1 Introduction

Throughout this work, one always supposes that C is a nonempty convex set in a Banach space X whose dual is denoted by \(X^{*}\). One denotes by the same notation, \(\|\cdot \|\), the norms of X and \(X^{*}\). A common problem in machine learning, automatic control, and utility-based bandwidth allocation problems consists of finding a solution of some equation satisfying some constraints. This common problem is called the convex feasibility problem, which can be characterized via the following model: \(x\in \bigcap_{i\in I}C_{i}\), where I denotes some index set, \(C_{i}\) is a convex set in X.

Next, one employs \(J_{q}:X\to 2^{X^{*}}\), where \(q>1\) is real number, to denote the duality mapping, which is defined by \(J_{q}(x):=\{ \phi \in X^{*}:\langle x,\phi \rangle =\|x\| ^{q},\|x\|^{q-1}=\| \phi \|\}\), \(\forall x\in X\). Let \(A_{1},A_{2}:C\to X\) be two nonlinear non-self mappings. Consider the problem of finding \((x^{*},y^{*}) \in C\times C\) such that

$$ \textstyle\begin{cases} \langle x^{*}-y^{*}+\mu _{1}A_{1}y^{*},J(x-x^{*})\rangle \geq 0, & \forall x\in C, \\ \langle y^{*}-x^{*}+\mu _{2}A_{2}x^{*},J(x-y^{*})\rangle \geq 0, & \forall x\in C, \end{cases} $$
(1.1)

with two positive real constants \(\mu _{1}\) and \(\mu _{2}\). This is called a system of generalized variational inequalities (SGVIs). This is a natural extension of the generalized variational inequality considered by Aoyama, Iiduka and Takahashi [1] in uniformly convex and 2-uniformly smooth Banach spaces; see [1] for more details. In Hilbert spaces, the system is reduced to the system of variational inequalities considered by Ceng et al. [2]. Problem (1.1) and its special cases are now under the spotlight of research because of their connections to other real convex and set optimization problems; see, e.g., [3–8] and the references therein. Recently, a fixed point method has been studied for solving convex and non-convex optimization problems since the equivalence between fixed point problems and zero point problems; see, e.g., [9–13] and the references therein. Indeed, one can transfer zero point problems (inclusion problems) to some fixed point problem of nonexpansive operators. The core is the resolvent of original operators. For example, one can show that the resolvent operator of m-accretive or maximally accretive operators is nonexpansive. Hence, Mann-like algorithms are applicable, however, they are only weakly convergent. Strong convergence is desirable in lots of situations, such as, image recovery, optimal control and quantum physics since they are in infinite-dimensional spaces. In this paper, we study, in the framework of Banach spaces, a convex feasibility problem with the constraints of the generalized system of monotone variational inequalities, a variational inclusion and a countable family of nonexpansive operators. Strong convergence theorems are obtained without any compact assumption on operators. Our rule is based on the Korpelevich extragradient method, the perturbation mapping, and the W-mappings constructed by \(\{S_{n}\}^{\infty }_{n=0}\). The main results extend and improve some recent results in [14–17].

2 Preliminaries

Next, one uses \(\rho _{X}:[0,\infty )\to [0,\infty )\) to stand for the smoothness modulus of space X which is defined by \(\rho _{X}(t)= \sup \{(\|x+y\|+\|x-y\|) /2-1:x\in U, \|y\|\leq t\}\). One says that X is uniformly smooth if \(\lim_{t\to 0^{+}}\rho _{X}(t)/t=0\). Let \(q\in (1,2]\) be a fixed real number. A Banach space X is said to be q-uniformly smooth if \(\rho _{X}(t)\leq t^{q}d\), \(\forall t>0\), where d is some constant. It is well known that Hilbert spaces, \(L^{p}\) and \(\ell _{p}\) are uniformly smooth where \(p>1\). More precisely, each Hilbert space is 2-uniformly smooth, while \(L^{p}\) and \(\ell _{p}\) are \(\min \{p,2\}\)-uniformly smooth for each \(p>1\).

Let \(A:C\to 2^{X}\) be a set-valued operator with \(Ax\neq \emptyset \), \(\forall x\in C\). An operator A is said to be accretive if, \(\forall x,y\in C\), \(\langle u-v,j_{q}(x-y)\rangle \geq 0\), \(\forall u\in Ax\), \(v\in Ay\), where \(j_{q}(x-y)\in J_{q}(x-y)\). A single-valued accretive operator A is said to be α-inverse-strongly accretive of order q if, \(\forall x,y \in C\), there exist \(\alpha >0\) and \(j_{q}(x-y)\in J_{q}(x-y)\) such that \(\langle u-v,j_{q}(x-y)\rangle \geq \alpha \|Ax-Ay\|^{q}\), \(\forall u \in Ax\), \(v\in Ay\). Back to Hilbert spaces, A is called the inverse-strongly monotone. This class of mappings is a key component in projection-based approximation methods; see, e.g., [18–22]. An accretive operator A is said to be m-accretive if and only if A is accretive and satisfies the range condition: \((I+\lambda A)C=X\) for all \(\lambda >0\). For an accretive operator A, we define the mapping \(J^{A}_{\lambda }:(I+\lambda A)C \to C\) by \(J^{A}_{\lambda }= (I+\lambda A)^{-1}\) for each \(\lambda >0\). Such \(J^{A}_{\lambda }\) is called the resolvent of A; see, e.g., [23–25] and the references therein. Recall now that a single-valued mapping \(F:C\to X\) is called η-strongly accretive if \(\langle Fx-Fy,j(x-y)\rangle \geq \eta \|x-y\|^{2}\) for some \(\eta \in (0,1)\) and \(j(x-y)\in J(x-y)\). Moreover, F is called ξ-strictly pseudocontractive if, \(\forall x,y\in C\), \(\langle Fx-Fy,j(x-y) \rangle \leq \|x-y\|^{2}-\xi \|x-y-(Fx-Fy)\|^{2}\) for some \(\xi \in (0,1)\), where \(j(x-y)\in J(x-y)\).

Let \(F:C\to X\) be a mapping. Then (i) if \(F:C\to X\) is η-strongly accretive and ξ-strictly pseudocontractive with \(\eta +\xi \geq 1\), then \(I-F\) is nonexpansive, and F is Lipschitz continuous with constant \(1+\frac{1}{\xi }\); (ii) if \(F:C\to X\) is η-strongly accretive and ξ-strictly pseudocontractive with \(\eta +\xi \geq 1\), then, for any fixed \(\tau \in (0,1)\), \(I-\tau F\) is a contraction with constant \(1-\tau (1-\sqrt{\frac{1-\eta }{\xi }})\).

From now on, one employs Π to denote a mapping from C onto its subset D. One says that Π is sunny if, whenever \({\varPi }(x)+t(x- {\varPi }(x))\in C\) for \(x\in C\), \({\varPi }[{\varPi }(x)+t(x-{\varPi }(x))]= {\varPi }(x)\). A mapping Π defined on C is called a retraction if \(\varPi =\varPi ^{2}\). One says that subset D is a sunny nonexpansive retract of the set C if there exists a sunny nonexpansive retraction from C onto D.

Let \(\{S_{n}\}^{\infty }_{n=0}\) be a countable family of nonexpansive mappings defined on C, which is a convex and closed subset of a strictly convex Banach space, and let \(\{\zeta _{n}\}^{\infty }_{n=0}\) be a sequence in \([0,1]\). For any \(n\geq 0\), define a mapping \(W_{n}:C\to C\) as follows:

$$ \textstyle\begin{cases} U_{n,n+1}=I, \\ U_{n,n}=\zeta _{n}S_{n}U_{n,n+1}+(1-\zeta _{n})I, \\ \cdots \\ U_{n,1}=\zeta _{1}S_{1}U_{n,2}+(1-\zeta _{1})I, \\ W_{n}=U_{n,0}=\zeta _{0}S_{0}U_{n,1}+(1-\zeta _{0})I. \end{cases} $$
(2.1)

Lemma 2.1

([25, 26])

Suppose that\(\{S_{n}\}^{\infty }_{n=0}\)is a countable family of nonexpansive mappings defined on a subsetCof a strictly convex spaceX. Suppose that\(\bigcap^{\infty }_{n=0}{\operatorname{Fix}}(S _{n})\neq \emptyset \), and\(\{\zeta _{n}\}^{\infty }_{n=0}\)is a real sequence such that\(0<\zeta _{n}\leq b<1\), \(\forall n\geq 0\). Then

  1. (i)

    \(W_{n}\)is nonexpansive and\(\operatorname{Fix}(W_{n})=\bigcap^{n}_{i=0} {\operatorname{Fix}}(S_{i})\), \(\forall n\geq 0\);

  2. (ii)

    the limit\(\lim_{n\to \infty }U_{n,k}x\)exists for all\(x\in C\)and\(k\geq 0\);

  3. (iii)

    the mapping\(W:C\to C\)defined by\(Wx:=\lim_{n\to \infty }W_{n}x= \lim_{n\to \infty }U_{n,0}x\), \(\forall x\in C\), is a nonexpansive mapping satisfying\(\operatorname{Fix}(W)=\bigcap^{\infty }_{n=0}{\operatorname{Fix}}(S _{n})\)and it is called theW-mapping. IfDis any bounded subset ofC, then\(\lim_{n\to \infty }\sup_{x\in D}\|W_{n}x-Wx\|=0\).

For our main strong convergence theorems, the following tools are also needed.

Lemma 2.2

([27])

LetXbe smooth, Dbe a nonempty subset ofCandΠbe a retraction ofContoD. Then the following are equivalent: (i) Πis sunny and nonexpansive; (ii) \(\|{\varPi }(x)- {\varPi }(y)\|^{2}\leq \langle x-y,J({\varPi }(x)-\varPi (y))\rangle\), \(\forall x,y\in C\); (iii) \(\langle x-{\varPi }(x),J(y-{\varPi }(x))\rangle \leq 0\), \(\forall x\in C\), \(y\in D\).

Lemma 2.3

([28])

Let\(q\in (1,2]\)a given real number and letXbeq-uniformly smooth. Then\(\|x+y\|^{q}\leq q\langle y,J_{q}(x)\rangle +\|x\|^{q}+\kappa _{q}\|y\|^{q}\), \(\forall x,y\in X\), where\(\kappa _{q}\)is theq-uniformly smooth constant ofX. For any given\(x,y\in X\), one has\(\|x+y\|^{q}\leq \|x\|^{q}+q\langle y,j_{q}(x+y)\rangle \), \(\forall j_{q}(x+y)\in J_{q}(x+y)\).

Lemma 2.4

([28, 29])

LetXbe a uniformly convex andq-uniformly, where\(1< q\leq 2\), smooth Banach space. Let\(A:C\to X\)be anα-inverse-strongly accretive mapping of orderqand\(B:C\to 2^{X}\)be anm-accretive operator. In the sequel, we will use the notation\(T_{\lambda }:=J^{B}_{\lambda }(I-\lambda A)=(I+\lambda B)^{-1}(I-\lambda A)\), \(\forall \lambda >0\). The following statements hold:

  1. (i)

    the resolvent identity: \(J_{\lambda }x=J_{\mu }(\frac{\mu }{ \lambda }x+(1-\frac{\mu }{\lambda })J_{\lambda }x)\), \(\forall \lambda ,\mu >0\), \(x\in X\);

  2. (ii)

    if\(J^{A}_{\lambda }\)is a resolvent ofAfor\(\lambda >0\), then\(J^{A}_{\lambda }\)is a single-valued nonexpansive mapping with\(\operatorname{Fix}(J^{A}_{\lambda })=A^{-1}0\), where\(A^{-1}0=\{x\in C:0 \in Ax\}\);

  3. (iii)

    \(\operatorname{Fix}(T_{\lambda })=(A+B)^{-1}0\), \(\forall \lambda >0\);

  4. (iv)

    \(\|x-T_{\lambda }x\|\leq 2\|x-T_{s}x\|\)for\(0<\lambda \leq s\)and\(x\in X\);

  5. (v)

    \(\|T_{\lambda }x-T_{\lambda }y\|\leq \|x-y\|\);

  6. (vi)

    \(\|(I-\lambda A)x-(I-\lambda A)y\|^{q}\leq \|x-y\|^{q}-\lambda (q \alpha -\kappa _{q}\lambda ^{q-1})\|Ax-Ay\|^{q}\), \(\forall x,y\in C\). In particular, if\(0<\lambda \leq (\frac{q\alpha }{\kappa _{q}})^{ \frac{1}{q-1}}\), then\(I-\lambda A\)is nonexpansive.

Lemma 2.5

([30])

Let\(T:C\to C\)be nonexpansive with\(\operatorname{Fix}(T) \neq \emptyset \), and let\(f:C\to C\)be a fixed contraction mapping, whereCis convex and closed set in a real reflexive Banach space with the uniformly Gâteaux differentiable norm and the normal structure. Let\(z_{t}\in C\), where\(t\in (0,1)\), be the unique fixed point of the contraction\(C\ni z\mapsto (1-t)Tz+tf(z)\)onC, that is, \(z_{t}=(1-t)Tz _{t}+tf(z_{t})\). Then\(\{z_{t}\}\)converges to\(x^{*}\in {\operatorname{Fix}}(T)\)in norm. This convergent point also solves\(\langle (f-I)x^{*}, J(p-x^{*})\rangle \leq 0\), \(\forall p\in {\operatorname{Fix}}(T)\).

Lemma 2.6

([14])

Suppose that\({\varPi }_{C}\)is a sunny nonexpansive retraction from aq-uniformly smoothXonto its convex closed subsetC. Let the mapping\(A_{i}:C\to X\)be\(\alpha _{i}\)-inverse-strongly accretive of orderqfor\(i=1,2\). Let the mapping\(G:C\to C\)be defined as\(Gx:={\varPi }_{C}(I-\mu _{1}A_{1}){\varPi }_{C}(I-\mu _{2}A_{2})\), \(\forall x\in C\). If\(0<\mu _{i}\leq (\frac{q\alpha _{i}}{\kappa _{q}})^{ \frac{1}{q-1}}\)for\(i=1,2\), then\(G:C\to C\)is a Lipschitz mapping. More precisely, it is nonexpansive. Let\(A_{1},A_{2}:C\to X\)be two nonlinear mappings. For given\((x^{*},y^{*})\in C\times C\), \((x^{*},y^{*})\)is a solution of SVIs (1.1) iff\(x^{*}={\varPi } _{C}(y^{*}-\mu _{1}A_{1}y^{*})\), where\(y^{*}={\varPi }_{C}(x^{*}-\mu _{2}A _{2}x^{*})\).

Lemma 2.7

([31])

Let\(\{a_{n}\}\)be a sequence defined by\(a_{n+1}\leq \gamma _{n}\lambda _{n}+a_{n}(1-\lambda _{n})\), \(\forall n\geq 0\), where\(\{\lambda _{n}\}\)and\(\{\gamma _{n}\}\)are sequences of real numbers such that (i) \(\limsup_{n\to \infty }\gamma _{n}\leq 0\)or\(\sum^{ \infty }_{n=0}|\lambda _{n}\gamma _{n}|<\infty \); (ii) \(\{\lambda _{n}\} \subset [0,1]\)and\(\sum^{\infty }_{n=0}\lambda _{n}=\infty \). Then\(\lim_{n\to \infty }a_{n}=0\).

Lemma 2.8

([28])

Let\(B_{r}=\{x\in X:\|x\|\leq r\}\), \(r>0\), whereXis a uniformly convex Banach space. Then there exists a continuous, strictly increasing and convex function\(g:[0,\infty )\to [0,\infty )\), \(g(0)=0\)such that, with\(p>1\),

$$ \Vert \alpha x+\beta y+\gamma z \Vert ^{p}+\frac{\alpha ^{p}\beta +\beta ^{p} \alpha }{(\alpha +\beta )^{p}}g \bigl( \Vert x-y \Vert \bigr)\leq \alpha \Vert x \Vert ^{p}+ \beta \Vert y \Vert ^{p}+\gamma \Vert z \Vert ^{p} $$

for all\(x,y,z\in B_{r}\)and\(\alpha ,\beta ,\gamma \in [0,1]\)with\(\alpha +\beta +\gamma =1\).

Lemma 2.9

([32])

Suppose that\(\{x_{n}\}\)is a sequence defined by\(x_{n+1}=\alpha _{n}x_{n}+(1-\alpha _{n})y_{n}\), \(\forall n\geq 0\), where\(\{y_{n}\}\)is bounded sequences in Banach spaceXand let\(\{\alpha _{n}\}\)be a real sequence such that\(0<\liminf_{n\to \infty }\alpha _{n}\leq \limsup_{n\to \infty }\alpha _{n}<1\). If\(\limsup_{n\to \infty }(\|y_{n+1}-y_{n}\|- \|x_{n+1}-x_{n}\|)\leq 0\), then\(\lim_{n\to \infty }\|y_{n}-x_{n}\|=0\).

3 Iterative algorithms and convergence criteria

Theorem 3.1

LetXbe a both uniformly convex andq-uniformly smooth space with\(1< q\leq 2\)and let\(B:C\to 2^{X}\)be anm-accretive operator. Let\(A_{i}:C\to X\)be an\(\alpha _{i}\)-inverse-strongly accretive operator of orderqfor each\(i=1,2\)and\(A:C\to X\)be anα-inverse-strongly accretive of orderq. Assume that\({\varOmega }=\bigcap^{\infty }_{n=0}{\operatorname{Fix}}(S_{n})\cap {\mathrm{SVI}}(C,A _{1}, A_{2})\cap (A+B)^{-1}0\neq \emptyset \), where\(\mathrm{SVI}(C,A _{1},A_{2})\)is the fixed point set of\(G:={\varPi }_{C}(I-\mu _{1}A_{1}) {\varPi }_{C}(I-\mu _{2}A_{2})\)with\(0<\mu _{i}<(\frac{q\alpha _{i}}{\kappa _{q}})^{\frac{1}{q-1}}\)for\(i=1,2\). Let\(f:C\to C\)be aδ-contraction with constant\(\delta \in (0,1)\)and let\(F:C\to X\)beη-strongly accretive andξ-strictly pseudocontractive with\(\eta +\xi \geq 1\). For arbitrarily given\(x_{0}\in C\), let\(\{x_{n}\}\)be a sequence generated by

$$ \textstyle\begin{cases} v_{n}={\varPi }_{C}(I-\mu _{1}A_{1}){\varPi }_{C}(y_{n}-\mu _{2}A_{2}y_{n}), \\ y_{n}=\beta _{n}x_{n}+\gamma _{n}{\varPi }_{C}(I-\sigma _{n}F)(t_{n}x_{n}+(1-t _{n})W_{n}v_{n})+\alpha _{n}f(y_{n}), \\ x_{n+1}=\delta _{n}x_{n}+(1-\delta _{n})J^{B}_{\lambda _{n}}(y_{n}- \lambda _{n}Ay_{n}),\quad n\geq 0, \end{cases} $$
(3.1)

where\({\varPi }_{C}\)is the sunny nonexpansive retraction fromXontoC, \(\{W_{n}\}\)is the sequence defined by (2.1), \(\{\lambda _{n}\}\subset (0,(\frac{q\alpha }{\kappa _{q}})^{\frac{1}{q-1}})\), \(\{\sigma _{n}\}\subset [0,1)\)and\(\{\alpha _{n}\}\), \(\{\beta _{n}\}\), \(\{\gamma _{n}\},\{\delta _{n}\},\{t_{n}\}\subset (0,1)\)satisfy the following conditions:

  1. (i)

    \(\alpha _{n}+\beta _{n}+\gamma _{n}=1\), \(\sum^{\infty }_{n=0}\alpha _{n}=\infty \)and\(\lim_{n\to \infty }\alpha _{n}=0\);

  2. (ii)

    \(\lim_{n\to \infty }\frac{\sigma _{n}}{\alpha _{n}}=0\), \(\lim_{n\to \infty }|\gamma _{n}-\gamma _{n-1}|=0\)and\(\lim_{n\to \infty }|\beta _{n}-\beta _{n-1}|=0\);

  3. (iii)

    \(\lim_{n\to \infty }|t_{n}-t_{n-1}|=0\), \(\limsup_{n\to \infty } \gamma _{n}t_{n}(1-t_{n})<1\)and\(\liminf_{n\to \infty }\gamma _{n}(1-t _{n})>0\);

  4. (iv)

    \(\liminf_{n\to \infty }\beta _{n}\gamma _{n}>0\), \(\limsup_{n\to \infty }\delta _{n}<1\)and\(\liminf_{n\to \infty }\delta _{n}>0\);

  5. (v)

    \(0<\bar{\lambda }\leq \lambda _{n}\), \(\forall n\geq 0\)and\(\lim_{n\to \infty }\lambda _{n}=\lambda <(\frac{q\alpha }{\kappa _{q}}) ^{\frac{1}{q-1}}\).

Then\(x_{n}\to x^{*}\in {\varOmega }\), which is a unique solution to the generalized variational inequality (GVI) \(\langle (I-f)x^{*},J(x^{*}-p) \rangle \leq 0\), \(\forall p\in {\varOmega }\).

Proof

Put \(u_{n}={\varPi }_{C}(y_{n}-\mu _{2}A_{2}y_{n})\). It is easy to see that scheme (3.1) can be rewritten as

$$ \textstyle\begin{cases} y_{n}=\beta _{n}x_{n}+\gamma _{n}{\varPi }_{C}(I-\sigma _{n}F)(t_{n}x_{n}+(1-t _{n})W_{n}Gy_{n})+\alpha _{n}f(y_{n}), \\ x_{n+1}=\delta _{n}x_{n}+(1-\delta _{n})T_{n}y_{n},\quad n\geq 0, \end{cases} $$
(3.2)

where \(T_{n}:=J^{B}_{\lambda _{n}}(I-\lambda _{n}A)\). From \(\eta + \xi \geq 1\), \(\{\sigma _{n}\}\subset [0,1)\), one asserts that \({\varPi } _{C}(I-\sigma _{n}F):C\to C\) is a nonexpansive mapping for each \(n\geq 0\). Because of the situation \(\alpha _{n}+\beta _{n}+\gamma _{n}=1\), one knows that

$$ \alpha _{n}\delta +\gamma _{n}(1-t_{n})+ \beta _{n}+\gamma _{n}t_{n}=\alpha _{n}\delta +\gamma _{n}+\beta _{n}=1- \alpha _{n}(1-\delta ) \quad \forall n\geq 0. $$

One now shows that the sequence \(\{x_{n}\}\) generated by (3.2) is well defined. Define a mapping \(F_{n}:C\to C\) by \(F_{n}(x)=\beta _{n}x _{n}+\gamma _{n}{\varPi }_{C}(I-\sigma _{n}F)(t_{n}x_{n}+(1-t_{n})W_{n} Gx)+ \alpha _{n}f(x)\), \(\forall x\in C\). Then

$$\begin{aligned} \bigl\Vert F_{n}(x)-F_{n}(y) \bigr\Vert \leq& \gamma _{n} \bigl\Vert {\varPi }_{C}(I- \sigma _{n}F) \bigl(t _{n}x_{n}+(1-t_{n})W_{n}Gx \bigr) \\ &{}-{\varPi }_{C}(I-\sigma _{n}F) \bigl(t_{n}x_{n}+(1-t _{n})W_{n}Gy \bigr) \bigr\Vert \\ &{} +\alpha _{n} \bigl\Vert f(x)-f(y) \bigr\Vert \\ \leq &\gamma _{n}(1-t_{n}) \Vert W_{n}Gx-W_{n}Gy \Vert +\alpha _{n}\delta \Vert x-y \Vert \\ \leq& \bigl(1-\alpha _{n}(1-\delta )\bigr) \Vert x-y \Vert . \end{aligned}$$

This guarantees the result that \(F_{n}\) is a contraction mapping. Hence there is a unique fixed point \(y_{n}\in C\) satisfying

$$ y_{n}=\beta _{n}x_{n}+\gamma _{n}{\varPi }_{C}(I-\sigma _{n}F) \bigl((1-t_{n})W _{n}Gy_{n}+t_{n}x_{n} \bigr)+\alpha _{n}f(y_{n}). $$

One next divides the rest of the proof into several steps.

Step 1. Show that \(\{x_{n}\}\) is bounded.

From \(\{\lambda _{n}\}\subset (0,(\frac{q\alpha }{\kappa _{q}})^{ \frac{1}{q-1}})\), one observes that \(T_{n}: C\to C\) is a nonexpansive mapping for each \(n\geq 0\). Take a fixed \(p\in {\varOmega }=\bigcap^{\infty }_{n=0}{\operatorname{Fix}}(S_{n})\cap {\mathrm{SVI}}(C,A_{1},A_{2})\cap (A+B)^{-1}0\) arbitrarily. From Lemmas 2.4 and 2.6, we know that \(W_{n}p=p\), \(Gp=p\) and \(T_{n}p=p\). Moreover, using the nonexpansivity of \(W_{n}\) and G yields

$$\begin{aligned} \Vert y_{n}-p \Vert \leq& \beta _{n} \Vert x_{n}-p \Vert +\gamma _{n} \bigl\Vert { \varPi }_{C}(I-\sigma _{n}F) \bigl(t_{n}x_{n} +(1-t_{n})W_{n}Gy_{n}\bigr)-{\varPi }_{C}(I-\sigma _{n}F)p \bigr\Vert \\ &{} +\gamma _{n} \bigl\Vert {\varPi }_{C}(I-\sigma _{n}F)p-p \bigr\Vert +\alpha _{n}\bigl( \bigl\Vert f(y _{n})-f(p) \bigr\Vert + \bigl\Vert f(p)-p \bigr\Vert \bigr) \\ \leq& \beta _{n} \Vert x_{n}-p \Vert +\alpha _{n}\bigl(\delta \Vert y_{n}-p \Vert + \bigl\Vert f(p)-p \bigr\Vert \bigr)+ \gamma _{n}\bigl[t_{n} \Vert x_{n}-p \Vert \\ &{}+(1-t_{n}) \Vert W_{n}Gy_{n}-p \Vert \bigr] \\ &{} +\gamma _{n}\sigma _{n} \Vert Fp \Vert \\ \leq& (\beta _{n}+\gamma _{n}t_{n}) \Vert x_{n}-p \Vert +\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert + \sigma _{n} \Vert Fp \Vert +\bigl(\alpha _{n}\delta +\gamma _{n}(1-t_{n})\bigr) \Vert y_{n}-p \Vert , \end{aligned}$$

which hence implies that

$$ \Vert y_{n}-p \Vert \leq \frac{\alpha _{n} \Vert f(p)-p \Vert +\sigma _{n} \Vert Fp \Vert }{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}+ \frac{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))-\alpha _{n}(1-\delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t _{n}))} \Vert x_{n}-p \Vert . $$
(3.3)

Since \(\lim_{n\to \infty }\frac{\sigma _{n}}{\alpha _{n}}=0\), one may suppose \(\sigma _{n}\leq \alpha _{n}\). Thus, from (3.2), (3.3) and the nonexpansivity of \(T_{n}\), we find that

$$\begin{aligned} \Vert x_{n+1}-p \Vert \leq& \delta _{n} \Vert x_{n}-p \Vert +(1-\delta _{n}) \Vert T_{n}y_{n}-p \Vert \\ \leq& \delta _{n} \Vert x_{n}-p \Vert +(1-\delta _{n}) \Vert y_{n}-p \Vert \\ \leq& \delta _{n} \Vert x_{n}-p \Vert +(1-\delta _{n})\biggl\{ \biggl(1-\frac{\alpha _{n}(1- \delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\biggr) \Vert x_{n}-p \Vert \\ &{}+\frac{ \alpha _{n} \Vert f(p)-p \Vert +\alpha _{n} \Vert Fp \Vert }{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\biggr\} \\ =&\biggl[1-\frac{(1-\delta _{n})(1-\delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t _{n}))}\alpha _{n}\biggr] \Vert x_{n}-p \Vert \\ &{} +\frac{(1-\delta _{n})(1-\delta )}{{1-( \alpha _{n}\delta +\gamma _{n}(1-t_{n}))}}\alpha _{n}\frac{ \Vert f(p)-p \Vert + \Vert Fp \Vert }{1-\delta } \\ \leq& \max \biggl\{ \frac{ \Vert f(p)-p \Vert + \Vert Fp \Vert }{1-\delta }, \Vert x_{n}-p \Vert \biggr\} . \end{aligned}$$

It immediately follows that \(\{x_{n}\}\) is a bounded vector in set C.

Step 2. One shows that \(\|x_{n+1}-x_{n}\|\to 0\) as \(n\to \infty \).

Indeed,

$$\begin{aligned} z_{n}-z_{n-1} =&(t_{n}-t_{n-1}) (x_{n-1}-W_{n-1}Gy_{n-1})+(1-t_{n}) (W _{n}Gy_{n}-W_{n-1}Gy_{n-1}) \\ &{}+t_{n}(x_{n}-x_{n-1}) \end{aligned}$$

and

$$\begin{aligned} y_{n}-y_{n-1} =&(\alpha _{n}-\alpha _{n-1})f(y_{n-1})+\beta _{n}(x_{n}-x _{n-1})+\alpha _{n}\bigl(f(y_{n})-f(y_{n-1})\bigr) \\ &{} +(\beta _{n}-\beta _{n-1})x_{n-1}+\gamma _{n}\bigl({\varPi }_{C}(I-\sigma _{n}F)z _{n}-{\varPi }_{C}(I-\sigma _{n-1}F)z_{n-1} \bigr) \\ &{} +(\gamma _{n}-\gamma _{n-1}){\varPi }_{C}(I-\sigma _{n-1}F)z_{n-1}. \end{aligned}$$
(3.4)

Utilizing Lemmas 2.1 and 2.4 yields

$$\begin{aligned} & \Vert T_{n}y_{n}-T_{n-1}y_{n-1} \Vert \\ &\quad \leq \Vert T_{n}y_{n}-T_{n}y_{n-1} \Vert + \Vert T _{n}y_{n-1}-T_{n-1}y_{n-1} \Vert \\ &\quad \leq \bigl\Vert J^{B}_{\lambda _{n}}(I-\lambda _{n}A)y_{n-1}-J^{B}_{\lambda _{n-1}}(I- \lambda _{n}A)y_{n-1} \bigr\Vert + \Vert y_{n}-y_{n-1} \Vert \\ &\qquad {} + \bigl\Vert J^{B}_{\lambda _{n-1}}(I-\lambda _{n}A)y_{n-1}-J^{B}_{\lambda _{n-1}}(I- \lambda _{n-1}A)y_{n-1} \bigr\Vert \\ &\quad = \Vert y_{n}-y_{n-1} \Vert + \biggl\Vert J^{B}_{\lambda _{n-1}}\biggl(\frac{\lambda _{n-1}}{ \lambda _{n}}I+\biggl(1- \frac{\lambda _{n-1}}{\lambda _{n}}\biggr) J^{B}_{\lambda _{n}}\biggr) (I-\lambda _{n}A)y_{n-1} \\ &\qquad {} -J^{B}_{\lambda _{n-1}}(I-\lambda _{n}A)y_{n-1} \biggr\Vert + \bigl\Vert J^{B}_{\lambda _{n-1}}(I- \lambda _{n}A)y_{n-1} -J^{B}_{\lambda _{n-1}}(I- \lambda _{n-1}A)y_{n-1} \bigr\Vert \\ &\quad \leq \biggl\vert 1-\frac{\lambda _{n-1}}{\lambda _{n}} \biggr\vert \bigl\Vert J^{B}_{\lambda _{n}}(I- \lambda _{n}A)y_{n-1} -(I-\lambda _{n}A)y_{n-1} \bigr\Vert + \Vert y_{n}-y_{n-1} \Vert \\ &\qquad {} + \vert \lambda _{n}-\lambda _{n-1} \vert \Vert Ay_{n-1} \Vert \\ &\quad \leq \vert \lambda _{n}-\lambda _{n-1} \vert M_{1}+ \Vert y_{n}-y_{n-1} \Vert , \end{aligned}$$
(3.5)

where \(\sup_{n\geq 1}\{\frac{1}{\bar{\lambda }}\|J^{B}_{\lambda _{n}}(I- \lambda _{n}A)y_{n-1}-(I-\lambda _{n}A)y_{n-1}\| +\|Ay_{n-1}\|\}\leq M _{1}\) for some \(M_{1}>0\). Also, it follows from the nonexpansivity of \({\varPi }_{C}\) and \((I-\sigma _{n}F)\) that

$$\begin{aligned} & \bigl\Vert {\varPi }_{C}(I-\sigma _{n}F)z_{n}-{\varPi }_{C}(I-\sigma _{n-1}F)z_{n-1} \bigr\Vert \\ &\quad \leq \bigl\Vert {\varPi }_{C}(I-\sigma _{n}F)z_{n}-{\varPi }_{C}(I-\sigma _{n}F)z_{n-1} \bigr\Vert + \bigl\Vert {\varPi }_{C}(I-\sigma _{n}F)z_{n-1} -{\varPi }_{C}(I-\sigma _{n-1}F)z _{n-1} \bigr\Vert \\ &\quad \leq \Vert z_{n}-z_{n-1} \Vert + \vert \sigma _{n}-\sigma _{n-1} \vert \Vert Fz_{n-1} \Vert \\ &\quad \leq t_{n} \Vert x_{n}-x_{n-1} \Vert + \vert t_{n}-t_{n-1} \vert \Vert x_{n-1}-W_{n-1}Gy_{n-1} \Vert \\ &\qquad {} +(1-t_{n}) \Vert W_{n}Gy_{n}-W_{n-1}Gy_{n-1} \Vert + \vert \sigma _{n}-\sigma _{n-1} \vert \Vert Fz _{n-1} \Vert \\ &\quad \leq t_{n} \Vert x_{n}-x_{n-1} \Vert + \vert t_{n}-t_{n-1} \vert \Vert x_{n-1}-W_{n-1}Gy_{n-1} \Vert \\ &\qquad {} +(1-t_{n})\bigl[ \Vert y_{n}-y_{n-1} \Vert + \Vert W_{n}Gy_{n-1}-W_{n-1}Gy_{n-1} \Vert \bigr]+ \vert \sigma _{n}-\sigma _{n-1} \vert \Vert Fz_{n-1} \Vert . \end{aligned}$$

This together with (3.4) guarantees

$$\begin{aligned} \Vert y_{n}-y_{n-1} \Vert \leq& \alpha _{n}\delta \Vert y_{n}-y_{n-1} \Vert + \vert \alpha _{n}-\alpha _{n-1} \vert \bigl\Vert f(y_{n-1}) \bigr\Vert +\beta _{n} \Vert x_{n}-x_{n-1} \Vert \\ &{} + \vert \beta _{n}-\beta _{n-1} \vert \Vert x_{n-1} \Vert +\gamma _{n}\bigl\{ t_{n} \Vert x_{n}-x_{n-1} \Vert + \vert t_{n}-t_{n-1} \vert \Vert x_{n-1}-W_{n-1}Gy_{n-1} \Vert \\ &{} +(1-t_{n})\bigl[ \Vert y_{n}-y_{n-1} \Vert + \Vert W_{n}Gy_{n-1}-W_{n-1}Gy_{n-1} \Vert \bigr] \\ &{}+ \vert \sigma _{n}-\sigma _{n-1} \vert \Vert Fz_{n-1} \Vert \bigr\} \\ &{} + \vert \gamma _{n}-\gamma _{n-1} \vert \bigl\Vert {\varPi }_{C}(I-\sigma _{n-1}F)z_{n-1} \bigr\Vert \\ \leq& \bigl(\alpha _{n}\delta +\gamma _{n}(1-t_{n}) \bigr) \Vert y_{n}-y_{n-1} \Vert +(\beta _{n}+\gamma _{n}t_{n}) \Vert x_{n}-x_{n-1} \Vert +\bigl( \vert \alpha _{n}-\alpha _{n-1} \vert \\ &{} + \vert \beta _{n}-\beta _{n-1} \vert + \vert \gamma _{n}-\gamma _{n-1} \vert + \vert \sigma _{n}- \sigma _{n-1} \vert + \vert t_{n}-t_{n-1} \vert \bigr)M_{2} \\ &{} + \Vert W_{n}Gy_{n-1}-W_{n-1}Gy_{n-1} \Vert , \end{aligned}$$

where \(\sup_{n\geq 0}\{\|x_{n}\|+\|f(y_{n})\|+\|W_{n}Gy_{n}\|+\|Fz _{n}\|+\|{\varPi }_{C}(I-\sigma _{n}F)z_{n}\|\}\leq M_{2}\) for some \(M_{2}>0\). Then

$$\begin{aligned} \Vert y_{n}-y_{n-1} \Vert \leq& \frac{\beta _{n}+\gamma _{n}t_{n}}{1-(\alpha _{n} \delta +\gamma _{n}(1-t_{n}))} \Vert x_{n}-x_{n-1} \Vert \\ &{} +\frac{1}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\bigl( \vert \alpha _{n}-\alpha _{n-1} \vert + \vert \beta _{n}-\beta _{n-1} \vert \\ &{} + \vert \gamma _{n}-\gamma _{n-1} \vert + \vert \sigma _{n}-\sigma _{n-1} \vert + \vert t_{n}-t_{n-1} \vert \bigr)M _{2}+ \Vert W_{n}Gy_{n-1}-W_{n-1}Gy_{n-1} \Vert \bigr] \\ =&\biggl(1-\frac{\alpha _{n}(1-\delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t _{n}))}\biggr) \Vert x_{n}-x_{n-1} \Vert \\ &{} +\frac{1}{1-(\alpha _{n}\delta +\gamma _{n}(1-t _{n}))}\bigl[\bigl( \vert \alpha _{n}-\alpha _{n-1} \vert + \vert \beta _{n}-\beta _{n-1} \vert \\ &{} + \vert \gamma _{n}-\gamma _{n-1} \vert + \vert \sigma _{n}-\sigma _{n-1} \vert + \vert t_{n}-t_{n-1} \vert \bigr)M _{2} + \Vert W_{n}Gy_{n-1}-W_{n-1}Gy_{n-1} \Vert \bigr] \\ \leq& \Vert x_{n}-x_{n-1} \Vert \\ &{}+\frac{1}{1-(\alpha _{n}\delta +\gamma _{n}(1-t _{n}))}\bigl[\bigl( \vert \alpha _{n}-\alpha _{n-1} \vert + \vert \beta _{n}-\beta _{n-1} \vert + \vert \gamma _{n}-\gamma _{n-1} \vert \\ &{} + \vert \sigma _{n}-\sigma _{n-1} \vert + \vert t_{n}-t_{n-1} \vert \bigr)M_{2}+ \Vert W_{n}Gy_{n-1}-W _{n-1}Gy_{n-1} \Vert \bigr], \end{aligned}$$

which together with (3.5) asserts that

$$\begin{aligned} \Vert T_{n}y_{n}-T_{n-1}y_{n-1} \Vert - \Vert x_{n}-x_{n-1} \Vert \leq& \frac{1}{1-( \alpha _{n}\delta +\gamma _{n}(1-t_{n}))} \bigl[\bigl( \vert \alpha _{n}-\alpha _{n-1} \vert + \vert \beta _{n}-\beta _{n-1} \vert \\ &{} + \vert \gamma _{n}-\gamma _{n-1} \vert + \vert \sigma _{n}-\sigma _{n-1} \vert + \vert t_{n}-t_{n-1} \vert \bigr)M _{2} \\ &{} + \Vert W_{n}Gy_{n-1}-W_{n-1}Gy_{n-1} \Vert \bigr]+ \vert \lambda _{n}-\lambda _{n-1} \vert M_{1}. \end{aligned}$$

Since \(\lim_{n\to \infty }\sup_{x\in D}\|W_{n}x-Wx\|=0\) on bounded subset \(D=\{Gy_{n}:n\geq 0\}\) of C, one knows that

$$ \lim_{n\to \infty } \Vert W_{n}Gy_{n-1}-W_{n-1}Gy_{n-1} \Vert =0. $$

Note that \(\lim_{n\to \infty }\alpha _{n}=0\), \(\lim_{n\to \infty }\frac{ \sigma _{n}}{\alpha _{n}}=0\), \(\lim_{n\to \infty }\lambda _{n}= \lambda \) and \(\liminf_{n\to \infty }(1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n})))>0\). Thus, from \(|\beta _{n}-\beta _{n-1}|\to 0\), \(|\gamma _{n}-\gamma _{n-1}| \to 0\) and \(|t_{n}-t_{n-1}|\to 0\) as \(n\to \infty \) (due to conditions (ii), (iii)), we get

$$ \limsup_{n\to \infty }\bigl( \Vert T_{n}y_{n}-T_{n-1}y_{n-1} \Vert - \Vert x_{n}-x_{n-1} \Vert \bigr)\leq 0. $$

So it follows from condition (iv) and Lemma 2.9 that \(\lim_{n\to \infty }\|T_{n}y_{n}-x_{n}\|=0\). Hence

$$ \lim_{n\to \infty } \Vert x_{n+1}-x_{n} \Vert =\lim_{n\to \infty }(1-\delta _{n}) \Vert T_{n}y_{n}-x_{n} \Vert =0. $$
(3.6)

Step 3. One shows that \(\|x_{n}-y_{n}\|\to 0\) and \(\|x_{n}-Gx _{n}\|\to 0\) as \(n\to \infty \). Indeed, for simplicity, set \(\bar{p}:={\varPi }_{C} (I-\mu _{2}A_{2})p\). Note that \(u_{n}={\varPi }_{C}(I- \mu _{2}A_{2})y_{n}\) and \(v_{n}={\varPi }_{C}(I-\mu _{1}A_{1})u_{n}\). Then \(v_{n}=Gy_{n}\). An application of Lemma 2.4 yields

$$\begin{aligned} \Vert u_{n}-\bar{p} \Vert ^{q} \leq& \bigl\Vert (I-\mu _{2}A_{2})y_{n}-(I- \mu _{2}A_{2})p \bigr\Vert ^{q} \\ \leq& \Vert y_{n}-p \Vert ^{q}-\mu _{2}\bigl(q\alpha _{2}-\kappa _{q}\mu ^{q-1}_{2}\bigr) \Vert A _{2}y_{n}-A_{2}p \Vert ^{q}. \end{aligned}$$
(3.7)

One also has

$$ \Vert v_{n}-p \Vert ^{q}\leq \Vert u_{n}-\bar{p} \Vert ^{q}-\mu _{1} \bigl(q\alpha _{1}-\kappa _{q}\mu ^{q-1}_{1} \bigr) \Vert A_{1}u_{n}-A_{1}\bar{p} \Vert ^{q}. $$
(3.8)

By using (3.7) and (3.8), one reaches

$$\begin{aligned} \Vert v_{n}-p \Vert ^{q} \leq& \Vert y_{n}-p \Vert ^{q}-\mu _{2}\bigl(q\alpha _{2}-\kappa _{q} \mu ^{q-1}_{2}\bigr) \Vert A_{2}y_{n}-A_{2}p \Vert ^{q} \\ &{} -\mu _{1}\bigl(q\alpha _{1}-\kappa _{q}\mu ^{q-1}_{1}\bigr) \Vert A_{1}u_{n}-A_{1} \bar{p} \Vert ^{q}. \end{aligned}$$
(3.9)

Equations (3.2) and (3.9) further guarantee that \(\|z_{n}-p\|^{q}\leq t_{n}\|x_{n}-p\|^{q}+(1-t_{n})\|v_{n}-p\|^{q}\) and

$$\begin{aligned} \bigl\Vert {\varPi }_{C}(I-\sigma _{n}F)z_{n}-p \bigr\Vert ^{q} \leq& \Vert z_{n}-p-\sigma _{n}Fz _{n} \Vert ^{q} \\ \leq& \Vert z_{n}-p \Vert ^{q}-q\sigma _{n}\bigl\langle Fz_{n},J_{q}(z_{n}-p- \sigma _{n}Fz_{n})\bigr\rangle \\ \leq& \Vert z_{n}-p \Vert ^{q}+q\sigma _{n} \Vert Fz_{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert ^{q-1}. \end{aligned}$$

Thus

$$\begin{aligned} & \Vert y_{n}-p \Vert ^{q} \\ &\quad \leq \beta _{n} \Vert x_{n}-p \Vert ^{q}+\gamma _{n} \bigl\Vert {\varPi }_{C}(I-\sigma _{n}F)z _{n}-p \bigr\Vert ^{q}+q\alpha _{n}\bigl\langle f(p)-p,J_{q}(y_{n}-p) \bigr\rangle \\ &\qquad {}+\alpha _{n} \bigl\Vert f(y_{n})-f(p) \bigr\Vert ^{q} \\ &\quad \leq \beta _{n} \Vert x_{n}-p \Vert ^{q}+\gamma _{n}\bigl[ \Vert z_{n}-p \Vert ^{q}+q\sigma _{n} \Vert Fz_{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert ^{q-1}\bigr] \\ &\qquad {} +q\alpha _{n}\bigl\langle f(p)-p,J_{q}(y_{n}-p) \bigr\rangle +\alpha _{n} \bigl\Vert f(y_{n})-f(p) \bigr\Vert ^{q} \\ &\quad \leq \beta _{n} \Vert x_{n}-p \Vert ^{q}+\alpha _{n}\delta \Vert y_{n}-p \Vert ^{q}+\gamma _{n}\bigl[t_{n} \Vert x_{n}-p \Vert ^{q}+(1-t_{n}) \Vert v_{n}-p \Vert ^{q} \\ &\qquad {} +q\sigma _{n} \Vert Fz_{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert ^{q-1}\bigr]+q\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert ^{q-1} \\ &\quad \leq (\beta _{n}+\gamma _{n}t_{n}) \Vert x_{n}-p \Vert ^{q}+\alpha _{n} \delta \Vert y _{n}-p \Vert ^{q}+\gamma _{n}(1-t_{n})\bigl[ \Vert y_{n}-p \Vert ^{q} \\ &\qquad {} -\mu _{2}\bigl(q\alpha _{2}-\kappa _{q}\mu ^{q-1}_{2}\bigr) \Vert A_{2}y_{n}-A_{2}p \Vert ^{q}-\mu _{1}\bigl(q\alpha _{1}-\kappa _{q}\mu ^{q-1}_{1}\bigr) \Vert A_{1}u_{n}-A_{1} \bar{p} \Vert ^{q}\bigr] \\ &\qquad {} +q\sigma _{n} \Vert Fz_{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert ^{q-1}+q\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert ^{q-1}, \end{aligned}$$

which immediately yields

$$\begin{aligned} \Vert y_{n}-p \Vert ^{q} \leq& \biggl(1-\frac{\alpha _{n}(1-\delta )}{1-(\alpha _{n} \delta +\gamma _{n}(1-t_{n}))}\biggr) \Vert x_{n}-p \Vert ^{q} -\frac{\gamma _{n}(1-t _{n})}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))} \\ &{} \times \bigl[\mu _{2}\bigl(q\alpha _{2}-\kappa _{q}\mu ^{q-1}_{2}\bigr) \Vert A_{2}y_{n}-A _{2}p \Vert ^{q} +\mu _{1}\bigl(q\alpha _{1}-\kappa _{q}\mu ^{q-1}_{1}\bigr) \Vert A_{1}u_{n}-A _{1}\bar{p} \Vert ^{q}\bigr] \\ &{} +\frac{q\alpha _{n}}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[ \Vert Fz _{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert ^{q-1} \\ &{}+ \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert ^{q-1}\bigr]. \end{aligned}$$

On the other hand, (3.2) implies

$$\begin{aligned} & \Vert x_{n+1}-p \Vert ^{q} \\ &\quad \leq (1-\delta _{n}) \Vert y_{n}-p \Vert ^{q}+\delta _{n} \Vert x _{n}-p \Vert ^{q} \\ &\quad \leq \delta _{n} \Vert x_{n}-p \Vert ^{q}+(1-\delta _{n})\biggl\{ \biggl(1- \frac{\alpha _{n}(1- \delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\biggr) \Vert x_{n}-p \Vert ^{q} \\ &\qquad {}-\frac{ \gamma _{n}(1-t_{n})}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))} \\ &\qquad {} \times \bigl[\mu _{2}\bigl(q\alpha _{2}- \kappa _{q}\mu ^{q-1}_{2}\bigr) \Vert A_{2}y_{n}-A _{2}p \Vert ^{q} +\mu _{1}\bigl(q\alpha _{1}-\kappa _{q}\mu ^{q-1}_{1}\bigr) \Vert A_{1}u_{n}-A _{1}\bar{p} \Vert ^{q}\bigr] \\ &\qquad {} +\frac{q\alpha _{n}}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[ \Vert Fz _{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert ^{q-1} + \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert ^{q-1}\bigr] \biggr\} \\ &\quad =\biggl(1-\frac{\alpha _{n}(1-\delta _{n})(1-\delta )}{1-(\alpha _{n}\delta + \gamma _{n}(1-t_{n}))}\biggr) \Vert x_{n}-p \Vert ^{q} -\frac{\gamma _{n}(1-\delta _{n})(1-t _{n})}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\mu _{2}\bigl(q \alpha _{2}- \kappa _{q}\mu ^{q-1}_{2} \bigr) \\ &\qquad {} \times \Vert A_{2}y_{n}-A_{2}p \Vert ^{q}+\mu _{1}\bigl(q\alpha _{1}- \kappa _{q}\mu ^{q-1}_{1}\bigr) \Vert A_{1}u_{n}-A_{1}\bar{p} \Vert ^{q}\bigr] \\ &\qquad {} +\frac{q(1-\delta _{n})\alpha _{n}}{1-(\alpha _{n}\delta +\gamma _{n}(1-t _{n}))}\bigl[ \Vert Fz_{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert ^{q-1} + \bigl\Vert f(p)-p \bigr\Vert \Vert y _{n}-p \Vert ^{q-1}\bigr] \\ &\quad \leq \Vert x_{n}-p \Vert ^{q}- \frac{(1-\delta _{n})\gamma _{n}(1-t_{n})}{1-( \alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\mu _{2}\bigl(q\alpha _{2}- \kappa _{q}\mu ^{q-1}_{2}\bigr) \Vert A_{2}y_{n}-A_{2}p \Vert ^{q} \\ &\qquad {} +\mu _{1}\bigl(q\alpha _{1}-\kappa _{q}\mu ^{q-1}_{1}\bigr) \Vert A_{1}u_{n}-A_{1} \bar{p} \Vert ^{q}\bigr]+\alpha _{n}M_{3}, \end{aligned}$$
(3.10)

where

$$ \sup_{n\geq 0}\biggl\{ \frac{q(1-\delta _{n})}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[ \Vert Fz_{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert ^{q-1} + \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert ^{q-1}\bigr] \biggr\} \leq M_{3} $$

for some \(M_{3}>0\). So it follows from (3.10) that

$$\begin{aligned} &\frac{(1-\delta _{n})\gamma _{n}(1-t_{n})}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\mu _{2}\bigl(q\alpha _{2}-\kappa _{q}\mu ^{q-1}_{2} \bigr) \Vert A_{2}y _{n}-A_{2}p \Vert ^{q} \\ &\qquad {}+\mu _{1}\bigl(q\alpha _{1}-\kappa _{q}\mu ^{q-1}_{1}\bigr) \Vert A_{1}u _{n}-A_{1}\bar{p} \Vert ^{q}\bigr] \\ &\quad \leq \Vert x_{n}-p \Vert ^{q}- \Vert x_{n+1}-p \Vert ^{q}+\alpha _{n}M_{3} \\ &\quad \leq q \Vert x_{n}-x_{n+1} \Vert \Vert x_{n+1}-p \Vert ^{q-1}+\kappa _{q} \Vert x_{n}-x_{n+1} \Vert ^{q}+\alpha _{n}M_{3}. \end{aligned}$$

Thanks to \(0<\mu _{i}<(\frac{q\alpha _{i}}{\kappa _{q}})^{\frac{1}{q-1}}\) for \(i=1,2\), \(\liminf_{n\to \infty }\gamma _{n} (1-t_{n})>0\), \(\liminf_{n\to \infty }(1-\delta _{n})>0\) and \(\lim_{n\to \infty }\alpha _{n}=0\), one asserts

$$ \lim_{n\to \infty } \Vert A_{2}y_{n}-A_{2}p \Vert =0 \quad \text{and}\quad \lim_{n\to \infty } \Vert A_{1}u_{n}-A_{1}\bar{p} \Vert =0. $$
(3.11)

This further implies

$$\begin{aligned} \Vert u_{n}-\bar{p} \Vert ^{2} \leq{}& \bigl\langle (I-\mu _{2}A_{2})y_{n}-(I-\mu _{2}A _{2})p,J(u_{n}-\bar{p})\bigr\rangle \\ ={}&\bigl\langle y_{n}-p,J(u_{n}-\bar{p})\bigr\rangle +\mu _{2}\bigl\langle A_{2}p-A _{2}y_{n},J(u_{n}- \bar{p})\bigr\rangle \\ \leq{}& \mu _{2} \Vert A_{2}p-A_{2}y_{n} \Vert \Vert u_{n}-\bar{p} \Vert \\ &{}+\frac{1}{2}\bigl[ \Vert y _{n}-p \Vert ^{2}+ \Vert u_{n}-\bar{p} \Vert ^{2}-g_{1}\bigl( \bigl\Vert y_{n}-u_{n}-(p- \bar{p}) \bigr\Vert \bigr)\bigr], \end{aligned}$$

from which one concludes

$$ \Vert u_{n}-\bar{p} \Vert ^{2} \leq \Vert y_{n}-p \Vert ^{2}-g_{1} \bigl( \bigl\Vert y_{n}-u_{n}-(p- \bar{p}) \bigr\Vert \bigr)+2\mu _{2} \Vert A_{2}p-A_{2}y_{n} \Vert \Vert u_{n}-\bar{p} \Vert . $$
(3.12)

One also derives that

$$ \Vert v_{n}-p \Vert ^{2}\leq \Vert u_{n}-\bar{p} \Vert ^{2}-g_{2} \bigl( \bigl\Vert u_{n}-v_{n}+(p- \bar{p}) \bigr\Vert \bigr)+2\mu _{1} \Vert A_{1} \bar{p}-A_{1}u_{n} \Vert \Vert v_{n}-p \Vert . $$
(3.13)

Employing (3.12) and (3.13), one arrives at

$$\begin{aligned} \Vert v_{n}-p \Vert ^{2} \leq& \Vert y_{n}-p \Vert ^{2}-g_{1} \bigl( \bigl\Vert y_{n}-u_{n}-(p-\bar{p}) \bigr\Vert \bigr)-g_{2}\bigl( \bigl\Vert u_{n}-v_{n}+(p- \bar{p}) \bigr\Vert \bigr) \\ &{} +2\mu _{2} \Vert A_{2}p-A_{2}y_{n} \Vert \Vert u_{n}-\bar{p} \Vert +2\mu _{1} \Vert A_{1} \bar{p}-A_{1}u_{n} \Vert \Vert v_{n}-p \Vert . \end{aligned}$$
(3.14)

Utilizing Lemma 2.8, we obtain from (3.2) and (3.14)

$$\begin{aligned} \Vert z_{n}-p \Vert ^{2} \leq& t_{n} \Vert x_{n}-p \Vert ^{2}+(1-t_{n}) \Vert W_{n}Gy_{n}-p \Vert ^{2}-t_{n}(1-t_{n})g_{3} \bigl( \Vert x_{n}-W_{n}Gy_{n} \Vert \bigr) \\ \leq& t_{n} \Vert x_{n}-p \Vert ^{2}+(1-t_{n}) \Vert v_{n}-p \Vert ^{2}-t_{n}(1-t_{n})g _{3} \bigl( \Vert x_{n}-W_{n}Gy_{n} \Vert \bigr), \end{aligned}$$

and hence

$$\begin{aligned} \Vert y_{n}-p \Vert ^{2} \leq& \beta _{n} \Vert x_{n}-p \Vert ^{2}+\alpha _{n} \bigl\Vert f(y_{n})-f(p) \bigr\Vert ^{2}+\gamma _{n} \bigl\Vert {\varPi }_{C}(I-\sigma _{n}F)z_{n}-p \bigr\Vert ^{2} \\ &{} -\beta _{n}\gamma _{n} g_{4}\bigl( \bigl\Vert x_{n}-{\varPi }_{C}(I-\sigma _{n}F)z_{n} \bigr\Vert \bigr)+2 \alpha _{n}\bigl\langle f(p)-p,J(y_{n}-p)\bigr\rangle \\ \leq& \beta _{n} \Vert x_{n}-p \Vert ^{2}+\alpha _{n}\delta \Vert y_{n}-p \Vert ^{2}+\gamma _{n}\bigl[t_{n} \Vert x_{n}-p \Vert ^{2}+(1-t_{n}) \Vert v_{n}-p \Vert ^{2} \\ &{} -t_{n}(1-t_{n})g_{3}\bigl( \Vert x_{n}-W_{n}Gy_{n} \Vert \bigr)+2\sigma _{n} \Vert Fz_{n} \Vert \Vert z _{n}-p-\sigma _{n}Fz_{n} \Vert \bigr] \\ &{} +2\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert -\beta _{n}\gamma _{n} g_{4}\bigl( \bigl\Vert x_{n}- {\varPi }_{C}(I-\sigma _{n}F)z_{n} \bigr\Vert \bigr) \\ \leq& \beta _{n} \Vert x_{n}-p \Vert ^{2}+\alpha _{n}\delta \Vert y_{n}-p \Vert ^{2}+\gamma _{n}\bigl\{ t_{n} \Vert x_{n}-p \Vert ^{2}+(1-t_{n}) \bigl[ \Vert y_{n}-p \Vert ^{2} \\ &{} -g_{1}\bigl( \bigl\Vert y_{n}-u_{n}-(p- \bar{p}) \bigr\Vert \bigr)-g_{2}\bigl( \bigl\Vert u_{n}-v_{n}+(p-\bar{p}) \bigr\Vert \bigr) \\ &{} +2\mu _{2} \Vert A_{2}p-A_{2}y_{n} \Vert \Vert u_{n}-\bar{p} \Vert +2\mu _{1} \Vert A_{1} \bar{p}-A_{1}u_{n} \Vert \Vert v_{n}-p \Vert \bigr] \\ &{} -t_{n}(1-t_{n})g_{3}\bigl( \Vert x_{n}-W_{n}Gy_{n} \Vert \bigr)+2\sigma _{n} \Vert Fz_{n} \Vert \Vert z _{n}-p-\sigma _{n}Fz_{n} \Vert \bigr\} \\ &{} +2\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert -\beta _{n}\gamma _{n} g_{4}\bigl( \bigl\Vert x_{n}- {\varPi }_{C}(I-\sigma _{n}F)z_{n} \bigr\Vert \bigr) \\ \leq& (\beta _{n}+\gamma _{n}t_{n}) \Vert x_{n}-p \Vert ^{2}+\bigl(\alpha _{n}\delta + \gamma _{n}(1-t_{n})\bigr) \Vert y_{n}-p \Vert ^{2} \\ &{} -\gamma _{n}(1-t_{n})\bigl[g_{1} \bigl( \bigl\Vert y_{n}-u_{n}-(p-\bar{p}) \bigr\Vert \bigr)+g_{2}\bigl( \bigl\Vert u_{n}-v _{n}+(p-\bar{p}) \bigr\Vert \bigr)\bigr] \\ &{} +2\mu _{2} \Vert A_{2}p-A_{2}y_{n} \Vert \Vert u_{n}-\bar{p} \Vert +2\mu _{1} \Vert A_{1} \bar{p}-A_{1}u_{n} \Vert \Vert v_{n}-p \Vert \\ &{} +2\sigma _{n} \Vert Fz_{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert +2\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert \\ &{} -\gamma _{n}t_{n}(1-t_{n})g_{3} \bigl( \Vert x_{n}-W_{n}Gy_{n} \Vert \bigr)-\beta _{n}\gamma _{n} g_{4}\bigl( \bigl\Vert x_{n}-{\varPi }_{C}(I-\sigma _{n}F)z_{n} \bigr\Vert \bigr), \end{aligned}$$

which immediately yields

$$\begin{aligned} \Vert y_{n}-p \Vert ^{2} \leq& \biggl(1-\frac{\alpha _{n}(1-\delta )}{1-(\alpha _{n} \delta +\gamma _{n}(1-t_{n}))}\biggr) \Vert x_{n}-p \Vert ^{2} \\ &{} -\frac{\gamma _{n}(1-t _{n})}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[g_{1}\bigl( \bigl\Vert y_{n}-u_{n}-(p- \bar{p}) \bigr\Vert \bigr) \\ &{} +g_{2}\bigl( \bigl\Vert u_{n}-v_{n}+(p- \bar{p}) \bigr\Vert \bigr)\bigr] \\ &{}+\frac{2}{1-(\alpha _{n} \delta +\gamma _{n}(1-t_{n}))}\bigl[\mu _{2} \Vert A_{2}p-A_{2}y_{n} \Vert \Vert u_{n}- \bar{p} \Vert \\ &{} +\mu _{1} \Vert A_{1}\bar{p}-A_{1}u_{n} \Vert \Vert v_{n}-p \Vert +\alpha _{n} \Vert Fz _{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert \\ &{}+\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert \bigr] \\ &{} -\frac{1}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\gamma _{n}t_{n}(1-t _{n})g_{3}\bigl( \Vert x_{n}-W_{n}Gy_{n} \Vert \bigr) \\ &{}+\beta _{n}\gamma _{n} g_{4}\bigl( \bigl\Vert x_{n}- {\varPi }_{C}(I-\sigma _{n}F)z_{n} \bigr\Vert \bigr)\bigr]. \end{aligned}$$

This guarantees

$$\begin{aligned} \Vert x_{n+1}-p \Vert ^{2} \leq& \delta _{n} \Vert x_{n}-p \Vert ^{2}+(1- \delta _{n})\biggl\{ \biggl(1-\frac{ \alpha _{n}(1-\delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\biggr) \Vert x _{n}-p \Vert ^{2} \\ &{} - \frac{\gamma _{n}(1-t_{n})}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[g _{1}\bigl( \bigl\Vert y_{n}-u_{n}-(p-\bar{p}) \bigr\Vert \bigr) \\ &{} +g_{2}\bigl( \bigl\Vert u_{n}-v_{n}+(p- \bar{p}) \bigr\Vert \bigr)\bigr] \\ &{} +\frac{2}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\mu _{2} \Vert A_{2}p-A _{2}y_{n} \Vert \Vert u_{n}-\bar{p} \Vert \\ &{} +\mu _{1} \Vert A_{1}\bar{p}-A_{1}u_{n} \Vert \Vert v _{n}-p \Vert \\ &{} +\alpha _{n} \Vert Fz_{n} \Vert \Vert z_{n}-p-\sigma _{n}Fz_{n} \Vert +\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert \bigr] \\ &{} -\frac{1}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))} \\ &{} \times \bigl[\gamma _{n}t_{n}(1-t_{n})g_{3} \bigl( \Vert x_{n}-W_{n}Gy_{n} \Vert \bigr)+\beta _{n}\gamma _{n} g_{4}\bigl( \bigl\Vert x_{n}-{\varPi }_{C}(I-\sigma _{n}F)z_{n} \bigr\Vert \bigr)\bigr]\biggr\} \\ \leq& \biggl(1-\frac{\alpha _{n}(1-\delta _{n})(1-\delta )}{1-(\alpha _{n} \delta +\gamma _{n}(1-t_{n}))}\biggr) \Vert x_{n}-p \Vert ^{2} \\ &{} -\frac{1-\delta _{n}}{1-( \alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\gamma _{n}(1-t_{n}) \bigl(g_{1}\bigl( \bigl\Vert y _{n}-u_{n}-(p- \bar{p}) \bigr\Vert \bigr) \\ &{} +g_{2}\bigl( \bigl\Vert u_{n}-v_{n}+(p- \bar{p}) \bigr\Vert \bigr)\bigr)+\gamma _{n}t_{n}(1-t_{n})g_{3} \bigl( \Vert x _{n}-W_{n}Gy_{n} \Vert \bigr) \\ &{} +\beta _{n}\gamma _{n}g_{4}\bigl( \bigl\Vert x_{n}-{\varPi }_{C}(I-\sigma _{n}F)z_{n} \bigr\Vert \bigr)\bigr] \\ &{} +\frac{2}{1-( \alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\mu _{2} \Vert A_{2}p-A_{2}y_{n} \Vert \Vert u_{n}-\bar{p} \Vert \\ &{} +\mu _{1} \Vert A_{1}\bar{p}-A_{1}u_{n} \Vert \Vert v_{n}-p \Vert +\alpha _{n} \Vert Fz_{n} \Vert \Vert z _{n}-p-\sigma _{n}Fz_{n} \Vert \\ &{}+\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert \bigr] \\ \leq& \Vert x_{n}-p \Vert ^{2}- \frac{1-\delta _{n}}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\gamma _{n}(1-t_{n}) \bigl(g_{1}\bigl( \bigl\Vert y_{n}-u_{n}-(p- \bar{p}) \bigr\Vert \bigr) \\ &{} +g_{2}\bigl( \bigl\Vert u_{n}-v_{n}+(p- \bar{p}) \bigr\Vert \bigr)\bigr)+\gamma _{n}t_{n}(1-t_{n})g_{3} \bigl( \Vert x _{n}-W_{n}Gy_{n} \Vert \bigr) \\ &{} +\beta _{n}\gamma _{n}g_{4}\bigl( \bigl\Vert x_{n}-{\varPi }_{C}(I-\sigma _{n}F)z_{n} \bigr\Vert \bigr)\bigr] \\ &{}+\frac{2}{1-( \alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\mu _{2} \Vert A_{2}p-A_{2}y_{n} \Vert \Vert u_{n}-\bar{p} \Vert \\ &{} +\mu _{1} \Vert A_{1}\bar{p}-A_{1}u_{n} \Vert \Vert v_{n}-p \Vert +\alpha _{n} \Vert Fz_{n} \Vert \Vert z _{n}-p \\ &{}-\sigma _{n}Fz_{n} \Vert +\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert \bigr], \end{aligned}$$

which immediately yields

$$\begin{aligned} &\frac{1-\delta _{n}}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\gamma _{n}(1-t_{n}) \bigl(g_{1}\bigl( \bigl\Vert y_{n}-u_{n}-(p- \bar{p}) \bigr\Vert \bigr) +g_{2}\bigl( \bigl\Vert u_{n}-v_{n}+(p- \bar{p}) \bigr\Vert \bigr)\bigr) \\ &\qquad {} +\gamma _{n}t_{n}(1-t_{n})g_{3} \bigl( \Vert x_{n}-W_{n}Gy_{n} \Vert \bigr)+\beta _{n}\gamma _{n}g_{4}\bigl( \bigl\Vert x_{n}-{\varPi }_{C}(I-\sigma _{n}F)z_{n} \bigr\Vert \bigr)\bigr] \\ &\quad \leq \Vert x_{n}-p \Vert ^{2}- \Vert x_{n+1}-p \Vert ^{2}+\frac{2}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\mu _{2} \Vert A_{2}p-A_{2}y_{n} \Vert \Vert u_{n}-\bar{p} \Vert \\ &\qquad {} +\mu _{1} \Vert A_{1} \bar{p}-A_{1}u_{n} \Vert \Vert v_{n}-p \Vert +\alpha _{n} \Vert Fz_{n} \Vert \Vert z _{n}-p-\sigma _{n}Fz_{n} \Vert +\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert \bigr] \\ &\quad \leq \bigl( \Vert x_{n}-p \Vert + \Vert x_{n+1}-p \Vert \bigr) \Vert x_{n}-x_{n+1} \Vert \\ &\qquad {}+\frac{2}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\bigl[\mu _{2} \Vert A_{2}p-A_{2}y_{n} \Vert \Vert u _{n}-\bar{p} \Vert \\ &\qquad {} +\mu _{1} \Vert A_{1} \bar{p}-A_{1}u_{n} \Vert \Vert v_{n}-p \Vert +\alpha _{n} \Vert Fz_{n} \Vert \Vert z _{n}-p-\sigma _{n}Fz_{n} \Vert +\alpha _{n} \bigl\Vert f(p)-p \bigr\Vert \Vert y_{n}-p \Vert \bigr]. \end{aligned}$$

Utilizing (3.6) and (3.11), we asserts from \(\liminf_{n\to \infty }(1-\delta _{n})>0\), \(\liminf_{n\to \infty }\gamma _{n}t_{n}(1-t_{n})>0\) and \(\liminf_{n\to \infty }\beta _{n}\gamma _{n}>0\) that \(\lim_{n\to \infty }g_{1}(\|y_{n}-u_{n}-(p-\bar{p})\|)=0\), \(\lim_{n\to \infty }g_{2}(\|u_{n}-v_{n}+(p-\bar{p})\|)=0\), \(\lim_{n\to \infty }g_{3}(\|x_{n}-W_{n}Gy_{n}\|)=0\) and \(\lim_{n\to \infty } g_{4}(\|x_{n}-{\varPi }_{C}(I-\sigma _{n}F)z_{n}\|)=0\). So, \(\lim_{n\to \infty }\|y_{n}-u_{n}-(p-\bar{p})\|=\lim_{n\to \infty }\|u_{n}-v_{n}+(p-\bar{p})\|=0\) and

$$ \lim_{n\to \infty } \Vert x_{n}-W_{n}Gy_{n} \Vert =\lim_{n\to \infty } \bigl\Vert x_{n}- {\varPi }_{C}(I-\sigma _{n}F)z_{n} \bigr\Vert =0. $$
(3.15)

Furthermore, one has

$$\begin{aligned} \Vert y_{n}-Gy_{n} \Vert &= \Vert y_{n}-v_{n} \Vert \\ &\leq \bigl\Vert y_{n}-u_{n}-(p-\bar{p}) \bigr\Vert + \bigl\Vert u _{n}-v_{n}+(p-\bar{p}) \bigr\Vert \to 0\quad (n\to \infty ). \end{aligned}$$
(3.16)

Since \(y_{n}-x_{n}=\alpha _{n}(f(y_{n})-x_{n})+\gamma _{n}({\varPi }_{C}(I- \sigma _{n}F)z_{n}-x_{n})\), we see from (3.15) that \(\|y_{n}-x _{n}\|\leq \|{\varPi }_{C}(I-\sigma _{n}F)z_{n}-x_{n}\|+\alpha _{n}\|x_{n}-f(y _{n})\|\to 0\) (\(n\to \infty \)). With the aid of (3.16), one asserts

$$\begin{aligned} \Vert x_{n}-Gx_{n} \Vert &\leq \Vert x_{n}-y_{n} \Vert + \Vert y_{n}-Gy_{n} \Vert + \Vert Gy_{n}-Gx _{n} \Vert \\ &\leq 2 \Vert x_{n}-y_{n} \Vert + \Vert y_{n}-Gy_{n} \Vert \to 0 \quad (n\to \infty ). \end{aligned}$$
(3.17)

Step 4. One shows that \(\|x_{n}-Wx_{n}\|\to 0\), \(\|x_{n}-T_{ \lambda }x_{n}\|\to 0\) and \(\|x_{n}-{\varGamma }x_{n}\|\to 0\) as \(n\to \infty \), where \(Wx=\lim_{n\to \infty }W_{n}x\), \(\forall x\in C\), \(T_{\lambda }=J^{B}_{\lambda }(I-\lambda A)\) and \({\varGamma }x= \theta _{1}Wx+\theta _{2}Gx+\theta _{3}T_{\lambda }x\), \(\forall x\in C\) for constants \(\theta _{1},\theta _{2},\theta _{3}\in (0,1)\) satisfying \(\theta _{1}+\theta _{2} +\theta _{3}=1\). Indeed, utilizing (3.15) and (3.17), one deduces that

$$\begin{aligned} \Vert Wx_{n}-x_{n} \Vert \leq {}& \Vert Wx_{n}-WGx_{n} \Vert + \Vert WGx_{n}-W_{n}Gx_{n} \Vert + \Vert W _{n}Gx_{n}-W_{n}Gy_{n} \Vert \\ &{}+ \Vert W_{n}Gy_{n}-x_{n} \Vert \\ \leq{}& \Vert x_{n}-Gx_{n} \Vert + \Vert WGx_{n}-W_{n}Gx_{n} \Vert + \Vert x_{n}-y_{n} \Vert \\ &{}+ \Vert W _{n}Gy_{n}-x_{n} \Vert \to 0 \quad (n\to \infty ). \end{aligned}$$
(3.18)

Furthermore, since \(x_{n+1}-x_{n}+x_{n}-y_{n}=\delta _{n}(x_{n}-y_{n})+(1- \delta _{n})(T_{n}y_{n}-y_{n})\), from \(x_{n}-x_{n+1}\to 0\) and \(x_{n}-y_{n}\to 0\), we have

$$\begin{aligned} \Vert T_{n}y_{n}-y_{n} \Vert &= \frac{1}{1-\delta _{n}} \bigl\Vert x_{n+1}-x_{n}+(1-\delta _{n}) (x_{n}-y_{n}) \bigr\Vert \\ &\leq \frac{ \Vert x_{n+1}-x_{n} \Vert + \Vert x_{n}-y_{n} \Vert }{1- \delta _{n}}\to 0 \quad (n\to \infty ). \end{aligned}$$

Also, utilizing similar arguments to those of (3.5), we obtain

$$\begin{aligned} \Vert T_{n}y_{n}-T_{\lambda }y_{n} \Vert &\leq \biggl\vert 1-\frac{\lambda }{\lambda _{n}} \biggr\vert \bigl\Vert J^{B}_{\lambda _{n}}(I-\lambda _{n}A)y_{n} -(I-\lambda _{n}A)y_{n} \bigr\Vert + \vert \lambda _{n}-\lambda \vert \Vert Ay_{n} \Vert \\ &= \biggl\vert 1-\frac{\lambda }{\lambda _{n}} \biggr\vert \bigl\Vert T_{n}y_{n}-(I-\lambda _{n}A)y_{n} \bigr\Vert + \vert \lambda _{n}-\lambda \vert \Vert Ay_{n} \Vert . \end{aligned}$$

Since \(\lim_{n\to \infty }\lambda _{n}=\lambda \) and the sequences \(\{y_{n}\}\), \(\{T_{n}y_{n}\}\), \(\{Ay_{n}\}\) are bounded, we get

$$ \lim_{n\to \infty } \Vert T_{n}y_{n}-T_{\lambda }y_{n} \Vert =0. $$
(3.19)

Taking into account condition (v), i.e., \(0<\bar{\lambda }\leq \lambda _{n}\), \(\forall n\geq 0\) and \(\lim_{n\to \infty }\lambda _{n}=\lambda \), where \(\kappa _{q}\lambda ^{q-1}< q\alpha \), we know that \(0<\kappa _{q}\bar{ \lambda }^{q-1}\leq \kappa _{q}\lambda ^{q-1}< q\alpha \). So \(\operatorname{Fix}(T_{\lambda })=(A+B)^{-1}0\) and \(T_{\lambda }:C\to C\) is nonexpansive. Therefore, we infer from (3.19) and \(x_{n}-y_{n} \to 0\) that

$$\begin{aligned} \Vert T_{\lambda }x_{n}-x_{n} \Vert &\leq \Vert T_{\lambda }x_{n}-T_{\lambda }y _{n} \Vert + \Vert T_{\lambda }y_{n}-T_{n}y_{n} \Vert + \Vert T_{n}y_{n}-y_{n} \Vert + \Vert y_{n}-x _{n} \Vert \\ &\leq 2 \Vert x_{n}-y_{n} \Vert + \Vert T_{\lambda }y_{n}-T_{n}y_{n} \Vert + \Vert T_{n}y_{n}-y _{n} \Vert \to 0 \quad (n\to \infty ). \end{aligned}$$
(3.20)

One now defines the mapping \({\varGamma }x=\theta _{1}Wx+\theta _{2}Gx+ \theta _{3}T_{\lambda }x\), \(\forall x\in C\) with constants \(\theta _{1}, \theta _{2},\theta _{3}\in (0,1)\) satisfying \(\theta _{1}+\theta _{2}+ \theta _{3}=1\). One gets \(\operatorname{Fix}({\varGamma })=\operatorname{Fix}(W) \mathrel{\cap} \operatorname{Fix}(G)\mathrel{\cap} \operatorname{Fix}(T_{\lambda })={\varOmega }\). Observe that

$$ \Vert {\varGamma }x_{n}-x_{n} \Vert \leq \theta _{1} \Vert x_{n}-Wx_{n} \Vert +\theta _{2} \Vert x _{n}-Gx_{n} \Vert +\theta _{3} \Vert x_{n}-T_{\lambda }x_{n} \Vert . $$
(3.21)

From (3.17), (3.18), (3.20) and (3.21), one gets

$$ \lim_{n\to \infty } \Vert {\varGamma }x_{n}-x_{n} \Vert =0. $$
(3.22)

Step 5. Letting \(x_{t}\) is the unique fixed point of \(x\mapsto (1-t){\varGamma }x+tf(x)\) for each \(t\in (0,1)\), one shows that

$$ \limsup_{n\to \infty }\bigl\langle f \bigl(x^{*}\bigr)-x^{*},J\bigl(x_{n}-x^{*} \bigr)\bigr\rangle \leq 0, $$
(3.23)

where \(x^{*}=\mbox{s-}\lim_{n\to \infty }x_{t}\). By Lemmas 2.3 and 2.5, one asserts

$$\begin{aligned} \Vert x_{t}-x_{n} \Vert ^{2} \leq& 2t\bigl\langle f(x_{t})-x_{n},J(x_{t}-x_{n}) \bigr\rangle +(1-t)^{2} \Vert {\varGamma }x_{t}-x_{n} \Vert ^{2} \\ \leq& \Vert {\varGamma }x_{n}-x_{n} \Vert )^{2}+2t\bigl\langle f(x_{t})-x_{n},J(x_{t}-x _{n})\bigr\rangle +(1-t)^{2}( \Vert {\varGamma }x_{t}-{\varGamma }x_{n} \Vert \\ \leq& \bigl(1-2t+t^{2}\bigr) \Vert x_{t}-x_{n} \Vert ^{2}+f_{n}(t)+2t\bigl\langle f(x_{t})-x _{t},J(x_{t}-x_{n}) \bigr\rangle \\ &{} +2t \Vert x_{t}-x_{n} \Vert ^{2}, \end{aligned}$$
(3.24)

where

$$ f_{n}(t)=(1-t)^{2} \Vert x_{n}-{\varGamma }x_{n} \Vert \bigl(2 \Vert x_{t}-x_{n} \Vert + \Vert x_{n}- { \varGamma }x_{n} \Vert \bigr)\to 0 \quad (n\to \infty ). $$
(3.25)

It follows from (3.24) that

$$ 2\bigl\langle x_{t}-f(x_{t}),J(x_{t}-x_{n}) \bigr\rangle \leq t \Vert x_{t}-x_{n} \Vert ^{2}+\frac{f_{n}(t)}{t}. $$
(3.26)

Letting \(n\to \infty \) and employing (3.25), one derives

$$ 2\limsup_{n\to \infty }\bigl\langle x_{t}-f(x_{t}),J(x_{t}-x_{n}) \bigr\rangle \leq tM_{4}, $$
(3.27)

where \(\sup \{\|x_{t}-x_{n}\|^{2}:t\in (0,1) \text{ and } n\geq 0\} \leq M_{4}\) for some \(M_{4}>0\). Taking \(t\to 0\) in (3.27), we have

$$ \limsup_{t\to 0}\limsup_{n\to \infty }\bigl\langle x_{t}-f(x_{t}),J(x_{t}-x _{n})\bigr\rangle \leq 0. $$

On the other hand, we have

$$\begin{aligned} &\bigl\langle f\bigl(x^{*}\bigr)-x^{*},J \bigl(x_{n}-x^{*}\bigr)\bigr\rangle \\ &\quad =\bigl\langle f\bigl(x^{*}\bigr)-x^{*},J\bigl(x _{n}-x^{*}\bigr)\bigr\rangle -\bigl\langle f \bigl(x^{*}\bigr)-x^{*}, J(x_{n}-x_{t}) \bigr\rangle \\ &\qquad {} +\bigl\langle f\bigl(x^{*}\bigr)-x^{*},J(x_{n}-x_{t}) \bigr\rangle -\bigl\langle f\bigl(x^{*}\bigr)-x_{t},J(x _{n}-x_{t})\bigr\rangle \\ &\qquad {} +\bigl\langle f \bigl(x^{*}\bigr)-x_{t},J(x_{n}-x_{t}) \bigr\rangle \\ &\qquad {} -\bigl\langle f(x_{t})-x_{t},J(x_{n}-x_{t}) \bigr\rangle +\bigl\langle f(x_{t})-x_{t},J(x _{n}-x_{t})\bigr\rangle \\ &\quad =\bigl\langle f\bigl(x^{*}\bigr)-x^{*},J \bigl(x_{n}-x^{*}\bigr)-J(x_{n}-x_{t}) \bigr\rangle +\bigl\langle x_{t}-x^{*},J(x_{n}-x_{t}) \bigr\rangle \\ &\qquad {} +\bigl\langle f\bigl(x^{*}\bigr)-f(x_{t}),J(x_{n}-x_{t}) \bigr\rangle +\bigl\langle f(x_{t})-x _{t},J(x_{n}-x_{t}) \bigr\rangle . \end{aligned}$$

So, it follows that

$$\begin{aligned} &\limsup_{n\to \infty }\bigl\langle f\bigl(x^{*} \bigr)-x^{*},J\bigl(x_{n}-x^{*}\bigr)\bigr\rangle \\ &\quad \leq \limsup_{n\to \infty }\bigl\langle f\bigl(x^{*} \bigr)-x^{*},J\bigl(x_{n}-x^{*} \bigr)-J(x_{n}-x _{t})\bigr\rangle \\ &\qquad {} +(1+\delta ) \bigl\Vert x_{t}-x^{*} \bigr\Vert \limsup_{n\to \infty } \Vert x_{n}-x_{t} \Vert +\limsup_{n\to \infty }\bigl\langle f(x_{t})-x_{t},J(x_{n}-x_{t}) \bigr\rangle . \end{aligned}$$

Taking into account that \(x_{t}\to x^{*}\) as \(t\to 0\), we have

$$\begin{aligned} &\limsup_{n\to \infty }\bigl\langle f \bigl(x^{*}\bigr)-x^{*},J\bigl(x_{n}-x^{*} \bigr)\bigr\rangle \\ &\quad =\limsup_{t\to 0}\limsup_{n\to \infty }\bigl\langle f\bigl(x^{*}\bigr)-x^{*},J\bigl(x_{n}-x ^{*}\bigr)\bigr\rangle \\ & \quad \leq \limsup_{t\to 0}\limsup_{n\to \infty } \bigl\langle f\bigl(x^{*}\bigr)-x^{*},J \bigl(x_{n}-x ^{*}\bigr)-J(x_{n}-x_{t}) \bigr\rangle . \end{aligned}$$
(3.28)

Thanks to the space (q-uniformly smooth), one knows that the two limits can be interchangeable. Equation (3.23) therefore holds. Note that \(x_{n}-y_{n}\to 0\) implies \(J(y_{n}-x^{*})-J(x_{n}-x^{*})\to 0\). Thus, we conclude from (3.23) that

$$\begin{aligned} &\limsup_{n\to \infty }\bigl\langle f \bigl(x^{*}\bigr)-x^{*},J\bigl(y_{n}-x^{*} \bigr)\bigr\rangle \\ &\quad = \limsup_{n\to \infty }\bigl\{ \bigl\langle f \bigl(x^{*}\bigr)-x^{*},J\bigl(x_{n}-x^{*} \bigr)\bigr\rangle +\bigl\langle f\bigl(x^{*}\bigr)-x^{*},J \bigl(y_{n}-x^{*}\bigr)-J\bigl(x_{n}-x^{*} \bigr)\bigr\rangle \bigr\} \\ &\quad = \limsup_{n\to \infty }\bigl\langle f\bigl(x^{*} \bigr)-x^{*},J\bigl(x_{n}-x^{*}\bigr)\bigr\rangle \leq 0. \end{aligned}$$
(3.29)

Step 6. One shows \(\|x_{n}-x^{*}\|\to 0\) as \(n\to \infty \).

$$\begin{aligned} \bigl\Vert y_{n}-x^{*} \bigr\Vert ^{2} =& \bigl\Vert \alpha _{n}\bigl(f(y_{n})-f \bigl(x^{*}\bigr)\bigr)+\beta _{n}\bigl(x_{n}-x ^{*}\bigr)+\gamma _{n}\bigl({\varPi }_{C}(I- \sigma _{n}F)z_{n}-x^{*}\bigr) \\ &{} +\alpha _{n}\bigl(f\bigl(x^{*} \bigr)-x^{*}\bigr) \bigr\Vert ^{2} \\ \leq& \alpha _{n} \bigl\Vert f(y_{n})-f \bigl(x^{*}\bigr) \bigr\Vert ^{2}+\beta _{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+\gamma _{n}\bigl[ \bigl\Vert z_{n}-x^{*} \bigr\Vert ^{2} \\ &{}+2\sigma _{n} \Vert Fz_{n} \Vert \bigl\Vert z_{n}-x ^{*}-\sigma _{n}Fz_{n} \bigr\Vert \bigr] \\ &{} +2\alpha _{n}\bigl\langle f\bigl(x^{*} \bigr)-x^{*},J\bigl(y_{n}-x^{*}\bigr)\bigr\rangle \\ \leq& \alpha _{n}\delta \bigl\Vert y_{n}-x^{*} \bigr\Vert ^{2}+\beta _{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+\gamma _{n}\bigl(t_{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-t_{n}) \bigl\Vert y_{n}-x^{*} \bigr\Vert ^{2}\bigr) \\ &{} +2\sigma _{n} \Vert Fz_{n} \Vert \bigl\Vert z_{n}-x^{*}-\sigma _{n}Fz_{n} \bigr\Vert +2\alpha _{n} \bigl\langle f\bigl(x^{*} \bigr)-x^{*},J\bigl(y_{n}-x^{*}\bigr)\bigr\rangle , \end{aligned}$$

which hence yields

$$\begin{aligned} \bigl\Vert y_{n}-x^{*} \bigr\Vert ^{2} \leq& \biggl(1-\frac{\alpha _{n}(1-\delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\biggr) \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} + \frac{2\alpha _{n}}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))} \\ &{} \times \biggl[\frac{\sigma _{n}}{\alpha _{n}} \Vert Fz_{n} \Vert \bigl\Vert z_{n}-x^{*}-\sigma _{n}Fz_{n} \bigr\Vert +\bigl\langle f\bigl(x^{*}\bigr)-x^{*},J \bigl(y_{n}-x^{*}\bigr)\bigr\rangle \biggr]. \end{aligned}$$

Due to the convexity of \(\|\cdot \|^{2}\), and the nonexpansivity of \(T_{n}\), one asserts

$$\begin{aligned} \bigl\Vert x_{n+1}-x^{*} \bigr\Vert ^{2} \leq& \delta _{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-\delta _{n}) \bigl\Vert y_{n}-x^{*} \bigr\Vert ^{2} \\ \leq& \delta _{n} \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2}+(1-\delta _{n})\biggl\{ \biggl(1- \frac{\alpha _{n}(1-\delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\biggr) \bigl\Vert x_{n}-x ^{*} \bigr\Vert ^{2} \\ &{} +\frac{2\alpha _{n}}{1-(\alpha _{n}\delta +\gamma _{n}(1-t _{n}))} \\ &{} \times \biggl[\frac{\sigma _{n}}{\alpha _{n}} \Vert Fz_{n} \Vert \bigl\Vert z_{n}-x^{*}-\sigma _{n}Fz_{n} \bigr\Vert +\bigl\langle f\bigl(x^{*}\bigr)-x^{*},J \bigl(y_{n}-x^{*}\bigr)\bigr\rangle \biggr]\biggr\} \\ =&\biggl[1-\frac{\alpha _{n}(1-\delta _{n})(1-\delta )}{1-(\alpha _{n}\delta + \gamma _{n}(1-t_{n}))}\biggr] \bigl\Vert x_{n}-x^{*} \bigr\Vert ^{2} +\frac{\alpha _{n}(1-\delta _{n})(1-\delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))} \\ &{}\times \frac{2[\frac{ \sigma _{n}}{\alpha _{n}} \Vert Fz_{n} \Vert \Vert z_{n}-x^{*}-\sigma _{n}Fz_{n} \Vert + \langle f(x^{*})-x^{*},J(y_{n}-x^{*})\rangle ]}{1-\delta }. \end{aligned}$$
(3.30)

Since \(\liminf_{n\to \infty }\frac{(1-\delta _{n})(1-\delta )}{1-( \alpha _{n}\delta +\gamma _{n}(1-t_{n}))}>0\), \(\{\frac{\alpha _{n}(1- \delta )}{1-(\alpha _{n}\delta +\gamma _{n}(1-t_{n}))}\}\subset (0,1)\) and \(\sum^{\infty }_{n=0}\alpha _{n}=\infty \), we know

$$ \biggl\{ \frac{\alpha _{n}(1-\delta _{n})(1-\delta )}{1-(\alpha _{n}\delta + \gamma _{n}(1-t_{n}))}\biggr\} \subset (0,1) $$

and

$$ \sum^{\infty }_{n=0}\frac{\alpha _{n}(1-\delta _{n})(1-\delta )}{1-( \alpha _{n}\delta +\gamma _{n}(1-t_{n}))}= \infty . $$

Utilizing (3.29) and Lemma 2.7, we conclude from (3.30) that \(\|x_{n}-x^{*}\|\to 0\) as \(n\to \infty \). This completes the proof. □

Remark 3.1

Comparing with the corresponding results in and Chang et al. [8], we have the following aspects. The problem of solving a HVI with the constraints of SGVIs (1.1) and a countable family of nonexpansive mappings in [8, Theorem 3.1] is extended to our problem of solving a HVI with the constraints of SGVIs (1.1), a variational inclusion (VI) and a countable family of nonexpansive mappings. The modified relaxed extragradient method in[8, Theorem 3.1] is extended to our composite extragradient implicit rule (3.1). That is, two iterative steps \(y_{n}=(1-\beta _{n})x_{n}+ \beta _{n}Gx_{n}\) and \(x_{n+1}={\varPi }_{C}[\gamma _{n}x_{n}+((1-\gamma _{n})I-\alpha _{n}\rho F)S_{n}y_{n}+\alpha _{n}\gamma f(x_{n})]\) in [8, Theorem 3.1] are extended to our two iterative steps \(y_{n}=\beta _{n}x_{n}+\gamma _{n}{\varPi }_{C}(I-\sigma _{n}F)(t_{n}x_{n}+(1-t _{n})W_{n}Gy_{n})+\alpha _{n}f(y_{n})\) and \(x_{n+1}=\delta _{n}x_{n} +(1- \delta _{n})T_{n}y_{n}\), where \(T_{n}=J^{B}_{\lambda _{n}}(I-\lambda _{n}A)\).