Open Access

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

Fixed Point Theory and Applications20152015:15

https://doi.org/10.1186/s13663-015-0264-y

Received: 11 September 2014

Accepted: 2 January 2015

Published: 1 February 2015

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.

Keywords

fixed point q-uniformly smooth Banach space variational inequality iterative algorithm inverse strongly accretive operator

MSC

47H09 47H10 49J05

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:
  1. (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
    $$\bigl\langle (I-T)x -(I-T)y , j_{q}(x-y)\bigr\rangle \geq\lambda\bigl\| (I-T)x -(I-T)y\bigr\| ^{q}. $$
     
  2. (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 [59].
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:
  1. (a)

    Q is sunny and nonexpansive.

     
  2. (b)

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

     
  3. (c)

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

     
  4. (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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (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:
  1. (i)

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

     
  2. (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:
  1. (i)

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

     
  2. (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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (iii)

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

     
  4. (iv)

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

     
  5. (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.

Declarations

Acknowledgements

The authors wish to thank the referees for their helpful comments, which notably improved the presentation of this manuscript.

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors’ Affiliations

(1)
Department of Mathematics, Tafresh University

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