Open Access

Mann-type hybrid steepest-descent method for three nonlinear problems

  • Abdul Latif1Email author,
  • Abdulaziz SM Alofi1,
  • Abdullah E Al-Mazrooei2 and
  • Jen-Chih Yao1, 3
Journal of Inequalities and Applications20152015:282

https://doi.org/10.1186/s13660-015-0807-0

Received: 12 August 2015

Accepted: 30 August 2015

Published: 17 September 2015

Abstract

We introduce a Mann-type hybrid steepest-descent iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of solutions of finitely many variational inequalities, and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical variational inequality.

Keywords

Mann-type hybrid steepest-descent method general mixed equilibrium general system of variational inequalities nonexpansive mapping strict pseudocontraction inverse-strongly monotone mapping

MSC

49J30 47H09 47J20 49M05

1 Introduction

Let H be a real Hilbert space with inner product \(\langle\cdot,\cdot \rangle\) and norm \(\|\cdot\|\), C be a nonempty closed convex subset of H and \(P_{C}\) be the metric projection of H onto C. Let \(S:C\to C\) be a self-mapping on C. We denote by \(\operatorname{Fix}(S)\) the set of fixed points of S and by R the set of all real numbers. A mapping \(A:C\to H\) is called L-Lipschitz continuous if there exists a constant \(L\geq 0\) such that
$$\|Ax-Ay\|\leq L\|x-y\|,\quad \forall x,y\in C. $$
A mapping \(T:C\to C\) is called ξ-strictly pseudocontractive if there exists a constant \(\xi\in[0,1)\) such that
$$\|Tx-Ty\|^{2}\leq\|x-y\|^{2}+\xi \bigl\Vert (I-T)x-(I-T)y \bigr\Vert ^{2},\quad \forall x,y\in C. $$
Let \(A:C\to H\) be a nonlinear mapping on C. We consider the following variational inequality problem (VIP) [1]: find a point \(\bar{x}\in C\) such that
$$ \langle A\bar{x},y-\bar{x}\rangle\geq0, \quad \forall y\in C. $$
(1.1)
The solution set of VIP (1.1) is denoted by \(\operatorname{VI}(C,A)\).
The general mixed equilibrium problem (GMEP) (see, e.g., [2]) is to find \(x\in C\) such that
$$ {\varTheta }(x,y)+h(x,y)\geq0, \quad \forall y\in C, $$
(1.2)
where \({\varTheta },h:C\times C\to{\mathbf{R}}\) are two bi-functions. We denote the set of solutions of GMEP (1.2) by \(\operatorname{GMEP}({\varTheta },h)\). We assume as in [3] that \({\varTheta }:C\times C\to{\mathbf{R}}\) is a bi-function satisfying conditions (θ1)-(θ3) and \(h:C\times C\to{\mathbf{R}}\) is a bi-function with restrictions (h1)-(h3), where
(θ1): 

\({\varTheta }(x,x)=0\) for all \(x\in C\);

(θ2): 
Θ is monotone (i.e., \({\varTheta }(x,y)+{\varTheta }(y,x)\leq0\), \(\forall x,y\in C\)) and upper hemicontinuous in the first variable, i.e., for each \(x,y,z\in C\),
$$\limsup_{t\to0^{+}}{\varTheta } \bigl(tz+(1-t)x,y \bigr)\leq{ \varTheta }(x,y); $$
(θ3): 

Θ is lower semicontinuous and convex in the second variable;

(h1): 

\(h(x,x)=0\) for all \(x\in C\);

(h2): 

h is monotone and weakly upper semicontinuous in the first variable;

(h3): 

h is convex in the second variable.

For \(r>0\) and \(x\in H\), let \(T_{r}:H\to2^{C}\) be a mapping defined by
$$T_{r}x= \biggl\{ z\in C:{\varTheta }(z,y)+h(z,y)+\frac{1}{r}\langle y-z,z-x\rangle \geq0,\forall y\in C \biggr\} $$
called the resolvent of Θ and h.
Let \(F_{1},F_{2}:C\to H\) be two mappings. Consider the following general system of variational inequalities (GSVI) of finding \((x^{*},y^{*})\in C\times C\) such that
$$ \left \{ \textstyle\begin{array}{l} \langle\nu_{1}F_{1}y^{*}+x^{*}-y^{*},x-x^{*}\rangle\geq0,\quad \forall x\in C,\\ \langle\nu_{2}F_{2}x^{*}+y^{*}-x^{*},x-y^{*}\rangle\geq0,\quad \forall x\in C, \end{array}\displaystyle \right . $$
(1.3)
where \(\nu_{1}>0\) and \(\nu_{2}>0\) are two constants. The solution set of GSVI (1.3) is denoted by \(\operatorname{GSVI}(C,F_{1},F_{2})\).
If C is the fixed point set \(\operatorname{Fix}(T)\) of a nonexpansive mapping T and S is another nonexpansive mapping (not necessarily with fixed points), then VIP (1.1) becomes the variational inequality problem of finding \(x^{*}\in\operatorname{Fix}(T)\) such that
$$ \bigl\langle (I-S)x^{*},x-x^{*} \bigr\rangle \geq0,\quad \forall x\in \operatorname{Fix}(T). $$
(1.4)
This problem, introduced by Mainge and Moudafi [4, 5], is called the hierarchical fixed point problem. It is clear that if S has fixed points, then they are solutions of VIP (1.4).

During the 1980s and 1990s, the system of variational inequalities used as tools to solve Nash equilibrium problems. See, for example, [68] and the references therein. On the similar lines, the results of this paper can be applicable to solve Nash equilibrium problem for two person game. In the recent past, several iterative methods have been proposed and analyzed to three nonlinear problems, namely, system of variational inequalities, generalized mixed equilibrium problems and variational inequalities; see, for example, [911] and the references therein.

In this paper, we will introduce a Mann-type hybrid steepest-descent iterative algorithm for finding a common element of the solution set \(\operatorname{GMEP}({\varTheta },h)\) of GMEP (1.2), the solution set \(\operatorname{GSVI}(C,F_{1},F_{2})\) (i.e., Ξ) of GSVI (1.3), the solution set \(\bigcap^{M}_{k=1}\operatorname{VI}(C,A_{k})\) of finitely many variational inequalities for inverse-strongly monotone mappings \(A_{k}:C\to H\), \(k=1,\ldots,M\), and the common fixed point set \(\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\operatorname{Fix}(T)\) of finitely many nonexpansive mappings \(S_{i}:C\to C\), \(i=1,\ldots,N\) and a strictly pseudocontractive mapping \(T:C\to C\), in the setting of the infinite-dimensional Hilbert space. The iterative algorithm is based on Korpelevich’s extragradient method, the viscosity approximation method [12], Mann’s iteration method, and the hybrid steepest-descent method. Our aim is to prove that the iterative algorithm converges strongly to a common element of these sets, which also solves some hierarchical variational inequality. We observe that related results have been derived in [4, 5, 13, 14].

2 Preliminaries

Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by \(\langle\cdot, \cdot\rangle\) and \(\|\cdot\|\), respectively. Let C be a nonempty, closed, and convex subset of H. We write \(x_{n}\rightharpoonup x\) to indicate that the sequence \(\{x_{n}\}\) converges weakly to x and \(x_{n}\to x\) to indicate that the sequence \(\{x_{n}\}\) converges strongly to x. Moreover, we use \(\omega_{w}(x_{n})\) to denote the weak ω-limit set of the sequence \(\{x_{n}\}\) and \(\omega_{s}(x_{n})\) to denote the strong ω-limit set of the sequence \(\{x_{n}\}\), i.e.,
$$\omega_{w}(x_{n}):= \bigl\{ x\in H:x_{n_{i}} \rightharpoonup x \mbox{ for some subsequence }\{x_{n_{i}}\} \mbox{ of } \{x_{n}\} \bigr\} $$
and
$$\omega_{s}(x_{n}):= \bigl\{ x\in H:x_{n_{i}}\to x \mbox{ for some subsequence }\{ x_{n_{i}}\} \mbox{ of } \{x_{n} \} \bigr\} . $$
The metric (or nearest point) projection from H onto C is the mapping \(P_{C}:H\to C\) which assigns to each point \(x\in H\) the unique point \(P_{C}x\in C\) satisfying the property
$$\|x-P_{C}x\|=\inf_{y\in C}\|x-y\|=:d(x,C). $$

The following properties of projections are useful for our purpose.

Proposition 2.1

Given any \(x\in H\) and \(z\in C\). One has
  1. (i)

    \(z=P_{C}x \Leftrightarrow\langle x-z,y-z\rangle\leq0\), \(\forall y\in C\);

     
  2. (ii)

    \(z=P_{C}x \Leftrightarrow\|x-z\|^{2}\leq\|x-y\|^{2}-\|y-z\|^{2}\), \(\forall y\in C\);

     
  3. (iii)

    \(\langle P_{C}x-P_{C}y,x-y\rangle\geq\|P_{C}x-P_{C}y\|^{2}\), \(\forall y\in H\), which hence implies that \(P_{C}\) is nonexpansive and monotone.

     

Definition 2.1

A mapping \(T:H\to H\) is said to be
  1. (a)
    nonexpansive if
    $$\|Tx-Ty\|\leq\|x-y\|,\quad \forall x,y\in H; $$
     
  2. (b)
    firmly nonexpansive if \(2T-I\) is nonexpansive, or equivalently, if T is 1-inverse-strongly monotone (1-ism),
    $$\langle x-y,Tx-Ty\rangle\geq\|Tx-Ty\|^{2}, \quad \forall x,y\in H; $$
    alternatively, T is firmly nonexpansive if and only if T can be expressed as
    $$T=\frac{1}{2}(I+S), $$
    where \(S:H\to H\) is nonexpansive; projections are firmly nonexpansive.
     

Definition 2.2

A mapping \(A:C\to H\) is said to be
  1. (i)
    monotone if
    $$\langle Ax-Ay,x-y\rangle\geq0,\quad \forall x,y\in C; $$
     
  2. (ii)
    η-strongly monotone if there exists a constant \(\eta>0\) such that
    $$\langle Ax-Ay,x-y\rangle\geq\eta\|x-y\|^{2}, \quad \forall x,y\in C; $$
     
  3. (iii)
    α-inverse-strongly monotone if there exists a constant \(\alpha>0\) such that
    $$\langle Ax-Ay,x-y\rangle\geq\alpha\|Ax-Ay\|^{2},\quad \forall x,y\in C. $$
     
It is obvious that if \(A:C\to H\) is α-inverse-strongly monotone, then A is monotone and \(\frac{1}{\alpha}\)-Lipschitz continuous. Moreover, we also have, for all \(u,v\in C\) and \(\lambda>0\),
$$ \bigl\Vert (I-\lambda A)u-(I-\lambda A)v \bigr\Vert ^{2}\leq\|u-v \|^{2}+\lambda(\lambda-2\alpha )\|Au-Av\|^{2}. $$
(2.1)

Proposition 2.2

(see [15])

For given \(\bar{x},\bar{y}\in C\), \((\bar{x},\bar{y})\) is a solution of the GSVI (1.3) if and only if is a fixed point of the mapping \(G:C\to C\) defined by
$$Gx=P_{C}(I-\nu_{1}F_{1})P_{C}(I- \nu_{2}F_{2})x,\quad \forall x\in C, $$
where \(\bar{y}=P_{C}(I-\nu_{2}F_{2})\bar{x}\).

In particular, if the mapping \(F_{j}:C\to H\) is \(\zeta _{j}\)-inverse-strongly monotone for \(j=1,2\), then the mapping G is nonexpansive provided \(\nu_{j}\in(0,2\zeta_{j}]\) for \(j=1,2\). We denote by Ξ the fixed point set of the mapping G.

We need some facts and tools in a real Hilbert space H which are listed as lemmas below.

Lemma 2.1

Let X be a real inner product space. Then there holds the following inequality
$$\|x+y\|^{2}\leq\|x\|^{2}+2\langle y,x+y\rangle,\quad \forall x,y\in X. $$

Lemma 2.2

Let H be a real Hilbert space. Then the following hold:
  1. (a)

    \(\|x-y\|^{2}=\|x\|^{2}-\|y\|^{2}-2\langle x-y,y\rangle\) for all \(x,y\in H\);

     
  2. (b)

    \(\|\lambda x+\mu y\|^{2}=\lambda\|x\|^{2}+\mu\|y\|^{2}-\lambda\mu\|x-y\| ^{2}\) for all \(x,y\in H\) and \(\lambda,\mu\in[0,1]\) with \(\lambda+\mu=1\);

     
  3. (c)
    if \(\{x_{n}\}\) is a sequence in H such that \(x_{n}\rightharpoonup x\), it follows that
    $$\limsup_{n\to\infty}\|x_{n}-y\|^{2}=\limsup _{n\to\infty}\|x_{n}-x\|^{2}+\|x-y\| ^{2},\quad \forall y\in H. $$
     
Let C be a nonempty, closed, and convex subset of a real Hilbert space H. We introduce some notations. Let λ be a number in \((0,1]\) and let \(\mu>0\). Associating with a nonexpansive mapping \(T:C\to C\), we define the mapping \(T^{\lambda}:C\to H\) by
$$T^{\lambda}x:=Tx-\lambda\mu F(Tx),\quad \forall x\in C, $$
where \(F:C\to H\) is an operator such that, for some positive constants \(\kappa,\eta>0\), F is κ-Lipschitzian and η-strongly monotone on C; that is, F satisfies the conditions:
$$\|Fx-Fy\|\leq\kappa\|x-y\| \quad \textit{and}\quad \langle Fx-Fy,x-y\rangle\geq \eta\| x-y\|^{2} $$
for all \(x,y\in C\).

In the sequel, we let \(\operatorname{GMEP}({\varTheta },h)\) denote the solution set of GMEP (1.2).

Lemma 2.3

(see [3])

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let \({\varTheta }:C\times C\to{\mathbf{R}}\) be a bi-function satisfying conditions (θ1)-(θ3) and \(h:C\times C\to{\mathbf{R}}\) is a bi-function with restrictions (h1)-(h3). Moreover, let us suppose that
  1. (H)

    for fixed \(r>0\) and \(x\in C\), there exist a bounded \(K\subset C\) and \(\hat{x}\in K\) such that for all \(z\in C\setminus K\), \(-{\varTheta } (\hat{x},z)+h(z,\hat{x})+\frac{1}{r}\langle\hat{x}-z,z-x\rangle<0\).

     
For \(r>0\) and \(x\in H\), the mapping \(T_{r}:H\to2^{C}\) (i.e., the resolvent of Θ and h) has the following properties:
  1. (i)

    \(T_{r}x\neq\emptyset\);

     
  2. (ii)

    \(T_{r}x\) is a singleton;

     
  3. (iii)

    \(T_{r}\) is firmly nonexpansive;

     
  4. (iv)

    \(\operatorname{GMEP}({\varTheta },h)=\operatorname{Fix}(T_{r})\) and it is closed and convex.

     
Recall that a set-valued mapping \(T:D(T)\subset H\to2^{H}\) is called monotone if for all \(x,y\in D(T)\), \(f\in Tx\) and \(g\in Ty\) imply
$$\langle f-g,x-y\rangle\geq0. $$
A set-valued mapping T is called maximal monotone if T is monotone and \((I+\lambda T)D(T)=H\) for each \(\lambda>0\), where I is the identity mapping of H. We denote by \(G(T)\) the graph of T. It is known that a monotone mapping T is maximal if and only if, for \((x,f)\in H\times H\), \(\langle f-g,x-y\rangle\geq0\) for every \((y,g)\in G(T)\) implies \(f\in Tx\).

3 Main results

We now propose the following Mann-type hybrid steepest-descent iterative scheme:
$$ \left \{ \textstyle\begin{array}{l} {\varTheta }(u_{n},y)+h(u_{n},y)+\frac{1}{r_{n}}\langle y-u_{n},u_{n}-x_{n}\rangle\geq0,\quad \forall y\in C, \\ y_{n,1}=\beta_{n,1}S_{1}u_{n}+(1-\beta_{n,1})u_{n}, \\ y_{n,i}=\beta_{n,i}S_{i}u_{n}+(1-\beta_{n,i})y_{n,i-1},\quad i=2,\ldots,N, \\ y_{n}=P_{C}[\alpha_{n}\gamma f(y_{n,N})+(I-\alpha_{n}\mu F)Gy_{n,N}], \\ x_{n+1}=\beta_{n}x_{n}+\gamma_{n}{\varLambda }^{M}_{n}y_{n}+\delta_{n}T{\varLambda }^{M}_{n}y_{n} \end{array}\displaystyle \right . $$
(3.1)
for all \(n\geq0\), where
  • \(F:C\to H\) is a κ-Lipschitzian and η-strongly monotone operator with positive constants \(\kappa,\eta>0\) and \(f:C\to C\) is an l-Lipschitzian mapping with constant \(l\geq0\);

  • \(A_{k}:C\to H\) is \(\eta_{k}\)-inverse-strongly monotone, \(\{\lambda_{k,n}\} \subset[a_{k},b_{k}]\subset(0,2\eta_{k})\), \(\forall k\in \{1,\ldots,M\}\), and \({\varLambda }^{M}_{n}:=P_{C}(I-\lambda_{M,n}A_{M})\cdots P_{C}(I-\lambda_{1,n}A_{1})\);

  • \(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse-strongly monotone and \(G:=P_{C}(I-\nu _{1}F_{1})P_{C}(I-\nu_{2}F_{2})\) with \(\nu_{j}\in(0,2\zeta_{j})\) for \(j=1,2\);

  • \(T:C\to C\) is a ξ-strict pseudocontraction and \(S_{i}:C\to C\) is a nonexpansive mapping for each \(i=1,\ldots,N\);

  • \({\varTheta },h:C\times C\to{\mathbf{R}}\) are two bi-functions satisfying the hypotheses of Lemma 2.3;

  • \(0<\mu<2\eta/\kappa^{2}\) and \(0\leq\gamma l<\tau\) with \(\tau:=1-\sqrt {1-\mu(2\eta-\mu\kappa^{2})}\);

  • \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) are sequences in \((0,1)\) with \(0<\liminf_{n\to\infty}\beta_{n}\leq\limsup_{n\to\infty}\beta_{n}<1\);

  • \(\{\gamma_{n}\}\), \(\{\delta_{n}\}\) are sequences in \([0,1]\) with \(\beta _{n}+\gamma_{n}+\delta_{n}=1\), \(\forall n\geq0\);

  • \(\{\beta_{n,i}\}^{N}_{i=1}\) are sequences in \((0,1)\) and \((\gamma _{n}+\delta_{n})\xi\leq\gamma_{n}\), \(\forall n\geq0\);

  • \(\{r_{n}\}\) is a sequence in \((0,\infty)\) with \(\liminf_{n\to\infty }r_{n}>0\) and \(\liminf_{n\to\infty}\delta_{n}>0\).

We start our main result from the following series of propositions.

Proposition 3.1

Let us suppose that \({\varOmega }=\operatorname{Fix}(T)\cap\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\cap\bigcap^{M}_{k=1}\operatorname{VI} (C,A_{k})\cap \operatorname{GMEP}({\varTheta },h)\cap{ \varXi }\neq\emptyset\). Then the sequences \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{y_{n,i}\}\) for all i, \(\{u_{n}\}\) are bounded.

Proof

Since \(0<\liminf_{n\to\infty}\beta_{n}\leq\limsup_{n\to \infty}\beta_{n}<1\), we may assume, without loss of generality, that \(\{\beta_{n}\}\subset[c,d]\subset(0,1)\). For simplicity, we write
$$v_{n}=\alpha_{n}\gamma f(y_{n,N})+(I- \alpha_{n}\mu F)Gy_{n,N} $$
for all \(n\geq0\). Then \(y_{n}=P_{C}v_{n}\). Also, we set \(\tilde{y}_{n}={ \varLambda }^{M}_{n}y_{n}\),
$${\varLambda }^{k}_{n}=P_{C}(I-\lambda_{k,n}A_{k})P_{C}(I- \lambda _{k-1,n}A_{k-1})\cdots P_{C}(I- \lambda_{1,n}A_{1}) $$
for all \(k\in\{1,2,\ldots,M\}\) and \(n\geq0\), and \({\varLambda }^{0}_{n}=I\), where I is the identity mapping on H.
First of all, take a fixed \(p\in{ \varOmega }\) arbitrarily. We observe that
$$\|y_{n,1}-p\|\leq\|u_{n}-p\|\leq\|x_{n}-p\|. $$
For all i from \(i=2\) to \(i=N\), by induction, one proves that
$$\begin{aligned} \|y_{n,i}-p\| \leq&\beta_{n,i}\|u_{n}-p\|+(1- \beta_{n,i})\|y_{n,i-1}-p\| \\ \leq& \|u_{n}-p\|\leq \|x_{n}-p\|. \end{aligned}$$
Thus we obtain that for every \(i=1,\ldots,N\),
$$ \|y_{n,i}-p\|\leq\|u_{n}-p\|\leq\|x_{n}-p \|. $$
(3.2)
Since for each \(k\in\{1,\ldots,M\}\), \(I-\lambda_{k,n}A_{k}\) is nonexpansive and \(p=P_{C}(I-\lambda_{k,n}A_{k})p\), we have
$$\begin{aligned} \Vert \tilde{y}_{n}-p\Vert =& \bigl\Vert P_{C}(I- \lambda_{M,n}A_{M}){\varLambda }^{M-1}_{n}y_{n}-P_{C}(I- \lambda_{M,n}A_{M}){ \varLambda }^{M-1}_{n}p \bigr\Vert \\ \leq& \bigl\Vert (I-\lambda_{M,n}A_{M}){ \varLambda }^{M-1}_{n}y_{n}-(I-\lambda _{M,n}A_{M}){\varLambda }^{M-1}_{n}p \bigr\Vert \\ \leq& \bigl\Vert {\varLambda }^{M-1}_{n}y_{n}-{ \varLambda }^{M-1}_{n}p \bigr\Vert \\ &\cdots \\ \leq& \bigl\Vert {\varLambda }^{0}_{n}y_{n}-{ \varLambda }^{0}_{n}p \bigr\Vert \\ =&\Vert y_{n}-p\Vert . \end{aligned}$$
(3.3)
For simplicity, we write \(\tilde{p}=P_{C}(p-\nu_{2}F_{2}p)\), \(\tilde{y}_{n,N}=P_{C}(y_{n,N}-\nu_{2}F_{2}y_{n,N})\) and \(z_{n}=P_{C}(\tilde{y}_{n,N}-\nu_{1}F_{1}\tilde{y}_{n,N})\) for each \(n\geq0\). Then \(z_{n}=Gy_{n,N}\) and
$$p=P_{C}(I-\nu_{1}F_{1})\tilde{p}=P_{C}(I- \nu_{1}F_{1})P_{C}(I-\nu_{2}F_{2})p=Gp. $$
Since \(F_{j}:C\to H\) is \(\zeta_{j}\)-inverse-strongly monotone and \(0<\nu _{j}<2\zeta_{j}\) for each \(j=1,2\), we know that, for all \(n\geq0\),
$$\begin{aligned} \Vert z_{n}-p\Vert ^{2} =&\Vert Gy_{n,N}-p \Vert ^{2} \\ =& \bigl\Vert P_{C}(I-\nu_{1}F_{1})P_{C}(I- \nu_{2}F_{2})y_{n,N}-P_{C}(I- \nu_{1}F_{1})P_{C}(I-\nu _{2}F_{2})p \bigr\Vert ^{2} \\ \leq& \bigl\Vert (I-\nu_{1}F_{1})P_{C}(I- \nu_{2}F_{2})y_{n,N}-(I-\nu_{1}F_{1})P_{C}(I- \nu _{2}F_{2})p \bigr\Vert ^{2} \\ =& \bigl\Vert \bigl[P_{C}(I-\nu_{2}F_{2})y_{n,N}-P_{C}(I- \nu_{2}F_{2})p \bigr] \\ &{}-\bigl.\bigl.\nu_{1} \bigl[F_{1}P_{C}(I- \nu _{2}F_{2})y_{n,N}-F_{1}P_{C}(I- \nu_{2}F_{2})p \bigr] \bigr\Vert \bigr.^{2} \\ \leq& \bigl\Vert P_{C}(I-\nu_{2}F_{2})y_{n,N}-P_{C}(I- \nu_{2}F_{2})p \bigr\Vert ^{2} \\ &{} +\nu_{1}(\nu_{1}-2\zeta_{1}) \bigl\Vert F_{1}P_{C}(I-\nu_{2}F_{2})y_{n,N}-F_{1}P_{C}(I- \nu _{2}F_{2})p \bigr\Vert ^{2} \\ \leq& \bigl\Vert P_{C}(I-\nu_{2}F_{2})y_{n,N}-P_{C}(I- \nu_{2}F_{2})p \bigr\Vert ^{2} \\ \leq& \bigl\Vert (I-\nu_{2}F_{2})y_{n,N}-(I- \nu_{2}F_{2})p \bigr\Vert ^{2} \\ =& \bigl\Vert (y_{n,N}-p)-\nu_{2}(F_{2}y_{n,N}-F_{2}p) \bigr\Vert ^{2} \\ \leq&\Vert y_{n,N}-p\Vert ^{2}+\nu_{2}( \nu_{2}-2\zeta_{2})\Vert F_{2}y_{n,N}-F_{2}p \Vert ^{2} \\ \leq&\Vert y_{n,N}-p\Vert ^{2}\leq \Vert u_{n}-p\Vert ^{2}\leq \Vert x_{n}-p\Vert ^{2}. \end{aligned}$$
(3.4)
Also, since \(Gp=p\) and G is nonexpansive, utilizing Lemma 3.1 of [16] we have from (3.1) and (3.4)
$$\begin{aligned} \Vert y_{n}-p\Vert =&\Vert P_{C}v_{n}-p \Vert \\ \leq& \bigl\Vert \alpha_{n}\gamma \bigl(f(y_{n,N})-f(p) \bigr)+(I-\alpha_{n}\mu F)Gy_{n,N}-(I-\alpha_{n}\mu F)p+\alpha_{n}(\gamma f-\mu F)p \bigr\Vert \\ \leq&\alpha_{n}\gamma \bigl\Vert f(y_{n,N})-f(p) \bigr\Vert + \bigl\Vert (I-\alpha_{n}\mu F)Gy_{n,N}-(I- \alpha_{n}\mu F)p \bigr\Vert +\alpha_{n} \bigl\Vert ( \gamma f-\mu F)p \bigr\Vert \\ \leq&\alpha_{n}\gamma l\Vert y_{n,N}-p\Vert +(1- \alpha_{n}\tau)\Vert y_{n,N}-p\Vert +\alpha _{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \\ =& \bigl(1-\alpha_{n}(\tau-\gamma l) \bigr)\Vert y_{n,N}-p \Vert +\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \\ =& \bigl(1-\alpha_{n}(\tau-\gamma l) \bigr)\Vert y_{n,N}-p \Vert +\alpha_{n}(\tau-\gamma l)\frac {\Vert (\gamma f-\mu F)p\Vert }{\tau-\gamma l} \\ \leq&\max \biggl\{ \Vert y_{n,N}-p\Vert ,\frac{\Vert (\gamma f-\mu F)p\Vert }{\tau-\gamma l} \biggr\} \\ \leq&\max \biggl\{ \Vert x_{n}-p\Vert ,\frac{\Vert (\gamma f-\mu F)p\Vert }{\tau-\gamma l} \biggr\} . \end{aligned}$$
(3.5)
Taking into account \((\gamma_{n}+\delta_{n})\xi\leq\gamma_{n}\) and utilizing [17], we obtain from (3.1), (3.3), and (3.5) that
$$\begin{aligned} \Vert x_{n+1}-p\Vert &= \bigl\Vert \beta_{n}(x_{n}-p)+ \gamma_{n}(\tilde{y}_{n}-p)+\delta_{n}(T \tilde{y}_{n}-p) \bigr\Vert \\ &\leq\beta_{n}\Vert x_{n}-p\Vert + \bigl\Vert \gamma_{n}(\tilde{y}_{n}-p)+\delta_{n}(T \tilde{y}_{n}-p) \bigr\Vert \\ &\leq\beta_{n}\Vert x_{n}-p\Vert +( \gamma_{n}+\delta_{n})\Vert \tilde{y}_{n}-p \Vert \\ &\leq\beta_{n}\Vert x_{n}-p\Vert +( \gamma_{n}+\delta_{n})\Vert y_{n}-p\Vert \\ &\leq\beta_{n}\Vert x_{n}-p\Vert +( \gamma_{n}+\delta_{n})\max \biggl\{ \Vert x_{n}-p \Vert ,\frac{\Vert (\gamma f-\mu F)p\Vert }{\tau-\gamma l} \biggr\} \\ &\leq\max \biggl\{ \Vert x_{n}-p\Vert ,\frac{\Vert (\gamma f-\mu F)p\Vert }{\tau-\gamma l} \biggr\} . \end{aligned}$$
(3.6)
By induction, we get
$$\|x_{n}-p\|\leq\max \biggl\{ \|x_{0}-p\|,\frac{\|(\gamma f-\mu F)p\|}{\tau-\gamma l} \biggr\} ,\quad \forall n\geq0. $$
This implies that \(\{x_{n}\}\) is bounded and so are \(\{F_{2}y_{n,N}\}\), \(\{ F_{1}\tilde{y}_{n,N}\}\), \(\{\tilde{y}_{n,N}\}\), \(\{z_{n}\}\), \(\{u_{n}\}\), \(\{v_{n}\}\), \(\{y_{n}\}\), \(\{y_{n,i}\}\) for each \(i=1,\ldots,N\). Since \(\| T\tilde{y}_{n}-p\|\leq\frac{1+\xi}{1-\xi}\|\tilde{y}_{n}-p\|\), \(\{T\tilde{y}_{n}\}\) is also bounded. □

Proposition 3.2

Let us suppose that \({\varOmega }\neq\emptyset\). Moreover, let us suppose that the following hold:
  1. (H0)

    \(\lim_{n\to\infty}\alpha_{n}=0\) and \(\sum^{\infty}_{n=0}\alpha _{n}=\infty\);

     
  2. (H1)

    \(\sum^{\infty}_{n=1}|\lambda_{k,n}-\lambda_{k,n-1}|<\infty\) or \(\lim_{n\to\infty}\frac{|\lambda_{k,n}-\lambda_{k,n-1}|}{ \alpha_{n}}=0\) for each \(k=1,\ldots,M\);

     
  3. (H2)

    \(\sum^{\infty}_{n=1}|\alpha_{n}-\alpha_{n-1}|<\infty\) or \(\lim_{n\to \infty}\frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}}=0\);

     
  4. (H3)

    \(\sum^{\infty}_{n=1}|\beta_{n,i}-\beta_{n-1,i}|<\infty\) or \(\lim_{n\to\infty}\frac{|\beta_{n,i}-\beta_{n-1,i}|}{\alpha_{n}}=0\) for each \(i=1,\ldots,N\);

     
  5. (H4)

    \(\sum^{\infty}_{n=1}|r_{n}-r_{n-1}|<\infty\) or \(\lim_{n\to\infty}\frac {|r_{n}-r_{n-1}|}{\alpha_{n}}=0\);

     
  6. (H5)

    \(\sum^{\infty}_{n=1}|\beta_{n}-\beta_{n-1}|<\infty\) or \(\lim_{n\to \infty}\frac{|\beta_{n}-\beta_{n-1}|}{\alpha_{n}}=0\);

     
  7. (H6)

    \(\sum^{\infty}_{n=1}|\frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma _{n-1}}{1-\beta_{n-1}}|<\infty\) or \(\lim_{n\to\infty}\frac{1}{\alpha_{n}}|\frac{\gamma_{n}}{1-\beta_{n}}-\frac {\gamma_{n-1}}{1-\beta_{n-1}}|=0\).

     
Then \(\lim_{n\to\infty}\|x_{n+1}-x_{n}\|=0\), i.e., \(\{x_{n}\}\) is asymptotically regular.

Proof

First, it is known that \(\{\beta_{n}\}\subset[c,d]\subset (0,1)\) as in the proof of Proposition 3.1. Taking into account \(\liminf_{n\to\infty}r_{n}>0\), we may assume, without loss of generality, that \(\{r_{n}\}\subset[\epsilon,\infty)\) for some \(\epsilon>0\). First, we write \(x_{n}=\beta_{n-1}x_{n-1}+(1-\beta _{n-1})w_{n-1}\), \(\forall n\geq1\), where \(w_{n-1}= \frac{x_{n}-\beta_{n-1}x_{n-1}}{1-\beta_{n-1}}\). It follows that for all \(n\geq1\),
$$\begin{aligned} w_{n}-w_{n-1} =&\frac{x_{n+1}-\beta_{n}x_{n}}{1-\beta_{n}}-\frac {x_{n}-\beta_{n-1}x_{n-1}}{1-\beta_{n-1}} \\ =&\frac{\gamma_{n}\tilde{y}_{n}+\delta_{n}T\tilde{y}_{n}}{1-\beta_{n}}-\frac{\gamma _{n-1}\tilde{y}_{n-1}+\delta_{n-1}T\tilde{y}_{n-1}}{1-\beta_{n-1}} \\ =&\frac{\gamma_{n}(\tilde{y}_{n}-\tilde{y}_{n-1})+\delta_{n}(T\tilde{y}_{n}-T\tilde{y}_{n-1})}{1-\beta_{n}} \\ &{}+ \biggl(\frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma_{n-1}}{1-\beta_{n-1}} \biggr) \tilde{y}_{n-1} + \biggl(\frac{\delta_{n}}{1-\beta_{n}}-\frac{\delta_{n-1}}{1-\beta _{n-1}} \biggr)T \tilde{y}_{n-1}. \end{aligned}$$
(3.7)
Since \((\gamma_{n}+\delta_{n})\xi\leq\gamma_{n}\) for all \(n\geq0\), we have
$$ \bigl\Vert \gamma_{n}(\tilde{y}_{n}- \tilde{y}_{n-1})+\delta_{n}(T\tilde{y}_{n}-T \tilde{y}_{n-1}) \bigr\Vert \leq(\gamma_{n}+ \delta_{n})\|\tilde{y}_{n}-\tilde{y}_{n-1} \|. $$
(3.8)
Next, we estimate \(\|y_{n}-y_{n-1}\|\). Observe that
$$\begin{aligned} \Vert z_{n}-z_{n-1}\Vert ^{2} =& \bigl\Vert P_{C}(I-\nu_{1}F_{1})P_{C}(I- \nu_{2}F_{2})y_{n,N}-P_{C}(I- \nu_{1}F_{1})P_{C}(I-\nu _{2}F_{2})y_{n-1,N} \bigr\Vert ^{2} \\ \leq& \bigl\Vert (I-\nu_{1}F_{1})P_{C}(I- \nu_{2}F_{2})y_{n,N}-(I-\nu_{1}F_{1})P_{C}(I- \nu _{2}F_{2})y_{n-1,N} \bigr\Vert ^{2} \\ =& \bigl\Vert \bigl[P_{C}(I-\nu_{2}F_{2})y_{n,N}-P_{C}(I- \nu_{2}F_{2})y_{n-1,N} \bigr] \\ &{} -\bigl.\bigl.\nu_{1} \bigl[F_{1}P_{C}(I- \nu_{2}F_{2})y_{n,N}-F_{1}P_{C}(I- \nu_{2}F_{2})y_{n-1,N} \bigr] \bigr\Vert \bigr.^{2} \\ \leq& \bigl\Vert P_{C}(I-\nu_{2}F_{2})y_{n,N}-P_{C}(I- \nu_{2}F_{2})y_{n-1,N} \bigr\Vert ^{2} \\ &{} -\bigl.\bigl.\nu_{1}(2\zeta_{1}-\nu_{1}) \bigl\Vert F_{1}P_{C}(I-\nu_{2}F_{2})y_{n,N}-F_{1}P_{C}(I- \nu _{2}F_{2})y_{n-1,N} \bigr\Vert \bigr.^{2} \\ \leq& \bigl\Vert P_{C}(I-\nu_{2}F_{2})y_{n,N}-P_{C}(I- \nu_{2}F_{2})y_{n-1,N} \bigr\Vert ^{2} \\ \leq& \bigl\Vert (I-\nu_{2}F_{2})y_{n,N}-(I- \nu_{2}F_{2})y_{n-1,N} \bigr\Vert ^{2} \\ =& \bigl\Vert (y_{n,N}-y_{n-1,N})-\nu_{2}(F_{2}y_{n,N}-F_{2}y_{n-1,N}) \bigr\Vert ^{2} \\ \leq&\Vert y_{n,N}-y_{n-1,N}\Vert ^{2}- \nu_{2}(2\zeta_{2}-\nu_{2})\Vert F_{2}y_{n,N}-F_{2}y_{n-1,N}\Vert ^{2} \\ \leq&\Vert y_{n,N}-y_{n-1,N}\Vert ^{2}. \end{aligned}$$
(3.9)
Also, we observe that
$$\left \{ \textstyle\begin{array}{l} v_{n}=\alpha_{n}\gamma f(y_{n,N})+(I-\alpha_{n}\mu F)z_{n}, \\ v_{n-1}=\alpha_{n-1}\gamma f(y_{n-1,N})+(I-\alpha_{n-1}\mu F)z_{n-1},\quad \forall n\geq1. \end{array}\displaystyle \right . $$
Simple calculations show that
$$\begin{aligned} v_{n}-v_{n-1} =&(I-\alpha_{n}\mu F)z_{n}-(I-\alpha_{n}\mu F)z_{n-1}+( \alpha_{n}-\alpha_{n-1}) \bigl(\gamma f(y_{n-1,N})-\mu Fz_{n-1} \bigr) \\ &{} +\alpha_{n}\gamma \bigl(f(y_{n,N})-f(y_{n-1,N}) \bigr). \end{aligned}$$
Then, passing to the norm we get from (3.9)
$$\begin{aligned} \Vert v_{n}-v_{n-1}\Vert \leq& \bigl\Vert (I- \alpha_{n}\mu F)z_{n}-(I-\alpha_{n}\mu F)z_{n-1} \bigr\Vert +|\alpha_{n}-\alpha _{n-1}| \bigl\Vert \gamma f(y_{n-1,N})-\mu Fz_{n-1} \bigr\Vert \\ &{} +\alpha_{n}\gamma \bigl\Vert f(y_{n,N})-f(y_{n-1,N}) \bigr\Vert \\ \leq&(1-\alpha_{n}\tau)\Vert z_{n}-z_{n-1} \Vert +\widetilde{M}|\alpha_{n}-\alpha _{n-1}|+ \alpha_{n}\gamma l\Vert y_{n,N}-y_{n-1,N}\Vert \\ \leq&(1-\alpha_{n}\tau)\Vert y_{n,N}-y_{n-1,N} \Vert +\widetilde{M}|\alpha_{n}-\alpha _{n-1}|+ \alpha_{n}\gamma l\Vert y_{n,N}-y_{n-1,N}\Vert \\ =& \bigl(1-\alpha_{n}(\tau-\gamma l) \bigr)\Vert y_{n,N}-y_{n-1,N}\Vert +\widetilde{M}|\alpha _{n}- \alpha_{n-1}|, \end{aligned}$$
(3.10)
where \(\sup_{n\geq0}\|\gamma f(y_{n,N})-\mu Fz_{n}\|\leq\widetilde{M}\) for some \(\widetilde{M}>0\). In the meantime, by the definition of \(y_{n,i}\) one obtains, for all \(i=N,\ldots,2\),
$$\begin{aligned} \begin{aligned}[b] \|y_{n,i}-y_{n-1,i}\|\leq{}&\beta_{n,i} \|u_{n}-u_{n-1}\|+\| S_{i}u_{n-1}-y_{n-1,i-1} \||\beta_{n,i}-\beta_{n-1,i}| \\ &{} +(1-\beta_{n,i}) \|y_{n,i-1}-y_{n-1,i-1}\|. \end{aligned} \end{aligned}$$
(3.11)
In the case \(i=1\), we have
$$\begin{aligned} \|y_{n,1}-y_{n-1,1}\| \leq&\beta_{n,1} \|u_{n}-u_{n-1}\|+\| S_{1}u_{n-1}-u_{n-1} \||\beta_{n,1}-\beta_{n-1,1}| \\ &{}+(1-\beta_{n,1}) \|u_{n}-u_{n-1}\| \\ =&\|u_{n}-u_{n-1}\|+\|S_{1}u_{n-1}-u_{n-1} \||\beta_{n,1}-\beta_{n-1,1}|. \end{aligned}$$
(3.12)
Substituting (3.12) in all (3.11)-type expressions one obtains for \(i=2,\ldots,N\),
$$\begin{aligned} \|y_{n,i}-y_{n-1,i}\| \leq&\|u_{n}-u_{n-1}\|+{ \sum^{i}_{k=2}}\| S_{k}u_{n-1}-y_{n-1,k-1} \||\beta_{n,k}-\beta_{n-1,k}| \\ &{}+\|S_{1}u_{n-1}-u_{n-1} \||\beta_{n,1}-\beta_{n-1,1}|. \end{aligned}$$
This together with (3.10) implies that
$$\begin{aligned} \|y_{n}-y_{n-1}\| =&\|P_{C}v_{n}-P_{C}v_{n-1} \| \\ \leq& \bigl(1-\alpha_{n}(\tau-\gamma l) \bigr)\|y_{n,N}-y_{n-1,N} \|+\widetilde{M}|\alpha_{n}-\alpha_{n-1}| \\ \leq& \bigl(1-\alpha_{n}(\tau-\gamma l) \bigr) \Biggl[ \|u_{n}-u_{n-1}\|+{ \sum^{N}_{k=2}} \| S_{k}u_{n-1}-y_{n-1,k-1}\||\beta_{n,k}- \beta_{n-1,k}| \\ &{} +\|S_{1}u_{n-1}-u_{n-1}\||\beta_{n,1}- \beta_{n-1,1}| \Biggr]+\widetilde{M}|\alpha_{n}- \alpha_{n-1}| \\ \leq& \bigl(1-\alpha_{n}(\tau-\gamma l) \bigr)\|u_{n}-u_{n-1} \|+{ \sum^{N}_{k=2}}\| S_{k}u_{n-1}-y_{n-1,k-1} \||\beta_{n,k}-\beta_{n-1,k}| \\ &{} +\|S_{1}u_{n-1}-u_{n-1}\||\beta_{n,1}- \beta_{n-1,1}|+\widetilde{M}|\alpha_{n}-\alpha_{n-1}|. \end{aligned}$$
(3.13)
Furthermore, utilizing (2.1), we obtain
$$\begin{aligned} \Vert \tilde{y}_{n}-\tilde{y}_{n-1}\Vert =& \bigl\Vert {\varLambda }^{M}_{n}y_{n}-{ \varLambda }^{M}_{n-1}y_{n-1} \bigr\Vert \\ =& \bigl\Vert P_{C}(I-\lambda_{M,n}A_{M}){ \varLambda }^{M-1}_{n}y_{n}-P_{C}(I-\lambda _{M,n-1}A_{M}){\varLambda }^{M-1}_{n-1}y_{n-1} \bigr\Vert \\ \leq& \bigl\Vert P_{C}(I-\lambda_{M,n}A_{M}){ \varLambda }^{M-1}_{n}y_{n}-P_{C}(I-\lambda _{M,n-1}A_{M}){\varLambda }^{M-1}_{n}y_{n} \bigr\Vert \\ &{} + \bigl\Vert P_{C}(I-\lambda_{M,n-1}A_{M}){ \varLambda }^{M-1}_{n}y_{n}-P_{C}(I-\lambda _{M,n-1}A_{M}){\varLambda }^{M-1}_{n-1}y_{n-1} \bigr\Vert \\ \leq& \bigl\Vert (I-\lambda_{M,n}A_{M}){ \varLambda }^{M-1}_{n}y_{n}-(I-\lambda _{M,n-1}A_{M}){\varLambda }^{M-1}_{n}y_{n} \bigr\Vert \\ &{} + \bigl\Vert (I-\lambda_{M,n-1}A_{M}){ \varLambda }^{M-1}_{n}y_{n}-(I-\lambda _{M,n-1}A_{M}){\varLambda }^{M-1}_{n-1}y_{n-1} \bigr\Vert \\ \leq&|\lambda_{M,n}-\lambda_{M,n-1}| \bigl\Vert A_{M}{\varLambda }^{M-1}_{n}y_{n} \bigr\Vert + \bigl\Vert {\varLambda }^{M-1}_{n}y_{n}-{ \varLambda }^{M-1}_{n-1}y_{n-1} \bigr\Vert \\ \leq&|\lambda_{M,n}-\lambda_{M,n-1}| \bigl\Vert A_{M}{\varLambda }^{M-1}_{n}y_{n} \bigr\Vert +|\lambda_{M-1,n}-\lambda_{M,n-1}| \bigl\Vert A_{M-1}{\varLambda }^{M-2}_{n}y_{n} \bigr\Vert \\ &{} + \bigl\Vert {\varLambda }^{M-2}_{n}y_{n}-{ \varLambda }^{M-2}_{n-1}y_{n-1} \bigr\Vert \\ \leq&\cdots \\ \leq&|\lambda_{M,n}-\lambda_{M,n-1}| \bigl\Vert A_{M}{\varLambda }^{M-1}_{n}y_{n} \bigr\Vert \\ &{}+|\lambda_{M-1,n}-\lambda_{M-1,n-1}| \bigl\Vert A_{M-1}{\varLambda }^{M-2}_{n}y_{n} \bigr\Vert \\ &{} +\cdots+|\lambda_{1,n}-\lambda_{1,n-1}| \bigl\Vert A_{1}{\varLambda }^{0}_{n}y_{n} \bigr\Vert + \bigl\Vert {\varLambda }^{0}_{n}y_{n}-{ \varLambda }^{0}_{n-1}y_{n-1} \bigr\Vert \\ \leq&\widetilde{M}_{0} \sum^{M}_{k=1}| \lambda_{k,n}-\lambda_{k,n-1}|+\Vert y_{n}-y_{n-1} \Vert , \end{aligned}$$
(3.14)
where \(\sup_{n\geq1}\{\sum^{M}_{k=1}\|A_{k}{\varLambda }^{k-1}_{n}y_{n}\|\}\leq \widetilde{M}_{0}\) for some \(\widetilde{M}_{0}>0\).
By [3], we know that
$$ \|u_{n}-u_{n-1}\|\leq\|x_{n}-x_{n-1}\|+L \biggl\vert 1-\frac{r_{n-1}}{r_{n}} \biggr\vert , $$
(3.15)
where \(L=\sup_{n\geq0}\|u_{n}-x_{n}\|\). So, substituting (3.15) in (3.13), we obtain
$$\begin{aligned}& \|y_{n}-y_{n-1}\| \\& \quad \leq \bigl(1-\alpha_{n}(\tau-\gamma l) \bigr) \biggl( \|x_{n}-x_{n-1}\|+L \biggl\vert 1-\frac{r_{n-1}}{r_{n}} \biggr\vert \biggr) +{ \sum^{N}_{k=2}} \|S_{k}u_{n-1}-y_{n-1,k-1}\||\beta_{n,k}- \beta_{n-1,k}| \\& \qquad {} +\|S_{1}u_{n-1}-u_{n-1}\|| \beta_{n,1}-\beta_{n-1,1}|+\widetilde{M}|\alpha_{n}- \alpha_{n-1}| \\& \quad \leq \bigl(1-\alpha_{n}(\tau-\gamma l) \bigr) \|x_{n}-x_{n-1}\|+L\frac{|r_{n}-r_{n-1}|}{r_{n}} +{ \sum ^{N}_{k=2}}\|S_{k}u_{n-1}-y_{n-1,k-1} \||\beta_{n,k}-\beta_{n-1,k}| \\& \qquad {} +\|S_{1}u_{n-1}-u_{n-1}\|| \beta_{n,1}-\beta_{n-1,1}|+\widetilde{M}|\alpha_{n}- \alpha_{n-1}| \\& \quad \leq \bigl(1-\alpha_{n}(\tau-\gamma l) \bigr) \|x_{n}-x_{n-1}\|+\widetilde{M}_{1} \Biggl[ \frac {|r_{n}-r_{n-1}|}{r_{n}}+{ \sum^{N}_{k=2}}| \beta_{n,k}-\beta_{n-1,k}| \\& \qquad {} +|\beta_{n,1}-\beta_{n-1,1}|+|\alpha_{n}- \alpha_{n-1}| \Biggr] \\& \quad \leq \bigl(1-(\tau-\gamma l)\alpha_{n} \bigr) \|x_{n}-x_{n-1}\|+\widetilde{M}_{1} \Biggl[ \frac {|r_{n}-r_{n-1}|}{\epsilon} +{ \sum^{N}_{k=1}}| \beta_{n,k}-\beta_{n-1,k}|+|\alpha_{n}- \alpha_{n-1}| \Biggr], \end{aligned}$$
where \(\sup_{n\geq0}\{L+\widetilde{M}+\sum^{N}_{k=2}\|S_{k}u_{n}-y_{n,k-1}\|+\| S_{1}u_{n}-u_{n}\|\}\leq\widetilde{M}_{1}\) for some \(\widetilde{M}_{1}>0\). This together with (3.7), (3.8), and (3.14), implies that
$$\begin{aligned}& \|w_{n}-w_{n-1}\| \\& \quad \leq\frac{\|\gamma_{n}(\tilde{y}_{n}-\tilde{y}_{n-1})+\delta_{n}(T\tilde{y}_{n}-T\tilde{y}_{n-1})\|}{1-\beta_{n}} \\& \qquad {}+ \biggl\vert \frac{\gamma_{n}}{1-\beta_{n}}- \frac{\gamma_{n-1}}{1-\beta_{n-1}} \biggr\vert \| \tilde{y}_{n-1}\| + \biggl\vert \frac{\delta_{n}}{1-\beta_{n}}-\frac{\delta_{n-1}}{1-\beta_{n-1}} \biggr\vert \| T\tilde{y}_{n-1}\| \\& \quad \leq\frac{(\gamma_{n}+\delta_{n})\|\tilde{y}_{n}-\tilde{y}_{n-1}\|}{1-\beta_{n}} + \biggl\vert \frac{\gamma_{n}}{1-\beta_{n}}- \frac{\gamma_{n-1}}{1-\beta_{n-1}} \biggr\vert \| \tilde{y}_{n-1}\| + \biggl\vert \frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma_{n-1}}{1-\beta_{n-1}} \biggr\vert \| T\tilde{y}_{n-1}\| \\& \quad =\|\tilde{y}_{n}-\tilde{y}_{n-1}\|+ \biggl\vert \frac{\gamma_{n}}{1-\beta_{n}}-\frac {\gamma_{n-1}}{1-\beta_{n-1}} \biggr\vert \bigl(\Vert \tilde{y}_{n-1}\Vert +\|T\tilde{y}_{n-1}\| \bigr) \\& \quad \leq\widetilde{M}_{0}{ \sum^{M}_{k=1}}| \lambda_{k,n}-\lambda_{k,n-1}|+\| y_{n}-y_{n-1} \| + \biggl\vert \frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma_{n-1}}{1-\beta_{n-1}} \biggr\vert \bigl( \Vert \tilde{y}_{n-1}\Vert +\|T\tilde{y}_{n-1}\| \bigr) \\& \quad \leq\widetilde{M}_{0}{ \sum^{M}_{k=1}}| \lambda_{k,n}-\lambda_{k,n-1}| + \bigl(1-(\tau-\gamma l) \alpha_{n} \bigr)\|x_{n}-x_{n-1}\|+ \widetilde{M}_{1} \Biggl[\frac {|r_{n}-r_{n-1}|}{\epsilon} \\& \qquad {} +{ \sum^{N}_{k=1}}| \beta_{n,k}-\beta_{n-1,k}|+|\alpha_{n}- \alpha_{n-1}| \Biggr] + \biggl\vert \frac{\gamma_{n}}{1-\beta_{n}}- \frac{\gamma_{n-1}}{1-\beta_{n-1}} \biggr\vert \bigl(\Vert \tilde{y}_{n-1}\Vert + \|T \tilde{y}_{n-1}\| \bigr) \\& \quad \leq \bigl(1-(\tau-\gamma l)\alpha_{n} \bigr) \|x_{n}-x_{n-1}\|+\widetilde{M}_{2} \Biggl[ \frac {|r_{n}-r_{n-1}|}{\epsilon}+{ \sum^{M}_{k=1}}| \lambda_{k,n}-\lambda _{k,n-1}| \\& \qquad {} +{ \sum^{N}_{k=1}}| \beta_{n,k}-\beta_{n-1,k}|+|\alpha_{n}-\alpha _{n-1}|+ \biggl\vert \frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma_{n-1}}{1-\beta_{n-1}} \biggr\vert \Biggr], \end{aligned}$$
(3.16)
where \(\sup_{n\geq0}\{\widetilde{M}_{0}+\widetilde{M}_{1}+\|\tilde{y}_{n}\|+\| T\tilde{y}_{n}\|\}\leq\widetilde{M}_{2}\) for some \(\widetilde{M}_{2}>0\).
Further, we observe that
$$\left \{ \textstyle\begin{array}{l} x_{n+1}=\beta_{n}x_{n}+(1-\beta_{n})w_{n}, \\ x_{n}=\beta_{n-1}x_{n-1}+(1-\beta_{n-1})w_{n-1},\quad \forall n\geq1. \end{array}\displaystyle \right . $$
Simple calculations show that
$$x_{n+1}-x_{n}=(1-\beta_{n}) (w_{n}-w_{n-1})+( \beta_{n}-\beta _{n-1}) (x_{n-1}-w_{n-1})+ \beta_{n}(x_{n}-x_{n-1}). $$
Then, passing to the norm we get from (3.16)
$$\begin{aligned}& \|x_{n+1}-x_{n}\| \\& \quad \leq(1-\beta_{n})\|w_{n}-w_{n-1}\|+| \beta_{n}-\beta_{n-1}|\| x_{n-1}-w_{n-1}\|+ \beta_{n}\|x_{n}-x_{n-1}\| \\& \quad \leq(1-\beta_{n}) \Biggl\{ \bigl(1-\alpha_{n}(\tau- \gamma l) \bigr)\|x_{n}-x_{n-1}\|+\widetilde{M}_{2} \Biggl[\frac{|r_{n}-r_{n-1}|}{\epsilon}+{ \sum^{M}_{k=1}}| \lambda _{k,n}-\lambda_{k,n-1}| \\& \qquad {} +{ \sum^{N}_{k=1}}| \beta_{n,k}-\beta_{n-1,k}|+|\alpha_{n}- \alpha_{n-1}| + \biggl\vert \frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma_{n-1}}{1-\beta_{n-1}} \biggr\vert \Biggr] \Biggr\} \\& \qquad {} +|\beta_{n}-\beta_{n-1}|\|x_{n-1}-w_{n-1} \|+\beta_{n}\|x_{n}-x_{n-1}\| \\& \quad \leq \bigl(1-(\tau-\gamma l) (1-\beta_{n})\alpha_{n} \bigr)\|x_{n}-x_{n-1}\|+\widetilde{M}_{2} \Biggl[ \frac{|r_{n}-r_{n-1}|}{\epsilon}+{ \sum^{M}_{k=1}}|\lambda _{k,n}-\lambda_{k,n-1}| \\& \qquad {} +{ \sum^{N}_{k=1}}| \beta_{n,k}-\beta_{n-1,k}|+|\alpha_{n}- \alpha_{n-1}| + \biggl\vert \frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma_{n-1}}{1-\beta _{n-1}} \biggr\vert \Biggr] \\& \qquad {}+|\beta_{n}-\beta_{n-1}| \|x_{n-1}-w_{n-1} \| \\& \quad \leq \bigl(1-(\tau-\gamma l) (1-d)\alpha_{n} \bigr) \|x_{n}-x_{n-1}\|+\widetilde{M}_{3} \Biggl[ \frac{|r_{n}-r_{n-1}|}{\epsilon}+{ \sum^{M}_{k=1}}|\lambda _{k,n}-\lambda_{k,n-1}| \\& \qquad {} +{ \sum^{N}_{k=1}}| \beta_{n,k}-\beta_{n-1,k}|+|\alpha_{n}- \alpha_{n-1}| + \biggl\vert \frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma_{n-1}}{1-\beta_{n-1}} \biggr\vert +|\beta _{n}-\beta_{n-1}| \Biggr], \end{aligned}$$
(3.17)
where \(\sup_{n\geq0}\{\widetilde{M}_{2}+\|x_{n}-w_{n}\|\}\leq\widetilde{M}_{3}\) for some \(\widetilde{M}_{3}>0\). By hypotheses (H0)-(H6) and Lemma 2.1 of [16], we obtain the claim. □

Proposition 3.3

Let us suppose that \({\varOmega }\neq\emptyset\). Let us suppose that \(\{x_{n}\}\) is asymptotically regular. Then \(\|x_{n}-u_{n}\|=\|x_{n}-T_{r_{n}}x_{n}\|\to0\) and \(\|y_{n}-\tilde{y}_{n}\|\to0\) as \(n\to\infty\).

Proof

Take a fixed \(p\in{ \varOmega }\) arbitrarily. We recall that, by the firm nonexpansivity of \(T_{r_{n}}\), a standard calculation shows that for \(p\in\operatorname{GMEP}({\varTheta },h)\),
$$ \|u_{n}-p\|^{2}\leq\|x_{n}-p\|^{2}- \|x_{n}-u_{n}\|^{2}. $$
(3.18)
Observe that
$$\begin{aligned} \bigl\Vert {\varLambda }^{k}_{n}y_{n}-p \bigr\Vert ^{2} =& \bigl\Vert P_{C}(I-\lambda _{k,n}A_{k}){\varLambda }^{k-1}_{n}y_{n}-P_{C}(I- \lambda_{k,n}A_{k})p \bigr\Vert ^{2} \\ \leq& \bigl\Vert (I-\lambda_{k,n}A_{k}){ \varLambda }^{k-1}_{n}y_{n}-(I-\lambda _{k,n}A_{k})p \bigr\Vert ^{2} \\ \leq& \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-p \bigr\Vert ^{2}+\lambda_{k,n}(\lambda_{k,n}-2\eta _{k}) \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert ^{2} \\ \leq&\Vert y_{n}-p\Vert ^{2}+\lambda_{k,n}( \lambda_{k,n}-2\eta_{k}) \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert ^{2} \end{aligned}$$
(3.19)
for each \(k\in\{1,2,\ldots,M\}\).
Utilizing Lemma 2.1 and Lemma 3.1 of [16], we obtain from \(0\leq\gamma l<\tau\), (3.1), (3.4), and (3.18)
$$\begin{aligned}& \Vert y_{n}-p\Vert ^{2} \\& \quad = \bigl\Vert \alpha_{n}\gamma \bigl(f(y_{n,N})-f(p) \bigr)+(I-\alpha_{n}\mu F)z_{n}-(I-\alpha _{n}\mu F)p+\alpha_{n}(\gamma f-\mu F)p \bigr\Vert ^{2} \\& \quad \leq \bigl\Vert \alpha_{n}\gamma \bigl(f(y_{n,N})-f(p) \bigr)+(I-\alpha_{n}\mu F)z_{n}-(I-\alpha _{n} \mu F)p \bigr\Vert ^{2}+2\alpha_{n} \bigl\langle (\gamma f- \mu F)p,y_{n}-p \bigr\rangle \\& \quad \leq \bigl[\alpha_{n}\gamma \bigl\Vert f(y_{n,N})-f(p) \bigr\Vert + \bigl\Vert (I-\alpha_{n}\mu F)z_{n}-(I- \alpha_{n}\mu F)p \bigr\Vert \bigr]^{2}+2 \alpha_{n} \bigl\langle (\gamma f-\mu F)p,y_{n}-p \bigr\rangle \\& \quad \leq \bigl[\alpha_{n}\gamma l\Vert y_{n,N}-p\Vert +(1-\alpha_{n}\tau)\Vert z_{n}-p\Vert \bigr]^{2}+2\alpha_{n} \bigl\langle (\gamma f-\mu F)p,y_{n}-p \bigr\rangle \\& \quad = \biggl[\alpha_{n}\tau\frac{\gamma l}{\tau} \Vert y_{n,N}-p\Vert +(1-\alpha_{n}\tau)\Vert z_{n}-p\Vert \biggr]^{2}+2\alpha_{n} \bigl\langle (\gamma f-\mu F)p,y_{n}-p \bigr\rangle \\& \quad \leq\alpha_{n}\tau\frac{(\gamma l)^{2}}{\tau^{2}}\Vert y_{n,N}-p \Vert ^{2}+(1-\alpha _{n}\tau)\Vert z_{n}-p \Vert ^{2}+2\alpha_{n} \bigl\langle (\gamma f-\mu F)p,y_{n}-p \bigr\rangle \\& \quad \leq\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert z_{n}-p\Vert ^{2}+2 \alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \quad \leq\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert y_{n,N}-p\Vert ^{2}- \nu_{2}(2\zeta_{2}-\nu _{2})\Vert F_{2}y_{n,N}-F_{2}p\Vert ^{2} \\& \qquad {} -\nu_{1}(2\zeta_{1}-\nu_{1})\Vert F_{1}\tilde{y}_{n,N}-F_{1}\tilde{p}\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \quad \leq\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert u_{n}-p\Vert ^{2}- \nu_{2}(2\zeta_{2}-\nu_{2})\Vert F_{2}y_{n,N}-F_{2}p\Vert ^{2} \\& \qquad {} -\nu_{1}(2\zeta_{1}-\nu_{1})\Vert F_{1}\tilde{y}_{n,N}-F_{1}\tilde{p}\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \quad \leq\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert x_{n}-p\Vert ^{2}-\Vert x_{n}-u_{n}\Vert ^{2}-\nu_{2}(2\zeta _{2}-\nu_{2})\Vert F_{2}y_{n,N}-F_{2}p \Vert ^{2} \\& \qquad {} -\nu_{1}(2\zeta_{1}-\nu_{1})\Vert F_{1}\tilde{y}_{n,N}-F_{1}\tilde{p}\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert . \end{aligned}$$
(3.20)
Since \((\gamma_{n}+\delta_{n})\xi\leq\gamma_{n}\) for all \(n\geq0\), utilizing [17] we have from (3.19) and (3.20)
$$\begin{aligned}& \Vert x_{n+1}-p\Vert ^{2} \\& \quad = \bigl\Vert \beta_{n}(x_{n}-p)+ \gamma_{n}(\tilde{y}_{n}-p)+\delta_{n}(T \tilde{y}_{n}-p) \bigr\Vert ^{2} \\& \quad = \biggl\Vert \beta_{n}(x_{n}-p)+( \gamma_{n}+\delta_{n})\frac{1}{\gamma_{n}+\delta _{n}} \bigl[ \gamma_{n}(\tilde{y}_{n}-p)+\delta_{n}(T \tilde{y}_{n}-p) \bigr] \biggr\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+( \gamma_{n}+\delta_{n}) \biggl\Vert \frac{1}{\gamma_{n}+\delta _{n}} \bigl[ \gamma_{n}(\tilde{y}_{n}-p)+\delta_{n}(T \tilde{y}_{n}-p) \bigr] \biggr\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+( \gamma_{n}+\delta_{n})\Vert \tilde{y}_{n}-p \Vert ^{2} \\& \quad =\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1- \beta_{n})\Vert \tilde{y}_{n}-p\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl\Vert {\varLambda }^{k}_{n}y_{n}-p \bigr\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\Vert y_{n}-p\Vert ^{2}+\lambda_{k,n}(\lambda _{k,n}-2 \eta_{k}) \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert ^{2} \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert x_{n}-p\Vert ^{2}-\Vert x_{n}-u_{n}\Vert ^{2} \\& \qquad {} -\nu_{2}(2\zeta_{2}-\nu_{2})\Vert F_{2}y_{n,N}-F_{2}p\Vert ^{2}- \nu_{1}(2\zeta_{1}-\nu_{1})\Vert F_{1} \tilde{y}_{n,N}-F_{1}\tilde{p}\Vert ^{2} \\& \qquad {} +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert +\lambda_{k,n}(\lambda _{k,n}-2\eta_{k}) \bigl\Vert A_{k}{ \varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert ^{2} \bigr] \\& \quad \leq \Vert x_{n}-p\Vert ^{2}-(1- \beta_{n}) \bigl[\Vert x_{n}-u_{n}\Vert ^{2}+\lambda_{k,n}(2\eta _{k}-\lambda_{k,n}) \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert ^{2} \\& \qquad {} +\nu_{2}(2\zeta_{2}-\nu_{2})\Vert F_{2}y_{n,N}-F_{2}p\Vert ^{2}+ \nu_{1}(2\zeta_{1}-\nu_{1})\Vert F_{1} \tilde{y}_{n,N}-F_{1}\tilde{p}\Vert ^{2} \bigr] \\& \qquad {} +\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert . \end{aligned}$$
(3.21)
So, we deduce from \(\{\beta_{n}\}\subset[c,d]\subset(0,1)\) and \(\{\lambda _{k,n}\}\subset[a_{k},b_{k}]\subset(0,2\eta_{k})\), \(k=1,\ldots,M\), that
$$\begin{aligned}& (1-d) \bigl[\Vert x_{n}-u_{n}\Vert ^{2}+ \lambda_{k,n}(2\eta_{k}-\lambda _{k,n}) \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert ^{2} \\& \qquad {} +\nu_{2}(2\zeta_{2}-\nu_{2})\Vert F_{2}y_{n,N}-F_{2}p\Vert ^{2}+ \nu_{1}(2\zeta_{1}-\nu_{1})\Vert F_{1} \tilde{y}_{n,N}-F_{1}\tilde{p}\Vert ^{2} \bigr] \\& \quad \leq(1-\beta_{n}) \bigl[\Vert x_{n}-u_{n} \Vert ^{2}+\lambda_{k,n}(2\eta_{k}- \lambda_{k,n}) \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert ^{2} \\& \qquad {} +\nu_{2}(2\zeta_{2}-\nu_{2})\Vert F_{2}y_{n,N}-F_{2}p\Vert ^{2}+ \nu_{1}(2\zeta_{1}-\nu_{1})\Vert F_{1} \tilde{y}_{n,N}-F_{1}\tilde{p}\Vert ^{2} \bigr] \\& \quad \leq \Vert x_{n}-p\Vert ^{2}-\Vert x_{n+1}-p\Vert ^{2}+\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha _{n} \bigl\Vert ( \gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \quad \leq \Vert x_{n}-x_{n+1}\Vert \bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr)+ \alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2 \alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert . \end{aligned}$$
By Propositions 3.1 and 3.2 we know that the sequences \(\{x_{n}\}\), \(\{y_{n}\} \), and \(\{y_{n,N}\}\) are bounded, and that \(\{x_{n}\}\) is asymptotically regular. Therefore, from \(\alpha_{n}\to0\) we obtain
$$\begin{aligned} \lim_{n\to\infty}\|x_{n}-u_{n}\| =&\lim _{n\to\infty}\|F_{2}y_{n,N}-F_{2}p\|=\lim _{n\to\infty}\|F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\| \\ =&\lim_{n\to\infty} \bigl\Vert A_{k}{ \varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert =0 \end{aligned}$$
(3.22)
for each \(k\in\{1,\ldots,M\}\).
By Proposition 2.1(iii), we deduce that for each \(k\in\{1,2,\ldots,M\}\),
$$\begin{aligned} \bigl\Vert {\varLambda }^{k}_{n}y_{n}-p \bigr\Vert ^{2} =& \bigl\Vert P_{C}(I-\lambda_{k,n}A_{k}){ \varLambda }^{k-1}_{n}y_{n}-P_{C}(I-\lambda _{k,n}A_{k})p \bigr\Vert ^{2} \\ \leq& \bigl\langle (I-\lambda_{k,n}A_{k}){ \varLambda }^{k-1}_{n}y_{n}-(I-\lambda _{k,n}A_{k})p,{\varLambda }^{k}_{n}y_{n}-p \bigr\rangle \\ =&\frac{1}{2} \bigl( \bigl\Vert (I-\lambda_{k,n}A_{k}){ \varLambda }^{k-1}_{n}y_{n}-(I-\lambda _{k,n}A_{k})p \bigr\Vert ^{2}+ \bigl\Vert { \varLambda }^{k}_{n}y_{n}-p \bigr\Vert ^{2} \\ & {} - \bigl\Vert (I-\lambda_{k,n}A_{k}){ \varLambda }^{k-1}_{n}y_{n}-(I-\lambda _{k,n}A_{k})p- \bigl({\varLambda }^{k}_{n}y_{n}-p \bigr) \bigr\Vert ^{2} \bigr) \\ \leq&\frac{1}{2} \bigl( \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-p \bigr\Vert ^{2}+ \bigl\Vert {\varLambda }^{k}_{n}y_{n}-p \bigr\Vert ^{2} \\ &{}- \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n}-\lambda_{k,n} \bigl(A_{k}{ \varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr) \bigr\Vert ^{2} \bigr) \\ \leq&\frac{1}{2} \bigl(\Vert y_{n}-p\Vert ^{2}+ \bigl\Vert {\varLambda }^{k}_{n}y_{n}-p \bigr\Vert ^{2} \\ &{}- \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n}-\lambda_{k,n} \bigl(A_{k}{ \varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr) \bigr\Vert ^{2} \bigr), \end{aligned}$$
which immediately leads to
$$\begin{aligned} \bigl\Vert {\varLambda }^{k}_{n}y_{n}-p \bigr\Vert ^{2} \leq&\Vert y_{n}-p\Vert ^{2}- \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n}-\lambda _{k,n} \bigl(A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr) \bigr\Vert ^{2} \\ =&\Vert y_{n}-p\Vert ^{2}- \bigl\Vert { \varLambda }^{k-1}_{n}y_{n}-{\varLambda }^{k}_{n}y_{n} \bigr\Vert ^{2}-\lambda ^{2}_{k,n} \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert ^{2} \\ &{} +2\lambda_{k,n} \bigl\langle { \varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n},A_{k}{ \varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\rangle \\ \leq&\Vert y_{n}-p\Vert ^{2}- \bigl\Vert { \varLambda }^{k-1}_{n}y_{n}-{\varLambda }^{k}_{n}y_{n} \bigr\Vert ^{2} \\ &{} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert . \end{aligned}$$
(3.23)
From (3.4), (3.20), (3.21), and (3.23) we conclude that
$$\begin{aligned}& \Vert x_{n+1}-p\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n})\Vert \tilde{y}_{n}-p\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl\Vert {\varLambda }^{k}_{n}y_{n}-p \bigr\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\Vert y_{n}-p\Vert ^{2}- \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert ^{2} \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2} \\& \qquad {}+\Vert z_{n}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} - \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert ^{2} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{ \varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2} \\& \qquad {}+\Vert x_{n}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} - \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert ^{2} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{ \varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq \Vert x_{n}-p\Vert ^{2}-(1- \beta_{n}) \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert ^{2} \\& \qquad {}+2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{ \varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \\& \qquad {} +\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert , \end{aligned}$$
(3.24)
which together with \(\{\beta_{n}\}\subset[c,d]\subset(0,1)\) and \(\{\lambda _{k,n}\}\subset[a_{k},b_{k}]\subset(0,2\eta_{k})\), \(k=1,\ldots,M\), yields
$$\begin{aligned}& (1-d) \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert ^{2} \\& \quad \leq(1-\beta_{n}) \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert ^{2} \\& \quad \leq \Vert x_{n}-p\Vert ^{2}-\Vert x_{n+1}-p\Vert ^{2}+2\lambda_{k,n} \bigl\Vert { \varLambda }^{k-1}_{n}y_{n}-{\varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \\& \qquad {} +\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \quad \leq \Vert x_{n}-x_{n+1}\Vert \bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr) \\& \qquad {}+2b_{k} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \\& \qquad {} +\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert . \end{aligned}$$
Since \(\alpha_{n}\to0\), and \(\{x_{n}\}\), \(\{y_{n}\}\), and \(\{y_{n,N}\}\) are bounded, we obtain from (3.22) and the asymptotical regularity of \(\{x_{n}\}\) (due to Proposition 3.2),
$$ \lim_{n\to\infty} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert =0,\quad \forall k\in\{1,\ldots,M\}. $$
(3.25)
Note that
$$\begin{aligned} \Vert y_{n}-\tilde{y}_{n}\Vert &= \bigl\Vert { \varLambda }^{0}_{n}y_{n}-{\varLambda }^{M}_{n}y_{n} \bigr\Vert \\ &\leq \bigl\Vert {\varLambda }^{0}_{n}y_{n}-{ \varLambda }^{1}_{n}y_{n} \bigr\Vert + \bigl\Vert { \varLambda }^{1}_{n}y_{n}-{\varLambda }^{2}_{n}y_{n} \bigr\Vert +\cdots \bigl\Vert {\varLambda }^{M-1}_{n}y_{n}-{ \varLambda }^{M}_{n}y_{n} \bigr\Vert . \end{aligned}$$
Thus,
$$ \lim_{n\to\infty}\|y_{n}-\tilde{y}_{n} \|=0. $$
(3.26)
 □

Remark 3.1

By the last proposition we have \(\omega_{w}(x_{n})=\omega _{w}(u_{n})\) and \(\omega_{s}(x_{n})=\omega_{s}(u_{n})\), i.e., the sets of strong/weak cluster points of \(\{x_{n}\}\) and \(\{u_{n}\}\) coincide.

Of course, if \(\beta_{n,i}\to\beta_{i}\neq0\) as \(n\to\infty\), for all indices i, the assumptions of Proposition 3.2 are enough to assure that
$$\lim_{n\to\infty}\frac{\|x_{n+1}-x_{n}\|}{\beta_{n,i}}=0, \quad \forall i\in\{ 1, \ldots,N\}. $$

In the next proposition, we estimate the case in which at least one sequence \(\{\beta_{n,k_{0}}\}\) is a null sequence.

Proposition 3.4

Let us suppose that \({\varOmega }\neq\emptyset\). Let us suppose that (H0) holds. Moreover, for an index \(k_{0}\in\{1,\ldots,N\}\), \(\lim_{n\to\infty}\beta_{n,k_{0}}=0\) and the following hold:
  1. (H7)
    for each \(i\in\{1,\ldots,N\}\) and \(k\in\{1,\ldots,M\}\),
    $$\begin{aligned} { \lim_{n\to\infty}}\frac{|\beta_{n,i}-\beta _{n-1,i}|}{\alpha_{n}\beta_{n,k_{0}}} &={ \lim_{n\to\infty}} \frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}\beta_{n,k_{0}}} ={ \lim_{n\to\infty}}\frac{|\beta_{n}-\beta_{n-1}|}{\alpha_{n}\beta_{n,k_{0}}} ={ \lim _{n\to\infty}}\frac{|r_{n}-r_{n-1}|}{\alpha_{n}\beta_{n,k_{0}}} \\ &={ \lim_{n\to\infty}}\frac{1}{\alpha_{n}\beta_{n,k_{0}}}\biggl|\frac{\gamma _{n}}{1-\beta_{n}}- \frac{\gamma_{n-1}}{1-\beta_{n-1}}\biggr| \\ &={ \lim_{n\to\infty}}\frac{|\lambda_{k,n}-\lambda_{k,n-1}|}{\alpha _{n}\beta_{n,k_{0}}}=0; \end{aligned}$$
     
  2. (H8)

    there exists a constant \(b>0\) such that \(\frac{1}{\alpha_{n}}|\frac {1}{\beta_{n,k_{0}}}-\frac{1}{\beta_{n-1,k_{0}}}|< b\) for all \(n\geq1\).

     
Then
$$\lim_{n\to\infty}\frac{\|x_{n+1}-x_{n}\|}{\beta_{n,k_{0}}}=0. $$

Proof

We start by (3.17). Dividing both terms by \(\beta _{n,k_{0}}\) we have
$$\begin{aligned} \frac{\|x_{n+1}-x_{n}\|}{\beta_{n,k_{0}}} \leq& \bigl(1-(\tau-\gamma l) (1-d)\alpha_{n} \bigr)\frac{\|x_{n}-x_{n-1}\|}{\beta _{n,k_{0}}} \\ &{}+\widetilde{M}_{3} \biggl[\frac{|r_{n}-r_{n-1}|}{\epsilon\beta_{n,k_{0}}}+ \frac{{ \sum^{M}_{k=1}}|\lambda_{k,n}-\lambda_{k,n-1}|}{\beta_{n,k_{0}}} \\ &{} +\frac{{ \sum^{N}_{k=1}}|\beta_{n,k}-\beta_{n-1,k}|}{\beta _{n,k_{0}}}+\frac{|\alpha_{n}-\alpha_{n-1}|}{\beta_{n,k_{0}}} \\ &{}+\frac{|\frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma_{n-1}}{1-\beta _{n-1}}|}{\beta_{n,k_{0}}}+ \frac{|\beta_{n}-\beta_{n-1}|}{ \beta_{n,k_{0}}} \biggr]. \end{aligned}$$
(3.27)
So, by (H8) we have
$$\begin{aligned}& \frac{\|x_{n+1}-x_{n}\|}{\beta_{n,k_{0}}} \\& \quad \leq \bigl[1-(\tau-\gamma l) (1-d)\alpha_{n} \bigr] \frac{\|x_{n}-x_{n-1}\|}{\beta_{n-1,k_{0}}} \\& \qquad {}+ \bigl[1-(\tau-\gamma l) (1-d)\alpha_{n} \bigr] \|x_{n}-x_{n-1}\| \biggl\vert \frac{1}{\beta _{n,k_{0}}}- \frac{1}{\beta_{n-1,k_{0}}} \biggr\vert \\& \qquad {} +\widetilde{M}_{3} \biggl[\frac{|r_{n}-r_{n-1}|}{\epsilon\beta_{n,k_{0}}}+ \frac{{ \sum^{M}_{k=1}} |\lambda_{k,n}-\lambda_{k,n-1}|}{\beta_{n,k_{0}}}+\frac{{ \sum^{N}_{k=1}}|\beta_{n,k}-\beta_{n-1,k}|}{ \beta_{n,k_{0}}} \\& \qquad {}+\frac{|\frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma _{n-1}}{1-\beta_{n-1}}|}{\beta_{n,k_{0}}} +\frac{|\alpha_{n}-\alpha_{n-1}|}{\beta_{n,k_{0}}}+ \frac{|\beta_{n}-\beta _{n-1}|}{\beta_{n,k_{0}}} \biggr] \\& \quad \leq \bigl[1-(\tau-\gamma l) (1-d)\alpha_{n} \bigr] \frac{\|x_{n}-x_{n-1}\|}{\beta_{n-1,k_{0}}} +\|x_{n}-x_{n-1}\| \biggl\vert \frac{1}{\beta_{n,k_{0}}}-\frac{1}{\beta_{n-1,k_{0}}} \biggr\vert \\& \qquad {} +\widetilde{M}_{3} \biggl[\frac{|r_{n}-r_{n-1}|}{\epsilon\beta_{n,k_{0}}}+ \frac{{ \sum^{M}_{k=1}} |\lambda_{k,n}-\lambda_{k,n-1}|}{\beta_{n,k_{0}}}+\frac{{ \sum^{N}_{k=1}}|\beta_{n,k}-\beta_{n-1,k}|}{ \beta_{n,k_{0}}} \\& \qquad {}+\frac{|\frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma _{n-1}}{1-\beta_{n-1}}|}{\beta_{n,k_{0}}} +\frac{|\alpha_{n}-\alpha_{n-1}|}{\beta_{n,k_{0}}}+ \frac{|\beta_{n}-\beta _{n-1}|}{\beta_{n,k_{0}}} \biggr] \\& \quad \leq \bigl[1-(\tau-\gamma l) (1-d)\alpha_{n} \bigr] \frac{\|x_{n}-x_{n-1}\|}{\beta_{n-1,k_{0}}} +\alpha_{n}b\|x_{n}-x_{n-1}\| \\& \qquad {} +\widetilde{M}_{3} \biggl[\frac{|r_{n}-r_{n-1}|}{\epsilon\beta_{n,k_{0}}}+ \frac{{ \sum^{M}_{k=1}} |\lambda_{k,n}-\lambda_{k,n-1}|}{\beta_{n,k_{0}}}+\frac{{ \sum^{N}_{k=1}}|\beta_{n,k}-\beta_{n-1,k}|}{ \beta_{n,k_{0}}} \\& \qquad {}+\frac{|\frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma _{n-1}}{1-\beta_{n-1}}|}{\beta_{n,k_{0}}} +\frac{|\alpha_{n}-\alpha_{n-1}|}{\beta_{n,k_{0}}}+ \frac{|\beta_{n}-\beta _{n-1}|}{\beta_{n,k_{0}}} \biggr] \\& \quad = \bigl[1-\alpha_{n}(\tau-\gamma l) (1-d) \bigr] \frac{\|x_{n}-x_{n-1}\|}{\beta_{n-1,k_{0}}} \\& \qquad {}+\alpha_{n}(\tau-\gamma l) (1-d)\cdot \frac{1}{(\tau-\gamma l)(1-d)} \biggl\{ b\| x_{n}-x_{n-1}\| \\& \qquad {} +\widetilde{M}_{3} \biggl[\frac{|r_{n}-r_{n-1}|}{\epsilon\alpha_{n}\beta _{n,k_{0}}}+ \frac{{ \sum^{M}_{k=1}} |\lambda_{k,n}-\lambda_{k,n-1}|}{\alpha_{n}\beta_{n,k_{0}}}+\frac{{ \sum^{N}_{k=1}}|\beta_{n,k}-\beta_{n-1,k}|}{ \alpha_{n}\beta_{n,k_{0}}} \\& \qquad {}+\frac{|\frac{\gamma_{n}}{1-\beta_{n}}-\frac{\gamma _{n-1}}{1-\beta_{n-1}}|}{\alpha_{n}\beta_{n,k_{0}}} +\frac{|\alpha_{n}-\alpha_{n-1}|}{\alpha_{n}\beta_{n,k_{0}}}+ \frac{|\beta _{n}-\beta_{n-1}|}{\alpha_{n}\beta_{n,k_{0}}} \biggr] \biggr\} . \end{aligned}$$
Therefore, utilizing Lemma 2.1 of [16], from (H0), (H7), and the asymptotical regularity of \(\{x_{n}\}\) (due to Proposition 3.2), we deduce that
$$\lim_{n\to\infty}\frac{\|x_{n+1}-x_{n}\|}{\beta_{n,k_{0}}}=0. $$
 □

Proposition 3.5

Let us suppose that \({\varOmega }\neq\emptyset\). Let us suppose that (H0)-(H6) hold. Then \(\|z_{n}-y_{n,N}\|\to0\) as \(n\to\infty\).

Proof

Let \(p\in{ \varOmega }\). In terms of the firm nonexpansivity of \(P_{C}\) and the \(\zeta_{j}\)-inverse-strong monotonicity of \(F_{j}\) for \(j=1,2\), we obtain from \(\nu_{j}\in(0,2\zeta_{j})\), \(j=1,2\), and (3.4)
$$\begin{aligned} \Vert \tilde{y}_{n,N}-\tilde{p}\Vert ^{2} =& \bigl\Vert P_{C}(I-\nu_{2}F_{2})y_{n,N}-P_{C}(I- \nu_{2}F_{2})p \bigr\Vert ^{2} \\ \leq& \bigl\langle (I-\nu_{2}F_{2})y_{n,N}-(I- \nu_{2}F_{2})p,\tilde{y}_{n,N}-\tilde{p} \bigr\rangle \\ =&\frac{1}{2} \bigl[ \bigl\Vert (I-\nu_{2}F_{2})y_{n,N}-(I- \nu_{2}F_{2})p \bigr\Vert ^{2}+\Vert \tilde{y}_{n,N}-\tilde{p}\Vert ^{2} \\ &{} - \bigl\Vert (I-\nu_{2}F_{2})y_{n,N}-(I- \nu_{2}F_{2})p-(\tilde{y}_{n,N}-\tilde{p}) \bigr\Vert ^{2} \bigr] \\ \leq&\frac{1}{2} \bigl[\Vert y_{n,N}-p\Vert ^{2}+ \Vert \tilde{y}_{n,N}-\tilde{p}\Vert ^{2} \\ &{}- \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-\nu_{2}(F_{2}y_{n,N}-F_{2}p)-(p- \tilde{p}) \bigr\Vert ^{2} \bigr] \\ =&\frac{1}{2} \bigl[\Vert y_{n,N}-p\Vert ^{2}+ \Vert \tilde{y}_{n,N}-\tilde{p}\Vert ^{2}- \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert ^{2} \\ &{} +2\nu_{2} \bigl\langle (y_{n,N}-\tilde{y}_{n,N})-(p- \tilde{p}),F_{2}y_{n,N}-F_{2}p \bigr\rangle - \nu^{2}_{2}\Vert F_{2}y_{n,N}-F_{2}p \Vert ^{2} \bigr] \end{aligned}$$
and
$$\begin{aligned} \Vert z_{n}-p\Vert ^{2} =& \bigl\Vert P_{C}(I-\nu_{1}F_{1})\tilde{y}_{n,N}-P_{C}(I- \nu_{1}F_{1})\tilde{p} \bigr\Vert ^{2} \\ \leq& \bigl\langle (I-\nu_{1}F_{1})\tilde{y}_{n,N}-(I- \nu_{1}F_{1})\tilde{p},z_{n}-p \bigr\rangle \\ =&\frac{1}{2} \bigl[ \bigl\Vert (I-\nu_{1}F_{1}) \tilde{y}_{n,N}-(I-\nu_{1}F_{1})\tilde{p} \bigr\Vert ^{2}+\Vert z_{n}-p\Vert ^{2} \\ &{} - \bigl\Vert (I-\nu_{1}F_{1})\tilde{y}_{n,N}-(I- \nu_{1}F_{1})\tilde{p}-(z_{n}-p) \bigr\Vert ^{2} \bigr] \\ \leq&\frac{1}{2} \bigl[\Vert \tilde{y}_{n,N}-\tilde{p}\Vert ^{2}+\Vert z_{n}-p\Vert ^{2}- \bigl\Vert ( \tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert ^{2} \\ &{} +2\nu_{1} \bigl\langle F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p},(\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\rangle - \nu^{2}_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert ^{2} \bigr] \\ \leq&\frac{1}{2} \bigl[\Vert y_{n,N}-p\Vert ^{2}+ \Vert z_{n}-p\Vert ^{2}- \bigl\Vert ( \tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert ^{2} \\ &{} +2\nu_{1} \bigl\langle F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p},(\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\rangle \bigr]. \end{aligned}$$
Thus, we have
$$\begin{aligned} \|\tilde{y}_{n,N}-\tilde{p}\|^{2} \leq&\|y_{n,N}-p \|^{2}- \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p- \tilde{p}) \bigr\Vert ^{2} \\ &{} +2\nu_{2} \bigl\langle (y_{n,N}-\tilde{y}_{n,N})-(p- \tilde{p}),F_{2}y_{n,N}-F_{2}p \bigr\rangle \\ &{}- \nu^{2}_{2}\|F_{2}y_{n,N}-F_{2}p \|^{2} \end{aligned}$$
(3.28)
and
$$\begin{aligned} \|z_{n}-p\|^{2} \leq&\|y_{n,N}-p\|^{2}- \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert ^{2} \\ &{}+2\nu_{1}\|F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\| \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert . \end{aligned}$$
(3.29)
Consequently, from (3.4), (3.24) and (3.28), it follows that
$$\begin{aligned}& \Vert x_{n+1}-p\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert z_{n}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} - \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert ^{2} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{ \varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert \tilde{y}_{n,N}- \tilde{p}\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert y_{n,N}-p\Vert ^{2}- \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p- \tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{2} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert \Vert F_{2}y_{n,N}-F_{2}p \Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert x_{n}-p\Vert ^{2}- \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p- \tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{2} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert \Vert F_{2}y_{n,N}-F_{2}p \Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq \Vert x_{n}-p\Vert ^{2}+\alpha_{n} \tau \Vert y_{n,N}-p\Vert ^{2}-(1-\beta_{n}) \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{2} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert \Vert F_{2}y_{n,N}-F_{2}p \Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert , \end{aligned}$$
which yields
$$\begin{aligned}& (1-d) \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert ^{2} \\& \quad \leq(1-\beta_{n}) \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert ^{2} \\& \quad \leq \Vert x_{n}-p\Vert ^{2}+\alpha_{n} \tau \Vert y_{n,N}-p\Vert ^{2}-\Vert x_{n+1}-p \Vert ^{2} \\& \qquad {} +2\nu_{2} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert \Vert F_{2}y_{n,N}-F_{2}p \Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \\& \quad \leq \Vert x_{n}-x_{n+1}\Vert \bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr)+ \alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2} \\& \qquad {} +2\nu_{2} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert \Vert F_{2}y_{n,N}-F_{2}p \Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert . \end{aligned}$$
Since \(\lim_{n\to\infty}\alpha_{n}=0\), \(\lim_{n\to\infty}\|x_{n+1}-x_{n}\| =0\), and \(\{x_{n}\}\), \(\{y_{n}\}\), \(\{y_{n,N}\}\), and \(\{\tilde{y}_{n,N}\}\) are bounded, we deduce from (3.22) that
$$ \lim_{n\to\infty} \bigl\Vert (y_{n,N}- \tilde{y}_{n,N})-(p- \tilde{p}) \bigr\Vert =0. $$
(3.30)
Furthermore, from (3.4), (3.24), and (3.29), it follows that
$$\begin{aligned}& \Vert x_{n+1}-p\Vert ^{2} \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert z_{n}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert y_{n,N}-p\Vert ^{2}- \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1-\beta_{n}) \bigl[\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert x_{n}-p\Vert ^{2}- \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \bigr] \\& \quad \leq \Vert x_{n}-p\Vert ^{2}+\alpha_{n} \tau \Vert y_{n,N}-p\Vert ^{2}-(1-\beta_{n}) \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert ^{2} \\& \qquad {} +2\nu_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert , \end{aligned}$$
which leads to
$$\begin{aligned}& (1-d) \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert ^{2} \\& \quad \leq(1-\beta_{n}) \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert ^{2} \\& \quad \leq \Vert x_{n}-p\Vert ^{2}+\alpha_{n} \tau \Vert y_{n,N}-p\Vert ^{2}-\Vert x_{n+1}-p \Vert ^{2} \\& \qquad {} +2\nu_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert \\& \quad \leq \Vert x_{n}-x_{n+1}\Vert \bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr)+ \alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2} \\& \qquad {} +2\nu_{1}\Vert F_{1}\tilde{y}_{n,N}-F_{1} \tilde{p}\Vert \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert +2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +2\lambda_{k,n} \bigl\Vert {\varLambda }^{k-1}_{n}y_{n}-{ \varLambda }^{k}_{n}y_{n} \bigr\Vert \bigl\Vert A_{k}{\varLambda }^{k-1}_{n}y_{n}-A_{k}p \bigr\Vert . \end{aligned}$$
Since \(\lim_{n\to\infty}\alpha_{n}=0\), \(\lim_{n\to\infty}\|x_{n+1}-x_{n}\| =0\), and \(\{x_{n}\}\), \(\{z_{n}\}\), \(\{y_{n}\}\), \(\{y_{n,N}\}\), and \(\{\tilde{y}_{n,N}\}\) are bounded, we deduce from (3.22) that
$$ \lim_{n\to\infty} \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p- \tilde{p}) \bigr\Vert =0. $$
(3.31)
Note that
$$\Vert y_{n,N}-z_{n}\Vert \leq \bigl\Vert (y_{n,N}-\tilde{y}_{n,N})-(p-\tilde{p}) \bigr\Vert + \bigl\Vert (\tilde{y}_{n,N}-z_{n})+(p-\tilde{p}) \bigr\Vert . $$
Hence from (3.30) and (3.31) we get
$$ \lim_{n\to\infty}\|y_{n,N}-z_{n}\|=\lim _{n\to\infty}\|y_{n,N}-Gy_{n,N}\| =0. $$
(3.32)
 □

Proposition 3.6

Let us suppose that \({\varOmega }\neq\emptyset\). Let us suppose that \(0<\liminf_{n\to\infty}\beta_{n,i} \leq\limsup_{n\to\infty}\beta_{n,i}<1\) for each \(i=1,\ldots,N\). Moreover, suppose that (H0)-(H6) are satisfied. Then, \(\lim_{n\to \infty}\|S_{i}u_{n}-u_{n}\|=0\) for each \(i=1,\ldots,N\) provided \(\|Ty_{n}-y_{n}\|\to \) as \(n\to\infty\).

Proof

First of all, observe that
$$\begin{aligned} x_{n+1}-x_{n} =&\gamma_{n}(\tilde{y}_{n}-x_{n})+ \delta_{n}(T\tilde{y}_{n}-x_{n}) \\ =&\gamma_{n}(\tilde{y}_{n}-y_{n})+ \gamma_{n}(y_{n}-x_{n})+\delta_{n}(T \tilde{y}_{n}-Ty_{n}) \\ &{}+\delta_{n}(Ty_{n}-y_{n})+ \delta_{n}(y_{n}-x_{n}) \\ =&(\gamma_{n}+\delta_{n}) (y_{n}-x_{n})+ \gamma_{n}(\tilde{y}_{n}-y_{n})+\delta _{n}(T\tilde{y}_{n}-Ty_{n})+ \delta_{n}(Ty_{n}-y_{n}) \\ =&(1-\beta_{n}) (y_{n}-x_{n})+ \gamma_{n}(\tilde{y}_{n}-y_{n})+ \delta_{n}(T\tilde{y}_{n}-Ty_{n})+ \delta_{n}(Ty_{n}-y_{n}). \end{aligned}$$
By Proposition 3.2 we know that \(\{x_{n}\}\) is asymptotically regular. Utilizing [17] we have from \((\gamma_{n}+\delta_{n}) \xi\leq\gamma_{n}\),
$$\begin{aligned} (1-d)\Vert y_{n}-x_{n}\Vert &\leq(1- \beta_{n})\Vert y_{n}-x_{n}\Vert \\ &=\bigl\Vert x_{n+1}-x_{n}-\gamma_{n}( \tilde{y}_{n}-y_{n})-\delta_{n}(T \tilde{y}_{n}-Ty_{n})-\delta_{n}(Ty_{n}-y_{n}) \bigr\Vert \\ &\leq \Vert x_{n+1}-x_{n}\Vert +\bigl\Vert \gamma_{n}(\tilde{y}_{n}-y_{n})+ \delta_{n}(T\tilde{y}_{n}-Ty_{n})\bigr\Vert + \delta_{n}\Vert Ty_{n}-y_{n}\Vert \\ &\leq \Vert x_{n+1}-x_{n}\Vert +(\gamma_{n}+ \delta_{n})\Vert \tilde{y}_{n}-y_{n}\Vert + \delta_{n}\Vert Ty_{n}-y_{n}\Vert \\ &\leq \Vert x_{n+1}-x_{n}\Vert +\Vert \tilde{y}_{n}-y_{n}\Vert +\Vert Ty_{n}-y_{n} \Vert , \end{aligned}$$
which together with (3.26) and \(\|Ty_{n}-y_{n}\|\to0\), implies that
$$ \lim_{n\to\infty}\|x_{n}-y_{n} \|=0. $$
(3.33)
Let us show that for each \(i\in\{1,\ldots,N\}\), one has \(\| S_{i}u_{n}-y_{n,i-1}\|\to0\) as \(n\to\infty\). Let \(p\in{ \varOmega }\). When \(i=N\), by Lemma 2.2(b) we have from (3.2), (3.4), and (3.20)
$$\begin{aligned} \Vert y_{n}-p\Vert ^{2} \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+\Vert z_{n}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\ \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert +\Vert y_{n,N}-p\Vert ^{2} \\ =&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2 \alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert +\beta_{n,N}\Vert S_{N}u_{n}-p \Vert ^{2} \\ &{} +(1-\beta_{n,N})\Vert y_{n,N-1}-p\Vert ^{2}- \beta_{n,N}(1-\beta_{n,N})\Vert S_{N}u_{n}-y_{n,N-1} \Vert ^{2} \\ \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert +\beta_{n,N}\Vert u_{n}-p \Vert ^{2} \\ &{} +(1-\beta_{n,N})\Vert u_{n}-p\Vert ^{2}- \beta_{n,N}(1-\beta_{n,N})\Vert S_{N}u_{n}-y_{n,N-1} \Vert ^{2} \\ =&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2 \alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert +\Vert u_{n}-p\Vert ^{2} \\ &{} -\beta_{n,N}(1-\beta_{n,N})\Vert S_{N}u_{n}-y_{n,N-1} \Vert ^{2} \\ \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert +\Vert x_{n}-p\Vert ^{2} \\ &{} -\beta_{n,N}(1-\beta_{n,N})\Vert S_{N}u_{n}-y_{n,N-1} \Vert ^{2}. \end{aligned}$$
So, we have
$$\begin{aligned}& \beta_{n,N}(1-\beta_{n,N})\Vert S_{N}u_{n}-y_{n,N-1} \Vert ^{2} \\& \quad \leq\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {}+\Vert x_{n}-p\Vert ^{2}-\Vert y_{n}-p\Vert ^{2} \\& \quad \leq\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {} +\Vert x_{n}-y_{n} \Vert \bigl(\Vert x_{n}-p\Vert +\Vert y_{n}-p\Vert \bigr). \end{aligned}$$
Since \(\alpha_{n}\to0\), \(0<\liminf_{n\to\infty}\beta_{n,N}\leq\limsup_{n\to \infty}\beta_{n,N}<1\) and \(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\) (due to (3.33)), it is known that \(\{\|S_{N}u_{n}-y_{n,N-1}\|\}\) is a null sequence.
Let \(i\in\{1,\ldots,N-1\}\). Then one has
$$\begin{aligned} \Vert y_{n}-p\Vert ^{2} \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert +\Vert y_{n,N}-p\Vert ^{2} \\ \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert +\beta_{n,N}\Vert S_{N}u_{n}-p\Vert ^{2} \\ &{} +(1-\beta_{n,N})\Vert y_{n,N-1}-p\Vert ^{2} \\ \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert +\beta_{n,N}\Vert x_{n}-p \Vert ^{2} \\ &{} +(1-\beta_{n,N})\Vert y_{n,N-1}-p\Vert ^{2} \\ \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert +\beta_{n,N}\Vert x_{n}-p \Vert ^{2} \\ &{} +(1-\beta_{n,N}) \bigl[\beta_{n,N-1}\Vert S_{N-1}u_{n}-p\Vert ^{2}+(1-\beta_{n,N-1}) \Vert y_{n,N-2}-p\Vert ^{2} \bigr] \\ \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\ &{} + \bigl(\beta_{n,N}+(1-\beta_{n,N})\beta_{n,N-1} \bigr)\Vert x_{n}-p\Vert ^{2} \\ &{}+{ \prod ^{N}_{k=N-1}}(1-\beta_{n,k})\Vert y_{n,N-2}-p\Vert ^{2}, \end{aligned}$$
and so, after \((N-i+1)\) iterations,
$$\begin{aligned} \Vert y_{n}-p\Vert ^{2} \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n}\bigl\Vert (\gamma f-\mu F)p\bigr\Vert \Vert y_{n}-p\Vert \\ &{} +\Biggl(\beta_{n,N}+{ \sum^{N}_{j=i+2}} \Biggl({ \prod^{N}_{l=j}}(1- \beta_{n,l})\Biggr)\beta _{n,j-1}\Biggr)\Vert x_{n}-p \Vert ^{2} \\ &{}+{ \prod^{N}_{k=i+1}}(1- \beta_{n,k})\Vert y_{n,i}-p\Vert ^{2} \\ \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n}\bigl\Vert (\gamma f-\mu F)p\bigr\Vert \Vert y_{n}-p\Vert \\ &{} +\Biggl(\beta_{n,N}+{ \sum^{N}_{j=i+2}} \Biggl({ \prod^{N}_{l=j}}(1- \beta_{n,l})\Biggr)\beta _{n,j-1}\Biggr)\Vert x_{n}-p \Vert ^{2} \\ &{}+{ \prod^{N}_{k=i+1}}(1- \beta_{n,k})\bigl[\beta_{n,i}\Vert S_{i}u_{n}-p \Vert ^{2} \\ &{} +(1-\beta_{n,i})\Vert y_{n,i-1}-p\Vert ^{2}- \beta_{n,i}(1-\beta_{n,i})\Vert S_{i}u_{n}-y_{n,i-1} \Vert ^{2}\bigr] \\ \leq&\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n}\bigl\Vert (\gamma f-\mu F)p\bigr\Vert \Vert y_{n}-p\Vert +\Vert x_{n}-p\Vert ^{2} \\ &{} -\beta_{n,i}{ \prod^{N}_{k=i}}(1- \beta_{n,k})\Vert S_{i}u_{n}-y_{n,i-1} \Vert ^{2}. \end{aligned}$$
(3.34)
Again we obtain
$$\begin{aligned}& \beta_{n,i}{ \prod^{N}_{k=i}}(1- \beta_{n,k})\Vert S_{i}u_{n}-y_{n,i-1} \Vert ^{2} \\& \quad \leq\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {}+\Vert x_{n}-p\Vert ^{2}-\Vert y_{n}-p\Vert ^{2} \\& \quad \leq\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\& \qquad {}+\Vert x_{n}-y_{n} \Vert \bigl(\Vert x_{n}-p\Vert +\Vert y_{n}-p\Vert \bigr). \end{aligned}$$
Since \(\alpha_{n}\to0\), \(0<\liminf_{n\to\infty}\beta_{n,i}\leq\limsup_{n\to \infty}\beta_{n,i}<1\) for each \(i=1,\ldots,N-1\), and \(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\) (due to (3.33)), it is known that
$$\lim_{n\to\infty}\|S_{i}u_{n}-y_{n,i-1} \|=0. $$

Obviously for \(i=1\), we have \(\|S_{1}u_{n}-u_{n}\|\to0\).

To conclude, we have
$$\|S_{2}u_{n}-u_{n}\|\leq\|S_{2}u_{n}-y_{n,1} \|+\|y_{n,1}-u_{n}\|=\|S_{2}u_{n}-y_{n,1} \| +\beta_{n,1}\|S_{1}u_{n}-u_{n}\|, $$
from which \(\|S_{2}u_{n}-u_{n}\|\to0\). Thus by induction \(\|S_{i}u_{n}-u_{n}\|\to0\) for all \(i=2,\ldots,N\) since it is enough to observe that
$$\begin{aligned} \|S_{i}u_{n}-u_{n}\|&\leq\|S_{i}u_{n}-y_{n,i-1} \|+\| y_{n,i-1}-S_{i-1}u_{n}\|+\|S_{i-1}u_{n}-u_{n} \| \\ &\leq\|S_{i}u_{n}-y_{n,i-1}\|+(1- \beta_{n,i-1})\|S_{i-1}u_{n}-y_{n,i-2}\|+\| S_{i-1}u_{n}-u_{n}\|. \end{aligned}$$
 □

Remark 3.2

As an example, we consider \(M=1\), \(N=2\) and the sequences:
  1. (a)

    \(\lambda_{1,n}=\eta_{1}-\frac{1}{n}\), \(\forall n>\frac{1}{\eta_{1}}\);

     
  2. (b)

    \(\alpha_{n}=\frac{1}{\sqrt{n}}\), \(r_{n}=2-\frac{1}{n}\), \(\forall n>1\);

     
  3. (c)

    \(\beta_{n}=\beta_{n,1}=\frac{1}{2}-\frac{1}{n}\), \(\beta_{n,2}=\frac {1}{2}-\frac{1}{n^{2}}\), \(\forall n>2\).

     
Then they satisfy the hypotheses on the parameter sequences in Proposition 3.6.

Proposition 3.7

Let us suppose that \({\varOmega }\neq\emptyset\) and \(\beta_{n,i}\to\beta_{i}\) for all i as \(n\to\infty\). Suppose there exists \(k\in\{1,\ldots,N\}\) such that \(\beta_{n,k}\to0\) as \(n\to\infty\). Let \(k_{0} \in\{1,\ldots,N\}\) be the largest index such that \(\beta_{n,k_{0}}\to0\) as \(n\to\infty\). Suppose that
  1. (i)

    \(\frac{\alpha_{n}}{\beta_{n,k_{0}}}\to0\) as \(n\to\infty\);

     
  2. (ii)

    if \(i\leq k_{0}\) and \(\beta_{n,i}\to0\) then \(\frac{\beta _{n,k_{0}}}{\beta_{n,i}}\to0\) as \(n\to\infty\);

     
  3. (iii)

    if \(\beta_{n,i}\to\beta_{i}\neq0\) then \(\beta_{i}\) lies in \((0,1)\).

     
Moreover, suppose that (H0), (H7), and (H8) hold. Then, \(\lim_{n\to \infty}\|S_{i}u_{n}-u_{n}\|=0\) for each \(i=1,\ldots,N\) provided \(\|Ty_{n}-y_{n}\|\to0\) as \(n\to\infty\).

Proof

First of all we note that if (H7) holds then also (H1)-(H6) are satisfied. So \(\{x_{n}\}\) is asymptotically regular.

Let \(k_{0}\) be as in the hypotheses. As in Proposition 3.6, for every index \(i\in\{1,\ldots,N\}\) such that \(\beta_{n,i}\to\beta_{i} \neq0\) (which leads to \(0<\liminf_{n\to\infty}\beta_{n,i}\leq\limsup_{n\to\infty}\beta_{n,i}<1\)), one has \(\|S_{i}u_{n}-y_{n,i-1}\| \to0\) as \(n\to\infty\).

For all the other indices \(i\leq k_{0}\), we can prove that \(\| S_{i}u_{n}-y_{n,i-1}\|\to0\) as \(n\to\infty\) in a similar manner. By the relation (due to (3.21) and (3.34))
$$\begin{aligned} \Vert x_{n+1}-p\Vert ^{2} \leq&\beta_{n} \Vert x_{n}-p\Vert ^{2}+(1-\beta_{n})\Vert \tilde{y}_{n}-p\Vert ^{2} \\ \leq&\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1- \beta_{n})\Vert y_{n}-p\Vert ^{2} \\ \leq&\beta_{n}\Vert x_{n}-p\Vert ^{2}+(1- \beta_{n}) \\ &{}\times\Biggl[\alpha_{n}\tau \Vert y_{n,N}-p \Vert ^{2}+2\alpha _{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\ &{} +\Vert x_{n}-p\Vert ^{2}-\beta_{n,i}{ \prod^{N}_{k=i}}(1-\beta_{n,k}) \Vert S_{i}u_{n}-y_{n,i-1}\Vert ^{2} \Biggr] \\ \leq&\Vert x_{n}-p\Vert ^{2}+\alpha_{n}\tau \Vert y_{n,N}-p\Vert ^{2}+2\alpha_{n} \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \\ &{} -(1-\beta_{n})\beta_{n,i}{ \prod ^{N}_{k=i}}(1-\beta_{n,k})\Vert S_{i}u_{n}-y_{n,i-1}\Vert ^{2}, \end{aligned}$$
we immediately obtain that
$$\begin{aligned}& (1-d){ \prod^{N}_{k=i}(1- \beta_{n,k})}\Vert S_{i}u_{n}-y_{n,i-1} \Vert ^{2} \\& \quad \leq(1-\beta_{n}){ \prod^{N}_{k=i}(1- \beta_{n,k})}\Vert S_{i}u_{n}-y_{n,i-1} \Vert ^{2} \\& \quad \leq\frac{\alpha_{n}}{\beta_{n,i}} \bigl[\tau \Vert y_{n,N}-p\Vert ^{2}+2 \bigl\Vert (\gamma f-\mu F)p \bigr\Vert \Vert y_{n}-p\Vert \bigr] \\& \qquad {}+\frac{\Vert x_{n}-x_{n+1}\Vert }{\beta_{n,i}} \bigl(\Vert x_{n}-p\Vert +\Vert x_{n+1}-p\Vert \bigr). \end{aligned}$$
By Proposition 3.4 or by hypothesis (ii) on the sequences, we have
$$\frac{\|x_{n}-x_{n+1}\|}{\beta_{n,i}}=\frac{\|x_{n}-x_{n+1}\|}{\beta _{n,k_{0}}}\cdot\frac{\beta_{n,k_{0}}}{\beta_{n,i}}\to0. $$
So, the conclusion follows. □

Remark 3.3

Let us consider \(M=1\), \(N=3\) and the following sequences:
  1. (a)

    \(\alpha_{n}=\frac{1}{n^{1/2}}\), \(r_{n}=2-\frac{1}{n^{2}}\), \(\forall n>1\);

     
  2. (b)

    \(\lambda_{1,n}=\eta_{1}-\frac{1}{n^{2}}\), \(\forall n>\frac{1}{\eta^{1/2}_{1}}\);

     
  3. (c)

    \(\beta_{n,1}=\frac{1}{n^{1/4}}\), \(\beta_{n}=\beta_{n,2}=\frac {1}{2}-\frac{1}{n^{2}}\), \(\beta_{n,3}=\frac{1}{n^{1/3}}\), \(\forall n>1\).

     
It is easy to see that all hypotheses (i)-(iii), (H0), (H7), and (H8) of Proposition 3.7 are satisfied.

Remark 3.4

Under the hypotheses of Proposition 3.7, analogously to Proposition 3.6, one can see that
$$\lim_{n\to\infty}\|S_{i}u_{n}-y_{n,i-1} \|=0,\quad \forall i\in\{2,\ldots,N\}. $$

Corollary 3.1

Let us suppose that the hypotheses of either Proposition  3.6 or Proposition  3.7 are satisfied. Then \(\omega_{w}(x_{n})=\omega_{w}(u_{n})=\omega_{w}(y_{n,1})\), \(\omega_{s}(x_{n})=\omega _{s}(u_{n})=\omega_{s}(y_{n,1})\), and \(\omega_{w}(x_{n})\subset{ \varOmega }\).

Proof

By Remark 3.1, we have \(\omega_{w}(x_{n})=\omega_{w}(u_{n})\) and \(\omega_{s}(x_{n})=\omega_{s}(u_{n})\). Note that by Remark 3.4,
$$\lim_{n\to\infty}\|S_{N}u_{n}-y_{n,N-1} \|=0. $$
In the meantime, it is known that
$$\lim_{n\to\infty}\|S_{N}u_{n}-u_{n} \|=\lim_{n\to\infty}\|u_{n}-x_{n}\|=\lim _{n\to \infty}\|x_{n}-y_{n}\|=0. $$
Hence we have
$$ \lim_{n\to\infty}\|S_{N}u_{n}-y_{n} \|=0. $$
(3.35)
Furthermore, it follows from (3.1) that
$$\lim_{n\to\infty}\|y_{n,N}-y_{n,N-1}\|=\lim _{n\to\infty}\beta_{n,N}\| S_{N}u_{n}-y_{n,N-1} \|=0, $$
which, together with \(\lim_{n\to\infty}\|S_{N}u_{n}-y_{n,N-1}\|=0\), yields
$$ \lim_{n\to\infty}\|S_{N}u_{n}-y_{n,N} \|=0. $$
(3.36)
Combining (3.35) and (3.36), we conclude that
$$ \lim_{n\to\infty}\|y_{n}-y_{n,N} \|=0, $$
(3.37)
which, together with \(\lim_{n\to\infty}\|x_{n}-y_{n}\|=0\), leads to
$$ \lim_{n\to\infty}\|x_{n}-y_{n,N} \|=0. $$
(3.38)
Now we observe that
$$\|x_{n}-y_{n,1}\|\leq\|x_{n}-u_{n}\|+ \|y_{n,1}-u_{n}\|=\|x_{n}-u_{n}\|+ \beta_{n,1}\| S_{1}u_{n}-u_{n}\|. $$
By Propositions 3.3 and 3.6, \(\|x_{n}-u_{n}\|\to0\) and \(\|S_{1}u_{n}-u_{n}\|\to0\) as \(n\to\infty\), and hence
$$\lim_{n\to\infty}\|x_{n}-y_{n,1}\|=0. $$
So we get \(\omega_{w}(x_{n})=\omega_{w}(y_{n,1})\) and \(\omega_{s}(x_{n})=\omega _{s}(y_{n,1})\).
Let \(p\in\omega_{w}(x_{n})\). Then there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that \(x_{n_{i}}\rightharpoonup p\). Since \(p\in\omega_{w}(u_{n})\), by Proposition 3.6 and [18] (demiclosedness principle), we have \(p\in\operatorname{Fix}(S_{i})\) for each \(i=1,\ldots,N\), i.e., \(p\in\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i})\). Taking into account \(p\in\in\omega_{w}(y_{n,N})\) (due to (3.38)) and \(\|y_{n,N}-Gy_{n,N}\|\to0\) (due to (3.32)), by [18] we know that \(p\in\operatorname{Fix}(G)=: {\varXi }\). Also, since \(p\in\omega_{w}(y_{n})\) (due to (3.33)), in terms of \(\|Ty_{n}-y_{n}\|\to0\) and Proposition 2.1 of [19], we get \(p\in\operatorname{Fix}(T)\). Moreover, by [20] and Proposition 3.3 we know that \(p\in\operatorname{GMEP}({\varTheta },h)\). Next we prove that \(p\in\bigcap^{M}_{m=1}\operatorname{VI}(C,A_{m})\). As a matter of fact, from (3.25) and (3.33) we know that \(y_{n_{i}}\rightharpoonup p\) and \({\varLambda }^{m}_{n_{i}}y_{n_{i}}\rightharpoonup p\) for each \(m=1,\ldots,M\). Let
$$\widetilde{T}_{m}v=\left \{ \textstyle\begin{array}{l@{\quad}l} A_{m}v+N_{C}v, &v\in C, \\ \emptyset,& v\notin C, \end{array}\displaystyle \right . $$
where \(m\in\{1,2,\ldots,M\}\). Let \((v,u)\in G(\widetilde{T}_{m})\). Since \(u-A_{m}v\in N_{C}v\) and \({\varLambda }^{m}_{n}y_{n}\in C\), we have
$$\bigl\langle v-{\varLambda }^{m}_{n}y_{n},u-A_{m}v \bigr\rangle \geq0. $$
On the other hand, from \({\varLambda }^{m}_{n}y_{n}=P_{C}(I-\lambda_{m,n}A_{m}){ \varLambda }^{m-1}_{n}y_{n}\) and \(v\in C\), we have
$$\bigl\langle v-{\varLambda }^{m}_{n}y_{n},{ \varLambda }^{m}_{n}y_{n}- \bigl({\varLambda }^{m-1}_{n}y_{n}-\lambda_{m,n}A_{m}{ \varLambda }^{m-1}_{n}y_{n} \bigr) \bigr\rangle \geq0, $$
and hence
$$\biggl\langle v-{\varLambda }^{m}_{n}y_{n}, \frac{{\varLambda }^{m}_{n}y_{n}-{\varLambda }^{m-1}_{n}y_{n}}{\lambda_{m,n}}+A_{m}{\varLambda }^{m-1}_{n}y_{n} \biggr\rangle \geq0. $$
Therefore, we have
$$\begin{aligned}& \bigl\langle v-{\varLambda }^{m}_{n_{i}}y_{n_{i}},u \bigr\rangle \\& \quad \geq \bigl\langle v-{\varLambda }^{m}_{n_{i}}y_{n_{i}},A_{m}v \bigr\rangle \\& \quad \geq \bigl\langle v-{\varLambda }^{m}_{n_{i}}y_{n_{i}},A_{m}v \bigr\rangle - \biggl\langle v-{\varLambda }^{m}_{n_{i}}y_{n_{i}}, \frac{{\varLambda }^{m}_{n_{i}}y_{n_{i}}-{\varLambda }^{m-1}_{n_{i}}y_{n_{i}}}{\lambda_{m,{n_{i}}}} +A_{m}{\varLambda }^{m-1}_{n_{i}}y_{n_{i}} \biggr\rangle \\& \quad = \bigl\langle v-{\varLambda }^{m}_{n_{i}}y_{n_{i}},A_{m}v-A_{m}{ \varLambda }^{m}_{n_{i}}y_{n_{i}} \bigr\rangle + \bigl\langle v-{\varLambda }^{m}_{n_{i}}y_{n_{i}},A_{m}{ \varLambda }^{m}_{n_{i}}y_{n_{i}}-A_{m}{ \varLambda }^{m-1}_{n_{i}}y_{n_{i}} \bigr\rangle \\& \qquad {} - \biggl\langle v-{\varLambda }^{m}_{n_{i}}y_{n_{i}}, \frac{{\varLambda }^{m}_{n_{i}}y_{n_{i}}-{\varLambda }^{m-1}_{n_{i}}y_{n_{i}}}{\lambda _{m,{n_{i}}}} \biggr\rangle \\& \quad \geq \bigl\langle v-{\varLambda }^{m}_{n_{i}}y_{n_{i}},A_{m}{ \varLambda }^{m}_{n_{i}}y_{n_{i}}-A_{m}{ \varLambda }^{m-1}_{n_{i}}y_{n_{i}} \bigr\rangle - \biggl\langle v-{\varLambda }^{m}_{n_{i}}y_{n_{i}}, \frac{{\varLambda }^{m}_{n_{i}}y_{n_{i}}-{\varLambda }^{m-1}_{n_{i}}y_{n_{i}}}{\lambda _{m,{n_{i}}}} \biggr\rangle . \end{aligned}$$
From (3.25) and since \(A_{m}\) is Lipschitz continuous, we obtain
$$\lim_{n\to\infty} \bigl\Vert A_{m}{\varLambda }^{m}_{n}y_{n}-A_{m}{ \varLambda }^{m-1}_{n}y_{n} \bigr\Vert =0. $$
From \({\varLambda }^{m}_{n_{i}}y_{n_{i}}\rightharpoonup p\), \(\{\lambda_{m,n}\} \subset[a_{m},b_{m}]\subset(0,2\eta_{m})\), \(\forall m\in\{1,2,\ldots,M\}\) and (3.25), we have
$$\langle v-p,u\rangle\geq0. $$
Since \(\widetilde{T}_{m}\) is maximal monotone, we have \(p\in\widetilde{T}^{-1}_{m}0\) and hence \(p\in\operatorname{VI}(C,A_{m})\), \(m=1,2,\ldots,M\), which implies \(p\in\bigcap^{M}_{m=1}\operatorname{VI}(C,A_{m})\). Consequently, it is known that \(p\in \operatorname{Fix}(T)\cap\bigcap^{N}_{i=1}\operatorname{Fix}(S_{i}) \cap\bigcap^{M}_{m=1}\operatorname{VI}(C,A_{m})\cap\operatorname{GMEP}({\varTheta },h)\cap{ \varXi }=:{\varOmega }\). □

Theorem 3.1

Let us suppose that \({\varOmega }\neq\emptyset\). Let \(\{\alpha_{n}\}\), \(\{\beta_{n,i}\}\), \(i=1,\ldots,N\), be sequences in \((0,1)\) such that \(0<\liminf_{n\to\infty}\beta_{n,i}\leq\limsup_{n\to\infty}\beta_{n,i}<1\) for each index i. Moreover, let us suppose that (H0)-(H6) hold. Then the sequences \(\{x_{n}\}\), \(\{y_{n}\} \), and \(\{u_{n}\}\) defined by scheme (3.1), all converge strongly to \(x^{*}=P_{\varOmega }(I-(\mu F-\gamma f))x^{*}\) if and only if \(\|y_{n}-Ty_{n}\|\to0\) as \(n\to\infty\), where \(x^{*}=P_{ \varOmega }(I-(\mu F-\gamma f))x^{*}\) is the unique solution of the hierarchical VIP
$$ \bigl\langle (\gamma f-\mu F)x^{*},x-x^{*} \bigr\rangle \leq0,\quad \forall x\in{ \varOmega }. $$
(3.39)

Proof

First of all, we note that \(F:C\to H\) is η-strongly monotone and κ-Lipschitzian on C and \(f:C \to C\) is an l-Lipschitz continuous mapping with \(0\leq\gamma l<\tau\). Observe that
$$\mu\eta\geq\tau\quad \Leftrightarrow\quad \kappa\geq\eta. $$
It is clear that
$$\bigl\langle (\mu F-\gamma f)x-(\mu F-\gamma f)y,x-y \bigr\rangle \geq(\mu\eta - \gamma l)\|x-y\|^{2}, \quad \forall x,y\in C. $$
Hence we deduce that \(\mu F-\gamma f\) is \((\mu\eta-\gamma l)\)-strongly monotone. In the meantime, it is easy to see that \(\mu F- \gamma f\) is \((\mu\kappa+\gamma l)\)-Lipschitz continuous with constant \(\mu\kappa+\gamma l>0\). Thus, there exists a unique solution \(x^{*}\) in Ω to the VIP (3.39).
Now, observe that there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\} \) such that
$$ \limsup_{n\to\infty} \bigl\langle (\gamma f-\mu F)x^{*},x_{n}-x^{*} \bigr\rangle =\lim_{i\to \infty} \bigl\langle (\gamma f-\mu F)x^{*},x_{n_{i}}-x^{*} \bigr\rangle . $$
(3.40)
Since \(\{x_{n_{i}}\}\) is bounded, there exists a subsequence \(\{ x_{n_{i_{j}}}\}\) of \(\{x_{n_{i}}\}\) which converges weakly to some \(p\in H\). Without loss of generality, we may assume that \(x_{n_{i}}\rightharpoonup p\). Then by Corollary 3.1, we get \(p\in\omega_{w} (x_{n})\subset{ \varOmega }\). Hence, from (3.39) and (3.40), we have
$$ \limsup_{n\to\infty} \bigl\langle (\gamma f-\mu F)x^{*},x_{n}-x^{*} \bigr\rangle = \bigl\langle (\gamma f-\mu F)x^{*},p-x^{*} \bigr\rangle \leq0. $$
(3.41)
Since (H1)-(H6) hold, the sequence \(\{x_{n}\}\) is asymptotically regular (according to Proposition 3.2). In terms of (3.33) and Proposition 3.3, \(\|x_{n}-y_{n}\|\to0\) and \(\|x_{n}-u_{n}\|\to0\) as \(n\to\infty\).
Let us show that \(\|x_{n}-x^{*}\|\to0\) as \(n\to\infty\). Indeed, putting \(p=x^{*}\), we deduce from (3.3), (3.4), (3.20), and (3.21) that
$$\begin{aligned}& \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2} \\& \quad \leq\beta_{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-\beta_{n}) \bigl\Vert \tilde{y}_{n}-x^{*} \bigr\Vert ^{2} \\& \quad \leq\beta_{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-\beta_{n}) \bigl\Vert y_{n}-x^{*} \bigr\Vert ^{2} \\& \quad \leq\beta_{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-\beta_{n}) \biggl[\alpha_{n}\tau \frac{(\gamma l)^{2}}{\tau^{2}} \bigl\Vert y_{n,N}-x^{*} \bigr\Vert ^{2}+(1- \alpha_{n}\tau) \bigl\Vert z_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {} +2\alpha_{n} \bigl\langle (\gamma f-\mu F)x^{*},y_{n}-x^{*} \bigr\rangle \biggr] \\& \quad \leq\beta_{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-\beta_{n}) \biggl[\alpha_{n} \frac{(\gamma l)^{2}}{\tau } \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1- \alpha_{n}\tau) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {} +2\alpha_{n} \bigl\langle (\gamma f-\mu F)x^{*},y_{n}-x^{*} \bigr\rangle \biggr] \\& \quad =\beta_{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-\beta_{n}) \biggl[ \biggl(1-\alpha_{n} \frac{\tau^{2}-(\gamma l)^{2}}{\tau} \biggr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {}+2\alpha_{n} \bigl\langle (\gamma f-\mu F)x^{*},y_{n}-x^{*} \bigr\rangle \biggr] \\& \quad = \biggl(1-\alpha_{n}(1-\beta_{n})\frac{\tau^{2}-(\gamma l)^{2}}{\tau} \biggr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} +2 \alpha_{n}(1-\beta_{n}) \bigl\langle (\gamma f-\mu F)x^{*},y_{n}-x^{*} \bigr\rangle \\& \quad \leq \biggl(1-\alpha_{n}(1-\beta_{n}) \frac{\tau^{2}-(\gamma l)^{2}}{\tau} \biggr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} \\& \qquad {}+\alpha_{n}(1-\beta_{n})\frac{\tau^{2}-(\gamma l)^{2}}{\tau} \cdot\frac{2\tau}{\tau^{2}-(\gamma l)^{2}} \bigl\langle (\gamma f-\mu F)x^{*},y_{n}-x^{*} \bigr\rangle . \end{aligned}$$
(3.42)
Since \(\sum^{\infty}_{n=0}\alpha_{n}=\infty\), \(\{\beta_{n}\}\subset[c,d]\subset (0,1)\) and \(\|x_{n}-y_{n}\|\to0\), we obtain \(\sum^{\infty}_{n=0}\alpha_{n}(1-\beta_{n})\frac {\tau^{2}-(\gamma l)^{2}}{ \tau}\geq\sum^{\infty}_{n=0}\alpha_{n}(1-d)\frac{\tau^{2}-(\gamma l)^{2}}{\tau }=\infty\) and
$$\begin{aligned}& { \limsup_{n\to\infty}}\frac{2\tau}{\tau^{2}-(\gamma l)^{2}} \bigl\langle (\gamma f-\mu F)x^{*},y_{n}-x^{*} \bigr\rangle \\& \quad ={ \limsup_{n\to\infty}}\frac{2\tau}{\tau^{2}-(\gamma l)^{2}} \bigl( \bigl\langle ( \gamma f-\mu F)x^{*},x_{n}-x^{*} \bigr\rangle + \bigl\langle (\gamma f-\mu F)x^{*},y_{n}-x_{n} \bigr\rangle \bigr) \\& \quad ={ \limsup_{n\to\infty}}\frac{2\tau}{\tau^{2}-(\gamma l)^{2}} \bigl\langle (\gamma f-\mu F)x^{*},x_{n}-x^{*} \bigr\rangle \leq0 \end{aligned}$$
(due to (3.41)). Applying Lemma 2.1 of [16] to (3.42), we infer that the sequence \(\{x_{n}\}\) converges strongly to \(x^{*}\). This completes the proof. □

In a similar way, we can conclude another theorem as follows.

Theorem 3.2

Let us suppose that \({\varOmega }\neq\emptyset\). Let \(\{\alpha_{n}\}\), \(\{\beta_{n,i}\}\), \(i=1,\ldots,N\), be sequences in \((0,1)\) such that \(\beta_{n,i}\to\beta_{i}\) for each index i as \(n\to\infty\). Suppose that there exists \(k\in\{1,\ldots,N\}\) for which \(\beta_{n,k}\to0\) as \(n\to\infty\). Let \(k_{0}\in\{1,\ldots,N\}\) the largest index for which \(\beta_{n,k_{0}}\to0\). Moreover, let us suppose that (H0), (H7), and (H8) hold and
  1. (i)

    \(\frac{\alpha_{n}}{\beta_{n,k_{0}}}\to0\) as \(n\to\infty\);

     
  2. (ii)

    if \(i\leq k_{0}\) and \(\beta_{n,i}\to\beta_{i}\) then \(\frac{\beta _{n,k_{0}}}{\beta_{n,i}}\to0\) as \(n\to\infty\);

     
  3. (iii)

    if \(\beta_{n,i}\to\beta_{i}\neq0\) then \(\beta_{i}\) lies in \((0,1)\).

     
Then the sequences \(\{x_{n}\}\), \(\{y_{n}\}\), and \(\{u_{n}\}\) defined by scheme (3.1) all converge strongly to \(x^{*}=P_{\varOmega }(I-(\mu F-\gamma f))x^{*}\) if and only if \(\|y_{n}-Ty_{n}\| \to0\) as \(n\to\infty\), where \(x^{*}=P_{\varOmega } (I-(\mu F-\gamma f))x^{*}\) is the unique solution of the hierarchical VIP
$$\bigl\langle (\gamma f-\mu F)x^{*},x-x^{*} \bigr\rangle \leq0,\quad \forall x\in{ \varOmega }. $$

Remark 3.5

According to the above argument for Theorems 3.1 and 3.2, we can readily see that if, in scheme (3.1), the iterative step \(y_{n}=P_{C}[\alpha_{n}\gamma f(y_{n,N})+(I-\alpha _{n}\mu F)Gy_{n,N}]\) is replaced by the iterative one, \(y_{n}=P_{C}[\alpha_{n}\gamma f(x_{n})+(I-\alpha_{n}\mu F)Gy_{n,N}]\), then Theorems 3.1 and 3.2 remain valid.

Declarations

Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under Grant No. (34-130-36-HiCi). The authors, therefore, acknowledge with thanks DSR technical and financial support. Finally, the authors thank the honorable reviewers and respectable editor for their valuable comments and suggestions.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University
(2)
Department of Mathematics, University of Jeddah
(3)
Center for Fundamental Science, China Medical University

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