Approximation of common solutions for variational inequalities and fixed point of strict pseudo-contractions in q-uniformly smooth Banach spaces

Abstract

In the present paper, we introduce a general iterative algorithm for finding a common element of the set of common fixed points of an infinite family of strict pseudo-contractions and the set of solutions of the variational inequalities for finite family of strongly accretive mappings in a q-uniformly smooth Banach space. Furthermore, we prove strong convergence of the iterative sequence under suitable conditions. Our results generalize some recent results.

1 Introduction

Throughout this paper, we always assume that X is a real Banach space with the dual $X^{*}$. Let C be a subset of X, and T be a self-mapping of C. We use $F(T)$ to denote the fixed points of T. For $q>1$, the generalized duality mapping $J_{q}: X\rightarrow2^{X^{*}}$ is defined by

$$J_{q}(x)=\bigl\{ f\in X^{*} : \langle x,f\rangle=\| x\|^{q} , \| f\|=\| x\|^{q-1}\bigr\} , $$

where $\langle\cdot,\cdot\rangle$ denotes the duality pairing between X and $X^{*}$. In particular, $J_{q} = J_{2}$ is called the normalized duality mapping and $J_{q}(x)= \|x\|^{q-2}J_{2}(x)$ for $x\neq0$. If $X:= H$ is a real Hilbert space, then $J = I$ where I is the identity mapping. It is well known that if X is smooth, then $J_{q}$ is single-valued, which is denoted by $j_{q}$ [1].

Let $U=\{ x\in X : \| x \|=1\}$. A Banach space X is said to be strictly convex if $\frac{\| x+y\|}{2} \leq1$ for all $x,y \in X$ with $\| x\|=\| y\|=1$ and $x\neq y$. It is also called uniformly convex if $\lim\| x_{n} -y_{n} \|=0$ for any two sequences $\{ x_{n}\}$, $\{ y_{n}\}$ in X such that $\| x_{n}\|=\| y_{n}\|=1$ and $\lim\|\frac{x_{n} + y_{n}}{2} \|=1$. A Banach space X is said to be Gâteaux differentiable if the limit

$$ \lim_{t\rightarrow0} \frac{\| x+ty\|- \| x\|}{t} $$

(1)

exists for all $x,y\in U$. In this case X is smooth. Also, we define a function $\rho_{X}:[0, \infty) \rightarrow[0, \infty)$ called the modulus of smoothness of X as follows:

$$\rho_{X}(t)=\sup\biggl\{ \frac{1}{2}\bigl(\|x+y\|+\|x-y\|\bigr)-1:x\in U, \|y\|< t\biggr\} . $$

A Banach space X is said to be uniformly smooth if $\frac{\rho_{X}(t)}{t}\rightarrow0$ as $t\rightarrow0$. Suppose that $q > 1$, then X is said to be q-uniformly smooth if there exists $c > 0$ such that $\rho_{X}(t)\leq ct^{q}$. It is easy to see that if X is q-uniformly smooth, then $q\leq2$ and X is uniformly smooth.

Let C be a nonempty, closed, and convex subset of a Banach space X and D be a nonempty subset of C, then a mapping $Q:C\rightarrow D$ is said to be sunny provided

$$Q\bigl(Qx+t(x-Qx)\bigr)= Qx, $$

whenever $Qx+ t(x- Qx)\in C$ for $x\in C$, and $t\geq0$. A mapping $Q:C\rightarrow D$ is called a retraction if $Qx=x$ for all $x\in D$. Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is a retraction from C onto D which is also sunny and nonexpansive.

A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. In real Hilbert space, a sunny nonexpansive retraction $Q_{C}$ coincides with the metric projection from X onto C.

Definition 1.1

A mapping $T:C\rightarrow C$ is said to be:

• (i) λ-strictly pseudo contractive [2], if for all $x,y \in C$ there exist $\lambda>0$ and $j_{q}(x-y)\in J_{q}(x-y)$ such that

  $$\bigl\langle Tx -Ty , j_{q}(x-y)\bigr\rangle \leq\| x-y \|^{q} -\lambda\bigl\| (I-T)x -(I-T)y\bigr\| ^{q}, $$

  or equivalently

  $$\bigl\langle (I-T)x -(I-T)y , j_{q}(x-y)\bigr\rangle \geq\lambda\bigl\| (I-T)x -(I-T)y\bigr\| ^{q}. $$

• (ii) L-Lipschitzian if for all $x,y\in C$, there exists a constant $L>0$ such that

  $$\| Tx-Ty \|\leq L \| x-y \|. $$

  If $0< L<1$, then T is a contraction, and if $L=1$, then T is a nonexpansive mapping.

Remark 1.2

Let C be a nonempty subset of a real Hilbert space H and $T: C \rightarrow C$ be a mapping. Then T is said to be k-strictly pseudocontractive [2], if for all $x, y\in C$, there exists constant $k\in[0,1)$ such that

$$\|Tx-Ty\|^{2}\leq\|x-y\|+k\bigl\| (I-T)x-(I-T)y\bigr\| ^{2}. $$

Definition 1.3

A mapping $F:C\rightarrow X$ is said to be accretive if for all $x,y \in C$ there exists $j_{q}(x-y)\in J_{q}(x-y)$ such that

$$\bigl\langle Fx -Fy , j_{q}(x-y)\bigr\rangle \geq0. $$

For some $\eta>0$, $F:C\rightarrow X$ is said to be η-strongly accretive if for all $x,y \in C$ there exists $j_{q}(x-y)\in J_{q}(x-y)$ such that

$$\bigl\langle Fx -Fy , j_{q}(x-y)\bigr\rangle \geq\eta\| x-y \|^{q} . $$

For some $\mu>0$, the mapping $F:C\rightarrow X$ is said to be μ-inverse strongly accretive if for all $x,y \in C$ there exists $j_{q}(x-y)\in J_{q}(x-y)$ such that

$$\bigl\langle Fx -Fy , j_{q}(x-y)\bigr\rangle \geq\mu\|Fx-Fy \|^{q}. $$

Note that if $X:=H$ is a real Hilbert space, accretive and strongly accretive operators coincide with monotone and strongly monotone operators, respectively.

Let C be a nonempty, closed, and convex subset of X, and $A:C\rightarrow X$ be a mapping. The classical variational inequality problem is to find $x^{*}\in C$ such that

$$ \bigl\langle Ax^{*},j_{q}\bigl(x-x^{*} \bigr)\bigr\rangle \geq0,\quad \forall x\in C, $$

(2)

where $j_{q}(x-x^{*})\in J_{q}(x-x^{*})$. The solution set of a variational inequality is denoted by $VI(C,A)$. If $X=:H$ is a real Hilbert space, the variational inequality problem reduces to find $x^{*}\in C$ such that

$$ \bigl\langle Ax^{*},x-x^{*}\bigr\rangle \geq0, \quad\forall x\in C. $$

(3)

For more details of the variational inequality and its applications, we recommend the reader [3, 4]. On the other hand, we note that the iterative approximations of fixed points for nonexpansive mappings have been extensively studied by many authors [5–9].

In order to find the common element of the solution set of a variational inclusion (3) and the set of fixed points of a nonexpansive mapping, Takahashi and Toyoda [10] introduced the following iterative scheme in a Hilbert space H. Starting with an arbitrary point $x_{1}=x\in H$, define sequences $\{x_{n}\}$ by

$$ x_{n+1}=\alpha_{n}x_{n}+(1- \alpha_{n})SP_{C}(x_{n}-\lambda_{n}Ax_{n}), $$

(4)

where $A:H\rightarrow H$ is an α-inverse-strongly monotone mapping, $S:C\rightarrow C$ is a nonexpansive mapping and $\{\alpha_{n}\}$ is a sequence in $[0,1]$. Under mild conditions, they obtained a weak convergence theorem.

On the other hand, Aoyama et al. [11] considered the following algorithm in a uniformly convex and 2-uniformly smooth Banach spaces. For $x_{1}=x\in C$,

$$ x_{n+1}=\alpha_{n}x_{n} +(1- \alpha_{n})Q_{C}(x_{n}- \lambda_{n}Ax_{n}), $$

(5)

where $Q_{C}: X\rightarrow C$ is a sunny nonexpansive retraction, and A is a β-Lipschitzian and η-inverse strongly accretive operator. They proved that $\{x_{n}\}$ generated by (5) converges weakly to a unique element z of $VI(C,A)$.

Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth uniformly convex Banach space X. Assume the mapping $A_{m}:C\rightarrow X$ be a $\mu_{m}$-inverse-strongly accretive mapping for each $1\leq m\leq r$, where r is a positive integer. Let $\{T_{n}\}_{n=1}^{\infty}:C\rightarrow C$ be a family of λ-strict pseudo-contractions with $0 <\lambda< 1$. Define a mapping $S_{n}x:=(1 -\gamma_{n})x +\gamma_{n}T_{n}x$ for all $x\in C$ and $n\geq1$.

In this paper, motivated by the works mentioned above, we consider the following iteration:

$$ \begin{cases} x_{1}\in C, \\ y_{n} = \alpha_{n}x_{n}+(1-\alpha_{n})\sum_{m=1}^{r}\eta _{n}^{m}Q_{C}(x_{n}-\lambda_{m}A_{m}x_{n}),\\ x_{n+1}=Q_{C}[\beta_{n}\gamma fx_{n} +(I-\beta_{n}\mu F) S_{n}y_{n}],\quad n\geq1, \end{cases} $$

(6)

and we prove that the proposed iterative algorithm is strongly convergent under some mild conditions imposed on the algorithm parameters. The results proved in this paper represent a refinement and improvement of the previously found results in the earlier and recent literature.

2 Preliminaries

In order to prove our main results, we need the following lemmas.

Lemma 2.1

[12, 13]

Let C be a closed convex subset of a smooth Banach space X. Let D be a nonempty subset of C. Let $Q:C\rightarrow D$ be a retraction and J be the normalized duality mapping on X. Then the following are equivalent:

• (a) Q is sunny and nonexpansive.

• (b) $\|Qx-Qy\|^{2}\leq\langle x-y,J(Qx-Qy)\rangle$, $\forall x,y\in C$.

• (c) $\langle x-Qx,J(y-Qx)\rangle\leq0$, $\forall x\in C$, $y\in D$.

• (d) $\langle x-Qx,J_{q}(y-Qx)\rangle\leq0$, $\forall x\in C$, $y\in D$.

Lemma 2.2

[14]

Let C be a closed convex subset of a strictly convex Banach space X. Let $T_{1}$ and $T_{2}$ be two nonexpansive mappings from C into itself with $F(T_{1})\cap F(T_{2})\neq\emptyset$. Define a mapping S by

$$Sx =kT_{1}x +(1-k)T_{2}x, \quad\forall x\in C, $$

where k is a constant in $(0, 1)$. Then S is nonexpansive and $F(S)=F(T_{1})\cap F(T_{2})$.

Lemma 2.3

[15]

Let $\{ s_{n}\}$ be a sequence of nonnegative real numbers satisfying

$$s_{n+1}=(1-a_{n})s_{n} + a_{n} b_{n} + c_{n}, $$

where $\{a_{n}\}$, $\{b_{n}\}$, $\{c_{n}\}$ satisfy the restrictions:

• (i) $\lim_{n\rightarrow\infty}a_{n}=0$, $\sum_{n=1}^{\infty}a_{n} = \infty$,

• (ii) $c_{n}\geq0$, $\sum_{n=1}^{\infty}c_{n} < \infty$,

• (iii) $\limsup_{n\rightarrow\infty} b_{n} \leq0$.

Then $\lim_{n\rightarrow\infty} s_{n}=0$.

Lemma 2.4

[16]

Suppose that $q>1$. Then the following inequality holds:

$$ab \leq\frac{1}{q} a^{q} +\biggl(\frac{q-1}{q} \biggr)b ^{\frac{q}{q-1}}, $$

for arbitrary positive real numbers a, b.

Lemma 2.5

[17]

Let X be a real q-uniformly smooth Banach space, then there exists a constant $C_{q}>0$ such that

$$\| x+y\|^{q} \leq\| x\|^{q} +q\bigl\langle y,J_{q}(x)\bigr\rangle +c_{q} \| y \|^{q}, $$

for all $x,y\in X$. In particular, if X is real 2-uniformly smooth Banach space, then there exists a best smooth constant $K > 0$ such that

$$\| x+y\|^{2} \leq\| x\|^{2} +2\bigl\langle y,J(x)\bigr\rangle +2K \| y \|^{2} $$

for all $x,y\in C$.

Lemma 2.6

[18]

Let X a real smooth and uniformly convex Banach space and let $r >0$. Then there exists a strictly increasing, continuous, and convex function $g:[0,2 r] \rightarrow R$ such that $g(0)=0$ and $g(\| x-y\| )\leq\| x \|^{2}-2\langle x,Jy\rangle+\| y \|^{2}$, for all $x,y \in B_{r}$, where $B_{r}=\{ z\in X : \| z \|\leq r \}$.

Definition 2.7

[11]

Let ${T_{n}}$ be a family of mappings from a subset C of a Banach space X into itself with $\bigcap_{n=1}^{\infty} F(T_{n})\neq\emptyset$. We say that $\{T_{n}\}$ satisfies the AKTT-condition if for each bounded subset B of C,

$$ \sum_{n=1}^{\infty} \sup_{\omega\in B} \|T_{n+1}\omega-T_{n}\omega\|< \infty. $$

(7)

Lemma 2.8

[11]

Suppose that $\{T_{n}\}$ satisfies the AKTT-condition such that:

• (i) For each $x\in C$, $\{T_{n}x\}$ is converge strongly to some point in C.

• (ii) Let the mapping $T:C\rightarrow C$ defined by $Tx= \lim_{n\rightarrow\infty} T_{n}x$, for all $x\in C$.

Then $\lim_{n\rightarrow\infty}\sup_{\omega\in B}\|T\omega-T_{n}\omega\|=0$, for each bounded subset B of C.

Lemma 2.9

[7, 8]

Let C be a closed and convex subset of a smooth Banach space X. Suppose that $\{T_{n}\}_{n=1}^{\infty}:C\rightarrow X$ is a family of λ-strictly pseudocontractive mappings; $\{\mu_{m}\}_{m=1}^{\infty}$ is a real sequence in $(0,1)$ such that $\sum_{n=1}^{\infty}\mu_{m}=1$. Then the following conclusions hold:

• (i) A mapping $G:C\rightarrow X$ defined by $G:=\sum_{n=1}^{\infty}\mu_{n}T_{n}$ is a λ-strictly pseudocontractive mapping.

• (ii) $F(G)=\bigcap_{n=1}^{\infty}F(T_{n})$.

Lemma 2.10

[19]

Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X which admits weakly sequentially continuous generalized duality mapping $j_{q}$ from X into $X^{*}$. Let $T: C\rightarrow C$ be a nonexpansive mapping. Then, for all $\{x_{n}\}\subset C$, if $x_{n}\rightharpoonup x$ and $x_{n}- Tx_{n}\rightarrow0$, then $x=Tx$.

Lemma 2.11

[19]

Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X. Let $F:C\rightarrow E$ be a k-Lipschitzian and η-strongly accretive operator with constants $k, \eta>0$. Let $0 < \mu< ( \frac{q\eta}{C_{q}k^{q}} )^{\frac{1}{q-1}}$ and $\tau=\mu(\eta -\frac{C_{q}\mu^{q-1} k^{q}}{q})$. Then for $t\in (0,\min\{1,\frac{1}{\tau}\})$, the mapping $S:C\rightarrow E$ defined by $S:=(I-t\mu F)$ is a contraction with a constant $1-t\tau$.

Lemma 2.12

[19]

Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X. Let $Q_{C}$ be a sunny nonexpansive retraction from X onto C. Let $F:C\rightarrow X$ be a k-Lipschitzian and η-strongly accretive operator with constants $k,\eta>0$, $f:C\rightarrow X$ be an L-Lipschitzian mapping with a constant $L\geq0$ and $S:C\rightarrow C$ be a nonexpansive mapping such that $F(S)\neq \emptyset$. Let $0<\mu<(\frac{q\eta}{C_{q} k^{q}})^{\frac{1}{q-1}}$ and $0\leq\gamma L<\tau$, where $\tau= \mu (\eta-\frac{C_{q} \mu^{q-1} k^{q} }{q})$. Then $\{x_{t}\}$ defined by

$$ x_{t}=Q_{C}\bigl[t\gamma fx_{t}+(I-t\mu F)Sx_{t}\bigr] $$

(8)

has the following properties:

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

• (ii) $\lim_{t\rightarrow0 }\| x_{t} -Sx_{t}\|=0$.

• (iii) $\{ x_{t} \}$ defines a continuous curve from $(0,\min\{1,\frac{1}{\tau}\})$ into C.

Lemma 2.13

[13]

Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X which admits a weakly sequentially continuous generalized duality mapping $j_{q}$ from X into $X^{*}$. Let $Q_{C}$ be a sunny nonexpansive retraction from X onto C. Let $F:C\rightarrow X$ be a k-Lipschitzian and η-strongly accretive operator with constants $k,\eta>0$, $f:C\rightarrow X$ be an L-Lipschitzian mapping with a constant $L\geq0$, and $S:C\rightarrow C$ be a nonexpansive mapping such that $F(S)\neq\emptyset$. Suppose that $0<\mu <(\frac{q\eta}{C_{q} k^{q}})^{\frac{1}{q-1}}$ and $0\leq\gamma L<\tau$, where $\tau= \mu(\eta-\frac{C_{q} \mu^{q-1} k^{q} }{q})$. For each $t\in(0,\min\{1,\frac{1}{\tau}\})$, let $\{ x_{t} \}$ be defined by (8), then $\{ x_{t} \}$ converges strongly to $x^{*}\in F(S)$ as $t\rightarrow0$, in which $x^{*}$ is the unique solution of the variational inequality

$$ \bigl\langle (\mu F -\gamma V)x^{*} , j_{q}\bigl(x^{*} -p \bigr)\bigr\rangle \leq0 ,\quad \forall p\in F(S). $$

(9)

Lemma 2.14

[20]

Let X be a Banach space and J be a normality duality mapping. Then for any given $x,y\in X$, the following inequality holds:

$$\|x+y\|^{2}\leq\|x\|^{2}+2\bigl\langle y,j(x+y)\bigr\rangle , $$

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

3 Main results

Theorem 3.1

Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth, uniformly convex Banach space X. Let $Q_{C}$ be a sunny nonexpansive retraction from X onto C. Assume that the mapping $A_{m}:C\rightarrow H$ is a $\mu_{m}$-inverse-strongly accretive mapping for each $1\leq m\leq r$, where r is a positive integer. Let $F:C\rightarrow X$ be a k-Lipschitzian and η-strongly accretive operator with constants $k,\eta>0$, $f:C\rightarrow X$ be an L-Lipschitzian mapping with a constant $L\geq0$. Suppose that $0<\mu<(\frac{q\eta}{C_{q} k^{q}})^{\frac{1}{q-1}}$ and $0\leq\gamma L<\tau$, where $\tau= \mu (\eta-\frac{C_{q} \mu^{q-1} k^{q} }{q})$. Let $\{T_{n}\}_{n=1}^{\infty}:C\rightarrow C$ be a family of λ-strict pseudo-contractions with $0 <\lambda< 1$. Define a mapping $S_{n}x:=(1 -\gamma_{n})x +\gamma_{n}T_{n}x$, for all $x\in C$ and $n\geq1$. Assume that $F:=(\bigcap_{m=1}^{r}VI(C,A_{m}))\cap(\bigcap_{n=1}^{\infty}F(T_{n}))\neq \emptyset$. Let $\{x_{n}\}$ be a sequence generated by the following iterative algorithm:

$$\begin{cases} x_{1}\in C, \\ y_{n} = \alpha_{n}x_{n}+(1-\alpha_{n})\sum_{m=1}^{r}\eta _{n}^{m}Q_{C}(x_{n}-\lambda_{m}A_{m}x_{n}),\\ x_{n+1}=Q_{C}[\beta_{n}\gamma fx_{n} +(I-\beta_{n}\mu F) S_{n}y_{n}], \quad n\geq1, \end{cases} $$

where $\{\alpha_{n}\}, \{\beta_{n}\}, \{\eta_{n}^{1}\}, \{\eta_{n}^{2}\},\ldots$ and $\{\eta_{n}^{r}\}$ are sequences in $(0,1)$ and $\lambda_{m}$ is a real number such that $0<\lambda_{m}<(\frac{q\mu_{m}}{C_{q}})^{\frac{1}{q-1}}$, for each $1\leq m\leq r$. Assume that the above control sequences satisfy the following restrictions:

• (i) $\sum_{m=1}^{r}\eta_{n}^{m}=1$, $\forall n\geq1$, $\sum_{n=1}^{\infty}|\eta_{n+1}^{m}-\eta_{n}^{m}|<\infty$.

• (ii) $\lim_{n\rightarrow\infty}\eta_{n}^{m}=\eta^{m}\in(0,1)$, for each m, where $1\leq m\leq r$.

• (iii) $\sum_{n=1}^{\infty}\beta_{n}=\infty$, $\lim_{n\rightarrow \infty}\beta_{n}=0$, $\sum_{n=1}^{\infty}|\beta_{n+1}-\beta_{n}|<\infty$.

• (iv) $\sum_{n=1}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty$, $\liminf_{n\rightarrow\infty}\alpha_{n}>0$.

• (v) $0\leq\gamma_{n}\leq\delta$, $\delta=\min\{1,(\frac{q\lambda}{C_{q}})^{\frac{1}{q-1}}\}$, and $\sum_{n=1}^{\infty}|\gamma_{n+1}-\gamma_{n}|<\infty$.

Suppose in addition that $\{ T_{n}\} _{n=0}^{\infty}$ satisfies the AKTT-condition. Let $T:C\rightarrow C$ be the mapping defined by $Tx=\lim_{n\rightarrow\infty}T_{n}x$ for all $x\in C$ and suppose that $F(T)=\bigcap_{n=0}^{\infty}F(T_{n})$. Then the sequence $\{ x_{n} \}$ converges strongly to $x^{*}\in F$ as $n\rightarrow\infty$, in which $x^{*}$ is the unique solution of the variational inequality,

$$\bigl\langle (\mu F -\gamma f)x^{*} , j_{q} \bigl(x^{*} -p\bigr)\bigr\rangle \leq0 ,\quad \forall p\in F(S). $$

Proof

We divide the proof into several steps.

Step 1. We show that $I-\lambda_{m} A_{m}$ is nonexpansive for each m. Indeed, from Lemma 2.4, for all $x,y\in C$ we have

$$\begin{aligned} &\bigl\| (I-\lambda_{m} A_{m})x-(I-\lambda_{m} A_{m})y\bigr\| ^{q} \\ &\quad=\bigl\| (x-y)-\lambda_{m}( A_{m}x-A_{m}y) \bigr\| ^{q} \\ &\quad\leq\| x-y \|^{q}- q\lambda_{m} \bigl\langle A_{m}x-A_{m}y,j_{q}(x-y)\bigr\rangle +C_{q}\lambda_{m}^{q}\| A_{m}x-A_{m}y \|^{q} \\ &\quad\leq\| x-y \|^{q}-q \mu_{m}\lambda_{m} \| A_{m}x-A_{m}y \| ^{q}+C_{q} \lambda_{m}^{q}\| A_{m}x-A_{m}y \|^{q} \\ &\quad\leq \| x-y \|^{q} -\lambda_{m}\bigl(q \mu_{m}-C_{q}\lambda_{m}^{q-1}\bigr)\| A_{m}x-A_{m}y\|^{q}. \end{aligned}$$

It is clear that if $0<\lambda_{m}\leq (\frac{q\mu_{m}}{C_{q}})^{\frac{1}{q-1}}$, then $I-\lambda_{m} A_{m}$ is nonexpansive for each $1\leq m\leq r$.

Now, for each $1\leq m\leq r$, put

$$k_{n}^{m}=Q_{C}(x_{n}- \lambda_{m}A_{m}x_{n}),\qquad z_{n}= \sum_{m=1}^{r}\eta_{n}^{m}k_{n}^{m}. $$

Let $x^{*}\in F$, we have

$$\begin{aligned}[b] \bigl\| k_{n}^{m}-x^{*}\bigr\| &=\bigl\| Q_{C}(x_{n}- \lambda _{m}A_{m}x_{n})-Q_{C} \bigl(x^{*}-\lambda_{n}A_{m}x^{*}\bigr)\bigr\| \\ &\leq\bigl\| x_{n}-x^{*}\bigr\| \quad\forall m, 1\leq m\leq r. \end{aligned} $$

On the other hand we have

$$\begin{aligned} \bigl\| y_{n}-x^{*}\bigr\| &=\Biggl\| \alpha_{n}x_{n}+(1- \alpha_{n})\sum_{m=1}^{r}\eta _{n}^{m}k_{n}^{m}-x^{*}\Biggr\| \\ &\leq\alpha_{n}\bigl\| x_{n}-x^{*}\bigr\| +(1- \alpha_{n})\sum_{m=1}^{r}\eta _{n}^{m}\bigl\| x_{n}-x^{*}\bigr\| \\ &=\alpha_{n}\bigl\| x_{n}-x^{*}\bigr\| +(1- \alpha_{n})\bigl\| x_{n}-x^{*}\bigr\| =\bigl\| x_{n}-x^{*} \bigr\| . \end{aligned}$$

(10)

From (10) and the fact that $S_{n}$ is nonexpansive [19] we have

$$\begin{aligned} \bigl\| x_{n+1}-x^{*}\bigr\| &=\bigl\| Q_{C}\bigl( \beta_{n}\gamma fx_{n}+(I-\beta_{n}\mu F)S_{n}y_{n}\bigr)-Q_{C}x^{*}\bigr\| \\ &\leq\bigl\| \beta_{n}\gamma fx_{n}+(I-\beta_{n}\mu F)S_{n}y_{n}-x^{*}\bigr\| \\ &=\bigl\| \beta_{n}\bigl(\gamma fx_{n}-\mu Fx^{*} \bigr)+(I-\beta_{n}\mu F) \bigl(S_{n}y_{n}-x^{*} \bigr)\bigr\| \\ &\leq\beta_{n}\bigl\| \gamma fx_{n}-\mu Fx^{*}\bigr\| +(1- \beta_{n}\tau )\bigl\| S_{n}y_{n}-x^{*}\bigr\| \\ &\leq\beta_{n}\gamma\bigl\| fx_{n}-fx^{*}\bigr\| + \beta_{n}\bigl\| \gamma fx^{*}-\mu Fx^{*}\bigr\| +(1- \beta_{n}\tau)\bigl\| y_{n}-x^{*}\bigr\| \\ &\leq\beta_{n} L\gamma\bigl\| x_{n}-x^{*}\bigr\| + \beta_{n}\bigl\| \gamma fx^{*}-\mu Fx^{*}\bigr\| +(1- \beta_{n}\tau)\bigl\| x_{n}-x^{*}\bigr\| \\ &\leq\bigl(1-\beta_{n}(\tau- L\gamma)\bigr)\bigl\| x_{n}-x^{*} \bigr\| +\beta_{n}\bigl\| \gamma fx^{*}-\mu Fx^{*}\bigr\| \\ &\leq\max \bigl\{ \bigl\| x_{n}-x^{*}\bigr\| ,(\tau-\gamma L)^{-1}\bigl\| \gamma fx^{*}-\mu Fx^{*}\bigr\| \bigr\} . \end{aligned}$$

By induction, we find that

$$\bigl\| x_{n+1}-x^{*}\bigr\| \leq\max{ \bigl\{ \bigl\| x_{0}-x^{*} \bigr\| ,(\tau-\gamma L)^{-1}\bigl\| \gamma fx^{*}-\mu Fx^{*}\bigr\| \bigr\} }. $$

This shows that $\{x_{n}\}$ is bounded. Hence by (10), $\{y_{n}\}$ is also bounded.

Step 2: We show that $\lim_{n\rightarrow\infty}\| x_{n+1} -x_{n} \|=0$. Since

$$\begin{aligned} \bigl\| k_{n+1}^{m}-k_{n}^{m}\bigr\| = \bigl\| Q_{C}(I-\lambda _{m}A_{m})x_{n+1}-Q_{C}(I- \lambda_{m}A_{m})x_{n}\bigr\| \leq\|x_{n+1}-x_{n}\| \quad \forall1\leq m\leq r. \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \|z_{n+1}-z_{n}\|&=\Biggl\| \sum _{m=1}^{r}\eta_{n+1}^{m}k_{n+1}^{m}- \sum_{m=1}^{r}\eta_{n}^{m}k_{n}^{m} \Biggr\| \\ &\leq\Biggl\| \sum_{m=1}^{r}\eta_{n+1}^{m}k_{n+1}^{m}- \sum_{m=1}^{r}\eta _{n+1}^{m}k_{n}^{m} \Biggr\| + \Biggl\| \sum_{m=1}^{r}\eta_{n+1}^{m}k_{n}^{m}- \sum_{m=1}^{r}\eta _{n}^{m}k_{n}^{m} \Biggr\| \\ &\leq\sum_{m=1}^{r}\eta_{n+1}^{m} \bigl\| k_{n+1}^{m}-k_{n}^{m}\bigr\| + \sum _{m=1}^{r}\bigl|\eta_{n+1}^{m}- \eta_{n}^{m}\bigr| \bigl\| k_{n}^{m}\bigr\| \\ &\leq\|x_{n+1}-x_{n}\|+M\sum_{m=1}^{r}\bigl| \eta_{n+1}^{m}-\eta_{n}^{m}\bigr|, \end{aligned}$$

(11)

where M is an appropriate constant such that

$$M=\max\bigl\{ \sup\bigl\{ \bigl\| P_{C}(I-\lambda_{m}A_{m})x_{n} \bigr\| :n\geq1\bigr\} :1\leq m\leq r\bigr\} . $$

Observe that

$$y_{n+1}-y_{n}=(\alpha_{n+1}-\alpha_{n}) (x_{n+1}-z_{n})+\alpha _{n}(x_{n+1}-x_{n})+(1- \alpha_{n+1}) (z_{n+1}-z_{n}). $$

It follows from (11) that

$$\begin{aligned} \|y_{n+1}-y_{n}\|\leq{}& | \alpha_{n+1}-\alpha_{n}|\|x_{n+1}-z_{n}\|+ \alpha_{n+1}\|x_{n+1}-x_{n}\| +(1- \alpha_{n+1})\|z_{n+1}-z_{n}\| \\ \leq{}& |\alpha_{n+1}-\alpha_{n}|\|x_{n+1}-z_{n} \|+\alpha_{n+1}\|x_{n+1}-x_{n}\| \\ &{}+(1-\alpha_{n+1}) \Biggl(\|x_{n+1}-x_{n}\| +M \sum_{m=1}^{r}\bigl|\eta_{n+1}^{m}- \eta_{n}^{m}\bigr| \Biggr) \\ \leq{}&|\alpha_{n+1}-\alpha_{n}|\|x_{n+1}-z_{n} \|+\|x_{n+1}-x_{n}\| +M\sum_{m=1}^{r}\bigl| \eta_{n+1}^{m}-\eta_{n}^{m}\bigr|. \end{aligned}$$

(12)

Note that

$$\begin{aligned} \| S_{n+1}y_{n+1} -S_{n}y_{n} \|\leq{}&\| S _{n+1}y_{n+1}-S _{n+1}y_{n} \|+\| S _{n+1}y_{n}-S _{n}y_{n}\| \\ \leq{}&\|y_{n+1}-y_{n}\|+\bigl\| (1-\gamma_{n+1})y_{n}+ \gamma _{n+1}T_{n}y_{n}- \bigl[(1- \gamma_{n})y_{n}+\gamma_{n}T_{n}y_{n} \bigr]\bigr\| \\ \leq{}&\|y_{n+1}-y_{n}\|+\bigl\| (\gamma_{n+1}-\gamma _{n}) (T_{n+1}y_{n}-y_{n})+ \gamma_{n}(T_{n+1}y_{n}-T_{n}y_{n}) \bigr\| \\ \leq{}&\|y_{n+1}-y_{n}\|+|\gamma_{n+1}- \gamma_{n}|\|T_{n+1}y_{n}-y_{n}\| + \gamma_{n}\|T_{n+1}y_{n}-T_{n}y_{n} \| \\ \leq{}& |\alpha_{n+1}-\alpha_{n}|\|x_{n+1}-z_{n} \|+\|x_{n+1}-x_{n}\|+M\sum_{m=1}^{r}\bigl| \eta_{n+1}^{m}-\eta_{n}^{m}\bigr| \\ &{}+|\gamma_{n+1}-\gamma_{n}|\|T_{n+1}y_{n}-y_{n} \|+\gamma_{n}\| T_{n+1}y_{n}-T_{n}y_{n} \|. \end{aligned}$$

(13)

On the other hand,

$$\begin{aligned} &\|x_{n+1} -x_{n} \| \\ &\quad=\bigl\| Q_{C} \bigl(\beta_{n} \gamma fx_{n} +(I-\beta_{n} \mu F ) S_{n}y_{n} \bigr) - Q_{C} \bigl(\beta_{n-1} \gamma f x_{n-1} +(I- \beta_{n-1} \mu F) S_{n-1}y_{n-1} \bigr)\bigr\| \\ &\quad\leq\bigl\| \beta_{n} \gamma fx_{n} +(I- \beta_{n} \mu F ) S_{n}y_{n} - \bigl( \beta_{n-1} \gamma fx_{n-1} +(I-\beta_{n-1} \mu F) S_{n-1} y_{n-1} \bigr)\bigr\| \\ &\quad\leq\bigl\| \beta_{n}\gamma (fx_{n}-fx_{n-1})+( \beta_{n}-\beta_{n-1})\gamma fx_{n-1} \\ &\qquad{}+(I-\beta_{n}\mu F) (S_{n}y_{n}-S_{n-1}y_{n-1})+( \beta_{n}-\beta_{n-1})\mu FS_{n-1}y_{n-1}\bigr\| \\ &\quad\leq\beta_{n}\gamma L\|x_{n}-x_{n-1}\|+| \beta_{n}-\beta_{n-1}| \bigl(\gamma \|fx_{n-1}\|+\mu \|FS_{n-1}y_{n-1}\| \bigr) \\ &\qquad{}+(1-\beta_{n}\tau)\| S_{n}y_{n}-S_{n-1}y_{n-1} \|. \end{aligned}$$

(14)

Substituting (13) into (14), we obtain

$$\begin{aligned} &\|x_{n+1} -x_{n} \| \\ &\quad\leq\beta_{n}\gamma L\|x_{n}-x_{n-1}\| +| \beta_{n}-\beta_{n-1}| \bigl(\gamma \|fx_{n-1}\|+\mu \|FS_{n-1}y_{n-1}\| \bigr) \\ &\qquad{}+(1-\beta_{n}\tau) \Biggl(|\alpha_{n}- \alpha_{n-1}|\|x_{n}-z_{n-1}\|+\| x_{n}-x_{n-1}\| +M\sum_{m=1}^{r}\bigl| \eta_{n}^{m}-\eta_{n-1}^{m}\bigr| \\ &\qquad{}+|\gamma_{n}-\gamma_{n-1}|\|T_{n}y_{n-1}-y_{n-1} \|+\gamma_{n-1}\| T_{n}y_{n-1}-T_{n-1}y_{n-1} \| \Biggr) \\ &\quad\leq\bigl(1-\beta_{n}(\tau-\gamma L)\bigr)\|x_{n}-x_{n-1} \|+ \Biggl(|\beta_{n}-\beta_{n-1}|+|\alpha_{n}- \alpha _{n-1}|+|\gamma_{n}-\gamma_{n-1}| \\ &\qquad{}+\sum_{m=1}^{r}\bigl| \eta_{n}^{m}-\eta_{n-1}^{m}\bigr| \Biggr)M_{1}+\| T_{n}y_{n-1}-T_{n-1}y_{n-1}\|, \end{aligned}$$

(15)

where $M_{1}=\sup_{n\geq 0}\{\gamma\|fx_{n-1}\|+\mu\|FS_{n-1}y_{n-1}\|, \|x_{n}-z_{n-1}\|,\|T_{n}y_{n-1}-y_{n-1}\|, M\}$.

Since $\{T_{n}\}_{n=1}^{\infty}$ satisfies the AKTT-condition, we deduce that

$$ \sum_{n=0}^{\infty} \|T_{n}y_{n-1}-T_{n-1}y_{n-1}\|\leq\sum _{n=0}^{\infty} \sup_{\omega\in\{y_{n-1}\}} \|T_{n}\omega-T_{n-1}\omega\|< \infty. $$

(16)

From (14), (16), and Lemma 2.3, we deduce that

$$ \lim_{n\rightarrow\infty}\|x_{n+1}-x_{n} \|=0. $$

(17)

We observe that

$$\begin{aligned} \|S_{n}y_{n}-x_{n}\|&\leq\|x_{n+1}-x_{n} \|+\|x_{n+1}-S_{n}y_{n}\| \\ &=\|x_{n+1}-x_{n}\|+\bigl\| Q_{C} \bigl( \beta_{n}\gamma fx_{n} +(I-\beta_{n}\mu F)S_{n}y_{n} \bigr)-S_{n}y_{n}\bigr\| \\ &=\|x_{n+1}-x_{n}\|+\bigl\| \bigl(\beta_{n}\gamma fx_{n} +(I-\beta_{n}\mu F)S_{n}y_{n} \bigr)-S_{n}y_{n}\bigr\| \\ &=\|x_{n+1}-x_{n}\|+\beta_{n}\|\gamma fx_{n}-\mu FS_{n}y_{n}\|. \end{aligned}$$

From the condition (iii) and (17), we have

$$ \lim_{n\rightarrow\infty}\|S_{n}y_{n}-x_{n} \|=0. $$

(18)

Step 3. We prove that $\lim_{n\rightarrow \infty}\|T_{n}x_{n}-x_{n}\|=0$.

From Lemma 2.5, we have

$$\begin{aligned} \bigl\| k_{n}^{m}-x^{*}\bigr\| ^{q}&= \bigl\| Q_{C}(x_{n}-\lambda _{m}A_{m}x_{n})-Q_{c} \bigl(x^{*}-\lambda_{m}A_{m}x^{*}\bigr) \bigr\| ^{q} \\ &\leq\bigl\| (I-\lambda_{m}A_{m})x_{n}-(I- \lambda_{m}A_{m})x^{*}\bigr\| ^{q} \\ &\leq\bigl\| x_{n}-x^{*}\bigr\| ^{q}-\lambda_{m} \bigl(q\mu_{m}-C_{q}\lambda_{m}^{q-1} \bigr)\bigl\| A_{m}x_{n}-A_{m}x^{*} \bigr\| ^{q} \end{aligned}$$

and

$$\begin{aligned} \bigl\| z_{n}-x^{*}\bigr\| ^{q}&=\Biggl\| \sum _{m=1}^{r}\eta_{n}^{m}k_{n}^{m}-x^{*} \Biggr\| ^{q} \leq\sum_{m=1}^{r} \eta_{n}^{m}\bigl\| k_{n}^{m}-x^{*} \bigr\| ^{q} \\ &\leq\sum_{m=1}^{r}\eta_{n}^{m} \bigl(\bigl\| x_{n}-x^{*}\bigr\| ^{q} -\lambda_{m} \bigl(q\mu_{m}-C_{q}\lambda_{m}^{q-1} \bigr)\bigl\| A_{m}x_{n}-A_{m}x^{*}\bigr\| ^{q} \bigr) \\ &=\bigl\| x_{n}-x^{*}\bigr\| ^{q}-\sum _{m=1}^{r}\eta_{n}^{m} \lambda_{m}\bigl(q\mu_{m}-C_{q} \lambda_{m}^{q-1}\bigr)\bigl\| A_{m}x_{n}-A_{m}x^{*} \bigr\| ^{q}. \end{aligned}$$

By the convexity of $\|\cdot\|$, for all $q>1$, and Lemma 2.5, we obtain

$$\begin{aligned} &\bigl\| x_{n+1}-x^{*}\bigr\| ^{q} \\ &\quad=\bigl\| Q_{C} \bigl(\beta_{n}\gamma fx_{n} +(I-\beta_{n}\mu F) S_{n}y_{n} \bigr)-x^{*}\bigr\| ^{q} \\ &\quad\leq\bigl\| \bigl(\beta_{n}\gamma fx_{n} +(I- \beta_{n}\mu F) S_{n}y_{n} \bigr)-x^{*} \bigr\| ^{q} \\ &\quad=\bigl\| \beta_{n}(\gamma fx_{n}-\mu FS_{n}y_{n})+S_{n}y_{n}-x^{*} \bigr\| ^{q} \\ &\quad\leq\bigl\| S_{n}y_{n}-x^{*}\bigr\| ^{q}+q \bigl\langle \beta_{n}(\gamma fx_{n}-\mu FS_{n}y_{n}), J_{q}\bigl(S_{n}y_{n}-x^{*}\bigr)\bigr\rangle +C_{q}\bigl\| \beta_{n}(\gamma fx_{n}-\mu FS_{n}y_{n})\bigr\| ^{q} \\ &\quad\leq\bigl\| y_{n}-x^{*}\bigr\| ^{q}+q \beta_{n}\|\gamma fx_{n}-\mu FS_{n}y_{n} \|\bigl\| S_{n}y_{n}-x^{*}\bigr\| ^{q-1}+C_{q} \beta_{n}^{q}\|\gamma fx_{n}-\mu FS_{n}y_{n}\|^{q} \\ &\quad\leq\bigl\| \beta_{n}x_{n}+(1-\beta_{n})z_{n}-x^{*} \bigr\| ^{q}+\beta_{n}M_{2} \\ &\quad\leq\bigl\| \beta_{n}\bigl(x_{n}-x^{*}\bigr)+(1- \beta_{n}) \bigl(z_{n}-x^{*}\bigr) \bigr\| ^{q}+\beta _{n}M_{2} \\ &\quad\leq\beta_{n}\bigl\| x_{n}-x^{*} \bigr\| ^{q}+(1-\beta_{n})\bigl\| z_{n}-x^{*} \bigr\| ^{q}+\beta _{n}M_{2}, \\ &\quad\leq\beta_{n}\bigl\| x_{n}-x^{*} \bigr\| ^{q}+(1-\beta_{n}) \Biggl[\bigl\| x_{n}-x^{*} \bigr\| ^{q}- \sum_{m=1}^{r} \eta_{n}^{m}\lambda_{m}\bigl(q\mu_{m} \\ &\qquad{}-C_{q}\lambda_{m}^{q-1}\bigr) \bigl\| A_{m}x_{n}-A_{m}x^{*}\bigr\| ^{q} \Biggr]+\beta _{n}M_{2}, \\ &\quad\leq\bigl\| x_{n}-x^{*}\bigr\| ^{q}-(1- \beta_{n})\sum_{m=1}^{r} \eta_{n}^{m}\lambda _{m}\bigl(q\mu_{m} -C_{q}\lambda_{m}^{q-1}\bigr) \bigl\| A_{m}x_{n}-A_{m}x^{*} \bigr\| ^{q}+\beta_{n}M_{2}, \end{aligned}$$

where

$$M_{2}=\sup_{n\geq0}\bigl\{ q\|\gamma fx_{n}- \mu FS_{n}y_{n}\|\bigl\| S_{n}y_{n}-x^{*} \bigr\| ^{q-1}+C_{q}\beta_{n}^{q-1}\|\gamma fx_{n}-\mu FS_{n}y_{n}\|^{q}\bigr\} < \infty. $$

By the fact that $a^{r}-b^{r}\leq ra^{r-1}(a-b)$, $\forall r\geq1$, we get

$$\begin{aligned} &(1-\beta_{n})\sum_{m=1}^{r} \eta_{n}^{m}\lambda_{m}\bigl(q\mu _{m}-C_{q}\lambda_{m}^{q-1}\bigr) \bigl\| A_{m}x_{n}-A_{m}x^{*}\bigr\| ^{q} \\ &\quad\leq\bigl\| x_{n}-x^{*}\bigr\| ^{q}- \bigl\| x_{n+1}-x^{*}\bigr\| ^{q}+\beta_{n}M_{2} \\ &\quad\leq q\bigl\| x_{n}-x^{*}\bigr\| ^{q-1}\bigl( \bigl\| x_{n}-x^{*}\bigr\| -\bigl\| x_{n+1}-x^{*}\bigr\| \bigr)+ \beta _{n}M_{2} \\ &\quad\leq q\bigl\| x_{n}-x^{*}\bigr\| ^{q-1} \|x_{n}-x_{n+1}\|+\beta_{n}M_{2}. \end{aligned}$$

Since $0<\lambda_{m}<(\frac{q\mu_{m}}{C_{q}})^{\frac{1}{q-1}}$, from (17) and (iii) and the fact that $\{x_{n}\}$ is bounded we have

$$ \lim_{n\rightarrow\infty}\bigl\| A_{m}x_{n}-A_{m}x^{*} \bigr\| =0,\quad \forall m, 1\leq m\leq r. $$

(19)

Setting $r_{m}= \sup \{\| x_{n}-x^{*}\|,\| k_{n}^{m}-x^{*}\|\}$, we have from Lemmas 2.1 and 2.6

$$\begin{aligned} \bigl\| k_{n}^{m}-x^{*}\bigr\| ^{2}={}&\bigl\| Q_{C}(I-\lambda_{m}A_{m})x_{n}- Q_{C}(I-\lambda_{m}A_{m}) x^{*} \bigr\| ^{2} \\ \leq{}&\bigl\langle x_{n}-\lambda_{m}A_{m}x_{n}- \bigl(x^{*}-\lambda_{m}A_{m}x^{*}\bigr), j\bigl(k_{n}^{m}-x^{*}\bigr)\bigr\rangle \\ \leq{}&\bigl\langle x_{n}-x^{*}, j\bigl(k_{n}^{m}-x^{*} \bigr)\bigr\rangle +\lambda_{m}\bigl\langle A_{m}x^{*}-A_{m}x_{n}, j\bigl(k_{n}^{m}-x^{*}\bigr)\bigr\rangle \\ \leq{}&\frac{1}{2} \bigl[\bigl\| x_{n}-x^{*} \bigr\| ^{2}+\bigl\| k_{n}^{m}-x^{*}\bigr\| ^{2}-g_{m}\bigl(\bigl\| x_{n}-x^{*}-k_{n}^{m}+x^{*} \bigr\| \bigr) \bigr] \\ &{}+\lambda_{m}\bigl\langle A_{m}x^{*}-A_{m}x_{n}, j\bigl(k_{n}^{m}-x^{*}\bigr)\bigr\rangle , \end{aligned}$$

where $g_{m}:[0,2r_{m})\rightarrow[0,\infty)$ is a continuous, strictly increasing, and convex function such that $g_{m}(0)=0$ for all $1\leq m\leq r$. Hence, we have

$$ \bigl\| k_{n}^{m}-x^{*}\bigr\| ^{2} \leq\bigl\| x_{n}-x^{*}\bigr\| ^{2}-g_{m}\bigl( \bigl\| x_{n}-k_{n}^{m}\bigr\| \bigr) +2\lambda_{m} \bigl\| A_{m}x^{*}-A_{m}x_{n}\bigr\| \bigl\| k_{n}^{m}-x^{*}\bigr\| $$

(20)

for all m, with $1\leq m\leq r$. On the other hand, we have

$$\|z_{n}-x_{n}\|^{2}\leq\Biggl\| \sum _{m=1}^{r}\eta_{n}^{m}k_{n}^{m}-x_{n} \Biggr\| ^{2} \leq\sum_{m=1}^{r} \eta_{n}^{m}\bigl\| k_{n}^{m}-x_{n} \bigr\| ^{2}. $$

Since $g_{m}$ is increasing and convex by using (20) we have

$$\begin{aligned} &g_{m}\bigl(\|z_{n}-x_{n}\|^{2}\bigr) \\ &\quad\leq g_{m}\Biggl(\sum_{m=1}^{r} \eta_{n}^{m}\bigl\| k_{n}^{m}-x_{n} \bigr\| ^{2}\Biggr)\leq\sum_{m=1}^{r} \eta_{n}^{m}g_{m}\bigl(\bigl\| k_{n}^{m}-x_{n} \bigr\| ^{2}\bigr) \\ &\quad\leq\sum_{m=1}^{r} \eta_{n}^{m} \bigl[\bigl\| x_{n}-x^{*} \bigr\| ^{2}-\bigl\| k_{n}^{m}-x^{*} \bigr\| ^{2}+2\lambda_{m}\bigl\| A_{m}x^{*}-A_{m}x_{n} \bigr\| \bigl\| k_{n}^{m}-x^{*}\bigr\| \bigr] \\ &\quad=\bigl\| x_{n}-x^{*}\bigr\| ^{2}-\sum _{m=1}^{r}\eta_{n}^{m}\bigl\| k_{n}^{m}-x^{*}\bigr\| ^{2}+2\sum _{m=1}^{r}\eta_{n}^{m} \lambda_{m}\bigl\| A_{m}x^{*}-A_{m}x_{n} \bigr\| \bigl\| k_{n}^{m}-x^{*}\bigr\| . \end{aligned}$$

Thus we have

$$\sum_{m=1}^{r}\eta_{n}^{m} \bigl\| k_{n}^{m}-x^{*}\bigr\| ^{2}\leq \bigl\| x_{n}-x^{*}\bigr\| ^{2}-g_{m}\bigl( \bigl\| z_{n}-x_{n}\bigr\| ^{2}\bigr)+2\sum _{m=1}^{r}\eta_{n}^{m} \lambda_{m} \bigl\| A_{m}x^{*}-A_{m}x_{n} \bigr\| \bigl\| k_{n}^{m}-x^{*}\bigr\| . $$

Thanks to Lemma 2.5 we have

$$\begin{aligned} &\bigl\| x_{n+1}-x^{*}\bigr\| ^{2} \\ &\quad=\bigl\| Q_{C} \bigl(\beta_{n}\gamma fx_{n} +(I-\beta_{n}\mu F) S_{n}y_{n} \bigr)-x^{*}\bigr\| ^{2} \\ &\quad\leq\bigl\| \bigl(\beta_{n}\gamma fx_{n} +(I- \beta_{n}\mu F) S_{n}y_{n}\bigr)-x^{*} \bigr\| ^{2} \\ &\quad=\bigl\| \beta_{n}(\gamma fx_{n}-\mu FS_{n}y_{n})+S_{n}y_{n}-x^{*} \bigr\| ^{2} \\ &\quad\leq\bigl\| S_{n}y_{n}-x^{*}\bigr\| ^{2}+2 \bigl\langle \beta_{n}(\gamma fx_{n}-\mu FS_{n}y_{n}), j_{q} \bigl(\beta_{n}(\gamma fx_{n}-\mu FS_{n}y_{n})+ S_{n}y_{n}-x^{*} \bigr)\bigr\rangle \\ &\quad\leq\bigl\| y_{n}-x^{*}\bigr\| ^{2}+ \beta_{n}M_{3} \\ &\quad=\bigl\| \beta_{n}x_{n}+(1-\beta_{n})z_{n}-x^{*} \bigr\| ^{2}+\beta_{n}M_{3} \\ &\quad\leq\beta_{n}\bigl\| x_{n}-x^{*} \bigr\| ^{2}+(1-\beta_{n})\bigl\| z_{n}-x^{*} \bigr\| ^{2}+\beta _{n}M_{3} \\ &\quad\leq\beta_{n}\bigl\| x_{n}-x^{*} \bigr\| ^{2}+(1-\beta_{n}) \Biggl(\Biggl\| \sum _{m=1}^{r}\eta_{n}^{m}k_{n}^{m}-x^{*} \Biggr\| \Biggr)^{2} +\beta_{n}M_{3} \\ &\quad\leq\beta_{n}\bigl\| x_{n}-x^{*} \bigr\| ^{2}+(1-\beta_{n})\sum_{m=1}^{r} \eta _{n}^{m}\bigl\| k_{n}^{m}-x^{*} \bigr\| ^{2}+\beta_{n}M_{3} \\ &\quad\leq\beta_{n}\bigl\| x_{n}-x^{*} \bigr\| ^{2}+(1-\beta_{n}) \Biggl(\bigl\| x_{n}-x^{*} \bigr\| ^{2}-g_{m}\bigl(\|z_{n}-x_{n} \|^{2}\bigr) \\ &\qquad{}+2\sum_{m=1}^{r} \eta_{n}^{m}\lambda_{m} \bigl\| A_{m}x^{*}-A_{m}x_{n} \bigr\| \bigl\| k_{n}^{m}-x^{*}\bigr\| \Biggr)+ \beta_{n}M_{3} \\ &\quad\leq\bigl\| x_{n}-x^{*}\bigr\| ^{2}-(1- \beta_{n})g_{m}\bigl(\|z_{n}-x_{n}\| ^{2}\bigr)+2(1-\beta_{n})\sum_{m=1}^{r} \eta_{n}^{m}\lambda_{m} \\ &\qquad{}\times\bigl\| A_{m}x^{*}-A_{m}x_{n} \bigr\| \bigl\| k_{n}^{m}-x^{*}\bigr\| +\beta_{n}M_{3}, \end{aligned}$$

where $M_{3}=\sup_{n\geq0}\{2\langle\gamma fx_{n} -\mu FS_{n}y_{n}, j_{q} (\beta_{n}(\gamma fx_{n}-\mu fS_{n}y_{n})+S_{n}y_{n}-x^{*} )\rangle\}$.

This in turn implies that

$$\begin{aligned} (1-\beta_{n})g_{m}\bigl(\|z_{n}-x_{n} \|^{2}\bigr)\leq{}&\bigl\| x_{n}-x^{*}\bigr\| ^{2}-\bigl\| x_{n+1}-x^{*}\bigr\| ^{2} \\ &{}+2(1-\beta_{n})\sum_{m=1}^{r} \eta_{n}^{m}\lambda_{m} \bigl\| A_{m}x^{*}-A_{m}x_{n} \bigr\| \bigl\| k_{n}^{m}-x^{*}\bigr\| +\beta_{n}M_{3} \\ \leq{}&\|x_{n}-x_{n+1}\|\bigl(\bigl\| x_{n}-x^{*} \bigr\| +\bigl\| x_{n+1}-x^{*}\bigr\| \bigr) \\ &{}+2(1-\beta_{n})\sum_{m=1}^{r} \eta_{n}^{m}\lambda_{m} \bigl\| A_{m}x^{*}-A_{m}x_{n} \bigr\| \bigl\| k_{n}^{m}-x^{*}\bigr\| +\beta_{n}M_{3}. \end{aligned}$$

In view of (ii), (iii), (17), and (19) we have

$$\lim_{n\rightarrow\infty}g_{m}\bigl(\|z_{n}-x_{n} \|^{2}\bigr)=0. $$

By the properties of $g_{m}$, we get

$$ \lim_{n\rightarrow\infty}\|z_{n}-x_{n} \|^{2}=0. $$

(21)

On the other hand,

$$\begin{aligned} \|S_{n}x_{n}-x_{n}\|&\leq\|S_{n}x_{n}-S_{n}y_{n} \|+\|S_{n}y_{n}-x_{n}\| \\ &\leq\|x_{n}-y_{n}\|+\|S_{n}y_{n}-x_{n} \| \\ &\leq\|x_{n}-z_{n}\|+\|z_{n}-y_{n} \|+\|S_{n}y_{n}-x_{n}\| \\ &=\|x_{n}-z_{n}\|+\beta_{n} \|x_{n}-z_{n}\|+\|S_{n}y_{n}-x_{n} \|. \end{aligned}$$

It follows from (21), (18), and (iii) that

$$ \lim_{n\rightarrow\infty}\|S_{n}x_{n}-x_{n} \|=0. $$

(22)

Next, we show that $\|x_{n}-Sx_{n}\|\rightarrow0$ as $n\rightarrow\infty$. For any bounded subset B of C, we observe that

$$\begin{aligned} \sup\|S_{n+1}\omega-S_{n}\omega\|={}&\sup _{\omega\in B}\bigl\| \gamma_{n+1}\omega+(1-\gamma_{n+1})T_{n+1} \omega-\bigl(\gamma_{n}\omega +(1-\gamma_{n})T_{n} \omega\bigr)\bigr\| \\ \leq{}&|\gamma_{n+1}-\gamma_{n}|\sup_{\omega\in B}| \omega|+(1-\gamma_{n+1})\sup_{\omega\in B}\|T_{n+1} \omega-T_{n}\omega\| \\ &{}+|\gamma_{n+1}-\gamma_{n}|\sup_{\omega\in B} \|T_{n}\omega\| \\ \leq{}&|\gamma_{n+1}-\gamma_{n}|M_{3}+\sup _{\omega\in B}\|T_{n+1}\omega-T_{n}\omega\|, \end{aligned}$$

where $M_{3}=\sup_{n\geq1}\{\|\omega\|, \|T_{n}\omega\|\}$. By (v) and the fact that $\{T_{n}\}$ satisfies the AKTT-condition, we have

$$\sum_{n=1}^{\infty}\sup_{\omega\in B} \|S_{n+1}\omega-S_{n}\omega\|< \infty, $$

that is, $\{S_{n}\}$ satisfies the AKTT-condition. Now we define the nonexpansive mapping $S:C\rightarrow C$ by $Sx=\lim_{n\rightarrow \infty}S_{n}x$ for all $x\in C$. Since $\{\gamma_{n}\}$ is bounded, there exists a subsequence $\{\gamma_{n_{i}}\}$ of $\{\gamma_{n}\}$ such that $\gamma_{n_{i}}\rightarrow\nu$ as $i\rightarrow\infty$. It follows that

$$\begin{aligned} Sx=\lim_{i\rightarrow\infty}S_{n_{i}}x=\lim_{i\rightarrow\infty} \bigl[\gamma_{n_{i}}x+(1-\gamma_{n_{i}})T_{n_{i}}x\bigr] =\nu x+(1-\nu)Tx, \quad \forall x\in C. \end{aligned}$$

That is $F(S)=F(T)$. Hence $F(S)=\bigcap_{n=1}^{\infty}F(T_{n})=\bigcap_{n=1}^{\infty}F(S_{n})$. On the other hand we have

$$\begin{aligned} \|x_{n}-Sx_{n}\|&\leq\|x_{n}-S_{n}x_{n} \|+\|S_{n}x_{n}-Sx_{n}\| \\ &\leq\|x_{n}-S_{n}x_{n}\|+\sup _{\omega\in\{x_{n}\}}\|S_{n}\omega -S\omega\|. \end{aligned}$$

This implies by Lemma 2.8 and (22) that

$$ \lim_{n\rightarrow\infty}\|x_{n}-Sx_{n} \|=0. $$

(23)

Now we define a mapping $h:C\rightarrow C$ by

$$hx=\sum_{m=1}^{r}\eta^{m}P_{C}(I- \lambda_{m}A_{m})x,\quad \forall x\in C, $$

where $\eta^{m}=\lim_{n\rightarrow\infty}\eta_{n}^{m}$. From Lemma 2.9, h is nonexpansive such that

$$F(h)=\bigcap_{m=1}^{r}F \bigl(P_{C}(I-\lambda_{m}A_{m})\bigr)=\bigcap _{m=1}^{r}VI(C,A_{m})=\Omega. $$

Next, we define a mapping $U:C\rightarrow C$ by $Ux=\delta Sx+(1-\delta)hx$, where $\delta\in(0,1)$ is a constant. Then by Lemma 2.2, U is a nonexpansive and

$$F(U)=F(S)\cap F(h)=\bigcap_{n=1}^{\infty}F(T_{n}) \cap\Omega=F=F(T)\cap\Omega. $$

Note that

$$\begin{aligned} \|x_{n}-hx_{n}\|&\leq\|x_{n}-z_{n} \|+\|z_{n}-hx_{n}\| \\ &\leq\|x_{n}-z_{n}\|+\biggl\| \sum_{n=1}^{m} \eta_{n}^{m}P_{C}(I-\lambda_{m}A_{m})x_{n} -\sum_{m=1}^{r}\eta^{m}P_{C}(I- \lambda_{m}A_{m})x_{n}\biggr\| \\ &\leq\|x_{n}-z_{n}\|+M\sum_{m=1}^{r}\bigl| \eta_{n}^{m}-\eta^{m}\bigr|. \end{aligned}$$

In view of restriction (ii), we find from (21) that

$$ \lim_{n\rightarrow\infty}\|x_{n}-hx_{n} \|=0. $$

(24)

Setting $x_{t}=Q_{C}[t\gamma fx_{t}+(I-t\mu F)Ux_{t}]$, it follows from Lemma 2.13 that $\{x_{t}\}$ converges strongly to a point $x^{*}\in F(U)=F$, in which $x^{*}$ is the unique solution of the variational inequality (9). From (23) and (24), we have

$$\begin{aligned} \|x_{n}-Ux_{n}\|&=\bigl\| \delta(x_{n}-Sx_{n})+(1- \delta) (x_{n}-hx_{n})\bigr\| \\ &\leq\delta\|x_{n}-Sx_{n}\|+(1-\delta) \|x_{n}-hx_{n}\|\rightarrow0. \end{aligned}$$

Step 4. We show that

$$\limsup \bigl\langle (\gamma f -\mu F)x^{*} , j_{q}\bigl(x_{n} -x^{*}\bigr)\bigr\rangle \leq0, $$

where $x^{*}$ is a solution of the variational inequality (9). To show this, we can choose a subsequence $\{ x_{n_{j}}\}$ of $\{ x_{n}\}$ such that

$$\limsup_{n\rightarrow\infty} \bigl\langle (\gamma f -\mu F)x^{*} , j_{q}\bigl(x_{n} -x^{*}\bigr)\bigr\rangle =\lim _{j\rightarrow\infty} \bigl\langle (\gamma f-\mu F)x^{*} , j_{q} \bigl(x_{n_{j}} -x^{*}\bigr)\bigr\rangle . $$

By reflexivity of a Banach space X and since $\{ x_{n} \}$ is bounded, there exists a subsequence $\{ x_{n_{j}}\}$ of $\{x_{n}\}$ which converges weakly to z. Without loss of generality, we can assume that $x_{n_{j}} \rightharpoonup z$. Since $\| x_{n} -Ux_{n}\|\rightarrow0$ by step 3, we obtain $z=Uz$ and we have $z\in F(U)$. Since Banach space X has a weakly sequentially continuous generalized duality mapping, we obtain

$$\begin{aligned} \limsup_{n\rightarrow\infty}\bigl\langle (\gamma f -\mu F)x^{*} , j_{q}\bigl(x_{n} -x^{*}\bigr)\bigr\rangle &=\lim _{j\rightarrow\infty} \bigl\langle (\gamma f -\mu F)x^{*} , j_{q} \bigl(x_{n_{j}} -x^{*}\bigr)\bigr\rangle \\ &= \bigl\langle (\gamma f -\mu F)x^{*} , j_{q}\bigl(z -x^{*}\bigr)\bigr\rangle \leq0. \end{aligned}$$

Step 5. Finally, we show that $\lim_{n\rightarrow\infty} \| x_{n} -x^{*} \|=0$. Setting $h_{n}=\beta_{n}\gamma fx_{n}+(I-\beta_{n}\mu F)S_{n}y_{n}$, $\forall n\geq1$. Then we can rewrite $x_{n+1}=Q_{C}h_{n}$. It follows from Lemmas 2.1 and 2.4 that

$$\begin{aligned} &\bigl\| x_{n+1} -x^{*} \bigr\| ^{q} \\ &\quad=\bigl\langle Q_{C}h_{n}-h_{n},j_{q}\bigl(x_{n+1}-x^{*}\bigr) \bigr\rangle +\bigl\langle h_{n}-x^{*}, j_{q} \bigl(x_{n+1}-x^{*}\bigr)\bigr\rangle \\ &\quad\leq\bigl\langle h_{n}-x^{*}, j_{q}\bigl(x_{n+1}-x^{*}\bigr) \bigr\rangle \\ &\quad=\beta_{n}\bigl\langle \gamma fx_{n}-\mu Fx^{*}, j_{q}\bigl(x_{n+1}-x^{*}\bigr) \bigr\rangle +\bigl\langle (I-\beta_{n}\mu F) \bigl(S_{n}y_{n}-x^{*} \bigr), j_{q}\bigl(x_{n+1}-x^{*}\bigr)\bigr\rangle \\ &\quad=\beta_{n}\bigl\langle \gamma\bigl(fx_{n}-fx^{*} \bigr), j_{q}\bigl(x_{n+1}-x^{*}\bigr)\bigr\rangle + \beta_{n}\bigl\langle \gamma fx^{*}-\mu Fx^{*}, j_{q}\bigl(x_{n+1}-x^{*}\bigr)\bigr\rangle \\ &\qquad{}+\bigl\langle (I-\beta_{n}\mu F) \bigl(S_{n}y_{n}-x^{*} \bigr), j_{q}\bigl(x_{n+1}-x^{*}\bigr)\bigr\rangle \\ &\quad\leq\beta_{n}\gamma L\bigl\| x_{n}-x^{*}\bigr\| \bigl\| x_{n+1}-x^{*}\bigr\| ^{q-1} +\beta_{n}\bigl\langle \gamma fx^{*}-\mu Fx^{*}, j_{q} \bigl(x_{n+1}-x^{*}\bigr)\bigr\rangle \\ &\qquad{}+(1-\beta_{n}\tau)\bigl\| y_{n}-x^{*}\bigr\| \bigl\| x_{n+1}-x^{*}\bigr\| ^{q-1} \\ &\quad\leq\beta_{n}\gamma L\bigl\| x_{n}-x^{*}\bigr\| \bigl\| x_{n+1}-x^{*}\bigr\| ^{q-1} +\beta_{n}\bigl\langle \gamma fx^{*}-\mu Fx^{*}, j_{q} \bigl(x_{n+1}-x^{*}\bigr)\bigr\rangle \\ &\qquad{}+(1-\beta_{n}\tau)\bigl\| x_{n}-x^{*}\bigr\| \bigl\| x_{n+1}-x^{*}\bigr\| ^{q-1} \\ &\quad=\bigl(1-(\tau-\gamma L)\beta_{n}\bigr)\bigl\| x_{n}-x^{*} \bigr\| \bigl\| x_{n+1}-x^{*}\bigr\| ^{q-1} +\beta_{n}\bigl\langle \gamma fx^{*}-\mu Fx^{*}, j_{q}\bigl(x_{n+1}-x^{*}\bigr) \bigr\rangle \\ &\quad\leq\bigl(1-(\tau-\gamma L)\beta_{n}\bigr) \biggl[ \frac{1}{q}\bigl\| x_{n}-x^{*}\bigr\| ^{q}+ \frac{q-1}{q} \bigl\| x_{n+1}-x^{*}\bigr\| ^{q-1} \biggr] \\ &\qquad{} +\beta_{n}\bigl\langle \gamma fx^{*}-\mu Fx^{*}, j_{q}\bigl(x_{n+1}-x^{*}\bigr) \bigr\rangle , \end{aligned}$$

which implies that

$$\begin{aligned} \bigl\| x_{n+1}-x^{*}\bigr\| ^{q}\leq{}&\frac{1-(\tau-\gamma L)\beta_{n}}{1+(q-1)(\tau-\gamma)\beta_{n}} \bigl\| x_{n}-x^{*}\bigr\| ^{q} \\ &{}+\frac{q\beta_{n}}{1+(q-1)(\tau-\gamma L)\beta_{n}} +\bigl\langle \gamma fx^{*}-\mu Fx^{*}, j_{q}\bigl(x_{n+1}-x^{*}\bigr)\bigr\rangle \\ \leq{}&\bigl(1-(\tau-\gamma L)\beta_{n}\bigr)\bigl\| x_{n}-x^{*} \bigr\| ^{q} \\ &{}+\frac{q\beta_{n}}{1+(q-1)(\tau-\gamma L)\beta_{n}}+\bigl\langle \gamma fx^{*}-\mu Fx^{*}, j_{q}\bigl(x_{n+1}-x^{*}\bigr)\bigr\rangle . \end{aligned}$$

Put $a_{n}=\beta_{n}(\tau-\gamma L)$ and $b_{n}=\frac{q}{(1+(q-1)(\tau-\gamma L)\beta_{n})(\tau-\gamma L)}+\langle\gamma fx^{*}-\mu Fx^{*}, j_{q}(x_{n+1}-x^{*})\rangle$. Applying Lemma 2.3, we obtain $x_{n}\rightarrow x^{*}$ as $n\rightarrow\infty$. This completes the proof. □

Remark 3.2

Theorem 3.1 improves and extends Theorem 2.1; see Cho and Kang [21]. Especially, our results extend the above results from Hilbert space to a more general q-uniformly smooth Banach space.