{n} - T^{n}_{k}U_{n,k-1}z_{n}\Vert = 0$, then $\underset{n \rightarrow \infty }{lim} \Vert K_{n+1}z_{n} - K_{n}z_{n}\Vert = 0,$ for every bounded sequence $\{z_{n}\}$ in $E, k=1,2,\ldots ,N;$

• (iii) For every bounded sequence $\{z_n\}$ in C such that $\lim \nolimits _{n\rightarrow \infty }||K_{n}z_n-z_n||=0$, we have $\lim \nolimits _{n\rightarrow \infty }||T_{k}z_n-z_n||=0$ for any $k\in \{1,2,3,\ldots ,N\}$. Furthermore , we have $w_{w}(z_n)\subset \cap ^{N}_{k=1}F(T_{k})$ and $F(K_{n})=\overset{N}{\subset }\cap ^{N}_{k=1}F(T_{k}).$

Proof

(i) Let $x,y\in C$ then from (3.1), if $N=1$ the result follows. Assume $N\ne 1$ and $U_{n,0}=I$ (identity map), then for $k\in \{1,2,\ldots ,N-1\},$ we have

$$\begin{aligned}&\Vert U_{n,k}x - U_{n,k}y\Vert \le \gamma _{n,k}\Vert T^{n}_{k}U_{n,k-1}x - T^{n}_{k}U_{n,k-1}y\Vert \\&\qquad +(1 - \gamma _{n,k})\Vert U_{n,k-1}x - U_{n,k-1}y\Vert \\&\quad \le [\gamma _{n,k}\rho _{n,k}+(1-\gamma _{n,k})] \Vert U_{n,k-1}x - U_{n,k-1}y\Vert \\&\quad =[1+\gamma _{n,k}(\rho _{n,k}-1)] \Vert U_{n,k-1}x - U_{n,k-1}y\Vert \\&\quad \le [1+\gamma _{n,k}(\rho _{n,k}-1)] [\gamma _{n,k-1}\Vert T^{n}_{k-1}U_{n,k-2}x-T^{n}_{k-1}U_{n,k-2}y\Vert \\&\qquad +(1-\gamma _{n,k-1})\Vert U_{n,k-2}x-U_{n,k-2}y\Vert ]\\&\quad \le [1+\gamma _{n,k}(\rho _{n,k}-1)][1+\gamma _{n,k-1}(\rho _{n,k-1}-1)]\Vert U_{n,k-2}x-U_{n,k-2}y\Vert \\&\quad \vdots \quad \vdots \\&\quad \le [1+\gamma _{n,k}(\rho _{n,k}-1)][1+\gamma _{n,k-1}(\rho _{n,k-1}-1)]\ldots [1+\gamma _{n,2}(\rho _{n,2}-1)]\\&\qquad \Vert U_{n,1}x - U_{n,1}y\Vert \\&\quad \le [1+\gamma _{n,k}(\rho _{n,k}-1)][1+\gamma _{n,k-1}(\rho _{n,k-1}-1)]\ldots [1+\gamma _{n,2}(\rho _{n,2}-1)]\\&\qquad [1+\gamma _{n,1}(\rho _{n,1}-1)]\Vert x - y\Vert \\&\quad =\prod ^{k}_{j=1}[1+\gamma _{n,j}(\rho _{n,j}-1)]\Vert x-y\Vert \\&\quad =(1+\lambda _{n,k})\Vert x-y\Vert , \end{aligned}$$

where $\prod ^{k}_{j=1}\Big (1+\gamma _{n,j}(\rho _{n,j}-1)\Big )=(1+\lambda _{n,k})$, observe that $\lim \nolimits _{n\rightarrow \infty }\lambda _{n,k}=0$. Then,

$$\begin{aligned} \Vert K_{n}x - K_{n}y\Vert= & {} \Vert U_{n,N}x - U_{n,N}y\Vert \\\le & {} \gamma _{n,N}\Vert T^{n}_{N}U_{n,N-1}x - T^{n}_{N}U_{n,N-1}y\Vert + (1 - \gamma _{n,N})\Vert x -y\Vert \\\le & {} \gamma _{n,N}\rho _{n,N}\Vert U_{n,N-1}x-U_{n,N-1}y\Vert +(1-\gamma _{n,N})\Vert x-y\Vert \\ {}\le & {} \gamma _{n,N}\rho _{n,N}(1+\lambda _{n,N-1})\Vert x-y\Vert +(1-\gamma _{n,N})\Vert x-y\Vert \\ {}= & {} [1+\gamma _{n,N}(\rho _{n,N}(1+\lambda _{n,N-1})-1)]\Vert x-y\Vert \\ {}\le & {} [1+(\rho _{n,N}(1+\lambda _{n,N-1})-1)]\Vert x-y\Vert \\= & {} (1+v_{n})\Vert x - y\Vert , \end{aligned}$$

where $v_{n}=\rho _{n,N}(1+\lambda _{n,N-1})-1$, observe that $\lim \nolimits _{n\rightarrow \infty }v_{n}=0$.

Next we show (ii). For $k \in \{2,3,\ldots ,N-1\}$ and any bounded sequence $\{z_{n}\}\subset E$, letting $\delta _{n+1,k}:=[1+\gamma _{n+1,k}(\rho _{n+1,k}-1)]$, $M_{n,k}:=[\Vert T^{n+1}_{k}U_{n,k}z_{n}\Vert +\Vert U_{n,k}z_{n}\Vert ]$ and$P_{n,k}:=\Vert T^{n+1}_{k-1}U_{n,k}z_{n}-T^{n}_{k-1}U_{n,k}z_{n}\Vert$, we have

$$\begin{aligned}&\Vert U_{n+1,k}z_{n} - U_{n,k}z_{n}\Vert = \Vert \gamma _{n+1,k}T^{n+1}_{k}U_{n+1,k-1}z_{n}\nonumber \\&\qquad - \gamma _{n+1,k}T^{n+1}_{k}U_{n,k-1}z_{n} \nonumber \\&\qquad + [\gamma _{n+1,k}-\gamma _{n,k}]T^{n+1}_{k}U_{n,k-1}z_{n} \nonumber \\&\qquad + \gamma _{n,k}[T^{n+1}_{k}U_{n,k-1}z_{n}-T^{n}_{k}U_{n,k-1}z_{n}]\nonumber \\&\qquad +(1 - \gamma _{n+1,k})(U_{n+1,k-1}z_{n}-U_{n,k-1}z_{n})\nonumber \\&\qquad +[(1- \gamma _{n+1,k})-(1- \gamma _{n,k})]U_{n,k-1}z_{n}\Vert \nonumber \\&\quad \le [1+\gamma _{n+1,k}( \rho _{n+1,k}-1)]\Vert U_{n+1,k-1}z_{n} - U_{n,k-1}z_{n}\Vert \nonumber \\&\qquad + |\gamma _{n+1,k} - \gamma _{n,k}|\Big [\Vert T^{n+1}_{k}U_{n,k-1}z_{n}\Vert +\Vert U_{n,k-1}z_{n}\Vert \Big ]\nonumber \\&\qquad +\gamma _{n,k}\Vert T^{n+1}_{k}U_{n,k-1}z_{n}-T^{n}_{k}U_{n,k-1}z_{n}\Vert \nonumber \\&\quad \le \big [1+\gamma _{n+1,k}( \rho _{n+1,k}-1)\big ]\Big [\big [1+\gamma _{n+1,k-1}( \rho _{n+1,k-1}-1)\big ]\Vert U_{n+1,k-2}z_{n} - U_{n,k-2}z_{n}\Vert \nonumber \\&\qquad + |\gamma _{n+1,k-1} - \gamma _{n,k-1}|\big [\Vert T^{n+1}_{k-1}U_{n,k-2}z_{n}\Vert +\Vert U_{n,k-2}z_{n}\Vert \big ]\nonumber \\&\qquad + \gamma _{n,k-1}\Vert T^{n+1}_{k-1}U_{n,k-2}z_{n}-T^{n}_{k-1}U_{n,k-2}z_{n}\Vert \Big ]\nonumber \\&\qquad + |\gamma _{n+1,k} - \gamma _{n,k}|\big [\Vert T^{n+1}_{k}U_{n,k-1}z_{n}\Vert +\Vert U_{n,k-1}z_{n}\Vert \big ]\nonumber \\&\qquad + \gamma _{n,k}\Vert T^{n+1}_{k}U_{n,k-1}z_{n}-T^{n}_{k}U_{n,k-1}z_{n}\Vert \nonumber \\&\quad = \delta _{n+1,k}\delta _{n+1,k-1}\Vert U_{n+1,k-2}z_{n} - U_{n,k-2}z_{n}\Vert \nonumber \\&\qquad + \delta _{n+1,k}|\gamma _{n+1,k-1} - \gamma _{n,k-1}|M_{n,k-2}\nonumber \\&\qquad + \delta _{n+1,k}\gamma _{n,k-1}P_{n,k-2}\nonumber \\&\qquad + |\gamma _{n+1,k} - \gamma _{n,k}|M_{n,k-1}\nonumber \\&\qquad + \gamma _{n,k} P_{n,k-1}\nonumber \\&\quad \le \delta _{n+1,k}\delta _{n+1,k-1}\Big [\delta _{n+1,k-2}\Vert U_{n+1,k-3}z_{n} - U_{n,k-3}z_{n}\Vert \nonumber \\&\qquad + |\gamma _{n+1,k-2} - \gamma _{n,k-2}|[\Vert T^{n+1}_{k-2}U_{n,k-3}z_{n}\Vert +\Vert U_{n,k-3}z_{n}\Vert ]\nonumber \\&\qquad +\gamma _{n,k-1}\Vert T^{n+1}_{k-2}U_{n,k-3}z_{n}-T^{n}_{k-2}U_{n,k-2}z_{n}\Vert \Big ]\nonumber \\&\qquad + \delta _{n+1,k}|\gamma _{n+1,k-1} - \gamma _{n,k-1}|M_{n,k-2}\nonumber \\&\qquad + \delta _{n+1,k}\gamma _{n,k-1}P_{n,k-2}\nonumber \\&\qquad + |\gamma _{n+1,k} - \gamma _{n,k}|M_{n,k-1}\nonumber \\&\qquad + \gamma _{n,k} P_{n,k-1}\nonumber \\&\quad = \delta _{n+1,k}\delta _{n+1,k-1}\delta _{n+1,k-2}\Vert U_{n+1,k-3}z_{n} - U_{n,k-3}z_{n}\Vert \nonumber \\&\qquad + \delta _{n+1,k}\delta _{n+1,k-1}|\gamma _{n+1,k-2} - \gamma _{n,k-2}|M_{n,k-3}\nonumber \\&\qquad + \delta _{n+1,k}\delta _{n+1,k-1}\gamma _{n,k-1}P_{n,k-3}\nonumber \\&\qquad + \delta _{n+1,k}|\gamma _{n+1,k-1} - \gamma _{n,k-1}|M_{n,k-2}\nonumber \\&\qquad + \delta _{n+1,k}\gamma _{n,k-1}P_{n,k-2}\nonumber \\&\qquad + |\gamma _{n+1,k} - \gamma _{n,k}|M_{n,k-1}\nonumber \\&\qquad + \gamma _{n,k} P_{n,k-1}\nonumber \\&\quad \le \nonumber \\&\qquad \vdots \vdots \end{aligned}$$

(3.2)

$$\begin{aligned}&\quad \le \delta _{n+1,k}\delta _{n+1,k-1}\delta _{n+1,k-2}\ldots \delta _{n+1,3}\delta _{n+1,2}\Vert U_{n+1,1}z_{n}-U_{n,1}z_{n}\Vert \nonumber \\&\qquad + \Big (\delta _{n+1,k}\delta _{n+1,k-1}\delta _{n+1,k-2}\ldots \delta _{n+1,3}\gamma _{n+1,2}P_{n,1}\nonumber \\&\qquad +\cdots + \delta _{n+1,k}\delta _{n+1,k-1}\gamma _{n+1,k-2}P_{n,k-3}\nonumber \\&\qquad + \delta _{n+1,k}\gamma _{n+1,k-1}P_{n,k-2} +\gamma _{n+1,k}P_{n,k-1}\Big )\nonumber \\&\qquad + \Big (\delta _{n+1,k}\delta _{n+1,k-1}\delta _{n+1,k-2}\ldots \delta _{n+1,3}|\gamma _{n+1,2}-\gamma _{n,2}|M_{n,1}\nonumber \\&\qquad +\cdots + \delta _{n+1,k}\delta _{n+1,k-1}|\gamma _{n+1,k-2}-\gamma _{n,k-2}|M_{n,k-3}\nonumber \\&\qquad + \delta _{n+1,k}|\gamma _{n+1,k-1}-\gamma _{n,k-1}|M_{n,k-2}\nonumber \\&\qquad +\vert \gamma _{n+1,k}-\gamma _{n,k}\vert M_{n,k-1} \Big )\nonumber \\&\quad =\Vert U_{n+1,1}z_{n}-U_{n,1}z_{n}\Vert \prod ^{k}_{j=2}\delta _{n+1,j}\nonumber \\&\qquad +\sum ^{k}_{i=2}\gamma _{n+1,i}P_{n,i-1}\prod ^{k}_{j=i+1}\delta _{n+1,j}+\sum ^{k}_{i=2}|\gamma _{n+1,i}-\gamma _{n,i}|M_{n,i-1}\prod ^{k}_{j=i+1}\delta _{n+1,j}\nonumber \\&\quad \le \Big [\gamma _{n+1,1}||T^{n+1}_{1}z_n-T^{n}_{1}z_{n}||+|\gamma _{n+1,1}-\gamma _{n,1}|(||z_n||+||T^{n}_{1}z_{n}||)\Big ]\prod ^{k}_{j=2}\delta _{n+1,j}\nonumber \\&\qquad +\sum ^{k-1}_{i=2}\gamma _{n+1,i}P_{n,i-1}\prod ^{k}_{j=i+1}\delta _{n+1,j}+\sum ^{k}_{i=2}|\gamma _{n+1,i}-\gamma _{n,i}|M_{n,i-1}\prod ^{k}_{j=i+1}\delta _{n+1,j}\nonumber \\&\quad =\sum ^{k}_{i=1}\gamma _{n+1,i}P_{n,i}\prod ^{k}_{j=i+1}\delta _{n+1,j}+\sum ^{k}_{i=1}|\gamma _{n+1,i}-\gamma _{n,i}|M_{n,i}\prod ^{k}_{j=i+1}\delta _{n+1,j} \end{aligned}$$

(3.3)

Hence, we have

$$\begin{aligned} \Vert K_{n+1}z_{n} - K_{n}z_{n}\Vert= & {} \Vert U_{n+1,N}z_{n}-U_{n,N}z_{n}\Vert \nonumber \\\le & {} \gamma _{n+1,N}||T^{n+1}_{N}U_{n+1,N-1}z_{n}-T^{n+1}_{N}U_{n,N-1}z_{n}||\nonumber \\&+\gamma _{n+1,N}||T^{n+1}_{N}U_{n+1,N-1}z_{n}-T^{n}_{N}U_{n,N-1}z_{n}||\nonumber \\&+|\gamma _{n+1,N}-\gamma _{n,N}|[||T^{n}_{N}U_{n,N-1}z_{n}||+||z_n||]\nonumber \\\le & {} \gamma _{n+1,N}\rho _{n+1,N}||U_{n+1,N-1}z_{n}-U_{n,N-1}z_{n}||\nonumber \\&+\gamma _{n+1,N}||T^{n+1}_{N}U_{n+1,N-1}z_{n}-T^{n}_{N}U_{n,N-1}z_{n}||\nonumber \\&+|\gamma _{n+1,N}-\gamma _{n,N}|[||T^{n}_{N}U_{n,N-1}z_{n}||+||z_n||]\nonumber \\\le & {} \rho _{n+1,N}\left[ \sum ^{N-1}_{i=1}\gamma _{n+1,i}P_{n,i}\prod ^{N-1}_{j=i+1}\delta _{n+1,j}\right. \nonumber \\&\left. +\sum ^{N-1}_{i=1}|\gamma _{n+1,i}-\gamma _{n,i}|M_{n,i}\prod ^{N-1}_{j=i+1}\delta _{n+1,j}\right] \nonumber \\&+\gamma _{n+1,N}||T^{n+1}_{N}U_{n+1,N-1}z_{n}-T^{n}_{N}U_{n,N-1}z_{n}||\nonumber \\&+|\gamma _{n+1,N}-\gamma _{n,N}|\big [||T^{n}_{N}U_{n,N-1}z_{n}||+||z_n||\big ]. \end{aligned}$$

(3.4)

Therefore

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||K_{n+1}z_n-K_{n}z_{n}||=0. \end{aligned}$$

(3.5)

Hence (ii) is satisfied.

Next, we show (iii), let $\{z_n\}$ be a bounded sequence in E such that

${\lim \nolimits _{n\rightarrow \infty }}||K_nz_n-z_n||=0,$ then for $x^{*}\in \cap ^{N}_{k=1}F(T_{k})$, we obtain

$$\begin{aligned}&||K_{n}z_n-x^*||^2\le \gamma _{n,N}||T^{n}_{N}U_{n,N-1}z_n-x^*||^2+(1-\gamma _{n,N})||z_n-x^*||^2\nonumber \\&\quad \le \gamma _{n,N}\rho _{n,N}^2||U_{n,N-1}z_n-x^*||^{2}+(1-\gamma _{n,N})||z_{n}-x^*||^2\nonumber \\&\quad \le \gamma _{n,N}\rho ^2_{n,N}\big [\gamma _{n,N-1}||T^{n}_{N-1}U_{n,N-2}z_n-x^*||^2\nonumber \\&\qquad +(1-\gamma _{n,N-1})||U_{n,N-2}z_n-x^*||^2\big ]\nonumber \\&\qquad +(1-\gamma _{n,N})||z_{n}-x^*||^2 \end{aligned}$$

(3.6)

$$\begin{aligned}&\le \gamma _{n,N}\rho ^2_{n,N}\big [\gamma _{n,N-1}\rho ^{2}_{n,N-1}||U_{n,N-2}z_n-x^*||^2\nonumber \\&\qquad +(1-\gamma _{n,N-1})||U_{n,N-2}z_n-x^*||^2\big ]\nonumber \\&\qquad +(1-\gamma _{n,N})||z_{n}-x^*||^2\nonumber \\&\quad =\gamma _{n,N}\rho ^2_{n,N}\Big ([1+\gamma _{n,N-1}(\rho ^{2}_{n,N-1}-1)]||U_{n,N-2}z_n-x^*||^2\Big )\nonumber \\&\qquad +(1-\gamma _{n,N})||z_{n}-x^*||^2\nonumber \\&\quad \le \gamma _{n,N}\rho ^2_{n,N}\big [1+\gamma _{n,N-1}(\rho ^{2}_{n,N-1}-1)\big ]\big [1+\gamma _{n,N-2}(\rho ^{2}_{n,N-2}-1)\big ]\nonumber \\&\qquad \ldots \times [1+\gamma _{n,1}(\rho ^{2}_{n,1}-1)]||z_n-x^*||^2+(1-\gamma _{n,N})||z_{n}-x^*||^2\nonumber \\&\quad =\Big (1+\gamma _{n,N}\Big \{\rho ^2_{n,N}[1+\gamma _{n,N-1}(\rho ^{2}_{n,N-1}-1)][1+\gamma _{n,N-2}(\rho ^{2}_{n,N-2}-1)]\nonumber \\&\qquad \ldots \times [1+\gamma _{n,1}(\rho ^{2}_{n,1}-1)]-1\Big \}\Big )||z_{n}-x^*||^2\nonumber \\&\quad =(1+\vartheta _{n})||z_{n}-x^{*}||^2, \end{aligned}$$

(3.7)

where $\vartheta _{n}:=\gamma _{n,N}\Big \{\rho ^2_{n,N}[1+\gamma _{n,N-1}(\rho ^{2}_{n,N-1}-1)][1+\gamma _{n,N-2}(\rho ^{2}_{n,N-2}-1)]\ldots [1+\gamma _{n,2}(\rho ^{2} _{n,2}-1)][1+\gamma _{n,1}(\rho ^{2}_{n,1}-1)]-1\Big \}$ and observe that ${\lim \nolimits _{n\rightarrow \infty }}\vartheta _{n}=0.$

Then by using Lemma 2.3, (3.6) and (3.7), we have

$$\begin{aligned} ||K_nz_n-x^{*}||^2= & {} ||\gamma _{n,N}(T^{n}_{N}U_{n,N-1}z_n-x^*)+(1-\gamma _{n,N})(z_n-x^*)||^2\\\le & {} \gamma _{n,N}||T^{n}_{N}U_{n,N-1}z_n-x^*||^2+(1-\gamma _{n,N})||z_n-x^*||^2\\&-\gamma _{n,N}(1-\gamma _{n,N})g(||T^{n}_{N}U_{n,N-1}z_n-z_n||)\\\le & {} (1+\vartheta _{n})||z_n-x^*||^2-\gamma _{n,N}(1-\gamma _{n,N}) g(||T^{n}_{N}U_{n,N-1}z_n-z_n||), \end{aligned}$$

from this we obtain

$$\begin{aligned}&\gamma _{n,N}(1-\gamma _{n,N}) g(||T^{n}_{N}U_{n,N-1}z_n-z_n||)\le ||z_n-x^*||^2-||K_nz_n-x^*||^2+\vartheta _{n}||z_n-x^*||^2\\&\quad = (||z_n-x^*||-||K_nz_n-x^*||)(||z_n-x^*||+||K_nz_n-x^*||)+\vartheta _{n}||z_n-x^*||^2\\&\quad \le ||z_n-K_nz_n||(||z_n-x^*||+||K_nz_n-x^*||)+\vartheta _{n}||z_n-x^*||^2\\&\quad \le (||z_n-K_nz_n||+\vartheta _{n})M_{0}\rightarrow 0\quad \text {as}\quad n\rightarrow \infty , \end{aligned}$$

for some $M_{0}>0.$ Thus, by the property of g, we obtain that

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||T^{n}_{N}U_{n,N-1}z_{n}-z_{n}||=0. \end{aligned}$$

(3.8)

Moreover,

$$\begin{aligned}&||z_n-x^*||^2\le (||z_n-T^{n}_{N}U_{n,N-1}z_n||+||T^{n}_{N}U_{n,N-1}z_n-x^*||)^2\\&\quad =||z_n-T^{n}_{N}U_{n,N-1}z_n||(||z_{n}-T^{n}_{N}U_{n,N-1}z_{n}||+2||T^{n}_{N}U_{n,N-1}z_n-x^*||)\\&\qquad +||T^{n}_{N}U_{n,N-1}z_n-x^*||^2\\&\quad \le ||z_n-T^{n}_{N}U_{n,N-1}z_n||M_{1}+\rho _{n,N}^{2}||U_{n,N-1}z_{n}-x^*||^2\quad (\text {for some}\, M_{1}>0)\\&\quad \le ||z_n-T^{n}_{N}U_{n,N-1}z_n||M_{1}+\rho ^{2}_{n,N}[\gamma _{n,N-1}||T^{n,N-1}U_{n,N-2}z_{n}-x^*||^2\\&\qquad +(1-\gamma _{n,N-1})||U_{n,N-2}z_n-x^*||^2\\&\qquad -\gamma _{n,N-1}(1-\gamma _{n,N-1})g(||T^{n}_{N-1}U_{n,N-2}z_{n}-U_{n,N-2}z_{n}||)] \\&\quad \le ||z_n-T^{n}_{N}U_{n,N-1}z_n||M_{1}+\rho ^{2}_{n,N}[\gamma _{n,N-1}\rho ^{2}_{n,N-1}||U_{n,N-2}z_{n}-x^*||^2\\&\qquad +(1-\gamma _{n,N-1})||U_{n,N-2}z_n-x^*||^2\\&\qquad -\gamma _{n,N-1}(1-\gamma _{n,N-1})g(||T^{n}_{N-1}U_{n,N-2}z_{n}-U_{n,N-2}z_{n}||)]\\&\quad \le ||z_n-T^{n}_{N}U_{n,N-1}z_n||M_{1}+\rho ^{2}_{n,N}[(1+\gamma _{n,N-1}(\rho ^{2}_{n,N-1}-1))||U_{n,N-2}z_{n}-x^*||^2\\&\qquad -\gamma _{n,N-1}(1-\gamma _{n,N-1})g(||T^{n}_{N-1}U_{n,N-2}z_{n}-U_{n,N-2}z_{n}||)]\\&\qquad \le ||z_n-T^{n}_{N}U_{n,N-1}z_n||M_{1}+(1+\vartheta _{n})||z_{n}-x^*||^2\\&\qquad -\rho ^{2}_{n,N}\gamma _{n,N-1}(1-\gamma _{n,N-1})g(||T^{n}_{N-1}U_{n,N-2}z_{n}-U_{n,N-2}z_{n}||)],\\&g(||T^{n}_{N-1}U_{n,N-2}z_{n}-U_{n,N-2}z_{n}||)\nonumber \\&\quad \le \frac{\Big (||z_n-T^{n}_{N}U_{n,N-1}z_n||M_{1}+\vartheta _{n}||z_{n}-x^*||^2\Big )}{\rho _{n,N}^{2}{\gamma _{n,N-1}(1-\gamma _{n,N-1})}}\\&\quad \le \frac{\Big (||z_n-T^{n}_{N}U_{n,N-1}z_n||+\vartheta _{n}\Big )M}{\rho _{n,N}^{2}{\gamma _{n,N-1}(1-\gamma _{n,N-1})}}, \end{aligned}$$

for some $M>0$. Thus, using property of g, 

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||T^{n}_{N-1}U_{n,N-2}z_n-U_{n,N-2}z_{n}||=0. \end{aligned}$$

(3.9)

Continuing in this fashion we observe that for $k\in \{2,3,4,\ldots ,N-1\}$

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||T^{n}_{k}U_{n,k-1}z_n-U_{n,k-1}z_{n}||=0, \end{aligned}$$

(3.10)

and

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||T^{n}_{1}z_n-z_{n}||=0. \end{aligned}$$

(3.11)

Also

$$\begin{aligned} ||U_{n,k}z_{n}-z_{n}||\le & {} ||U_{n,k}z_n-T^{n}_{k}U_{n,k-1}z_n||+||T^{n}_{k}U_{n,k-1}z_n-U_{n,k-1}z_{n}||\\&+||U_{n,k-1}z_n-T^{n}_{k-1}U_{n,k-2}z_n||\\&+||T^{n}_{k-1}U_{n,k-2}z_n-U_{n,k-2}z_n||\\&+\cdots +||T^{n}_{2}U_{n,1}z_{n}-U_{n,1}z_{n}||+||U_{n,1}z_{n}-z_{n}||\\\le & {} (1-\gamma _{n,k})||U_{n,k-1}z_{n}-T^{n}_{k}U_{n,k-1}z_{n}||\\&+||T^{n}_{k}U_{n,k-1}z_n-U_{n,k-1}z_n||\\&+(1-\gamma _{n,k-1})||U_{n,k-2}z_n-T^{n}_{k-1}U_{n,k-2}z_{n}||\\&+\cdots +(1-\gamma _{n,2})||U_{n,1}z_n-T^{n}_{2}U_{n,1}z_n||\\&+\gamma _{n,1}||T^{n}_{1}z_{n}-z_{n}||\rightarrow 0\quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$

Thus

$$\begin{aligned} ||T^{n}_{k}U_{n,k-1}z_n-z_n||\le & {} ||T^{n}_{k}U_{n,k-1}z_n-U_{n,k-1}z_{n}||\\&+||U_{n,k-1}z_{n}-z_{n}||\rightarrow 0\quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$

So for any $k\in \{1,2,3,\ldots ,N\}$, we obtain

$$\begin{aligned} ||z_n-T^{n}_{k}z_{n}||\le & {} ||z_{n}-T^{n}_{k}U_{n,k-1}z_{n}||+||T^{n}_{k}U_{n,k-1}z_{n}-T^{n}_{k}z_{n}||\nonumber \\\le & {} ||z_{n}-T^{n}_{k}U_{n,k-1}z_{n}||\nonumber \\&+\rho _{n,k}||U_{n,k-1}z_{n}-z_{n}||\rightarrow 0\quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$

(3.12)

Hence

$$\begin{aligned} ||T_{k}z_{n}-z_{n}||\le & {} ||T_{k}z_{n}-T_{k}(T^{n}_{k})z_{n}||+||T_{k}(T^{n}_{k})z_{n}-T^{n}_{k}z_{n}||+||T^{n}_{k}z_{n}-z_{n}||\\\le & {} (L_{k}+1)||z_{n}-T^{n}_{k}z_{n}||+||T^{n+1}_{k}z_{n}-T^{n}_{k}z_{n}||. \end{aligned}$$

Therefore, from (3.12), for each $k\in \{1,2,3,\ldots ,N\}$, we obtain

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||T_{k}z_n-z_{n}||=0. \end{aligned}$$

(3.13)

Moreover, by Lemma 2.5, we have $w_{w}(x_{n})\subset \cap ^{N}_{k=1}F(T_{k})$, also since $\cap ^{N}_{k=1}F(T_{k})\subset F(K_n)$ is obvious, we only need to show that $F(K_{n})\subset \cap ^{N}_{k=1}F(T_{k}).$ Let $z^*\in F(K_n)$, and $z_n=z^*$, then, we have that $||z^*-T_{k}z^*||=0$ for each $k\in \{1,2,3,\ldots ,N\}$ that is $z^*=T_{k}z^*$, for each $k\in \{1,2,3,\ldots ,N\}$, so that $z^{*}\in \cap ^{N}_{k=1}F(T_{k})$. Hence (iii) is satisfied. $\square$

Theorem 3.2

Let E be a real uniformly convex Banach space with a uniformly G$\hat{a}$teaux differentiable norm, C a nonempty closed convex subset of E. Let $G : E \rightarrow E$ be an$\eta$-strongly accretive and $\mu$-strictly pseudocontractive with $\eta + \mu > 1$ and let $f : E \rightarrow E$ be a contraction with coefficient $\alpha \in (0,1)$. Let $\{T_{i}\}^{N}_{i=1}$ be a family of uniformly asymptotically regular asymptotically nonexpansive self mappings of C into itself and $F = \bigcap ^{N}_{i=1}F(T_{i}) \ne \emptyset .$ Let $K_{n}$ be as in Lemma 3.1. Assume that $0< \gamma < \frac{\tau }{\alpha }$, where $\tau := (1 - \sqrt{\frac{1 - \eta }{\mu }})$ and let $x_{0} \in C$. Let $\{\alpha _{n}\}^{\infty }_{n = 1}$ and $\{\beta _{n}\}^{\infty }_{n = 1}$ be sequences in (0, 1), and suppose that the following conditions are satisfied:

• (C1) $\alpha _{n} \rightarrow 0$ and $\frac{v_n}{\alpha _{n}} \rightarrow 0$ as $n \rightarrow \infty ,$ where $v_{n}$ is as in (i) of Lemma 3.1;

• (C2) $\sum ^{\infty }_{n=1}\alpha _{n} = \infty$

If $\{x_{n}\}^{\infty }_{n = 1}$ is a sequence defined by,

$$\begin{aligned} x_{n} = \alpha _{n}\gamma f(x_{n}) + (I - \alpha _{n}G)[ \beta _{n}x_{n} + (1 - \beta _{n})K_{n}x_{n}],\qquad n \ge 0, \end{aligned}$$

(3.14)

then $\{x_{n}\}^{\infty }_{n = 1}$ converges strongly to $p \in F$, which also solves the following variational inequality:

$$\begin{aligned} \langle \gamma f(p) - Gp, j(q - p)\rangle \le 0,\quad \forall q \in F. \end{aligned}$$

(3.15)

Proof

First, we show that $\{x_n\}$ defined by (3.14) is well defined. For all $n\in \mathbb {N}$, let us define the mapping

$$\begin{aligned} T^{f}_{n}x:=\alpha _{n}\gamma f(x)+(I-\alpha _{n}G)[\beta _{n}x+(1-\beta _{n})K_{n}x]. \end{aligned}$$

Indeed, for all $x,y\in E$, we have

$$\begin{aligned} ||T^{f}_{n}x-T^{f}_{n}y||= & {} ||\alpha _{n}\gamma (f(x)-f(y))+(1-\alpha _{n}G)[\beta _{n}(x-y)+(1-\beta _{n})(K_{n}x-K_{n}y)]||\\\le & {} \alpha _{n}\gamma \alpha ||x-y||+(1-\alpha _{n}\tau )[\beta _{n}||x-y||+(1-\beta _{n})(1+v_{n})||x-y||]\\\le & {} [\alpha _{n}\gamma \alpha +(1-\alpha _{n}\tau )(1+v_{n})]||x-y||\\= & {} \Big (1-\alpha _{n}[(\tau -\gamma \alpha )-(1-\alpha _{n}\tau )(v_{n}/\alpha _{n})]\Big )||x-y||. \end{aligned}$$

Since, ${\lim \nolimits _{n\rightarrow \infty }}(1-\alpha _{n}\tau )v_{n}/\alpha _{n}\rightarrow 0$, then there exist $n_{0}\in \mathbb {N}$ such that $(1-\alpha _{n}\tau )v_{n}/\alpha _{n}<(\tau -\gamma \alpha )/2$ for all $n\ge n_{0}$. Therefore, for $n\ge n_{0}$, we have

$$\begin{aligned} 1-\alpha _{n}[(\tau -\gamma \alpha )-(1-\alpha _{n}\tau )(v_{n}/\alpha _{n})]<1-\alpha _{n}[(\tau -\gamma \alpha )-(\tau -\gamma \alpha )/2]<1. \end{aligned}$$

Hence,

$$\begin{aligned} ||T^{f}_{n}x-T^{f}_{n}y||<||x-y||. \end{aligned}$$

Thus, $\{x_n\}$ defined by (3.14) is well defined. Therefore, by the contraction mapping principle, there exists a unique fixed point $x_{n}\in C$ of $T^{f}_{n}$ which satisfies (3.14).

From the choice of the parameter $\gamma ,$ it is easy to see that the mapping $(G-\gamma f):E\rightarrow E$ is strongly accretive and so the variational inequality (3.15) has unique solution in F. Let $p\in F$ then,

$$\begin{aligned} \Vert x_{n}-p\Vert ^{2}= & {} \alpha _{n}\langle \gamma f(p)-Gp, j(x_n-p) \rangle +\langle (I-\alpha _{n}G)[\beta _{n}x_{n}+(1-\beta _{n})K_{n}x_{n}]\\&-(I-\alpha _{n}G)p, j(x_n-p) \rangle +\alpha _{n}\langle \gamma f(x_{n})-\gamma f(p), j(x_n-p) \rangle \\\le & {} [1-\alpha _{n}(\tau -\gamma \alpha )+(1-\alpha _{n}\tau )v_{n}]\Vert x_{n}-p\Vert ^{2}+\alpha _{n}\langle (\gamma f-G)p, j(x_n-p) \rangle . \end{aligned}$$

Let $d_{n}=(1-\alpha _{n}\tau )(v_{n}/\alpha _{n})$. Since, ${\lim \nolimits _{n\rightarrow \infty }}(1-\alpha _{n}\tau )v_{n}/\alpha _{n}=0$, then there exist $n_{0}\in \mathbb {N}$ such that $(1-\alpha _{n}\tau )v_{n}/\alpha _{n}<(\tau -\gamma \alpha )/2$ for all $n\ge n_{0}$.

$$\begin{aligned} \Vert x_{n}-p\Vert ^{2} \le \frac{\langle (\gamma f-G)p, j(x_n-p) \rangle }{(\tau -\gamma \alpha )-d_{n}}, \end{aligned}$$

(3.16)

that is $||x_{n}-p||\le \frac{2||\gamma f(p)-G(p)||}{\tau -\gamma \alpha }$, for all $n\ge n_{0}$. Thus $\{x_{n}\}$ is bounded implies that $\{f(x_{n})\},$ $\{G(x_{n})\}$ and $\{K_{n}(x_{n})\}$ are also bounded. From (3.14) we also obtain

$$\begin{aligned} \Vert x_{n}-K_{n}x_{n}\Vert \le \beta _{n}\Vert x_{n}-K_{n}x_{n}\Vert +\alpha _{n}\Vert \gamma f(x_{n})-G(\beta _{n} x_{n}+(1-\beta _{n})K_{n}(x_{n}))\Vert \end{aligned}$$

and hence

$$\begin{aligned} \Vert x_{n}-K_{n}x_{n}\Vert \le \frac{\alpha _{n}}{1-\beta _{n}}\Vert \gamma f(x_{n})-G(\beta _{n} x_{n}+(1-\beta _{n})K_{n}(x_{n}))\Vert \,\rightarrow 0, \end{aligned}$$

(3.17)

as $n\rightarrow \infty .$ Since $\{x_n\}$ is bounded, using (3.17), it follows from (iii) of Lemma 3.1 that $F=F(K_{n})$.

We claim that the set $\{x_{n}\}$ is sequentially compact. Indeed, define a map $\phi :E\rightarrow \mathbb {R}$ by

$$\begin{aligned} \phi (y):=\mu _n||x_{n}-y||^2,\quad \forall y \in E. \end{aligned}$$

Then, $\phi (y)\rightarrow \infty$ as $||y||\rightarrow \infty$, $\phi$ is continuous and convex, so as E is reflexive, there exists $q \in E$ such that $\phi (q)=\min \nolimits _{u \in E} \phi (u)$. Hence, the set

$$\begin{aligned} K^*:=\{y \in E:\phi (y)=\underset{u \in E}{\min }\phi (u)\} \ne \emptyset . \end{aligned}$$

Since ${\lim \nolimits _{n\rightarrow \infty }}||x_{n}-K_nx_{n}||=0,$ ${\lim \nolimits _{n\rightarrow \infty }}||x_{n}-K^{m}_{n}x_{n}||=0,\, \mathrm{for\, any}\,m\ge 1$ by induction. Now let $v\in K^*$, we have

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\phi (K_nv)= & {} \underset{n\rightarrow \infty }{\lim }\mu _{n}||x_{n}-K_nv||^{2}\\= & {} \underset{n\rightarrow \infty }{\lim }\mu _{n}||x_{n}-K_nx_{n}+K_nx_{n}-K_nv||^{2}\\\le & {} \underset{n\rightarrow \infty }{\lim }\mu _{n}[(1+v_n)||x_{n}-v||]^{2}=\underset{n\rightarrow \infty }{\lim }\phi (v), \end{aligned}$$

and hence $K_{n}v\in K^{*}$.

Now let $z\in F,$ then $z=K_{n}z.$ Since $K^{*}$ is a closed convex set, there exists a unique $v^{*}\in K^{*}$ such that

$$\begin{aligned} ||z-v^{*}||=\underset{u\in K^{*}}{\min }||z-u||. \end{aligned}$$

But

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||z-K_{n}v^{*}||= & {} \underset{n\rightarrow \infty }{\lim }||K_{n}z-K_{n}v^{*}||\le \underset{n\rightarrow \infty }{\lim }(1+v_n) ||z-v^{*}||, \end{aligned}$$

which implies $v^{*}=K_{n}v^{*}$ and so $K^{*}\cap F\ne \emptyset .$

Let $p \in K^* \cap F$ and $\epsilon \in (0,1)$. Then, it follows that $\phi (p)\le \phi (p-\epsilon (G-\gamma f)p)$ and using Lemma 2.1, we obtain that

$$\begin{aligned} ||x_n-p+\epsilon (G-\gamma f)p||^2 \le ||x_n-p||^2+2\epsilon \langle (G-\gamma f)p,j(x_n-p+\epsilon (G-\gamma f)p)\rangle \end{aligned}$$

which implies

$$\begin{aligned} \mu _n\langle (\gamma f-G)p,j(x_n-p+\iota (G-\gamma f)p)\rangle \le 0. \end{aligned}$$

Moreover,

$$\begin{aligned} \mu _n\langle (\gamma f-G)p,j(x_n-p)\rangle= & {} \mu _n\langle (\gamma f-G)p,j(x_n-p)-j(x_n-p+\epsilon (G-\gamma f)p)\rangle \\&+\mu _n\langle (\gamma f-G)p,j(x_n-p+\epsilon (G-\gamma f)p)\rangle \\\le & {} \mu _n\langle (\gamma f-G)p,j(x_n-p)-j(x_n-p+\epsilon (G-\gamma f)p)\rangle . \end{aligned}$$

Since j is norm-to-$weak^{*}$ uniformly continuous on bounded subsets of E, we have that

$$\begin{aligned} \mu _n\langle (\gamma f-G)p, j(x_n-p) \rangle \le 0. \end{aligned}$$

It follows from (3.16) that

$$\begin{aligned} \Vert x_{n}-p\Vert ^{2} \le \frac{\langle (\gamma f-G)p, j(x_n-p) \rangle }{(\tau -\gamma \alpha )-d_{n}}, \end{aligned}$$

and so

$$\begin{aligned} \mu _{n}\Vert x_{n}-p\Vert ^{2} \le 0. \end{aligned}$$

Thus there exist a subsequence say $\{x_{n_{l}}\}$ of $\{x_{n}\}$ such that ${\lim \nolimits _{l\rightarrow \infty }}x_{n_{l}}=p.$

Define $S_n$ as $S_{n}x:=\beta _{n} x+ (1-\beta _{n})K_{n}x,$ then ${\lim \nolimits _{l\rightarrow \infty }}S_{n}x_{n_{l}}=p$ and $S_np=p.$ Thus for any $z\in F,$ using (3.14) we have

$$\begin{aligned} \langle G(x_{n_{l}})-\gamma f(x_{n_{l}}),j(x_{n_{l}}-z)\rangle= & {} \frac{-1}{\alpha _{n_{l}}}\langle (I-S_{n})x_{n_{l}}-(I-S_{n})p, j(x_{n_{l}}-z)\rangle \nonumber \\&+ \langle Gx_{n_{l}}-GS_{n}x_{n_{l}}, j(x_{n_{l}}-z)\rangle \nonumber \\\le & {} \langle Gx_{n_{l}}-GS_{n}x_{n_{l}}, j(x_{n_{l}}-z)\rangle \nonumber \\\le & {} (1+\frac{1}{\mu })||x_{n_l}-S_{n}x_{n_l}||||x_{n_l}-z||, \end{aligned}$$

(3.18)

since $\langle (I-S_{n})x_{n_{l}}-(I-S_{n})p, j(x_{n_{l}}-z)\rangle \ge 0$ and G is Lipschitzian. Using the fact that $\Vert x_{n_{l}}-S_{n}x_{n_{l}}\Vert =(1-\beta _{n_{l}})\Vert x_{n_{l}}-K_{n_{l}}x_{n_{l}}\Vert \rightarrow 0$ as $l\rightarrow \infty ,$ we have $\Vert x_{n_{l}}-S_{n}x_{n_{l}}\Vert \rightarrow 0~\mathrm{as}~l\rightarrow \infty .$ From (3.18), taking limit as $l\rightarrow \infty$ we obtain

$$\begin{aligned} \langle (G-\gamma f)p,j(p-z)\rangle \le 0. \end{aligned}$$

Hence p is the unique solution of the variational inequality (3.15). Now assume there exists another subsequence of $\{x_{n}\}$ say $\{x_{n_{k}}\}$ such that $\lim \nolimits _{k\rightarrow \infty }x_{n_{k}}=p^{*}.$ Then, using (3.17) we have $p^{*}\in F.$ Repeating the above argument with p replaced by $p^{*}$ we can easily obtain that $p^{*}$ also solved the variational inequality (3.15). By uniqueness of the solution of the variational inequality, we obtained that $p=p^{*}$ and this completes the proof. $\square$

Theorem 3.3

Let E be a real, uniformly convex Banach space with a uniformly G$\hat{a}$teaux differentiable norm,C a nonempty closed convex subset of E. Let $G : E \rightarrow E$ be an$\eta$-strongly accretive and $\mu$-strictly pseudocontractive with $\eta + \mu > 1$ and let $f : E \rightarrow E$ be a contraction with coefficient $\alpha \in (0,1)$. Let $\{T_{i}\}^{N}_{i=1}$ be family of uniformly asymptoticallyregular asymptotically nonexpansive self mappings of C into itself and $F = \bigcap ^{N}_{i=1}F(T_{i}) \ne \emptyset .$ Let $K_{n}$ be as in Lemma 3.1. Assume that $0< \gamma < \frac{\tau }{2\alpha }$, where $\tau := (1 - \sqrt{\frac{1 - \eta }{\mu }})$. Let $\{\alpha _{n}\}^{\infty }_{n = 1}$ and $\{\beta _{n}\}^{\infty }_{n = 1}$ be sequences in (0, 1), and suppose that the following conditions are satisfied:

• (C1) $\lim \nolimits _{n\rightarrow \infty }\alpha _{n}=0$ and $\lim \nolimits _{n\rightarrow \infty }\frac{v_n}{\alpha _{n}}=0$, where $v_{n}$ is as in (i) of Lemma 3.1;

• (C2) $\Sigma ^{\infty }_{n=0}\alpha _{n} = \infty$

• (C3) $0< \liminf _{n\rightarrow \infty }\beta _{n} \le \limsup _{n\rightarrow \infty }\beta _{n} < 1;$

Let $\{x_{n}\}^{\infty }_{n = 1}$ be a sequence defined iteratively by letting $x_{0}\in C$ arbitrary and,

$$\begin{aligned} y_{n}= & {} \beta _{n}x_{n} + (1 - \beta _{n})K_{n}x_{n}, \nonumber \\ x_{n+1}= & {} \alpha _{n}\gamma f(x_{n}) + (I - \alpha _{n}G)y_{n},\quad n \ge 0, \end{aligned}$$

(3.19)

then, the following holds

• (a) $\{x_{n}\}^{\infty }_{n = 1}$ is bounded;

• (b) $\lim \nolimits _{n\rightarrow \infty }||K_{n}x_{n}-x_{n}||=0$;

• (c) $F(K_n)=F$;

• (d) $\{x_n\}^{\infty }_{n = 1}$ converges strongly to $p \in F$, where p is a solution of the variational inequality:

  $$\begin{aligned} \langle \gamma f(p) - Gp, j(q - p)\rangle \le 0, \quad \forall q \in F. \end{aligned}$$

  (3.20)

Proof

First, we show that the sequence $\{x_{n}\}^{\infty }_{n=1}$ is bounded. Let $u \in F$ then, since $(1-\alpha _{n}\tau )(v_n/\alpha _n)\rightarrow 0$ as $n\rightarrow \infty ,$ there exists $n_0\in \mathbb {N}$ such that $(1-\alpha _{n}\tau )(v_n/\alpha _n)<(\tau -\gamma \alpha )/2$ for all $n\ge n_0.$ Hence, for $n\ge n_0$, we have the following.

$$\begin{aligned} \Vert y_{n} - u\Vert\le & {} \beta _{n}\Vert x_{n} - u\Vert + (1 -\beta _{n})\Vert K_{n}x_{n} - u\Vert \nonumber \\\le & {} \beta _{n}\Vert x_{n} - u\Vert + (1 -\beta _{n})(1+v_{n})\Vert x_{n} - u\Vert \nonumber \\\le & {} (1+v_{n}) \Vert x_{n} - u\Vert , \end{aligned}$$

(3.21)

so that,

$$\begin{aligned} \Vert x_{n+1} - u\Vert= & {} \Vert \alpha _{n}\gamma f(x_{n}) + (I - \alpha _{n}G)y_{n} - u\Vert \\= & {} \Vert \alpha _{n}\gamma f(x_{n}) \!- \alpha _{n}\gamma f(u) +\! \alpha _{n}\gamma f(u)\! - \alpha _{n}G(u) + \!\alpha _{n}G(u) + (I - \alpha _{n}G)y_{n} -\! u\Vert \\\le & {} \alpha _{n}\gamma \Vert f(x_{n}) - f(u)\Vert + \alpha _{n}\Vert \gamma f(u) - G(u)\Vert + \Vert (I - \alpha _{n}G)y_{n} - (I - \alpha _{n}G)u\Vert \\\le & {} \alpha _{n}\gamma \Vert f(x_{n}) \!- f(u)\Vert \!+ \alpha _{n}\Vert \gamma f(u) \!- G(u)\Vert \!+ (1 -\! \alpha _{n}\tau )\Vert y_{n} \!- u\Vert \\\le & {} \alpha _{n}\gamma \alpha \Vert x_{n} - u\Vert + \alpha _{n}\Vert \gamma f(u) - G(u)\Vert + (1 - \alpha _{n}\tau )(1+v_n)\Vert x_{n} - u\Vert \\= & {} \Big [1 - \alpha _{n}\Big ((\tau - \alpha \gamma )-(1-\alpha _n\tau )\frac{v_n}{\alpha _n}\Big )\Big ]\Vert x_{n} - u\Vert \\&+ \,\nonumber \alpha _{n}\Big ((\tau - \alpha \gamma )-(1-\alpha _n\tau )\frac{v_n}{\alpha _n}\Big ) \frac{2\Vert \gamma f(u) - G(u)\Vert }{\tau - \alpha \gamma }\\\le & {} \max \Big \{\Vert x_{n} - u\Vert , \frac{2\Vert \gamma f(u) - G(u)\Vert }{\tau - \alpha \gamma }\Big \}. \end{aligned}$$

Thus by induction, we’ve

$$\begin{aligned} \Vert x_{n} - u\Vert \le \max \Big \{\Vert x_{0} - u\Vert , \frac{2\Vert \gamma f(u) - G(u)\Vert }{\tau - \alpha \gamma }\Big \}, ~~\forall n\ge n_{0}. \end{aligned}$$

(3.22)

Hence, $\{x_{n}\}$ is bounded. As such $\{y_{n}\},$ $\{Gy_{n}\}$ and $\{f(x_{n})\}$ are also bounded. Next, we show that $\lim \nolimits _{n\rightarrow \infty }||x_{n+1}-x_{n}||=0$.

Let $z_{n}:=\frac{x_{n+1}-\beta _{n}x_{n}}{1-\beta _{n}}$, which implies

$$\begin{aligned} z_{n}= & {} \frac{\alpha _{n}\gamma f(x_{n})+(I-\alpha _{n}G)y_{n}-\beta _{n}x_{n}}{1-\beta _{n}}\\= & {} \frac{\alpha _{n}(\gamma f(x_{n})-Gy_{n})+y_{n}-\beta _{n}x_{n}}{1-\beta _{n}}\\= & {} \frac{\alpha _{n}(\gamma f(x_{n})-Gy_{n})+(1-\beta _{n})K_{n}x_{n}}{1-\beta _{n}}\\= & {} \frac{\alpha _{n}(\gamma f(x_{n})-Gy_{n})}{1-\beta _{n}}+K_{n}x_{n} \end{aligned}$$

then

$$\begin{aligned} z_{n+1}-z_{n}= & {} \frac{\alpha _{n+1}(\gamma f(x_{n+1})-Gy_{n+1})}{1-\beta _{n+1}}-\frac{\alpha _{n}(\gamma f(x_{n})-Gy_{n})}{1-\beta _{n}}\\&+K_{n+1}x_{n+1}-K_{n}x_{n}. \end{aligned}$$

Hence, by letting $M=\sup _{n}(||\gamma f(x_{n})||+||Gy_{n}||)$, we obtain

$$\begin{aligned} ||z_{n+1}-z_{n}||\le & {} \frac{\alpha _{n+1}}{1-\beta _{n+1}}(||\gamma f(x_{n+1})||+||Gy_{n+1}||)+\frac{\alpha _{n}}{1-\beta _{n}} (||\gamma f(x_{n})||+||Gy_{n}||)\\&+||K_{n+1}x_{n+1}-K_{n}x_{n}||\\\le & {} \Big (\frac{\alpha _{n+1}}{1-\beta _{n+1}}+\frac{\alpha _{n}}{1-\beta _{n}}\Big )M+||K_{n+1}x_{n+1}-K_{n+1}x_{n}||\\&+||K_{n+1}x_{n}-K_{n}x_{n}||\\\le & {} \Big (\frac{\alpha _{n+1}}{1-\beta _{n+1}}+\frac{\alpha _{n}}{1-\beta _{n}}\Big )M+(1+v_{n+1})||x_{n+1}-x_{n}||\\&+||K_{n+1}x_{n}-K_{n}x_{n}|| \end{aligned}$$

Therefore

$$\begin{aligned} ||z_{n+1}-z_{n}||-||x_{n+1}-x_{n}||\le & {} \Big (\frac{\alpha _{n+1}}{1-\beta _{n+1}}+\frac{\alpha _{n}}{1-\beta _{n}}\Big )M+v_{n+1}||x_{n+1}-x_{n}||\\&+||K_{n+1}x_{n}-K_{n}x_{n}|| \end{aligned}$$

which implies

$$\begin{aligned} \underset{n\rightarrow \infty }{\limsup }(||z_{n+1}-z_{n}||-||x_{n+1}-x_{n}||)\le 0. \end{aligned}$$

Hence, by Lemma 2.2, we obtain

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||z_{n}-x_{n}||=0 \end{aligned}$$

thus

$$\begin{aligned} ||x_{n+1}-x_{n}||=(1-\beta _{n})||z_{n}-x_{n}||\rightarrow 0\quad \text {as}\quad n\rightarrow \infty . \end{aligned}$$

From (3.19) it follows that,

$$\begin{aligned} \Vert x_{n+1} - y_{n}\Vert= & {} \Vert \alpha _{n}\gamma f(x_{n}) + (I - \alpha _{n}G)y_{n} - y_{n}\Vert \\\le & {} \Vert \alpha _{n}\gamma f(x_{n})\Vert + \Vert (I - \alpha _{n}G)y_{n} - y_{n}\Vert \\= & {} \alpha _{n}\Big \{\Vert \gamma f(x_{n})\Vert + \Vert G(y_{n})\Vert \Big \}, \end{aligned}$$

we have $\Vert x_{n+1} - y_{n}\Vert \rightarrow 0$ as $n \rightarrow \infty .$ As

$$\begin{aligned} \Vert x_{n} - y_{n}\Vert \le \Vert x_{n} - x_{n+1}\Vert + \Vert x_{n+1}- y_{n}\Vert , \end{aligned}$$

we also get

$$\begin{aligned} \Vert x_{n} - y_{n}\Vert \rightarrow 0~\mathrm{as}~ n \rightarrow \infty . \end{aligned}$$

(3.23)

On the other hand, we obtain

$$\begin{aligned} \Vert K_{n}x_{n} - x_{n}\Vert\le & {} \Vert x_{n} - y_{n}\Vert + \Vert y_{n}- K_{n}x_{n}\Vert \nonumber \\= & {} \Vert x_{n} - y_{n}\Vert + \Vert (\beta _{n}x_{n} + (1 - \beta _{n})K_{n}x_{n})- K_{n}x_{n}\Vert \nonumber \\= & {} \Vert x_{n} - y_{n}\Vert + \beta _{n}\Vert x_{n} - K_{n}x_{n}\Vert , \end{aligned}$$

(3.24)

which implies that $(1 - \beta _{n})\Vert K_{n}x_{n}- x_{n}\Vert \le \Vert x_{n} - y_{n}\Vert .$ From condition (C3) and (3.23) we obtain

$$\begin{aligned} \Vert K_{n}x_{n} - x_{n}\Vert \rightarrow 0~\mathrm{as}~ n\rightarrow \infty . \end{aligned}$$

(3.25)

Hence (b) is satisfied.

Next, we show that (c) is satisfied, that is $F(K_n)=\cap ^{N}_{i=1}F(T_i)$, but from (a), (b) above and (iii) of Lemma 3.1, (c) is satisfied.

Next, we show that

$$\begin{aligned} \limsup _{n \rightarrow \infty }\langle (\gamma f - G)p, j(x_{n} - p)\rangle \le 0, \end{aligned}$$

(3.26)

where p is the unique solution of the variational inequality (3.15). Let $z_m=\alpha _{m}\gamma f(z_m)+(1-\alpha _{m}G)y_m$, where $y_{m}=\beta _{m} z_{m}+(1-\beta _{m})K_{m}z_{m}$ and $\{\alpha _{m}\}$, $\{\beta _{m}\}$ satisfy the condition of Theorem 3.2. Then it follows from Theorem 3.2 that $p=\lim \nolimits _{m\rightarrow \infty }z_{m}$, so that

$$\begin{aligned} z_m-x_n=\alpha _{m}(\gamma f(z_m)-Gz_m)+\alpha _{m}(Gz_m-Gy_m)+y_m-x_n \end{aligned}$$

Hence

$$\begin{aligned} ||z_m-x_n||^2= & {} \alpha _{m}\langle \gamma f(z_m)-Gz_m,j(z_m-x_n)\rangle \\&+\alpha _{m}\langle Gz_m-Gy_m,j(z_m-x_n)\rangle \\&+\langle y_m-x_n,j(z_m-x_n)\rangle \\\le & {} \alpha _{m}\langle \gamma f(z_m)-Gz_m,j(z_m-x_n)\rangle \\&+\alpha _{m}||Gz_m-Gy_m||||z_m-x_n||\\&+||y_m-x_n||||z_m-x_n||\\\le & {} \alpha _{m}\langle \gamma f(z_m)-Gz_m,j(z_m-x_n)\rangle \\&+\alpha _{m}(1+\frac{1}{\mu })||z_m-y_m||||z_m-x_n||\\&+||y_m-x_n||||z_m-x_n||\\\le & {} \alpha _{m}\langle \gamma f(z_m)-Gz_m,j(z_m-x_n)\rangle \\&+\alpha _{m}(1+\frac{1}{\mu })(1-\beta _{m})||z_m-K_{m}z_m||||z_m-x_n||\\&+||x_n-z_m||^2+(1-\beta _{m})[v_{m}||z_m-x_n||\\&+||K_{m}x_n-x_n||]||z_m-x_n||. \end{aligned}$$

Therefore

$$\begin{aligned} \langle \gamma f(z_m)-Gz_m,j(x_n-z_m)\rangle\le & {} (1+\frac{1}{\mu })(1-\beta _{m})||z_m-K_{m}z_m||||z_m-x_n||\\&+(1-\beta _{m})[v_{m}/\alpha _{m}]||z_{m}-x_{n}||^{2}\\&+\frac{||K_{m}x_n-x_n||||z_m-x_n||}{\alpha _{m}}. \end{aligned}$$

Now, taking limit superior as $n\rightarrow \infty$ firstly, and then as $m\rightarrow \infty$, we have

$$\begin{aligned} \underset{m\rightarrow \infty }{\limsup }\underset{n\rightarrow \infty }{\limsup }\langle \gamma f(z_m)-Gz_m,j(x_n-z_m)\rangle \le 0 \end{aligned}$$

(3.27)

Moreover, we note that

$$\begin{aligned} \langle \gamma f(p)-Gp,j(x_n-p)\rangle= & {} \langle \gamma f(p)-Gp,j(x_n-p)\rangle -\langle \gamma f(p)-Gp,j(x_n-z_m)\rangle \nonumber \\&+\langle \gamma f(p)-Gp,j(x_n-z_m)\rangle -\langle \gamma f(p)-Gz_m,j(x_n-z_m)\rangle \nonumber \\&+\langle \gamma f(p)-Gz_m,j(x_n-z_m)\rangle -\langle \gamma f(z_m)-Gz_m,j(x_n-z_m)\rangle \nonumber \\&+\langle \gamma f(z_m)-Gz_m,j(x_n-z_m)\rangle \nonumber \\= & {} \langle \gamma f(p)-Gp,j(x_n-p)-j(x_n-z_m)\rangle \nonumber \\&+\langle Gz_m- Gp,j(x_n-z_m)\rangle \nonumber \\&+\langle \gamma f(z_m)-\gamma f(p),j(x_n-z_m)\rangle \nonumber \\&+\langle \gamma f(z_m)-Gz_m,j(x_n-z_m)\rangle \end{aligned}$$

(3.28)

Taking limit superior as $n\rightarrow \infty$ in (3.28), we have

$$\begin{aligned} \underset{n\rightarrow \infty }{\limsup }\langle \gamma f(p)-Gp,j(x_n-p)\rangle\le & {} \underset{n\rightarrow \infty }{\limsup }\langle \gamma f(p)-Gp,j(x_n-p)-j(x_n-z_m)\rangle \\&+|| Gz_m- Gp||\underset{n\rightarrow \infty }{\limsup }||x_n-z_m||\\&+||\gamma f(z_m)-\gamma f(p)||\underset{n\rightarrow \infty }{\limsup }||x_n-z_m||\\&+\underset{n\rightarrow \infty }{\limsup }\langle \gamma f(z_m)-Gz_m,j(x_n-z_m)\rangle \\\le & {} \underset{n\rightarrow \infty }{\limsup }\langle \gamma f(p)-Gp,j(x_n-p)-j(x_n-z_m)\rangle \\&+\Big ((1+\frac{1}{\mu })+\alpha \gamma \Big )\Vert z_{m}-p\Vert \underset{n\rightarrow \infty }{\limsup }||x_n-z_m||\\&+\underset{n\rightarrow \infty }{\limsup }\langle \gamma f(z_m)-Gz_m,j(x_n-z_m)\rangle \end{aligned}$$

By Theorem 3.2, $z_m\rightarrow p\in F$ as $m\rightarrow \infty$.

Since j is norm-to-$weak^{*}$ uniformly continuous on bounded subset of E, we obtain

$$\begin{aligned} \underset{m\rightarrow \infty }{\limsup }\underset{n\rightarrow \infty }{\limsup }\langle \gamma f(p)-Gp,j(x_n-p)-j(x_n-z_m)\rangle =0, \end{aligned}$$

therefore, from (3.27) we obtain

$$\begin{aligned} \underset{n\rightarrow \infty }{\limsup }\langle \gamma f(p)-Gp,j(x_n-p)\rangle \le 0 \end{aligned}$$

Finally, we show that (d) is satisfied, since ${\lim \nolimits _{n\rightarrow \infty }}(v_n/\alpha _n)=0$, if we denote by $\sigma _n$ the value of $2v_n+v_{n}^{2}$ then, clearly $\lim \nolimits _{n\rightarrow \infty }(\sigma _n/\alpha _n)=0.$ Let $N_0\in \mathbb {N}$ be large enough such that $(1-\alpha _n\tau )(\sigma _n/\alpha _n)<(\tau -2\gamma \alpha )/2$, for all $n\ge N_0.$ Then, using the recursion formula (3.19) and for all $n\ge N_0$, we obtain.

$$\begin{aligned} ||x_{n+1}-p||^{2}= & {} ||\alpha _{n}\gamma f(x_{n})-\alpha _{n}G(p)+(1-\alpha _n G)y_n-(1-\alpha _nG)p||^{2}\\\le & {} ||(1-\alpha _nG)y_n-(1-\alpha _nG)p||^{2}+2\alpha _n\langle \gamma f(x_{n})-G(p),j(x_{n+1}-p)\rangle \\\le & {} (1-\alpha _n\tau )^{2}||y_n-p||^2+2\alpha _n\langle \gamma f(x_n)-\gamma f(p),j(x_{n+1}-p)\rangle \\&+2\alpha _n\langle \gamma f(p)-G(p),j(x_{n+1}-p)\rangle \\\le & {} (1-\alpha _n\tau )||y_n-p||^2+2\alpha _n\gamma \alpha ||x_n-p||||x_{n+1}-p||\\&+2\alpha _n\langle \gamma f(p)-G(p),j(x_{n+1}-p)\rangle \\\le & {} (1-\alpha _n\tau )[\beta _{n}+(1-\beta _{n})(1+v_{n})^{2}]||x_n-p||^2\\&+\alpha _n\gamma \alpha ||x_n-p||^2+\alpha _n\gamma \alpha ||x_{n+1}-p||^2\\&+2\alpha _n\langle \gamma f(p)-G(p),j(x_{n+1}-p)\rangle \\\le & {} (1-\alpha _n\tau )[1+\sigma _{n}]||x_n-p||^2\\&+\alpha _n\gamma \alpha ||x_n-p||^2+\alpha _n\gamma \alpha ||x_{n+1}-p||^2\\&+2\alpha _n\langle \gamma f(p)-G(p),j(x_{n+1}-p)\rangle \\= & {} \Big (1-\alpha _n[(\tau -\alpha \gamma )-(1-\alpha _n\tau )(\sigma _{n}/\alpha _{n})]\Big )||x_n-p||^{2}\\&+ \alpha _n\gamma \alpha ||x_{n+1}-p||^2 +2\alpha _n\langle \gamma f(p)-G(p),j(x_{n+1}-p)\rangle . \end{aligned}$$

Therefore

$$\begin{aligned} ||x_{n+1}-p||^{2}\le & {} \Big (1-\alpha _n[\frac{(\tau -2\alpha \gamma )-(1-\alpha _{n}\tau )(\sigma _{n}/\alpha _{n})}{1-\alpha _n\alpha \gamma }]\Big )||x_{n}-p||^{2}\\&+\frac{2\alpha _n[(\tau -2\alpha \gamma )-(1-\alpha _{n}\tau )(\sigma _{n}/\alpha _{n})]\langle \gamma f(p)-G(p),j(x_{n+1}-p)\rangle }{(1-\alpha _{n}\gamma \alpha )[(\tau -2\alpha \gamma )-(1-\alpha _{n}\tau )(\sigma _{n}/\alpha _{n})]}. \end{aligned}$$

Observe that $\sum \alpha _{n}[(\tau -2\alpha \gamma )-(1-\alpha _{n}\tau )(\sigma _{n}/\alpha _{n})]=\infty$ and

$$\begin{aligned} \limsup \Big (\frac{2\alpha _n\langle \gamma f(p)-G(p),j(x_{n+1}-p)\rangle }{(1-\alpha _{n}\gamma \alpha )[(\tau -2\alpha \gamma )-(1-\alpha _{n}\tau )(\sigma _{n}/\alpha _{n})]} \Big )\le 0 \end{aligned}$$

Consequently, applying Lemma 2.4, we conclude that $x_{n} \rightarrow p~ \mathrm{as}~n\rightarrow \infty .$

Corollary 3.4

Let E be a real uniformly convex Banach space whose duality mapping J is weakly sequentially continuous. Let $G : H \rightarrow H,$ $f : E \rightarrow E,$ $\{T_{i}\}^{N}_{i=1}$ F, $\{\alpha _{n}\}^{\infty }_{n = 1}$, $\{\beta _{n}\}^{\infty }_{n =1}$ and $\{x_{n}\}^{\infty }_{n = 1}$ be as in Theorem (3.3), then $\{x_{n}\}^{\infty }_{n = 1}$ converges strongly to $p \in F$, which is also the unique solution of the variational inequality

$$\begin{aligned} \langle \gamma f(p) - Gp, j(q - p)\rangle \le 0,\quad \forall q \in F \end{aligned}$$

Corollary 3.5

Let H be a real Hilbert space, $\{z_{t}\}_{t\in (0,1)},$ be as in Theorem 3.2. Then $\{z_{t}\}$ converges strongly to a common fixed point of the family $\{T_{i}\}_{i=1}^{N}$ say p which is a unique solution of the variational inequality

$$\begin{aligned} \langle (G-\gamma f)p,q-p\rangle \ge 0,\quad \forall q\in F. \end{aligned}$$

Corollary 3.6

Let H be a real Hilbert space and let C a nonempty closed convex subset of H. Let $G : H \rightarrow H,$ $f : E \rightarrow E,$ $\{T_{i}\}^{N}_{i=1}$ F, $\{\alpha _{n}\}^{\infty }_{n = 1}$, $\{\beta _{n}\}^{\infty }_{n =1}$ and $\{x_{n}\}^{\infty }_{n = 1}$ be as in Theorem (3.3), then $\{x_{n}\}^{\infty }_{n = 1}$ converges strongly to $p \in F$, which is also the unique solution of the variational inequality

$$\begin{aligned} \langle \gamma f(p) - Gp, q - p\rangle \le 0, \quad \forall q \in F \end{aligned}$$