Modified general iterative algorithm for an infinite family of nonexpansive mappings in Banach spaces

Abstract

In this paper we introduce an iterative method for finding a common fixed point of an infinite family of nonexpansive mappings in q-uniformly real smooth Banach space which is also uniformly convex. We proved strong convergence of the proposed iterative algorithms to the unique solution of a variational inequality problem.

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, $x\in E,$

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

where $\langle .,.\rangle$ 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 $j_{\varphi }x$.

If however $\varphi (t)=t^{q-1}$, for some $q>1$, then $J_\varphi$ is still called the generalized duality mapping and is denoted by $J_{q}$ (see, for example [1, 5]).

Let $S(E):= \{x \in E:\Vert x\Vert = 1\}$ be the unit sphere of E. Then space E is said to have Gâteaux 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 norm, then $j:E\rightarrow E^*$ is uniformly continuous on bounded subsets of E to the weak$^*$ topology of $E^*$.

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

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

(1.2)

where D(G) denote the domain of G. G is called $\eta -strongly~accretive$ if for all $x,y\in D(G)$, there exists $j_q(x-y)\in J_q(x-y)$ and $\eta \in (0,1)$ such that

$$\begin{aligned} \langle Gx-Gy,j_q(x-y)\rangle \ge \eta ||x-y||^q, \end{aligned}$$

(1.3)

Let K be a nonempty, closed and convex subset of E and $G:K\rightarrow E$ be a nonlinear mapping. The variational inequality problem is to:

$$\begin{aligned} \text {find}~u\in K~\text {such that}~\langle Gu,j_q(v-u)\rangle \ge 0,~\forall v\in K, \end{aligned}$$

for some $j_q(v-u)\in J_q(v-u).$ The set of solution of variational inequality problem is denoted by VI(K, G). If $E:=H$, a real Hilbert space, the variational inequality problem reduces to:

$$\begin{aligned} \text {find}~u\in K~\text {such that}~\langle Gu,v-u\rangle \ge 0,~\forall v\in K, \end{aligned}$$

which was introduced and studied by Stampacchia [16].

Variational inequality theory has emerged as an important tool in studying a wide class of related problems in Mathematical, Physical, regional, engineering and nonlinear optimization sciences (see, for instance, [8, 9, 11, 15, 24–26]).

A mapping $T:E\rightarrow E$ is $L-Lipschitian$ if for some $L>0,||Tx-Ty||\le L||x-y||~\forall ~x,y\in E$. If $L\in [0,1)$, then T is called contraction mapping, but if $L\le 1$, then T is called nonexpansive mapping. A point $x\in E$ is called a fixed point of T if $Tx=x$. The set of fixed points of T is denoted by $F(T):=\{x\in E:Tx=x\}$. In Hilbert spaces H, accretive operators are called monotone where inequality (1.2) and (1.3) hold with $j_q$ replaced by the identity map on H.

In 2000, Moudafi [14] introduced the viscosity approximation method for nonexpansive mappings. Let f be a contraction on H, starting with an arbitrary $x_0\in H,$ define a sequence $\{x_n\}$ recursively by

$$\begin{aligned} x_{n+1}=\alpha _n f(x_n)+(1-\alpha _n)Tx_n,~~n\ge 0, \end{aligned}$$

(1.4)

where $\{\alpha _n\}$ is a sequence in (0,1). Xu [21] proved that under certain appropriate conditions on $\{\alpha _n\}$, the sequence $\{x_n\}$ generated by (1.4) strongly converges to the unique solution $x^*$ in F of the variational inequality

$$\begin{aligned} \langle (I-f)x^*,x-x^*\rangle \ge 0,~\text {for}~~x\in F. \end{aligned}$$

In [19], he proved, under some conditions on the real sequence $\{\alpha _n\}$, that the sequence $\{x_n\}$ defined by $x_0\in H$ chosen arbitrary,

$$\begin{aligned} x_{n+1}=\alpha _n b+(1-\alpha _n A)Tx_n,~~n\ge 0, \end{aligned}$$

(1.5)

converges strongly to $x^*\in F$ which is the unique solution of the minimization problem

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

where A is a strongly positive bounded linear operator. That is, there is a constant $\bar{\gamma }>0$ with the property

$$\begin{aligned} \langle Ax,x\rangle \ge \bar{\gamma }||x||^2,~~\forall x\in H. \end{aligned}$$

Combining the iterative method (1.4) and (1.5), Marino and Xu [13] consider the following general iterative method:

$$\begin{aligned} x_{n+1}=\alpha _n f(x_n)+(1-\alpha _n A)Tx_n,~~n\ge 0, \end{aligned}$$

(1.6)

they proved that if the sequence $\{\alpha _n\}$ of parameters satisfies appropriate conditions, then the sequence $\{x_n\}$ generated by (1.6) converges strongly to $x^*\in F$ which solves the variational inequality

$$\begin{aligned} \langle (\gamma f-A)x^*,x-x^*\rangle \le 0~~x\in F, \end{aligned}$$

which is the optimality condition for the minimization problem

$$\begin{aligned} \underset{x\in F}{\min }\frac{1}{2}\langle Ax,x\rangle -h(x), \end{aligned}$$

where h is a potential function for $\gamma f$ (i.e. $h'(x)=\gamma f(x)$ for $x\in H$).

On the other hand, Yamada [24] in 2001 introduced the following hybrid iterative method:

$$\begin{aligned} x_{n+1}=Tx_n-\lambda _n\mu GTx_n,~~n\ge 0, \end{aligned}$$

(1.7)

where G is a $\kappa$-Lipschitzian and $\eta$-strongly monotone operator with $\kappa>0,\eta >0$ and $0<\mu <2\eta /\kappa ^2$. Under some appropriate conditions, he proved that the sequence $\{x_n\}$ generated by (1.7) converges strongly to the unique solution of the variational inequality

$$\begin{aligned} \langle Gx^*,x-x^*\rangle \ge 0,~~\forall x\in F. \end{aligned}$$

Recently, combining (1.6) and (1.7), Tian [18] considered the following general iterative method:

$$\begin{aligned} x_{n+1}=\alpha _n\gamma f(x_n)+(I-\alpha _n\mu G)T(x_n), \end{aligned}$$

(1.8)

and proved that the sequence $\{x_n\}$ generated by (1.8) converges strongly to the unique solution $x^*\in F$ of the variational inequality

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

Most recently, Ali et al [4], extended the result of Tian [18] to q-uniformly smooth Banach space whose duality mapping is weakly sequentially continuous. Under some assumptions on $\{\alpha _n\},\gamma ,\mu$ and G being $\eta$-accretive mapping in (1.8), they proved that the sequence $\{x_n\}$ generated by (1.8) converges strongly to the unique solution $x^*\in F$ of the variational inequality

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

Let $\{T_i\}$ be countable family of nonexpansive mapping. We denote by a set $\textit{N}_{I}:=\{i\in \mathbb {N}:T_i\ne I\}$ (I being the identity mapping on E). Maingé [12] studied the Halpern-type scheme for approximation of a common fixed point of countable infinite family of nonexpansive mappings in a real Hilbert space. He proved the following theorems.

Theorem 1.1

(Maingé [12]) Let K be a nonempty closed convex subset of a real Hilbert space H. Let $\{T_i\}$ be countable family of nonexpansive self-mappings of K, $\{t_n\}$ and $\{\sigma _{i,t_n}\}$ be sequences in (0,1) satisfying the following conditions: (i) $\lim t_n=0,$ (ii) $\sum _{i\ge 1}\sigma _{i,t_n}=1-t_n$, (iii) $\forall i\in \textit{N}_{I},$  $\underset{n\rightarrow \infty }{\lim }\frac{t_n}{\sigma _{i,t_n}}=0.$ Define a fixed point sequence $\{x_{t_n}\}$ by

$$\begin{aligned} x_{t_n}=t_n Cx_{t_n}+\sum _{i\ge 1}\sigma _{i,t_{n}}T_ix_{t_n},~~n\ge 1, \end{aligned}$$

(1.9)

where $C:K\rightarrow K$ is a strict contraction. Assume $F:=\cap ^{\infty }_{i=1}F(T_i)\ne \emptyset$, the $\{x_{t_n}\}$ converges strongly to a unique fixed point of the contraction $P_{F}\circ C$, where $P_{F}$ is a metric projection from H onto F.

Theorem 1.2

(Maingé [12]) Let K be a nonempty closed convex subset of a real Hilbert space H. Let $\{T_i\}$ be countable family of nonexpansive self-mappings of K, $\{\alpha _n\}$ and $\{\sigma _{i,n}\}$ be sequences in (0,1) satisfying the following conditions:

• (i) $\sum \alpha _{n}=\infty ,$ $\sum _{i\ge 1}\sigma _{i,n}=1-\alpha _n,$

• (ii)
  $$\begin{aligned} \left\{ \begin{array}{ll} \frac{1}{\sigma _{i,n}}\Big \vert 1-\frac{\alpha _{n-1}}{\alpha _n}\Big \vert \rightarrow 0,~~\text {or}~~\sum _{n}\frac{1}{\sigma _{i,n}}|\alpha _{n-1}-\alpha _{n}|<\infty &{} \text { }\\ \frac{1}{\alpha _n}\Big \vert \frac{1}{\sigma _{i,n}}-\frac{1}{\sigma _{i,n-1}}\Big \vert \rightarrow 0,~~\text {or}~~\sum _n\Big \vert \frac{1}{\sigma _{i,n}}-\frac{1}{\sigma _{i,n-1}}\Big \vert<\infty &{} \text { }\\ \frac{1}{\sigma _{i,n}\alpha _n}\sum _{k\ge 0}|\sigma _{k,n}-\sigma _{k,n-1}|\rightarrow 0,~~\text {or}~~ \frac{1}{\sigma _{i,n}}\sum _{k\ge 0}|\sigma _{k,n}-\sigma _{k,n-1}|<\infty . \end{array} \right. \end{aligned}$$

• (iii) $\forall i\in \textit{N}_{I},$ $\underset{n\rightarrow \infty }{\lim }\frac{\alpha _n}{\sigma _{i,n}}=0.$

Then, the sequence $\{x_n\}$ define iteratively by $x_1\in K$,

$$\begin{aligned} x_{n+1}=\alpha _n Cx_n+\sum _{i\ge 1}\sigma _{i,n}T_ix_n,~~n\ge 1, \end{aligned}$$

(1.10)

where $C:K\rightarrow K$ is a strict contraction. Assume $F:=\cap ^{\infty }_{i=1}F(T_i)\ne \emptyset$, the $\{x_{n}\}$ converges strongly to a unique fixed point of the contraction $P_{F}\circ C$, where $P_{F}$ is a metric projection from H onto F.

Motivated by the results above, we introduce an iterative method for finding a common fixed point of an infinite family of nonexpansive mappings in q-uniformly real smooth Banach space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality problem.

2 Preliminaries

Let K be a nonempty, closed, convex and bounded subset of a Banach space E and let the diameter of K be defined by $d(K):= \sup \{\Vert x-y\Vert :x , y \in K \}$.   For each $x \in K$, let $r(x,K):=\sup \{\Vert x-y\Vert : y \in K \}$   and let $r(K):=\inf \{r(x,K): x \in K \}$  denote the Chebyshev radius of K relative to itself. The normal structure coefficient N(E) of E (introduced in 1980 by Bynum [3], see also Lim [10] and the references contained therein) is defined by N(E):=$\inf${$\frac{d(K)}{r(K)}$: K is a closed convex and bounded subset of E with $d(K)>0$}. A Banach space E such that $N(E)>1$ is said to have uniform normal structure . It is known that every Banach space with a uniform normal structure is reflexive, and that all uniformly convex and uniformly smooth Banach spaces have uniform normal structure (see e.g., [5, 23]).

Let E be a normed space with dimE $\ge 2$. The modulus of smoothness of E is the function $\rho _{E}:[0,\infty )\rightarrow [0,\infty )$ defined by

$$\begin{aligned} \rho _{E}(\tau ):=\sup \left\{ \frac{||x+y||+||x-y||}{2}-1:||x||=1;||y||=\tau \right\} . \end{aligned}$$

The space E is called uniformly smooth if and only if $\lim _{t\rightarrow 0^{+}}\frac{\rho _{E}(t)}{t}=0.$ For some positive constant $q\in E$ is called $q-uniformly~smooth$ if there exists a constant $c>0$ such that $\rho _{E}(t)\le ct^{q},t>0.$ It is well known that if E is smooth then the duality mapping is singled-valued, and if E is uniformly smooth then the duality mapping is norm-to-norm uniformly continuous on bounded subset of E.

Lemma 2.1

Let E be a real normed space. Then

$$\begin{aligned} ||x+y||^{2}\le ||x||^{2}+2\langle y,j(x+y)\rangle , \end{aligned}$$

for all $x,y\in E$ and for all $j(x+y)\in J(x+y).$

Lemma 2.2

(Xu, [22]) Let E be a real q-uniformly smooth Banach space for some $q>1$, then there exists some positive constant $d_{q}$ such that

$$\begin{aligned} ||x+y||^{q}\le ||x||^{q}+q\langle y,j_{q}(x)\rangle +d_{q}||y||^{q}~\forall x,y\in E~\text {and}~j_{q}\in J_{q}(x). \end{aligned}$$

Lemma 2.3

(Xu, [21]) 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 2.4

(Suzuki [17]) Let $\{x_{n}\}$ and $\{y_{n}\}$ be bounded sequence 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 ${\limsup }_{n\rightarrow \infty }(||y_{n+1}-y_{n}||-||x_{n+1}-x_{n}||)\le 0.$ Then, $\underset{n\rightarrow \infty }{\lim }||y_{n}-x_{n}||=0.$

Lemma 2.5

(See Lemma 2.1 of Ali [2]) Let E be a real smooth and uniformly convex Banach space and let $r>0$. Then there exists a strictly increasing, continuous and convex function $g:[0,2r]\rightarrow R$ such that $g(0)=0$ and $g(||x-y)\le ||x||^2-2\langle x,jy\rangle +||y||^2$ for all $B_{r}=\{x\in E:||x||\le r\}.$

Lemma 2.6

Let E be a real Banach space, $f:E\rightarrow E$ be contraction mapping with a coefficient $0<\beta<$ and let $G:E\rightarrow E$ be a $\kappa -$Lipschitzian and $\eta -$strongly accretive operator with $\kappa >0,\eta \in (0,1)$. Then for $\gamma \in (0,\frac{\mu \eta }{\beta })$,

$$\begin{aligned} \langle (\mu G-\gamma f)x-(\mu G-\gamma f)y,j(x-y)\rangle \ge (\mu \eta -\gamma \beta )||x-y||^2,~\forall x,y\in E. \end{aligned}$$

That is, $(\mu G-\gamma f)$ is strongly accretive with coefficient $(\mu \eta -\gamma \beta )$.

Let $\mu$ be a linear continuous functional on $l^{\infty }$ and let $a=(a_1,a_2,\cdots )\in l^{\infty }.$ We will sometimes write $\mu _{n}(a_n)$ in place of the value $\mu (a)$. A linear continuous functional $\mu$ such that $||\mu ||=1=\mu (1)$ and $\mu _{n}(a_n)=\mu _{n}(a_{n+1})$ for every $a=(a_1,a_2,\cdots )\in l^{\infty }$ is called a Banach limit. It is known that if $\mu$ is a Banach limit, then

$$\begin{aligned} \underset{n\rightarrow \infty }{\liminf }a_{n}\le \mu (a_n)\le \underset{n\rightarrow \infty }{\limsup }a_{n} \end{aligned}$$

for every $a=(a_1,a_2,\ldots )\in l^{\infty }$ (see, for example, [5, 6])

3 Main results

In the sequel we assume for each $\alpha \in (0,1)$, the sequence $\{\sigma _{i,\alpha }\}$ satisfies $\sum _{i\ge 1}\sigma _{i,\alpha }=1-\alpha$ and for the sequence $\{\alpha _{n}\}\subset (0,1),\{\sigma _{i,n}\}=1-\alpha _{n}$.

Lemma 3.1

Let E be a $q-$uniformly smooth real Banach space with constant $d_q, q>1$. Let $f:E\rightarrow E$ be a $\beta -$contraction mapping with a coefficient $\beta \in (0,1)$. Let $T_{i}:E\rightarrow E~i\in N,$ be a family of nonexpansive maps such that $F:=\cap ^{\infty }_{i=1}F(T_i)\ne \emptyset$ and $G:E\rightarrow E$ be an $\eta -$strongly accretive mapping which is also $\kappa -$Lipschizian. Let $\mu \in \Big (0,\min \Big \{1,(\frac{q\eta }{d_{q}\kappa ^{q}})^{\frac{1}{q-1}}\Big \}\Big )$ and $\tau :=\mu \Big (\eta -\frac{\mu ^{q-1}d_{q}\kappa ^{q}}{q}\Big )$. For each $t,\alpha \in (0,1)$ with $\alpha < t$ and $\beta \in (0,\frac{\tau }{\gamma })$. Assume that $S:=\alpha u+(1-\delta )(1-\alpha )I+\delta \sum _{i\ge 1}\sigma _{i,\alpha }T_i,$ where $\delta$ is some fixed number in (0, 1) and $u\in E$, $\alpha ,\sigma _{i,t}$ are in (0, 1). Define the following mapping $W_{t}$ on E by

$$\begin{aligned} W_tx:=t\gamma f(x)+(I-\mu t G)Sx. \end{aligned}$$

where t is in (0,1). Then $W_t$ is a strict contraction mapping. Furthermore, for any $x,y\in E,$

$$\begin{aligned} ||W_{t}x-W_{t}y||\le [1-t(\tau -\beta \gamma )]||x-y|| \end{aligned}$$

Proof

Observe that, for any $x,y\in E$

$$\begin{aligned} ||Sx-Sy||\le & {} (1-\delta )(1-\alpha )||x-y||+\delta \sum _{i\ge 1}\sigma _{i,\alpha }||T_{i}x-T_{i}y||\nonumber \\\le & {} (1-\delta )(1-\alpha )||x-y||+\delta (1-\alpha )||x-y||\nonumber \\\le & {} ||x-y||. \end{aligned}$$

(3.1)

Without loss of generality, assume $\eta <\frac{1}{q}$. Then, as $\mu <(\frac{q\eta }{d_{q}\kappa ^{q}})^{\frac{1}{q-1}}$, we have $0<q\eta -\mu ^{q-1}d_{q}\kappa ^{q}$. Furthermore, from $\eta <\frac{1}{q}$ we have $q\eta -\mu ^{q-1}d_{q}\kappa ^{q}<1$ so that $0<q\eta -\mu ^{q-1}d_{q}\kappa ^{q}<1$. Also as $\mu <1$ and $t\in (0,1)$, we obtain that $0<t\mu (q\eta -\mu ^{q-1}d_{q}\kappa ^{q})<1$.

For each $t,\alpha \in (0,1)$, then for any $x,y\in E$, define $K_{t}x=(1-t\mu G)Sx$, then from (3.1), we obtain

$$\begin{aligned} ||K_tx-K_ty||^{q}= & {} ||(1-t\mu G)Sx-(1-t\mu G)Sy||^q\\= & {} ||(Sx-Sy)-t\mu (G(Sx)-G(Sy))||^q\\\le & {} ||G(Sx)-G(Sy)||^q-qt\mu \langle G(Sx)-G(Sy),j_{q}(Sx-Sy)\rangle \\&+t^q\mu ^qd_q||Sx-Sy||^q\\\le & {} [1-t\mu (q\eta -t^{q-1}\mu ^{q-1}\kappa ^qd_q)]||x-y||^q\\\le & {} [1-qt\mu (\eta -\frac{\mu ^{q-1}\kappa ^qd_q}{q})]||x-y||^q\\\le & {} [1-t\mu (\eta -\frac{\mu ^{q-1}\kappa ^qd_q}{q})]^q||x-y||^q\\= & {} (1-t\tau )^q||x-y||^q, \end{aligned}$$

therefore

$$\begin{aligned} ||W_tx-W_ty||= & {} ||t\gamma (f(x)-f(y))+(K_t(Sx)-K_t(Sy))||\\\le & {} t\gamma ||f(x)-f(y)||+||K_t(Sx)-K_t(S_{t}y)||\\\le & {} t\beta \gamma ||x-y||+(1-t\tau )||x-y||\\= & {} [1-t(\tau -\beta \gamma )]||x-y||. \end{aligned}$$

Hence

$$\begin{aligned} ||W_tx-W_ty||\le [1-t(\tau -\beta \gamma )]||x-y||, \end{aligned}$$

(3.2)

which implies that $W_t$ is a strict contraction, by Banach contraction mapping principle, there exists a unique fixed point $x_t$ of $W_t$ in E. That is,

$$\begin{aligned} x_t=t\gamma f(x_t)+(1-t\mu G)Sx_t. \end{aligned}$$

(3.3)

$\square$

Theorem 3.2

Let E be a q-uniformly real smooth Banach space which is also uniformly convex. Let $T_{i},f,G,\mu ,\tau ,\beta ,\gamma ,S$ and F be as in Lemma 3.1. Let $\{t_n\},\{\alpha _n\}$ be sequences in (0,1), such that $\underset{n\rightarrow \infty }{\lim }t_n=0$ and $\underset{n\rightarrow \infty }{\lim }\frac{\alpha _n}{t_n}=0$. Let $\{x_{t_n}\}$ be a sequence satisfying (3.3), then

• (i) $\{x_{t_n}\}$ is bounded for $t_n\in (0,\frac{1}{\tau })$.

• (ii) $\underset{n\rightarrow \infty }{\lim }||x_{t_n}-T_{i}x_{t_n}||=0, \quad \forall i\in \mathbb {N}.$

• (iii) then $\{x_{t_0}\}$ converges strongly to a common fixed point p in F which is a unique solution of the variational inequality

  $$\begin{aligned} \langle (\mu G-\gamma f)p,j(p-x)\rangle \le 0,~~\forall x\in F. \end{aligned}$$

  (3.4)

Proof

Let $p\in F$ and $\alpha _n\le t_n$, then

$$\begin{aligned} ||x_{t_n}-p||^2= & {} \langle t_n\gamma f(x_{t_n})+(I-\mu t_n G)Sx_{t_n}-p,j(x_{t_n}-p)\rangle \\= & {} t_n\langle \gamma f(p)-\mu G(p),j(x_{t_n}-p)\rangle +t_n\gamma \langle f(x_{t_n})-f(p),j(x_{t_n}-p)\rangle \\&+\langle (I-t_n\mu G)Sx_{t_n}-(I-t_n\mu G)p,j(x_{t_n}-p)\rangle \\\le & {} t_n\langle \gamma f(p)-\mu G(p),j(x_{t_n}-p)\rangle +\beta \gamma t_n||x_{t_n}-p||^2\\&+(1-\tau t_n)||Sx_{t_n}-p||||x_{t_n}-p||\\\le & {} t_n\langle \gamma f(p)-\mu G(p),j(x_{t_n}-p)\rangle +\beta \gamma t_n||x_{t_n}-p||^2\\&+(1-\tau t_n)[\alpha _n||u-p||+(1-\alpha _n)||x_{t_n}-p||]||x_{t_n}-p||\\\le & {} t_n\langle \gamma f(p)-\mu G(p),j(x_{t_n}-p)\rangle +[1-t_n(\tau -\gamma \beta )||x_{t_n}-p||^2\\&+(1-\tau t_n)\alpha _n||u-p||||x_{t_n}-p||. \end{aligned}$$

Since $(1-\tau t_n)(\alpha _n/t_n)\rightarrow 0$ as $n\rightarrow \infty ,$ then there exists $n_0\in \mathbb {N}$ such that $(1-\tau t_{n})(\alpha _n/t_n)<(\tau -\gamma \beta )/2$ for $n\ge n_0.$ Furthermore

$$\begin{aligned} ||x_{t_n}-p||^2\le & {} \frac{\langle (\gamma f-\mu G)p,j(x_{t_n}-p)\rangle }{\tau -\gamma \beta }\nonumber \\&+\frac{(1-\tau t_n)\alpha _n}{t_n}\times \frac{||u-p||||x_{t_n}-p||}{\tau -\gamma \beta } \end{aligned}$$

(3.5)

for all $n\ge n_0$. That is, $||x_{t_n}-p||\le (\frac{||\gamma f(p)-\mu G(p)||}{\tau -\gamma \beta }+\frac{||u-p||}{2})$ for all $n\ge n_0$. Thus $\{x_{t_n}\}$ is bounded, so are $\{f(x_{t_n})\},\{G(x_{t_n})\},\{T_i(x_{t_n})\}$ and $\{G(T_ix_{t_n})\}$.

(ii) From (3.3), we have

$$\begin{aligned} ||x_{t_n}-Sx_{t_n}||=t_{n}||\gamma f(x_{t_n})-\mu G(Sx_{t_n})||\rightarrow 0~~\text {as}~~n\rightarrow \infty . \end{aligned}$$

(3.6)

Using Lemma 2.5, we have the following estimate

$$\begin{aligned} g(||T_ix_{t_n}-x_{t_n}||)= & {} g[||(p-T_ix_{t_n})-(p-x_{t_n})||]\nonumber \\\le & {} ||p-T_{i}x_{t_n}||^2-2\langle p-T_{i}x_{t_n},j(p-x_{t_n})\rangle +||p-x_{t_n}||^2 \nonumber \\\le & {} ||p-T_{i}x_{t_n}||^2-2\langle p-x_{t_n}+x_{t_n}-T_{i}x_{t_n},j(p-x_{t_n})\rangle +||p-x_{t_n}||^2\nonumber \\\le & {} 2||p-T_{i}x_{t_n}||^2-2\langle p-T_{i}x_{t_n},j(p-x_{t_n})\rangle +2\langle x_{t_n}-T_{i}x_{t_n},j(x_{t_n}-p)\rangle \nonumber \\\le & {} 2\langle x_{t_n}-T_{i}x_{t_n},j(x_{t_n}-p)\rangle \end{aligned}$$

(3.7)

Therefore

$$\begin{aligned} \frac{\delta }{2}\sum _{i\ge 1}\sigma _{i,n}g(||T_{i}x_{t_n}-x_{t_n}||)\le & {} \langle \delta (1-\alpha _n)x_{t_n}-\delta \sum _{i\ge 1}\sigma _{i,n}T_{i}x_{t_n},j(x_{t_n}-p)\rangle \\= & {} \langle \alpha _n(u-x_{t_n})+x_{t_n}-Sx_{t_n},j(x_{t_n}-p)\rangle \\\le & {} [\alpha _n||u-x_{t_n}||+||x_{t_n}-Sx_{t_n}||]||x_{t_n}-p||. \end{aligned}$$

Then we immediately obtain $\underset{n\rightarrow \infty }{\lim }\sum _{i\ge 1}\sigma _{i,n}g(||T_{i}x_{t_n}-x_{t_n}||)=0,$ it follows that $\underset{n\rightarrow \infty }{\lim }g(||T_{i}x_{t_n}-x_{t_n}||)=0~~\forall i\in \mathbb {N}.$ By the property of g we have that

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||T_{i}x_{t_n}-x_{t_n}||=0~~\forall i\in \mathbb {N}. \end{aligned}$$

(3.8)

(iii) By Lemma 2.6, $(\mu G-\gamma f)$ is strongly accretive, so the variational inequality (3.4) has a unique solution in F. Below we use $q\in F$ to denote the unique solution of (3.4). Next, we prove that $x_t\rightarrow q~(t\rightarrow 0).$

Let $\{t_{n}\}$ be a sequence in (0, 1) such that $\{x_{t_{n}}\}$ satisfies (3.3). By writing $\{x_{n}\}$ instead of $\{x_{t_{n}}\},$ define a map $\phi :E\rightarrow \mathbb {R}$ by

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

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 $\phi (q)=\underset{u \in E}{\min }\phi (u)$. Hence, the set

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

Since $\underset{n\rightarrow \infty }{\lim }||x_{n}-T_{i}x_{n}||=0,$ $\underset{n\rightarrow \infty }{\lim }||x_{n}-T_{i}^{m}x_{n}||=0,~ \mathrm{for~ any}~m\ge 1$ and $i\in \mathbb {N}$ by induction. Now let $v\in K^*$, we have for any $i\in \mathbb {N}$

$$\begin{aligned} \phi (T_{i}v)= & {} \mu _{n}||x_{n}-T_{i}v||^{2}=\mu _{n}||x_{n}-T_{i}x_{n} +T_{i}x_{n}-T_{i}v||^{2}\\\le & {} \mu _{n}||x_{n}-v||^2=\phi (v), \end{aligned}$$

and hence $T_{i}v\in K^{*}$.

Now let $z\in F,$ then $z=T_{i}z,$ for any $i\in \mathbb {N}$. 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, for any $i\in \mathbb {N}$

$$\begin{aligned} ||z-T_{i}v^{*}||= & {} ||T_{i}z-T_{i}v^{*}||\le ||z-v^{*}||, \end{aligned}$$

which implies $v^{*}=T_{i}v^{*}$ 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

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

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-norm uniformly continuous on bounded subsets of E, and $\epsilon \rightarrow 0$ we have that

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

Now from (3.5) and since $\underset{n\rightarrow \infty }{\lim }\frac{\alpha _n}{t_n}=0$, we have

$$\begin{aligned} \mu _n||x_{n}-p||^2\le & {} \mu _n\Big (\frac{\langle (\gamma f-\mu G)p,j(x_{n}-p)\rangle }{\tau -\gamma \beta }\Big )\\&+\mu _n\Big (\frac{(1-\tau t_n)\alpha _n}{t_n}\times \frac{||u-p||||x_{n}-p||}{\tau -\gamma \beta }\Big ) \end{aligned}$$

and so

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

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

By definition of $S_{\alpha }$ as $S_{\alpha _n}x_n:=\alpha _n u+ (1-\delta )(1-\alpha _n)x_n+\delta \sum _{i\ge 1}\sigma _{i,n}T_ix_n,$ which implies $S_{\alpha _{n_j}}x_{n_j}:=x_{n_j}+\alpha _{n_j} (u-x_{n_j})+ \delta \sum _{i\ge 1}\sigma _{i,n_j}(T_ix_{n_j}-x_{n_j}),$ then ${lim}_{j\rightarrow \infty }Sx_{n_{j}}={lim}_{j\rightarrow \infty }x_{n_{j}}=p$ and $S_{\alpha }p=p.$ Thus for any $z\in F,$ using (3.3) we have

$$\begin{aligned} \langle \mu G(x_{n_{j}})-\gamma f(x_{n_{j}}),j(x_{n_{j}}-z)\rangle= & {} \frac{-1}{t_{n_{j}}}\langle (I-S)x_{n_{j}}-(I-S)p, j(x_{n_{j}}-z)\rangle \nonumber \\+ & {} \mu \langle Gx_{n_{j}}-GSx_{n_{j}}, j(x_{n_{j}}-z)\rangle \nonumber \\\le & {} \mu \langle Gx_{n_{j}}-GSx_{n_{j}}, j(x_{n_{j}}-z)\rangle , \end{aligned}$$

(3.9)

since $\langle (I-S)x_{n_{j}}-(I-S)p, j(x_{n_{j}}-z)\rangle \ge 0.$ As G is Lipschitzian and the fact that $\Vert x_{n_{j}}-Sx_{n_{j}}\Vert \le \alpha _{n_j} \Vert u-x_{n_j}\Vert + \delta \sum _{i\ge 1}\sigma _{i,n_j}\Vert (T_ix_{n_j}-x_{n_j})\Vert \rightarrow 0$ as $j\rightarrow \infty ,$ we have $Gx_{n_{j}}-GSx_{n_{j}}\rightarrow 0~\mathrm{as}~j\rightarrow \infty .$ From this and (3.9), taking limit as $j\rightarrow \infty$ we obtain

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

Hence p is the unique solution of the variational inequality (3.4). Now assume there exists another subsequence of $\{z_{n}\}$ say $\{x_{n_{k}}\}$ such that ${lim}_{k\rightarrow \infty }x_{n_{k}}=p^{*}.$ Then, using (3.8) 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 (3.4). 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 q-uniformly real smooth Banach space which is also uniformly convex. Let $T_{i}:E\rightarrow E~~i\in \{1,2,\ldots \}$ be a family of nonexpansive mappings with $F:=\cap ^{\infty }_{i=1}F(T_{i})\ne \emptyset$. Let $G:E\rightarrow E$ be an $\eta$-strongly accretive map which is also $\kappa$-Lipschitzian. Let $f:E\rightarrow E$ be a contraction map with coefficient $0<\beta <1$. Let $\{\alpha _{n}\}$ and $\{\beta _{n}\}$ be sequences in (0,1) satisfying:

• (i) $\underset{n\rightarrow \infty }{\lim }\beta _{n}=0~$ and $~\sum ^{\infty }_{n=0}\beta _{n}=\infty ;$

• (ii) $\sum \alpha _{n}<\infty$ and $\underset{n\rightarrow \infty }{\lim }\frac{\alpha _n}{\beta _n}=0$;

• (iii) $\underset{n\rightarrow \infty }{\lim }\sum _{i\ge 1}|\sigma _{i,n+1}-\sigma _{i,n}|=0$ and $\sum _{i\ge 1}\sigma _{i,n}=1-\alpha _{n}.$

Let $\mu , \gamma ,$ and $\tau$ be as in Lemma 3.1 and $\delta \in (0,1)$ be fixed. define a sequence $\{x_{n}\}_{n=1}^{\infty }$ iteratively in E by $x_{0}\in E$

$$\begin{aligned} \left\{ \begin{array}{ll} y_{n}=\alpha _{n}u+(1-\delta )(1-\alpha _{n})x_{n}+\delta \sum _{i\ge 1}\sigma _{i,n}T_{i}x_{n} &{} \text { } \\ x_{n+1}=\beta _{n}\gamma f(x_{n})+(I-\beta _{n}\mu G)y_{n}. \end{array} \right. \end{aligned}$$

(3.10)

Then, $\{x_{n}\}^{\infty }_{n=1}$ converges strongly to $x^{*}\in F$ which is also a solution to the following variational inequality

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

(3.11)

Proof

Since $(\mu G-\gamma f)$ is strongly accretive, then the variational inequality (3.11) has a unique solution in F. Now we show that $\{x_{n}\}^{\infty }_{n=1}$ is bounded. Let $p\in F$ then, for every $i\in \mathbb {N}, T_{i}p=p$. From (3.10), we obtain

$$\begin{aligned} ||y_{n}-p||= & {} ||\alpha _nu+(1-\delta )(1-\alpha _n)x_n+\delta \sum _{i\ge 1}\sigma _{i,n}T_{i}x_n-p||\nonumber \\\le & {} \alpha _n||u-p||+(1-\delta )(1-\alpha _n)||x_n-p||\nonumber \\&+\delta \sum _{i\ge 1}\sigma _{i,n}||T_{i}x_n-p|| \nonumber \\\le & {} \alpha _n||u-p||+(1-\alpha _n)||x_n-p|| \nonumber \\\le & {} \alpha _n||u-p||+||x_n-p||. \end{aligned}$$

(3.12)

Also from (3.10) and (3.12), we obtain

$$\begin{aligned} ||x_{n+1}-p||= & {} ||\beta _{n}\gamma f(x_n)+(I-\beta _n\mu G)y_n-p|| \\\le & {} \beta \gamma \beta _n||x_n-p||+\beta _n||\gamma f(p)-\mu G(p)||\\&+||(I-\beta _n\mu G)y_n-(I-\beta _n\mu G)p|| \\\le & {} \beta \gamma \beta _n||x_n-p||+\beta _n||\gamma f(p)-\mu G(p)||\\&+(1-\tau \beta _n)||y_{n}-p|| \\\le & {} \beta \gamma \beta _n||x_n-p||+\beta _n||\gamma f(p)-\mu G(p)||\\&+(1-\tau \beta _n)[\alpha _n||u-p||+||x_n-p||] \\\le & {} [1-\beta _n(\tau -\gamma \beta )]||x_n-p||\\&+\beta _{n}[||\gamma f(p)-\mu G(p)||+||u-p||] \\\le & {} \max \Big \{||x_{n}-p||,\frac{||\gamma f(p)-\mu G(p)||+||u-p||}{\tau -\gamma \beta }\} \\\le & {} \cdots \le \max \Big \{||x_{1}-p||,\frac{||\gamma f(p)-\mu G(p)||+||u-p||}{\tau -\gamma \beta }\}. \end{aligned}$$

Therefore, $\{x_{n}\}$ is bounded. Hence $\{y_n\},\{T_ix_n\},\{Gy_n\},\{GT_iy_n\}$ and $\{f(y_n)\}$ are also bounded.

Next, we show that ${\lim }_{n\rightarrow \infty }||x_{n+1}-x_{n}||=0.$ Define two sequences $\{\lambda _{n}\}$ and $\{z_{n}\}$ by $\lambda _n:=(1-\delta )\alpha _n+\delta$ and

$$\begin{aligned} z_{n}:=\frac{x_{n+1}-x_{n}+\lambda _nx_{n}}{\lambda _n}. \end{aligned}$$

Observe that $\{z_n\}$ is bounded and that

$$\begin{aligned} ||z_{n+1}-z_{n}||-||x_{n+1}-x_{n}||\le & {} \Big (\frac{\beta _{n+1}}{\lambda _{n+1}} +\frac{\beta _{n}}{\lambda _{n}}\Big )M+\Big \vert \frac{\alpha _{n+1}}{\lambda _{n+1}} -\frac{\alpha _{n}}{\lambda _{n}}\Big \vert ||u||\\&+\Big [\frac{\delta (1-\alpha _{n+1})}{\lambda _{n+1}}-1\Big ]||x_{n+1}-x_{n}||\\&+\frac{\delta M}{\lambda _{n+1}}\sum _{i\ge 1}|\sigma _{i,n+1}-\sigma _{i,n}|\\&+\frac{\delta M}{\lambda _{n+1}}\sum _{i\ge 1}\sigma _{i,n}|\lambda _{n+1}-\lambda _{n}| \end{aligned}$$

for some real number $M:={\sup }_{n\ge 1}\{||\gamma f(x_n)-\mu G(y_n)||,||T_{i}x_n||,i=1,2,...\}$.

This implies

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

and by Lemma 2.4, we obtain

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

Hence

$$\begin{aligned} ||x_{n+1}-x_{n}||=\lambda _n||z_n-x_n||\rightarrow 0~~\text {as}~~n\rightarrow \infty . \end{aligned}$$

(3.13)

and from (3.10), we also obtain

$$\begin{aligned} ||x_{n+1}-y_{n}||=\beta _n||\gamma _nf(x_n)-\mu G(y_n)||\rightarrow 0~~\text {as}~~n\rightarrow \infty . \end{aligned}$$

(3.14)

from (3.13) and (3.14), we have

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

(3.15)

Next we show that $\underset{n\rightarrow \infty }{\lim }||T_ix_n-x_n||=0$ for all $i\in \mathbb {N}.$ Since $p\in F$, using the same argument in (3.7), we obtain

$$\begin{aligned} \frac{\delta }{2}\sum _{i\ge 1}\sigma _{i,n}g(||T_ix_n-x_n||)\le & {} \delta \sum _{i\ge 1}\sigma _{i,n}\langle x_n-T_ix_n,j(x_n-p)\rangle \\\le & {} \langle \delta (1-\alpha _n)x_n-\delta \sum _{i\ge 1}\sigma _{i,n}T_{i}x_n,j(x_n-p)\rangle \\\le & {} \langle \alpha _n(u-x_n)+x_n-y_n,j(x_{n}-p)\rangle \\\le & {} [\alpha _n||u-x_n||+||x_n-y_n||]||x_n-p||. \end{aligned}$$

From (3.15) and $\lim _{n\rightarrow \infty }\alpha _{n}=0$, we obtain

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\sum _{i\ge 1}\sigma _{i,n}||T_{i}x_n-x_n||=0, \end{aligned}$$

it follows that for every $i\in \mathbb {N},$

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||T_ix_{n}-x_n||=0. \end{aligned}$$

(3.16)

Let $z_t=t\gamma f(z_t)+(1-t\mu G)Sz_t$, where $S:=\alpha u+(1-\delta )(1-\alpha )I+\delta \sum _{i\ge 1}\sigma _{i,\alpha }T_i,$ as in Theorem 3.1. Then,

$$\begin{aligned} z_t-x_n=t(\gamma f(z_t)-Gz_t)+t\mu (Gz_t-G(Sz_t))+Sz_t-x_n \end{aligned}$$

Hence

$$\begin{aligned} ||z_t-x_n||^2= & {} \langle t(\gamma f(z_t)-Gz_t)+t\mu (Gz_t-G(Sz_t))+Sz_t-x_n,j(z_t-x_n)\rangle \\= & {} t\langle \gamma f(z_t)-\mu G(z_t),j(z_t-x_t)\rangle +t\mu \langle Gz_t-G(Sz_t),j(z_t-x_n)\rangle \\&+\langle Sz_t-x_n,j(z_t-x_n)\rangle \\\le & {} t\langle \gamma f(z_t)-\mu Gz_t,j(z_t-x_n)\rangle +t\mu \kappa ||z_t-Sz_t||||z_t-x_n||\\&+||Sz_t-x_n||||z_t-x_n||\\\le & {} t\langle \gamma f(z_t)-\mu Gz_t,j(z_t-x_n)\rangle +t(1+\mu )||z_t-Sz_t||||z_t-x_n||\\&+||z_t-x_n||^2+||Sx_n-x_n||||z_t-x_n||. \end{aligned}$$

Therefore

$$\begin{aligned} \langle \gamma f(z_t)-\mu Gz_t,j(x_n-z_t)\rangle\le & {} (1+\mu \kappa )||z_t-Sz_t||||z_t-x_n||\\&+\frac{1}{t}||Sx_n-x_n||||z_t-x_n|| \end{aligned}$$

Now, taking limit superior as $n\rightarrow \infty$ firstly, and then as $t\rightarrow 0$, we have

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

(3.17)

Moreover, we note that

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

(3.18)

Taking limit superior as $n\rightarrow \infty$ in (3.18), we have

$$\begin{aligned} \underset{n\rightarrow \infty }{\limsup }\langle \gamma f(p)-\mu Gp,j(x_n-p)\rangle\le & {} \underset{n\rightarrow \infty }{\limsup }\langle \gamma f(p)-\mu Gp,j(x_n-p)-j(x_n-z_t)\rangle \nonumber \\&+\mu || Gz_t- Gp||\underset{n\rightarrow \infty }{\limsup }||x_n-z_t||\nonumber \\&+||\gamma f(z_t)-\gamma f(p)||\underset{n\rightarrow \infty }{\limsup }||x_n-z_t||\nonumber \\&+\underset{n\rightarrow \infty }{\limsup }\langle \gamma f(z_t)-\mu Gz_t,j(x_n-z_t)\rangle \nonumber \\\le & {} \underset{n\rightarrow \infty }{\limsup }\langle \gamma f(p)-\mu Gp,j(x_n-p)-j(x_n-z_t)\rangle \nonumber \\&+((\mu +1)+\alpha \gamma )\Vert z_{t}-p\Vert \underset{n\rightarrow \infty }{\limsup }||x_n-z_t|| \nonumber \\&+\underset{n\rightarrow \infty }{\limsup }\langle \gamma f(z_t)-\mu Gz_t,j(x_n-z_t)\rangle \end{aligned}$$

(3.19)

since E has a uniformly Gâteaux differentiable norm, so j is norm-to-norm$^{*}$ uniformly continuous on bounded subset of E. Then, from Theorem 3.1 (i.e., $z_t\rightarrow p~~(t\rightarrow 0^{+}$)), we obtain

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

hence, using (3.17) in (3.19), we obtain

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

Finally, we show that $x_n\rightarrow p.$ From the recursion formula (3.10), by using (2.1) and taking $n\ge N$ where $N\in \mathbb {N}$ is large enough, we obtain

$$\begin{aligned} ||x_{n+1}-p||^2= & {} ||\beta _n\gamma f(x_n)-\beta _n\mu G(p)+(I-\beta _n\mu G)y_n-(I-\beta _n \mu G)p||^2 \\\le & {} ||(I\!-\!\beta _n\mu G)y_n\!-\!(I-\beta _n\mu G)p||^2+2\beta _n\langle \gamma f(x_n)-\mu G(p),j(x_{n+1}-p)\rangle \\\le & {} (1-\beta _n\tau )^2||y_n-p||^2+2\beta _n\langle \gamma f(x_{n})-\gamma f(p),j(x_{n+1}-p)\rangle \\&+2\beta _n\langle \gamma f(p)-\mu G(p),j(x_{n+1}-p)\rangle \\\le & {} \alpha _n||u-p||^2+(1-\beta _n\tau )^2||x_n-p||^2\\&+2\beta _n\langle \gamma f(x_{n})-\gamma f(p),j(x_{n+1}-p)\rangle \\&+2\beta _n\langle \gamma f(p)-\mu G(p),j(x_{n+1}-p)\rangle \end{aligned}$$

On the other hand

$$\begin{aligned} \langle \gamma f(x_{n})-\gamma f(p),j(x_{n+1}-p)\rangle\le & {} \gamma \beta ||x_n-p||||x_{n+1}-p||\\\le & {} \gamma \beta ||u-p||||x_{n}-p||\sqrt{\alpha _n}+\gamma \beta (1-\beta _n\tau )||x_n-p||^2\\&+\gamma \beta ||x_n-p||\sqrt{2|\langle \gamma f(x_n)-\gamma f(p),j(x_{n+1}-p)\rangle |}\sqrt{\beta _n} \\&+\gamma \beta ||x_n-p||\sqrt{2|\langle \gamma f(p)-\mu G(p),j(x_{n+1}-p)\rangle |}\sqrt{\beta _n}. \end{aligned}$$

Since $\{x_n\}$ and $\{f(x_n)\}$ are bounded, we pick a constant $G_0>0$ such that

$$\begin{aligned}&\sup \Big \{\gamma \beta ||x_n-p||||u-p||,\gamma \beta ||x_n-p||\Big (\sqrt{2|\langle \gamma f(x_n)-\gamma f(p),j(x_{n+1}-p)\rangle |}\\&\quad +\gamma \beta \sqrt{2|\langle \gamma f(p)-\mu G(p),j(x_{n+1}-p)\rangle |}\Big )\Big \}<G_{0}, \forall n\in \mathbb {N}. \end{aligned}$$

Therefore

$$\begin{aligned} \langle \gamma f(x_{n})-\gamma f(p),j(x_{n+1}-p)\rangle \le \gamma \beta (1-\beta _n\tau )||x_n-p||^2+G_{0}(\sqrt{\alpha _n}+\sqrt{\beta _n}) \end{aligned}$$

Hence

$$\begin{aligned} ||x_{n+1}-p||^2\le & {} \alpha _n||u-p||^2+(1-\beta _n\tau )^2||x_n-p||^2\\&+2\beta _n\gamma \beta (1-\beta _n\tau )||x_n-p||^2+2\beta _nG_{0} (\sqrt{\alpha _n}+\sqrt{\beta _n}) \\&+2\beta _n\langle \gamma f(p)-\mu G(p),j(x_{n+1}-p)\rangle \\= & {} \Big [1-2\beta _n(1-\beta _n\tau )(\tau -\gamma \beta )\Big ]||x_n-p||^2+\alpha _n||u-p||^2\\&+2\beta _nG_{0}(\sqrt{\alpha _n}+\sqrt{\beta _n}) +2\beta _n\langle \gamma f(p)-\mu G(p),j(x_{n+1}-p)\rangle \\\le & {} \Big [1-\beta _n(1-\beta _n\tau )(\tau -\gamma \beta )\Big ]||x_n-p||^2+\theta _n \end{aligned}$$

where $\theta _n:=\beta _n\Big (\alpha _n/\beta _n||u-p||^2+2G_{0}(\sqrt{\alpha _n}+\sqrt{\beta _n}). +2\langle \gamma f(p)-\mu G(p),j(x_{n+1}-p)\rangle \Big )$ By using Lemma 2.3 we obtain $x_n\rightarrow p$ as $n\rightarrow \infty$. This complete the proof. $\square$

Corollary 3.4

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 points of the family $\{T_{i}\}_{i=1}^{\infty }$ say p which is a unique solution of the variational inequality

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

Corollary 3.5

Let H be a real Hilbert space and let C a nonempty closed convex subset of H. Let $G:H \rightarrow H,$ $f:E \rightarrow E,$ $\{T_{i}\}^{\infty }_{i=1}$ F, $\{\alpha _{n}\}^{\infty }_{n = 1}$, $\{\beta _{n}\}^{\infty }_{n =1}$ and $\{x_{n}\}^{\infty }_{n = 1}$ be as in Theorem (3.1), then $\{x_{n}\}^{\infty }_{n = 1}$ converges strongly to $p \in F$, which is also the unique solution of the variational inequality

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