Fixed point solutions for variational inequalities in image restoration over q-uniformly smooth Banach spaces

Abstract

In this paper, we introduce new implicit and explicit iterative methods for finding a common fixed point set of an infinite family of strict pseudo-contractions by the sunny nonexpansive retractions in a real q-uniformly and uniformly convex Banach space which admits a weakly sequentially continuous generalized duality mapping. Then we prove the strong convergence under mild conditions of the purposed iterative scheme to a common fixed point of an infinite family of strict pseudo-contractions which is a solution of some variational inequalities. Furthermore, we apply our results to study some strong convergence theorems in $L_p$ and ${\ell }_p$ spaces with $1<p<\mathrm{\infty }$. Our results mainly improve and extend the results announced by Ceng et al. (Comput. Math. Appl. 61:2447-2455, 2011) and many authors from Hilbert spaces to Banach spaces. Finally, we give some numerical examples for support our main theorem in the end of the paper.

MSC:47H09, 47H10, 47H17, 47J25, 49J40.

1 Introduction

Let $C_1,C_2,\dots ,C_n$ be nonempty, closed, and convex subsets of a real Hilbert space H such that ${\bigcap }_{i=1}^n{C}_i\ne \mathrm{\varnothing }$. The problem of image recovery in a Hilbert space setting by using convex of metric projections $P_{C_i}$, may be stated as follows: the original unknown image z is known a priori to belong to the intersection of ${\{C_i\}}_{i=1}^n$; given only the metric projections $P_{C_i}$ of H onto $C_i$ for $i=1,2,\dots ,n$ recover z by an iterative scheme. Youla and Webb [1] first used iterative methods for applied in image restoration. The problems of image recovery have been studied in a Banach space setting by Kitahara and Takahashi [2] (see also [3, 4]) by using convex combinations of sunny nonexpansive retractions in uniformly convex Banach spaces. On the other hand, Alber [5] studied the problem of image recovery by the products of generalized projections in a uniformly convex and uniformly smooth Banach space whose duality mapping is weakly sequentially continuous (see also [6, 7]). Nakajo et al. [8] and Kimura et al. [9] considered this problem by the sunny nonexpansive retractions and proved convergence of the iterative sequence to a common point of countable nonempty, closed, and convex subsets in a uniformly convex and smooth Banach space, and in a strictly convex, smooth and reflexive Banach space having the Kadec-Klee property, respectively. Some iterative methods have been studied in problem of image recovery by numerous authors (see [2–5, 10–12]).

The problems of image recovery are connected with the convex feasibility problem, convex minimization problems, multiple-set split feasibility problems, common fixed point problems, and variational inequalities. In particular, variational inequality theory has been studied widely in several branches of pure and applied sciences. This field is dynamics and is experiencing an explosive growth in both theory and applications. Indeed, applications of the variational inequalities span as diverse disciplines as differential equations, time-optimal control, optimization, mathematical programming, mechanics, finance, and so on. Note that most of the variational problems, including minimization or maximization of functions, variational inequality problems, quasivariational inequality problems, decision and management sciences, and engineering sciences problems. Recently, some iterative methods have been developed for solving the fixed point problems and variational inequality problems in q-uniformly smooth Banach spaces by numerous authors (see [13–24]).

Let A be a strongly positive bounded linear operator on H, that is, there exists a constant $\overline{\gamma }>0$ such that

$$〈Ax,x〉\ge \overline{\gamma }{\parallel x\parallel }^2\text{for all }x\in H.$$

(1.1)

Remark 1.1 From the definition of operator A, we note that a strongly positive bounded linear operator A is a $\parallel A\parallel$-Lipschitzian and η-strongly monotone operator.

A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:

$$\underset{x\in C}{min}\frac{1}{2}〈Ax,x〉-〈x,u〉,$$

(1.2)

where C is the fixed point set of a nonexpansive mapping T on H and u is a given point in H.

In 2006, Marino and Xu [25] introduced and considered the following a general iterative method:

$$x_{n+1}={\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)T{x}_n,\mathrm{\forall }n\ge 0,$$

(1.3)

where A is a strongly positive bounded linear operator on a real Hilbert space H. They proved that if the sequence $\{{\alpha }_n\}$ satisfies appropriate conditions, then the sequence $\{x_n\}$ generated by (1.3) converges strongly to the unique solution of the variational inequality

$$〈(\gamma f-A)x^{\ast },x-x^{\ast }〉\le 0,\mathrm{\forall }x\in Fix(T),$$

(1.4)

which is the optimality condition for the minimization problem

$$\underset{x\in C}{min}\frac{1}{2}〈Ax,x〉-h(x),$$

(1.5)

where C is the fixed point set of a nonexpansive mapping T and h is a potential function for γf (i.e., $h^{\prime }(x)=\gamma f(x)$ for all $x\in H$).

On the other hand, Yamada [26] introduced a hybrid steepest descent method for a nonexpansive mapping T as follows:

$$x_{n+1}=T{x}_n-\mu {\lambda }_{n}F(T{x}_n),\mathrm{\forall }n\ge 0,$$

(1.6)

where F is a κ-Lipschitzian and η-strongly monotone operator on a real Hilbert space H with constants $\kappa ,\eta >0$ and $0<\mu <\frac{2\eta }{{\kappa }^2}$. He proved that if $\{{\lambda }_n\}$ satisfy the appropriate conditions, then the sequence $\{x_n\}$ generated by (1.6) converges strongly to the unique solution of the variational inequality

$$〈F{x}^{\ast },x-x^{\ast }〉\ge 0,\mathrm{\forall }x\in Fix(T).$$

(1.7)

Tian [27] combined the iterative method (1.3) with the Yamada method (1.6) and considered a general iterative method for a nonexpansive mapping T on a real Hilbert space H as follows:

$$x_{n+1}={\alpha }_n\gamma f(x_n)+(I-{\alpha }_n\mu F)T{x}_n,\mathrm{\forall }n\ge 0.$$

(1.8)

Then he proved that the sequence $\{x_n\}$ generated by (1.8) converges strongly to the unique solution of variational inequality

$$〈(\gamma f-\mu F)x^{\ast },x-x^{\ast }〉\le 0,\mathrm{\forall }x\in Fix(T).$$

(1.9)

In 2011, Ceng et al. [28] combined the iterative method (1.3) with Tian’s method (1.8) and consider the following a general composite iterative method:

$$x_{n+1}=(I-{\alpha }_{n}A)T{x}_n+{\alpha }_n[T{x}_n-{\beta }_n(\mu FT{x}_n-\gamma f(x_n))],\mathrm{\forall }n\ge 0,$$

(1.10)

where A is a strongly positive bounded linear operator on H with coefficient $\overline{\gamma }\in (1,2)$, and $\{{\alpha }_n\}\subset (0,1)$ and $\{{\beta }_n\}\subset (0,1]$ satisfy appropriate conditions. Then they proved that the sequence $\{x_n\}$ generated by (1.10) converges strongly to the unique solution $x^{\ast }\in C$ of the variational inequality

$$〈(I-A)x^{\ast },x-x^{\ast }〉\le 0,\mathrm{\forall }x\in C,$$

(1.11)

where $C=Fix(T)$.

In this paper, motivated by the above facts, we introduce new implicit and explicit iterative methods for finding a common fixed point set of an infinite family of strict pseudo-contractions by the sunny nonexpansive retractions in a real q-uniformly and uniformly convex Banach space X which admits a weakly sequentially continuous generalized duality mapping. Consequently, we prove the strong convergence under mild conditions of the purposed iterative scheme to a common fixed point of an infinite family of strict pseudo-contractions of nonempty, closed, and convex subsets of X which is a solution of some variational inequalities. Furthermore, we apply our results to the study of some strong convergence theorems in $L_p$ and ${\ell }_p$ spaces with $1<p<\mathrm{\infty }$. Our results extend the main result of Ceng et al. [28] in several aspects and the work of many authors from Hilbert spaces to Banach spaces. Finally, we give some numerical examples to support our main theorem in the end of the paper.

2 Preliminaries

Throughout this paper, we denote by X and $X^{\ast }$ a real Banach space and the dual space of X, respectively. Let $q>1$ be a real number. The generalized duality mapping $J_q:X\to 2^{X^{\ast }}$ is defined by

$$J_q(x)=\{f\in X^{\ast }:〈x,f〉={\parallel x\parallel }^q,\parallel f\parallel ={\parallel x\parallel }^{q-1}\},$$

where $〈\cdot ,\cdot 〉$ denotes the duality pairing between X and $X^{\ast }$. In particular, $J_q=J_2$ is called the normalized duality mapping and $J_q(x)={\parallel x\parallel }^{q-2}{J}_2(x)$ for $x\ne 0$. If $X:=H$ is a real Hilbert space, then $J=I$, where I is the identity mapping. It is well known that if X is smooth, then $J_q$ is single-valued, which is denoted by $j_q$ (see [29]).

A Banach space X is said to be strictly convex if $\frac{\parallel x+y\parallel }{2}<1$ for all $x,y\in X$ with $\parallel x\parallel =\parallel y\parallel =1$ and $x\ne y$. A Banach space X is said to be uniformly convex if, for each $ϵ>0$, there exists $\delta >0$ such that for $x,y\in X$ with $\parallel x\parallel ,\parallel y\parallel \le 1$ and $\parallel x-y\parallel \ge ϵ$, $\frac{\parallel x+y\parallel }{2}\le 1-\delta$ holds. Let $S(X)=\{x\in X:\parallel x\parallel =1\}$. The norm of X is said to be Gâteaux differentiable (or X is said to be smooth) if the limit

$$\underset{t\to 0}{lim}\frac{\parallel x+ty\parallel -\parallel x\parallel }{t}$$

exists for each $x,y\in S(X)$. The norm of X is said to be uniformly Gâteaux differentiable, if, for each $y\in S(X)$, the limit is attained uniformly for $x\in S(X)$.

Let ${\rho }_X:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ be the modulus of smoothness of X defined by

$${\rho }_X(\tau )=sup\{\frac{1}{2}(\parallel x+y\parallel +\parallel x-y\parallel )-1:x\in S(X),\parallel y\parallel \le \tau \}.$$

A Banach space X is said to be uniformly smooth if $\frac{{\rho }_X(t)}{t}\to 0$ as $t\to 0$. Suppose that $q>1$, then X is said to be q-uniformly smooth if there exists $c>0$ such that ${\rho }_X(t)\le c{t}^q$ for all $t>0$. It is shown in [30] (see also [31]) that there is no Banach space which is q-uniformly smooth with $q>2$. If X is q-uniformly smooth, then X is uniformly smooth. It is well known that each uniformly convex Banach space X is reflexive and strictly convex and every uniformly smooth Banach space X is a reflexive Banach space with uniformly Gâteaux differentiable norm (see [29]). Typical examples of both uniformly convex and uniformly smooth Banach spaces are $L_p$, where $p>1$. More precisely, $L_p$ is $min\{p,2\}$-uniformly smooth for every $p>1$.

Let C be a nonempty, closed, and convex subset of X and T be a self-mapping on C. We denote the fixed points set of the mapping T by $Fix(T)=\{x\in C:Tx=x\}$.

Definition 2.1 A mapping $T:C\to C$ is said to be:

• (i) λ-strictly pseudo-contractive [32] if, for all $x,y\in C$, there exist $\lambda >0$ and $j_q(x-y)\in J_q(x-y)$ such that

  $$〈Tx-Ty,j_q(x-y)〉\le {\parallel x-y\parallel }^q-\lambda {\parallel (I-T)x-(I-T)y\parallel }^q,$$

  (2.1)

  or equivalently

  $$〈(I-T)x-(I-T)y,j_q(x-y)〉\ge \lambda {\parallel (I-T)x-(I-T)y\parallel }^q.$$

  (2.2)

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

  $$\parallel Tx-Ty\parallel \le L\parallel x-y\parallel .$$

If $0<L<1$, then T is a contraction and if $L=1$, then T is a nonexpansive mapping. By the definition, we know that every λ-strictly pseudo-contractive mapping is $(\frac{1+\lambda }{\lambda })$-Lipschitzian (see [33]).

Remark 2.2 Let C be a nonempty subset of a real Hilbert space H and $T:C\to C$ be a mapping. Then T is said to be k-strictly pseudo-contractive [32] if, for all $x,y\in C$, there exists $k\in [0,1)$ such that

$${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+k{\parallel (I-T)x-(I-T)y\parallel }^2.$$

(2.3)

It is well known that (2.3) is equivalent to the following inequality:

$$〈Tx-Ty,x-y〉\le {\parallel x-y\parallel }^2-\frac{1-k}{2}{\parallel (I-T)x-(I-T)y\parallel }^2.$$

A mapping $F:C\to X$ is said to be accretive if, for all $x,y\in C$, there exists $j_q(x-y)\in J_q(x-y)$ such that

$$〈Fx-Fy,j_q(x-y)〉\ge 0.$$

For some $\eta >0$, $F:C\to X$ is said to be strongly accretive if, for all $x,y\in C$, there exists $j_q(x-y)\in J_q(x-y)$ such that

$$〈Fx-Fy,j_q(x-y)〉\ge \eta {\parallel x-y\parallel }^q.$$

Remark 2.3 If $X:=H$ is a real Hilbert space, accretive and strongly accretive mappings coincide with monotone and strongly monotone mappings, respectively.

Let D be a nonempty subset of C. A mapping $Q:C\to D$ is said to be sunny [34] if

$$Q(Qx+t(x-Qx))=Qx,$$

whenever $Qx+t(x-Qx)\in C$ for $x\in C$ and $t\ge 0$. A mapping $Q:C\to D$ is said to be retraction if $Qx=x$ for all $x\in D$. Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. It is well known that if $X:=H$ is a real Hilbert space, then a sunny nonexpansive retraction Q is coincident with the metric projection from X onto C.

Lemma 2.4 ([14])

Let C be a closed and convex subset of a smooth Banach space X. Let D be a nonempty subset of C. Let $Q:C\to D$ be a retraction and let j, $j_q$ be the normalized duality mapping and generalized duality mapping on X, respectively. Then the following are equivalent:

• (a) Q is sunny and nonexpansive.

• (b) ${\parallel Qx-Qy\parallel }^2\le 〈x-y,j(Qx-Qy)〉$ for all $x,y\in C$.

• (c) $〈x-Qx,j(y-Qx)〉\le 0$ for all $x\in C$ and $y\in D$.

• (d) $〈x-Qx,j_q(y-Qx)〉\le 0$ for all $x\in C$ and $y\in D$.

Lemma 2.5 ([35])

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

$$ab\le \frac{1}{q}{a}^q+\left(\frac{q-1}{q}\right)b^{\frac{q}{q-1}}$$

for arbitrary positive real numbers a, b.

In a real q-uniformly smooth Banach space, Xu [36] proved the following important inequality:

Lemma 2.6 ([36])

Let X be a real q-uniformly smooth Banach space. Then the following inequality holds:

$${\parallel x+y\parallel }^q\le {\parallel x\parallel }^q+q〈y,J_q(x)〉+C_q{\parallel y\parallel }^q$$

(2.4)

for all $x,y\in X$ and for some $C_q>0$.

Remark 2.7 The constant $C_q$ satisfying (2.4) is called the best q-uniform smoothness constant.

Lemma 2.8 ([21])

Let C be a nonempty and convex subset of a real q-uniformly smooth Banach space X and $T:C\to C$ be a λ-strict pseudo-contraction. For $\gamma \in (0,1)$, define $Sx=(1-\gamma )x+\gamma Tx$. Then, as $\gamma \in (0,\nu )$, $\nu =min\{1,{(\frac{q\lambda }{C_q})}^{\frac{1}{q-1}}\}$, $S:C\to C$ is nonexpansive and $Fix(S)=Fix(T)$, where $C_q$ is the best q-uniform smoothness constant.

Definition 2.9 ([37])

Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X. Let $T_{n,k}={\theta }_{n,k}{S}_k+(1-{\theta }_{n,k})I$, where $S_k:C\to C$ is ${\lambda }_k$-strict pseudo-contraction and $\{t_n\}$ be a nonnegative real sequence with $0\le t_n\le 1$, $\mathrm{\forall }n\in \mathbb{N}$. For $n\ge 1$, define a mapping $W_n:C\to C$ as follows:

$$\begin{array}{l}{U}_{n,n+1}=I,\\ U_{n,n}=t_n{T}_{n,n}{U}_{n,n+1}+(1-t_n)I,\\ ⋮\\ U_{n,k}=t_k{T}_{n,k}{U}_{n,k+1}+(1-t_k)I,\\ U_{n,k-1}=t_{k-1}{T}_{n,k-1}{U}_{n,k}+(1-t_{k-1})I,\\ ⋮\\ U_{n,2}=t_2{T}_{n,2}{U}_{n,3}+(1-t_2)I,\\ W_n=U_{n,1}=t_1{T}_{n,1}{U}_{n,2}+(1-t_1)I.\end{array}$$

(2.5)

Such a mapping $W_n$ is called the W-mapping generated by $T_{n,n},T_{n,n-1},\dots ,T_{n,1}$ and $t_n,t_{n-1},\dots ,t_1$.

Throughout this paper, we will assume that $\{{\theta }_{n,k}\}$ satisfies the following conditions:

(H1) ${\theta }_{n,k}\in (0,\nu ]$, $\nu =min\{1,{(\frac{q\overline{\lambda }}{C_q})}^{\frac{1}{q-1}}\}$ with $\overline{\lambda }=inf{\lambda }_k>0$, $\mathrm{\forall }n,k\in \mathbb{N}$;

(H2) $|{\theta }_{n+1,k}-{\theta }_{n,k}|\le a_n$, $\mathrm{\forall }n\in \mathbb{N}$ and $1\le k\le n$ with ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n<\mathrm{\infty }$;

The hypothesis (H2) secures the existence of ${lim}_{n\to \mathrm{\infty }}{\theta }_{n,k}$, $\mathrm{\forall }k\in \mathbb{N}$. Set ${\theta }_{1,k}:={lim}_{n\to \mathrm{\infty }}{\theta }_{n,k}$, $\mathrm{\forall }n\in \mathbb{N}$. Furthermore, we assume

(H3) ${\theta }_{1,k}>0$, $\mathrm{\forall }k\in \mathbb{N}$.

It is obvious that ${\theta }_{1,k}$ satisfies (H1). Using condition (H3), from $T_{n,k}={\theta }_{n,k}{S}_k+(1-{\theta }_{n,k})I$, we define mappings $T_{1,k}x:={lim}_{n\to \mathrm{\infty }}{T}_{n,k}x={\theta }_{1,k}{S}_{k}x+(1-{\theta }_{1,k})x$, $\mathrm{\forall }x\in C$.

Lemma 2.10 ([37])

Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth and strictly convex Banach space X. Let $T_{n,i}={\theta }_{n,i}{S}_i+(1-{\theta }_{n,i})I$, where $S_i:C\to C$ ($i=1,2,\dots$) is ${\lambda }_i$-strict pseudo-contraction with ${\bigcap }_{n=1}^{\mathrm{\infty }}Fix(S_n)\ne \mathrm{\varnothing }$ and $inf{\lambda }_i>0$. Let $t_1,t_2,\dots$ be nonnegative real numbers such that $0<t_n\le b<1$, $\mathrm{\forall }n\ge 1$. Assume the sequence $\{{\theta }_{n,k}\}$ satisfies (H1)-(H3). Then

• (1) $W_n$ is nonexpansive and $Fix(W_n)={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(S_n)$ for each $n\ge 1$;

• (2) for each $x\in C$ and for each positive integer k, the limit ${lim}_{n\to \mathrm{\infty }}{U}_{n,k}$ exists;

• (3) the mapping $W:C\to C$ defined by

  $$Wx:=\underset{n\to \mathrm{\infty }}{lim}{W}_{n}x=\underset{n\to \mathrm{\infty }}{lim}{U}_{n,1}x,\mathrm{\forall }x\in C,$$

is a nonexpansive mapping satisfying $Fix(W)={\bigcap }_{n=1}^{\mathrm{\infty }}Fix(S_n)$ and it is called the W-mapping generated by $S_1,S_2,\dots$ and $t_1,t_2,\dots$ and ${\theta }_{n,k}$, $\mathrm{\forall }n\in \mathbb{N}$ and $1\le k\le n$.

Lemma 2.11 ([37])

Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth and strictly convex Banach space X. Let $T_{n,i}={\theta }_{n,i}{S}_i+(1-{\theta }_{n,i})I$, where $S_i:C\to C$ ($i=1,2,\dots$) is ${\lambda }_i$-strict pseudo-contraction with ${\bigcap }_{n=1}^{\mathrm{\infty }}Fix(S_n)\ne \mathrm{\varnothing }$ and $inf{\lambda }_i>0$. Let $t_1,t_2,\dots$ be nonnegative real numbers such that $0<t_n\le b<1$, $\mathrm{\forall }n\ge 1$. Assume the sequence $\{{\theta }_{n,k}\}$ satisfies (H1)-(H3). If $\{{\omega }_n\}$ is a bounded sequence in C, then

$$\underset{n\to \mathrm{\infty }}{lim}\parallel W{\omega }_n-W_n{\omega }_n\parallel =0.$$

In the following, the notation ⇀ and → denote the weak and strong convergence, respectively. The duality mapping $J_q$ from a smooth Banach space X into $X^{\ast }$ is said to be weakly sequentially continuous generalized duality mapping if, for all $\{x_n\}\subset X$, $x_n\rightharpoonup x$ implies $J_q(x_n){\rightharpoonup }^{\ast }{J}_q(x)$.

A Banach space X is said to be satisfy Opial’s condition [38], that is, for any sequence $\{x_n\}$ in X, $x_n\rightharpoonup x$ implies that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_n-x\parallel <\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_n-y\parallel ,\mathrm{\forall }y\in X\text{ with }x\ne y.$$

By Theorem 3.2.8 in [39], it is well known that if X admits a weakly sequentially continuous generalized duality mapping, then X satisfies Opial’s condition.

Lemma 2.12 ([13])

Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X which admits weakly sequentially continuous generalized duality mapping $j_q$ from X into $X^{\ast }$. Let $T:C\to C$ be a nonexpansive mapping. Then, for all $\{x_n\}\subset C$, if $x_n\rightharpoonup x$ and $x_n-T{x}_n\to 0$, then $x=Tx$.

Lemma 2.13 ([40])

Let $\{a_n\}$, $\{{\mu }_n\}$, and $\{{\delta }_n\}$ be real sequences of nonnegative numbers such that

$$a_{n+1}\le (1-{\sigma }_n)a_n+{\mu }_n+{\delta }_n,\mathrm{\forall }n\ge 1,$$

where ${\sigma }_n\in (0,1)$, ${\sum }_{n=1}^{\mathrm{\infty }}{\sigma }_n=\mathrm{\infty }$, ${\mu }_n=\circ ({\sigma }_n)$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\delta }_n<\mathrm{\infty }$. Then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

3 Main results

In order to prove our main result, the following lemma is needed.

Lemma 3.1 Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X with the best q-uniform smoothness constant $C_q>0$. Let $F:C\to X$ be a κ-Lipschitzian and η-strongly accretive operator with constants $\kappa ,\eta >0$. Let $0<\mu <{(\frac{q\eta }{C_q{\kappa }^q})}^{\frac{1}{q-1}}$ and $\tau =\mu (\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q})$. Then for $t\in (0,min\{1,\frac{1}{q\tau }\})$, the mapping $S:C\to X$ defined by $S:=I-t\mu F$ is a contraction with constant $1-t\tau$.

Proof Since $0<\mu <{(\frac{q\eta }{C_q{\kappa }^q})}^{\frac{1}{q-1}}$ and $t\in (0,min\{1,\frac{1}{q\tau }\})$. This implies that $1-t\tau \in (0,1)$. From Lemma 2.6, for all $x,y\in C$, we have

$$\begin{array}{rcl}{\parallel Sx-Sy\parallel }^q& =& {\parallel (I-t\mu F)x-(I-t\mu F)y\parallel }^q\\ =& {\parallel (x-y)-t\mu (Fx-Fy)\parallel }^q\\ \le & {\parallel x-y\parallel }^q-qt\mu 〈Fx-Fy,j_q(x-y)〉+C_q{t}^q{\mu }^q{\parallel Fx-Fy\parallel }^q\\ \le & {\parallel x-y\parallel }^q-qt\mu \eta {\parallel x-y\parallel }^q+C_q{t}^q{\mu }^q{\kappa }^q{\parallel x-y\parallel }^q\\ \le & [1-t\mu (q\eta -C_q{\mu }^{q-1}{\kappa }^q)]{\parallel x-y\parallel }^q\\ =& [1-t\mu q(\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q})]{\parallel x-y\parallel }^q\\ \le & {[1-t\mu (\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q})]}^q{\parallel x-y\parallel }^q\\ =& {(1-t\tau )}^q{\parallel x-y\parallel }^q.\end{array}$$

It follows that

$$\parallel Sx-Sy\parallel \le (1-t\tau )\parallel x-y\parallel .$$

Hence, we have $S:=I-t\mu F$ is a contraction with constant $1-t\tau$. □

Lemma 3.2 Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X and $G:C\to X$ be a mapping.

• (i) If G is a δ-strongly accretive and λ-strictly pseudo-contractive mapping wit $\delta +\lambda >1$, then $I-G$ is a contraction with constant $L_{\delta ,\lambda }:={(\frac{1-\delta }{\lambda })}^{\frac{1}{q}}$.

• (ii) If G is a δ-strongly accretive and λ-strictly pseudo-contractive mapping with $\delta +\lambda >1$. For a fixed number $t\in (0,1)$, then $I-tG$ is a contraction with constant $1-(1-L_{\delta ,\lambda })t$.

Proof (i) For all $x,y\in C$, from (2.2), we have

$$\begin{array}{rcl}\lambda {\parallel (I-G)x-(I-G)y\parallel }^q& \le & {\parallel x-y\parallel }^q-〈Gx-Gy,j_q(x-y)〉\\ \le & (1-\delta ){\parallel x-y\parallel }^q.\end{array}$$

Observe that

$$\delta +\lambda >1⟺{\left(\frac{1-\delta }{\lambda }\right)}^{\frac{1}{q}}\in (0,1).$$

It follows that

$$\parallel (I-G)x-(I-G)y\parallel \le {\left(\frac{1-\delta }{\lambda }\right)}^{\frac{1}{q}}\parallel x-y\parallel :=L_{\delta ,\lambda }\parallel x-y\parallel .$$

Hence, $I-G$ is a contraction with constant $L_{\delta ,\lambda }$.

(ii) Since $I-G$ is a contraction with constant $L_{\delta ,\lambda }$. For all $t\in (0,1)$, we have

$$\begin{array}{rcl}\parallel (I-tG)x-(I-tG)y\parallel & =& \parallel (x-y)-t(Gx-Gy)\parallel \\ =& \parallel (1-t)(x-y)+t[(I-G)x-(I-G)y]\parallel \\ \le & (1-t)\parallel x-y\parallel +t\parallel (I-G)x-(I-G)y\parallel \\ \le & (1-(1-L_{\delta ,\lambda })t)\parallel x-y\parallel .\end{array}$$

Hence, $I-tG$ is a contraction with constant $1-(1-L_{\delta ,\lambda })t$. This completes the proof. □

3.1 Implicit iteration scheme

Let C be a nonempty, closed, and convex subset of a real reflexive and q-uniformly smooth Banach space X which admits a weakly sequentially continuous generalized duality mapping $j_q$. Let $Q_C$ be a sunny nonexpansive retraction from X onto C. Let $F:C\to X$ be a κ-Lipschitzian and η-strongly accretive operator with constants $\kappa ,\eta >0$, $G:C\to X$ be a δ-strongly accretive and λ-strictly pseudo-contractive mapping with $\delta +\lambda >1$, $V:C\to X$ be an L-Lipschitzian mapping with constant $L\ge 0$ and $T:C\to C$ be a nonexpansive mapping such that $Fix(T)\ne \mathrm{\varnothing }$. Let $0<\mu <{(\frac{q\eta }{C_q{\kappa }^q})}^{\frac{1}{q-1}}$ and $0\le \gamma L<\tau$, where $\tau =\mu (\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q})$. For each $\sigma \in (\frac{L_{\delta ,\lambda }}{\tau -\gamma L},min\{1,\frac{1}{q\tau },\frac{1+L_{\delta ,\lambda }}{\tau -\gamma L}\})$ and $t\in (0,1)$, we define a mapping $S_t:C\to C$ defined by

$$Sx:=Q_C[(I-tG)Tx+t(Tx-\sigma (\mu FTx-\gamma Vx))],\mathrm{\forall }x\in C.$$

It is easy to see immediately that $S_t$ is a contraction. Indeed, for all $x,y\in C$, from Lemmas 3.1 and 3.2(ii), we have

$$\begin{array}{rcl}\parallel S_{t}x-S_{t}y\parallel & =& \parallel Q_C[(I-tG)Tx+t(Tx-\sigma (\mu FTx-\gamma Vx))]\\ -Q_C[(I-tG)Ty+t(Ty-\sigma (\mu FTy-\gamma Vy))]\parallel \\ \le & \parallel (I-tG)(Tx-Ty)+t[(I-\sigma \mu F)(Tx-Ty)+\sigma \gamma (Vx-Vy)]\parallel \\ \le & (1-t(1-L_{\delta ,\lambda }))\parallel x-y\parallel +t[\sigma \gamma \parallel Vx-Vy\parallel +\parallel (I-\sigma \mu F)(Tx-Ty)\parallel ]\\ \le & (1-t(1-L_{\delta ,\lambda }))\parallel x-y\parallel +t(1-\sigma (\tau -\gamma L))\parallel x-y\parallel \\ =& [1-t(\sigma (\tau -\gamma L)-L_{\delta ,\lambda })]\parallel x-y\parallel \\ =& (1-t\theta )\parallel x-y\parallel ,\end{array}$$

(3.1)

where $\theta :=\sigma (\tau -\gamma L)-L_{\delta ,\lambda }$. Since $\tau -\gamma L>0$ and $L_{\delta ,\lambda }\in (0,1)$, observe that

$$\frac{L_{\delta ,\lambda }}{\tau -\gamma L}<\sigma <min\{1,\frac{1}{q\tau },\frac{1+L_{\delta ,\lambda }}{\tau -\gamma L}\}\le \frac{1+L_{\delta ,\lambda }}{\tau -\gamma L}.$$

It follows that

$$\sigma <\frac{1+L_{\delta ,\lambda }}{\tau -\gamma L}⟺\theta =\sigma (\tau -\gamma L)-L_{\delta ,\lambda }<1$$

and

$$\frac{L_{\delta ,\lambda }}{\tau -\gamma L}<\sigma ⟺\theta =\sigma (\tau -\gamma L)-L_{\delta ,\lambda }>0.$$

This implies that $\theta =\sigma (\tau -\gamma L)-L_{\delta ,\lambda }\in (0,1)$, which together with $t\in (0,1)$ gives

$$1-t(\sigma (\tau -\gamma L)-L_{\delta ,\lambda })\in (0,1).$$

Hence $S_t$ is a contraction. By the Banach contraction principle, $S_t$ has a unique fixed point, denote by $x_t$, which uniquely solves the fixed point equation

$$x_t=Q_C[(I-tG)T{x}_t+t(T{x}_t-\sigma (\mu FT{x}_t-\gamma V{x}_t))].$$

(3.2)

The following proposition summarizes the properties of the net $\{x_t\}$.

Proposition 3.3 Let $\{x_t\}$ be defined by (3.2). Then the following hold:

• (i) $\{x_t\}$ is bounded for each $t\in (0,1)$;

• (ii) ${lim}_{t\to 0}\parallel x_t-T{x}_t\parallel =0$;

• (iii) $\{x_t\}$ defines a continuous curve from $(0,1)$ into C.

Proof (i) Take $p\in Fix(T)$, and denote a mapping $S_t:C\to C$ by

$$S_{t}x:=Q_C[(I-tG)Tx+t(Tx-\sigma (\mu FTx-\gamma Vx))],\mathrm{\forall }x\in C.$$

From (3.1), we have

$$\begin{array}{rcl}\parallel x_t-p\parallel & \le & \parallel S_t{x}_t-S_{t}p\parallel +\parallel S_{t}p-p\parallel \\ \le & (1-t\theta )\parallel x_t-p\parallel +\parallel Q_C[(I-tG)Tp+t(Tp-\sigma (\mu FTp-\gamma Vp))]-Q_{C}p\parallel \\ \le & (1-t\theta )\parallel x_t-p\parallel +t\parallel -Gp+p-\sigma (\mu Fp-\gamma Vp)\parallel \\ \le & (1-t\theta )\parallel x_t-p\parallel +t[\parallel I-G\parallel \parallel p\parallel +\sigma \mu \parallel Fp\parallel +\sigma \gamma \parallel Vp\parallel ],\end{array}$$

where $\theta :=\sigma (\tau -\gamma L)-L_{\delta ,\lambda }$. It follows that

$$\parallel x_t-p\parallel \le \frac{\parallel I-G\parallel \parallel p\parallel +\sigma \mu \parallel Fp\parallel +\sigma \gamma \parallel Vp\parallel }{\theta }.$$

Hence $\{x_t\}$ is bounded, so are $\{V{x}_t\}$, $\{FT{x}_t\}$, and $\{GT{x}_t\}$.

• (ii) By definition of $\{x_t\}$, we have

  $$\begin{array}{rcl}\parallel x_t-T{x}_t\parallel & =& \parallel Q_C[(I-tG)T{x}_t+t(T{x}_t-\sigma (\mu FT{x}_t-\gamma V{x}_t))]-Q_{C}T{x}_t\parallel \\ \le & t\parallel (I-G)T{x}_t-\sigma (\mu FT{x}_t-\gamma V{x}_t)\parallel \to 0\text{as }t\to 0.\end{array}$$

• (iii) Take $t,t_0\in (0,1)$ and calculate

  $$\begin{array}{rcl}\parallel x_t-x_{t_0}\parallel & =& \parallel Q_C[(I-tG)T{x}_t+t(T{x}_t-\sigma (\mu FT{x}_t-\gamma V{x}_t))]\\ -Q_C[(I-t_{0}G)T{x}_{t_0}+t(T{x}_{t_0}-\sigma (\mu FT{x}_{t_0}-\gamma V{x}_{t_0}))]\parallel \\ \le & \parallel (t_0-t)GT{x}_t+(I-t_{0}G)(T{x}_t-T{x}_{t_0})+(t-t_0)[T{x}_t-\sigma (\mu FT{x}_t-\gamma V{x}_t)]\\ +t_0[T{x}_t-\sigma (\mu FT{x}_{t_0}-\gamma V{x}_{t_0})-[T{x}_{t_0}-\sigma (\mu FT{x}_{t_0}-\gamma V{x}_{t_0})]]\parallel \\ =& \parallel (t_0-t)GT{x}_t+(I-t_{0}G)(T{x}_t-T{x}_{t_0})+(t-t_0)[T{x}_t-\sigma (\mu FT{x}_t-\gamma V{x}_t)]\\ +t_0[\sigma \gamma (V{x}_t-V{x}_{t_0})+(I-\sigma \mu F)(T{x}_t-T{x}_{t_0})]\parallel \\ \le & |t-t_0|\parallel GT{x}_t\parallel +(1-(1-L_{\delta ,\lambda }))\parallel x_t-x_{t_0}\parallel \\ +|t-t_0|\parallel T{x}_t-\sigma (\mu FT{x}_t-\gamma V{x}_t)\parallel \\ +t_0(1-\sigma (\tau -\gamma L))\parallel x_t-x_{t_0}\parallel .\end{array}$$

It follows that

$$\parallel x_t-x_{t_0}\parallel \le \frac{\parallel GT{x}_t\parallel +\parallel T{x}_t-\sigma (\mu FT{x}_t-\gamma V{x}_t)\parallel }{t_0(\sigma (\tau -\gamma L)-L_{\delta ,\lambda })}|t-t_0|.$$

Since $\{V{x}_t\}$, $\{FT{x}_t\}$, and $\{GT{x}_t\}$ are bounded. Hence $\{x_t\}$ defines a continuous curve from $(0,1)$ into C. □

Theorem 3.4 Assume that $\{x_t\}$ is defined by (3.2), then $\{x_t\}$ converges strongly to $x^{\ast }\in Fix(T)$ as $t\to 0$, where $x^{\ast }$ is the unique solution of the variational inequality

$$〈(G-I+\sigma (\mu F-\gamma V))x^{\ast },j_q(x^{\ast }-v)〉\le 0,\mathrm{\forall }v\in Fix(T).$$

(3.3)

Proof We observe that

$$\begin{array}{rl}\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}>0& ⟺\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q}<\eta \\ ⟺\mu (\eta -\frac{C_q{\mu }^{q-1}{\kappa }^q}{q})<\mu \eta \\ ⟺\tau <\mu \eta .\end{array}$$

(3.4)

It follows that

$$0\le \gamma L<\tau <\mu \eta .$$

(3.5)

First, we show the uniqueness of solution of the variational inequality. Suppose that $\tilde{x},x^{\ast }\in Fix(T)$ are solutions of (3.3), then

$$〈(G-I+\sigma (\mu F-\gamma V))x^{\ast },j_q(x^{\ast }-\tilde{x})〉\le 0$$

(3.6)

and

$$〈(G-I+\sigma (\mu F-\gamma V))\tilde{x},j_q(\tilde{x}-x^{\ast })〉\le 0.$$

(3.7)

Adding up (3.6) and (3.7), and from Lemma 3.2(i), we obtain

$$\begin{array}{rcl}0& \ge & 〈(G-I+\sigma (\mu F-\gamma V))x^{\ast }-(G-I+\sigma (\mu F-\gamma V))\tilde{x},j_q(x^{\ast }-\tilde{x})〉\\ =& 〈(G-I)x^{\ast }-(G-I)\tilde{x},j_q(x^{\ast }-\tilde{x})〉+\sigma 〈(\mu F-\gamma V)x^{\ast }-(\mu F-\gamma V)\tilde{x},j_q(x^{\ast }-\tilde{x})〉\\ =& -〈(I-G)x^{\ast }-(I-G)\tilde{x},j_q(x^{\ast }-\tilde{x})〉+\sigma \mu 〈F{x}^{\ast }-F\tilde{x},j_q(x^{\ast }-\tilde{x})〉\\ -\sigma \gamma 〈V{x}^{\ast }-V\tilde{x},j_q(x^{\ast }-\tilde{x})〉\\ \ge & -L_{\delta ,\lambda }{\parallel x^{\ast }-\tilde{x}\parallel }^q+\sigma \mu \eta {\parallel x^{\ast }-\tilde{x}\parallel }^q-\sigma \gamma \parallel V{x}^{\ast }-V\tilde{x}\parallel {\parallel x^{\ast }-\tilde{x}\parallel }^{q-1}\\ \ge & (\sigma (\mu \eta -\gamma L)-L_{\delta ,\lambda }){\parallel x^{\ast }-\tilde{x}\parallel }^q.\end{array}$$

On the other hand, we observe from (3.5) that

$$\begin{array}{rl}\frac{L_{\delta ,\lambda }}{\tau -\gamma L}<\sigma & ⟺L_{\delta ,\lambda }<\sigma (\tau -\gamma L)\\ ⟺L_{\delta ,\lambda }<\sigma (\mu \eta -\gamma L)\\ ⟺0<\sigma (\mu \eta -\gamma L)-L_{\delta ,\lambda }.\end{array}$$

(3.8)

Note that (3.8) implies that $x^{\ast }=\tilde{x}$ and the uniqueness is proved. Below, we use $\tilde{x}$ to denote the unique solution of the variational inequality (3.3).

Next, we show that $x_t\to x^{\ast }$ as $t\to 0$. Set $x_t=Q_C{y}_t$, where $y_t=(I-tG)T{x}_t+t(T{x}_t-\sigma (\mu FT{x}_t-\gamma V{x}_t))$. Assume that $\{t_n\}\subset (0,1)$ is a sequence such that $t_n\to 0$ as $n\to \mathrm{\infty }$. Put $x_n:=x_{t_n}$ and $y_n:=y_{t_n}$. For $z\in Fix(T)$, we note that

$$\begin{array}{rcl}{x}_n-z& =& Q_C{y}_n-y_n+y_n-z\\ =& Q_C{y}_n-y_n+(I-t_{n}G)(T{x}_n-z)+t_n(T{x}_n-\sigma (\mu FT{x}_n-\gamma V{x}_n)-Gz)\\ =& Q_C{y}_n-y_n+(I-t_{n}G)(T{x}_n-z)+t_n[(I-\sigma \mu F)T{x}_n+\sigma \gamma V{x}_n-Gz]\\ =& Q_C{y}_n-y_n+(I-t_{n}G)(T{x}_n-z)+t_n[(I-\sigma \mu F)(T{x}_n-z)+\sigma \gamma (V{x}_n-Vz)]\\ +t_n[(I-\sigma \mu F)z+\sigma \gamma Vz-Gz].\end{array}$$

(3.9)

By Lemma 2.4, we have

$$〈Q_C{y}_n-y_n,j_q(Q_C{y}_n-z)〉\le 0.$$

(3.10)

It follows from (3.9) and (3.10) that

$$\begin{array}{rcl}{\parallel x_n-z\parallel }^q& =& 〈Q_C{y}_n-y_n,j_q(Q_C{y}_n-z)〉+〈y_n-z,j_q(x_n-z)〉\\ \le & 〈(I-t_{n}G)(T{x}_n-z),j_q(x_n-z)〉+t_n〈(I-\sigma \mu F)(T{x}_n-z),j_q(x_n-z)〉\\ +t_n\sigma \gamma 〈V{x}_n-Vz,j_q(x_n-z)〉+t_n〈(I-\sigma \mu F)z+\sigma \gamma Vz-Gz,j_q(x_n-z)〉\\ \le & [1-t_n(\sigma (\tau -\gamma L)-L_{\delta ,\lambda })]{\parallel x_n-z\parallel }^q\\ +t_n〈(I-\sigma \mu F)z+\sigma \gamma Vz-Gz,j_q(x_n-z)〉,\end{array}$$

which implies that

$${\parallel x_n-z\parallel }^q\le \frac{1}{\sigma (\tau -\gamma L)-L_{\delta ,\lambda }}〈(I-\sigma \mu F)z+\sigma \gamma Vz-Gz,j_q(x_n-z)〉.$$

In particular, we have

$${\parallel x_{n_i}-z\parallel }^q\le \frac{1}{\sigma (\tau -\gamma L)-L_{\delta ,\lambda }}〈(I-\sigma \mu F)z+\sigma \gamma Vz-Gz,j_q(x_{n_i}-z)〉.$$

(3.11)

By reflexivity of a Banach space X and boundedness of $\{x_n\}$, there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $x_{n_i}\rightharpoonup \tilde{x}$ as $i\to \mathrm{\infty }$. Since a Banach space X has a weakly sequentially continuous generalized duality mapping and by (3.11), we obtain $x_{n_i}\to \tilde{x}$. By Proposition 3.3(ii), we have $x_{n_i}-T{x}_{n_i}\to 0$ as $i\to \mathrm{\infty }$. Hence, it follows from Lemma 2.12 that $\tilde{x}\in Fix(T)$.

Next, we show that $\tilde{x}$ solves the variational inequality (3.3). We note that

$$x_t=Q_C{y}_t=Q_C{y}_t-y_t+(I-tG)Tx+t(T{x}_t-\sigma (\mu FT{x}_t-\gamma V{x}_t)),$$

we derive

$$(G-I+\sigma (\mu F-\gamma V))x_t=\frac{1}{t}(Q_C{y}_t-y_t)-\frac{1}{t}((I-tG)(I-T)x_t+t(I-\sigma \mu F)(I-T)x_t).$$

(3.12)

Since $I-T$ is accretive (i.e., $〈(I-T)x-(I-T)y,j_q(x-y)〉\ge 0$ for $x,y\in C$). For all $v\in Fix(T)$, it follows from (3.10) and (3.12) that

$$\begin{array}{c}〈(G-I+\sigma (\mu F-\gamma V))x_t,j_q(x_t-v)〉\hfill \\ =\frac{1}{t}〈Q_C{y}_t-y_t,j_q(Q_C{y}_t-v)〉-\frac{1}{t}〈(I-tG)(I-T)x_t,j_q(x_t-v)〉\hfill \\ -〈(I-\sigma \mu F)(I-T)x_t,j_q(x_t-v)〉\hfill \\ \le -\frac{1}{t}〈(I-T)x_t-(I-T)v,j_q(x_t-v)〉+〈G(I-T)x_t,j_q(x_t-v)〉\hfill \\ -〈(I-T)x_t-(I-T)v,j_q(x_t-v)〉+\sigma \mu 〈F(I-T)x_t,j_q(x_t-v)〉\hfill \\ \le 〈G(I-T)x_t,j_q(x_t-v)〉+\sigma \mu 〈F(I-T)x_t,j_q(x_t-v)〉\hfill \\ \le \parallel G\parallel \parallel x_t-T{x}_t\parallel {\parallel x_t-v\parallel }^{q-1}+\sigma \mu \parallel F\parallel \parallel x_t-T{x}_t\parallel {\parallel x_t-v\parallel }^{q-1}\hfill \\ \le \parallel x_t-T{x}_t\parallel M_1,\hfill \end{array}$$

(3.13)

where $M_1>0$ is an appropriate constant such that $M_1={sup}_{t\in (0,1)}\{\parallel G\parallel {\parallel x_t-v\parallel }^{q-1},\sigma \mu \parallel F\parallel {\parallel x_t-v\parallel }^{q-1}\}$. Now, replacing t in (3.13) with $t_n$ and taking the limit as $n\to \mathrm{\infty }$, we notice that $x_{t_n}-T{x}_{t_n}\to \tilde{x}-T\tilde{x}=0$, we obtain

$$〈(G-I+\sigma (\mu F-\gamma V))\tilde{x},j_q(\tilde{x}-v)〉\le 0.$$

That is, $\tilde{x}\in Fix(T)$ is the solution of the variational inequality (3.3). Consequently, $\tilde{x}=x^{\ast }$ by uniqueness. In a summary, we have shown that each cluster point of $\{x_t\}$ is equal to $x^{\ast }$. Therefore $x_t\to x^{\ast }$ as $t\to 0$. This completes the proof. □

3.2 Explicit iteration scheme

Theorem 3.5 Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth and uniformly convex Banach space X which admits a weakly sequentially continuous generalized duality mapping $j_q$. Let $Q_C$ be a sunny nonexpansive retraction such that X onto C. Let $F:C\to X$ be a κ-Lipschitzian and η-strongly accretive operator with constants $\kappa ,\eta >0$, $G:C\to X$ be a δ-strongly accretive and λ-strictly pseudo-contractive mapping with $\delta +\lambda >1$, $V:C\to X$ be an L-Lipschitzian mapping with constant $L\ge 0$. Let ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$ be an infinite family of ${\lambda }_i$-strictly pseudo-contractive mapping from C into itself such that $\mathcal{F}:={\bigcap }_{i=1}^{\mathrm{\infty }}Fix(S_i)\ne \mathrm{\varnothing }$. For given $x_1\in C$, define the sequence $\{x_n\}$ by

$$x_{n+1}=Q_C[(I-{\alpha }_{n}G)W_n{x}_n+{\alpha }_n(W_n{x}_n-\sigma (\mu F{W}_n{x}_n-\gamma V{x}_n))],\mathrm{\forall }n\ge 1,$$

(3.14)

where $\{{\alpha }_n\}$ is a sequence in $(0,1)$ which satisfies the following conditions:

(C1) ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

(C2) $|{\alpha }_{n+1}-{\alpha }_n|\le \circ ({\alpha }_n)+{\sigma }_n$ with ${\sum }_{n=1}^{\mathrm{\infty }}{\sigma }_n<\mathrm{\infty }$.

Suppose in addition that $\{{\theta }_{n,k}\}$ satisfies (H1)-(H3). Then the sequence $\{x_n\}$ defined by (3.14) converges strongly to $x^{\ast }\in \mathcal{F}$ as $n\to \mathrm{\infty }$, where $x^{\ast }$ is the unique solution of the variational inequality

$$〈(G-I+\sigma (\mu F-\gamma V))x^{\ast },j_q(x^{\ast }-v)〉\le 0,\mathrm{\forall }v\in \mathcal{F}.$$

(3.15)

Proof From the condition (C1), we may assume, without loss of generality, that ${\alpha }_n\le min\{1,\frac{1}{q\tau }\}$ for all $n\in \mathbb{N}$. First, we show that $\{x_n\}$ is bounded. Take $p\in \mathcal{F}$, and denote a mapping $S_n^{{\alpha }_n}:C\to C$ by

$$S_n^{{\alpha }_n}x:=Q_C[(I-{\alpha }_{n}G)W_{n}x+{\alpha }_n(W_{n}x-\sigma (\mu F{W}_{n}x-\gamma Vx))],\mathrm{\forall }x\in C.$$

Then we have

$$S_n^{{\alpha }_n}p=Q_C[(I-{\alpha }_{n}G)W_{n}p+{\alpha }_n(W_{n}p-\sigma (\mu F{W}_{n}p-\gamma Vp))].$$

From (3.1), we have

$$\begin{array}{rcl}\parallel x_{n+1}-p\parallel & \le & \parallel S_n^{{\alpha }_n}{x}_n-S_n^{{\alpha }_n}p\parallel +\parallel S_n^{{\alpha }_n}p-p\parallel \\ \le & (1-{\alpha }_n\theta )\parallel x_n-p\parallel +\parallel Q_C[(I-{\alpha }_{n}G)p+{\alpha }_n(p-\sigma (\mu Fp-\gamma Vp))]-Q_{C}p\parallel \\ \le & (1-{\alpha }_n\theta )\parallel x_n-p\parallel +{\alpha }_n\parallel -Gp+p-\sigma (\mu Fp-\gamma Vp)\parallel \\ \le & (1-{\alpha }_n\theta )\parallel x_n-p\parallel +{\alpha }_n(\parallel I-G\parallel \parallel p\parallel +\sigma \parallel \mu Fp-\gamma Vp\parallel )\\ \le & (1-{\alpha }_n\theta )\parallel x_n-p\parallel +{\alpha }_n\theta \frac{\parallel I-G\parallel \parallel p\parallel +\sigma \mu \parallel Fp\parallel +\gamma \parallel Vp\parallel }{\theta }\\ \le & max\{\parallel x_n-p\parallel ,\frac{\parallel I-G\parallel \parallel p\parallel +\sigma \mu \parallel Fp\parallel +\gamma \parallel Vp\parallel }{\theta }\},\end{array}$$

where $\theta :=\sigma (\tau -\gamma L)-L_{\delta ,\lambda }$. By induction, we obtain

$$\parallel x_n-p\parallel \le max\{\parallel x_1-p\parallel ,\frac{\parallel I-G\parallel \parallel p\parallel +\sigma \mu \parallel Fp\parallel +\gamma \parallel Vp\parallel }{\theta }\},\mathrm{\forall }n\ge 1.$$

Hence, $\{x_n\}$ is bounded, so are $\{V{x}_n\}$, $\{F{W}_n{x}_n\}$, and $\{G{W}_n{x}_n\}$.

Next, we show that $\parallel x_{n+1}-x_n\parallel \to 0$ as $n\to \mathrm{\infty }$. Set $S_n^{{\alpha }_n}{x}_n=Q_C{y}_n$, where $y_n=(I-{\alpha }_{n}G)W_n{x}_n+{\alpha }_n(W_n{x}_n-\sigma (\mu F{W}_n{x}_n-\gamma V{x}_n))$. From (2.5), we have

$$\begin{array}{c}\parallel W_{n+1}{x}_n-W_n{x}_n\parallel \hfill \\ =\parallel t_1{T}_{n+1,1}{U}_{n+1,2}{x}_n+(1-t_1)x_n-t_1{T}_{n,1}{U}_{n,2}-(1-t_1)x_n\parallel \hfill \\ =t_1\parallel T_{n+1,1}{U}_{n+1,2}{x}_n-T_{n,1}{U}_{n,2}{x}_n\parallel \hfill \\ =t_1\parallel ({\theta }_{n+1,1}{S}_1+(1-{\theta }_{n+1,1}))U_{n+1,2}{x}_n-T_{n,1}{U}_{n,2}{x}_n\parallel \hfill \\ =t_1\parallel ({\theta }_{n,1}{S}_1+(1-{\theta }_{n,1}))U_{n+1,2}{x}_n-T_{n,1}{U}_{n,2}{x}_n+({\theta }_{n+1,1}-{\theta }_{n,1})(S_1{U}_{n+1,2}{x}_n-U_{n+1,2}{x}_n)\parallel \hfill \\ \le t_1\parallel T_{n,1}{U}_{n+1,2}{x}_n-T_{n,1}{U}_{n,2}{x}_n\parallel +t_1|{\theta }_{n+1,2}-{\theta }_{n,1}|\parallel S_1{U}_{n+1,2}{x}_n-U_{n+1,2}{x}_n\parallel \hfill \\ \le t_1\parallel T_{n,1}{U}_{n+1,2}{x}_n-T_{n,1}{U}_{n,2}{x}_n\parallel +t_1|{\theta }_{n+1,2}-{\theta }_{n,1}|M^{\ast }\hfill \\ \le t_1\parallel T_{n,1}{U}_{n+1,2}{x}_n-T_{n,1}{U}_{n,2}{x}_n\parallel +t_1{a}_n{M}_1\hfill \\ ⋮\hfill \\ \le \prod _{i=1}^n{t}_i\parallel U_{n+1,n+1}{x}_n-U_{n,n+1}{x}_n\parallel +\left(a_n\sum _{j=1}^n\sum _{i=1}^j{t}_i\right)M_1\hfill \\ \le \prod _{i=1}^n{t}_i\parallel t_{n+1}{T}_{n+1,n+1}{x}_n+(1-t_{n+1})x_n-x_n\parallel +\frac{b}{1-b}{a}_n{M}_1\hfill \\ \le \prod _{i=1}^{n+1}{t}_i\parallel T_{n+1,n+1}{x}_n-x_n\parallel +\frac{b}{1-b}{a}_n{M}_1\hfill \\ \le (b^{n+1}+\frac{b}{1-b}{a}_n)M_1,\hfill \end{array}$$

(3.16)

where $M_1={inf}_{i=1,2,\dots }(\frac{1+2{\lambda }_i^{q-1}}{{\lambda }_i^{q-1}}){sup}_{n\ge 1}\{\parallel x_n-p\parallel \}$ with $p\in \mathcal{F}$.

On the other hand, we note that

$$\begin{array}{rcl}{y}_{n+1}-y_n& =& (I-{\alpha }_{n+1}G)W_{n+1}{x}_n+{\alpha }_{n+1}[W_{n+1}{x}_n-\sigma (\mu F{W}_{n+1}{x}_n-\gamma V{x}_n)]\\ -(I-{\alpha }_{n}G)W_n{x}_n-{\alpha }_n[W_n{x}_n-\sigma (\mu F{W}_n{x}_n-\gamma V{x}_n)]\\ =& (I-{\alpha }_{n+1}G)(W_{n+1}{x}_n-W_n{x}_n)+({\alpha }_n-{\alpha }_{n+1})G{W}_n{x}_n+({\alpha }_{n+1}-{\alpha }_n)W_{n+1}{x}_n\\ +{\alpha }_n(W_{n+1}{x}_n-W_n{x}_n)+\sigma ({\alpha }_n-{\alpha }_{n+1})(\mu F{W}_{n+1}{x}_n-\gamma V{x}_n)\\ -\sigma {\alpha }_n[\mu F{W}_{n+1}{x}_n-\gamma V{x}_n-(\mu F{W}_n{x}_n-\gamma V{x}_n)]\\ =& [(1+{\alpha }_n)I-{\alpha }_{n+1}G](W_{n+1}{x}_n-W_n{x}_n)+({\alpha }_{n+1}-{\alpha }_n)[W_{n+1}{x}_n-G{W}_n{x}_n]\\ +\sigma ({\alpha }_n-{\alpha }_{n+1})[\mu F{W}_{n+1}{x}_n-\gamma V{x}_n]-\sigma {\alpha }_n\mu F[W_{n+1}{x}_n-W_n{x}_n]\\ =& [(1+{\alpha }_n)I-{\alpha }_{n+1}G-\sigma {\alpha }_n\mu F](W_{n+1}{x}_n-W_n{x}_n)\\ +({\alpha }_{n+1}-{\alpha }_n)[W_{n+1}{x}_n-G{W}_n{x}_n]+\sigma ({\alpha }_n-{\alpha }_{n+1})[\mu F{W}_{n+1}{x}_n-\gamma V{x}_n].\end{array}$$

Hence, we have

$$\begin{array}{rcl}\parallel S_{n+1}^{{\alpha }_{n+1}}{x}_n-S_n^{{\alpha }_n}{x}_n\parallel & =& \parallel Q_C{y}_{n+1}-Q_C{y}_n\parallel \\ \le & \parallel y_{n+1}-y_n\parallel \\ \le & \parallel (1+{\alpha }_n)I-{\alpha }_{n+1}G-\sigma {\alpha }_n\mu F\parallel \parallel W_{n+1}{x}_n-W_n{x}_n\parallel \\ +|{\alpha }_{n+1}-{\alpha }_n|\parallel W_{n+1}{x}_n-G{W}_n{x}_n\parallel \\ +\sigma |{\alpha }_{n+1}-{\alpha }_n|\parallel \mu F{W}_{n+1}{x}_n-\gamma V{x}_n\parallel \\ \le & (\parallel W_{n+1}{x}_n-W_n{x}_n\parallel +|{\alpha }_{n+1}-{\alpha }_n|)M_2,\end{array}$$

where $M_2={sup}_{n\ge 1}\{\parallel (1+{\alpha }_n)I-{\alpha }_{n+1}G-\sigma {\alpha }_n\mu F\parallel ,\parallel W_{n+1}{x}_n-G{W}_n{x}_n\parallel ,\sigma \parallel \mu F{W}_{n+1}{x}_n-\gamma V{x}_n\parallel \}$. It follows from (3.1) and (3.16) that

$$\begin{array}{rcl}\parallel x_{n+2}-x_{n+1}\parallel & \le & \parallel S_{n+1}^{{\alpha }_{n+1}}{x}_{n+1}-S_{n+1}^{{\alpha }_{n+1}}{x}_n\parallel +\parallel S_{n+1}^{{\alpha }_{n+1}}{x}_n-S_n^{{\alpha }_n}{x}_n\parallel \\ \le & (1-{\alpha }_{n+1}\theta )\parallel x_{n+1}-x_n\parallel +(|{\alpha }_{n+1}-{\alpha }_n|+\parallel W_{n+1}{x}_n-W_n{x}_n\parallel )M_2\\ \le & (1-{\alpha }_{n+1}\theta )\parallel x_{n+1}-x_n\parallel +(\circ ({\alpha }_n)+{\sigma }_n)M_2+\parallel W_{n+1}{x}_n-W_n{x}_n\parallel M_2\\ \le & (1-{\alpha }_{n+1}\theta )\parallel x_{n+1}-x_n\parallel +\circ ({\alpha }_n)M_2+({\sigma }_n+b^{n+1}+\frac{b}{1-b}{a}_n)M,\end{array}$$

(3.17)

where $M=max\{M_1,M_2\}$. Then, by Lemma 2.13, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{n+1}-x_n\parallel =0.$$

(3.18)

Next, we show that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-W{x}_n\parallel =0$. Since

$$\begin{array}{rcl}\parallel x_n-W_n{x}_n\parallel & \le & \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-W_n{x}_n\parallel \\ =& \parallel x_n-x_{n+1}\parallel +\parallel Q_C[(I-{\alpha }_{n}G)W_n{x}_n+{\alpha }_n(W_n{x}_n-\sigma (\mu F{W}_n{x}_n-\gamma V{x}_n))]\\ -Q_C{W}_n{x}_n\parallel \\ \le & \parallel x_n-x_{n+1}\parallel +{\alpha }_n\parallel (I-G)W_n{x}_n-\sigma (\mu F{W}_n{x}_n-\gamma V{x}_n)\parallel .\end{array}$$

From (3.18) and the condition (C1), we obtain

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-W_n{x}_n\parallel =0.$$

(3.19)

At the same time, observe that

$$\parallel x_n-W{x}_n\parallel \le \parallel x_n-W_n{x}_n\parallel +\parallel W_n{x}_n-W{x}_n\parallel .$$

It follows from (3.19) and Lemma 2.11, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-W{x}_n\parallel =0.$$

(3.20)

Next, we show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈[I-G+\sigma (\gamma V-\mu F)]x^{\ast },j_q(x_n-x^{\ast })〉\le 0,$$

where $x^{\ast }$ is the same as in Theorem 3.4. Since $\{x_n\}$ is bounded, there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈[I-G+\sigma (\gamma V-\mu F)]x^{\ast },j_q(x_n-x^{\ast })〉\hfill \\ =\underset{i\to \mathrm{\infty }}{lim}〈[I-G+\sigma (\gamma V-\mu F)]x^{\ast },j_q(x_{n_i}-x^{\ast })〉.\hfill \end{array}$$

By reflexivity of a Banach space X and boundedness of $\{x_n\}$, without loss of generality, we may assume that $x_{n_i}\rightharpoonup v$ as $i\to \mathrm{\infty }$. It follows from (3.20) and Lemma 2.12 that $v\in \mathcal{F}$. Since a Banach space X has a weakly sequentially continuous generalized duality mapping, we obtain

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈[I-G+\sigma (\gamma V-\mu F)]x^{\ast },j_q(x_n-x^{\ast })〉\hfill \\ =\underset{i\to \mathrm{\infty }}{lim}〈[I-G+\sigma (\gamma V-\mu F)]x^{\ast },j_q(x_{n_i}-x^{\ast })〉\hfill \\ =〈[I-G+\sigma (\gamma V-\mu F)]x^{\ast }{j}_q(v-x^{\ast })〉\le 0.\hfill \end{array}$$

(3.21)

Finally, we show that $x_n\to x^{\ast }$ as $n\to \mathrm{\infty }$. Set $x_{n+1}=Q_C{y}_n$, where $y_n=(I-{\alpha }_{n}G)W_n{x}_n+{\alpha }_n(W_n{x}_n-\sigma (\mu W_n{x}_n-\gamma V{x}_n))$. From Lemmas 2.4 and 2.5, we have

$$\begin{array}{c}{\parallel x_{n+1}-x^{\ast }\parallel }^q\hfill \\ =〈y_n-x^{\ast },j_q(x_{n+1}-x^{\ast })〉+〈Q_C{y}_n-y_n,j_q(x_{n+1}-x^{\ast })〉\hfill \\ \le 〈y_n-x^{\ast },j_q(x_{n+1}-x^{\ast })〉\hfill \\ =〈(I-{\alpha }_{n}G)(W_n{x}_n-x^{\ast }),j_q(x_{n+1}-x^{\ast })〉+{\alpha }_n〈(I-\sigma \mu F)(W_n{x}_n-x^{\ast }),j_q(x_{n+1}-x^{\ast })〉\hfill \\ +{\alpha }_n\sigma \gamma 〈V{x}_n-V{x}^{\ast },j_q(x_{n+1}-x^{\ast })〉+{\alpha }_n〈(I-\sigma \mu F)x^{\ast }+\sigma \gamma V{x}^{\ast }-G{x}^{\ast },j_q(x_{n+1}-x^{\ast })〉\hfill \\ \le (1-{\alpha }_n(1-L_{\delta ,\lambda }))\parallel x_n-x^{\ast }\parallel {\parallel x_{n+1}-x^{\ast }\parallel }^{q-1}+{\alpha }_n(1-\sigma \tau )\parallel x_n-x^{\ast }\parallel {\parallel x_{n+1}-x^{\ast }\parallel }^{q-1}\hfill \\ +{\alpha }_n\sigma \gamma L\parallel x_n-x^{\ast }\parallel {\parallel x_{n+1}-x^{\ast }\parallel }^{q-1}+{\alpha }_n〈x^{\ast }-G{x}^{\ast }+\sigma (\gamma V{x}^{\ast }-\mu F{x}^{\ast }),j_q(x_{n+1}-x^{\ast })〉\hfill \\ =(1-{\alpha }_n(\sigma (\tau -\gamma L)-L_{\delta ,\lambda }))\parallel x_n-x^{\ast }\parallel {\parallel x_{n+1}-x^{\ast }\parallel }^{q-1}\hfill \\ +{\alpha }_n〈x^{\ast }-G{x}^{\ast }+\sigma (\gamma V{x}^{\ast }-\mu F{x}^{\ast }),j_q(x_{n+1}-x^{\ast })〉\hfill \\ \le (1-{\alpha }_n(\sigma (\tau -\gamma L)-L_{\delta ,\lambda }))[\frac{1}{q}{\parallel x_n-x^{\ast }\parallel }^q+\left(\frac{q-1}{q}\right){\parallel x_{n+1}-x^{\ast }\parallel }^q]\hfill \\ +{\alpha }_n〈x^{\ast }-G{x}^{\ast }+\sigma (\gamma V{x}^{\ast }-\mu F{x}^{\ast }),j_q(x_{n+1}-x^{\ast })〉,\hfill \end{array}$$

which implies that

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^q& \le & (1-{\alpha }_n(\sigma (\tau -\gamma L)-L_{\delta ,\lambda })){\parallel x_n-x^{\ast }\parallel }^q\\ +\frac{q{\alpha }_n}{1+(q-1)(\sigma (\tau -\gamma L)-L_{\delta ,\lambda })}\\ \times 〈x^{\ast }-G{x}^{\ast }+\sigma (\gamma V{x}^{\ast }-\mu F{x}^{\ast }),j_q(x_{n+1}-x^{\ast })〉.\end{array}$$

(3.22)

We can write (3.22) to the formula

$${\parallel x_{n+1}-x^{\ast }\parallel }^q\le (1-{\tau }_n){\parallel x_n-x\parallel }^q+{\xi }_n,$$

(3.23)

where ${\tau }_n:=(\sigma (\tau -\gamma L)-L_{\delta ,\lambda }){\alpha }_n$ and ${\xi }_n:=\frac{q{\alpha }_n}{1+(q-1)(\sigma (\tau -\gamma L)-L_{\delta ,\lambda })}〈x^{\ast }-G{x}^{\ast }+\sigma (\gamma V{x}^{\ast }-\mu F{x}^{\ast }),j_q(x_{n+1}-x^{\ast })〉$. Put $c_n=max\{0,{\xi }_n\}$, from (3.21), we have $c_n\to 0$ as $n\to \mathrm{\infty }$. Then we can rewrite (3.23) as

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^q& \le & (1-{\tau }_n){\parallel x_n-x^{\ast }\parallel }^q+c_n\\ \le & (1-{\tau }_n){\parallel x_n-x^{\ast }\parallel }^q+\circ ({\alpha }_n).\end{array}$$

Therefore, by Lemma 2.13, we conclude that $x_n\to x^{\ast }$ as $n\to \mathrm{\infty }$. This completes the proof. □

4 Some applications

In this section, we will utilize Theorems 3.4 and 3.5 to study some strong convergence theorems in $L_p$ (or ${\ell }_p$) spaces with $1<p<\mathrm{\infty }$. It well known that Hilbert spaces, $L_p$ (or ${\ell }_p$) spaces with $1<p<\mathrm{\infty }$ and the Sobolev spaces $W_m^p$ with $1<p<\mathrm{\infty }$ are q-uniformly smooth, i.e.,

$$L_p(\text{or }{\ell }_p)\text{ or }{W}_m^p\text{ is }\{\begin{array}{cc}2\text{-uniformly smooth},\hfill & \text{if }2\le p<\mathrm{\infty },\hfill \\ p\text{-uniformly smooth},\hfill & \text{if }1<p\le 2.\hfill \end{array}$$

Furthermore, we have the following properties of $L_p$ (or ${\ell }_p$) spaces with $1<p<\mathrm{\infty }$ (see [36, 39]):

• (1) For $2\le p<\mathrm{\infty }$, the spaces $L_p$ (or ${\ell }_p$) are 2-uniformly smooth with $C_q=C_2=p-1$.

• (2) For $1<p\le 2$, the spaces $L_p$ (or ${\ell }_p$) are p-uniformly smooth with $C_q=C_p=(1+t_p^{p-1}){(1+t_p)}^{1-p}$, where $t_p$ is the unique solution of the equation

  $$(p-2)t^{p-1}+(p-1)t^{p-2}-1=0,0<t<1.$$

• (3) Every Hilbert spaces are 2-uniformly smooth with $C_q=C_2=1$.

• (4) Every $L_p$ (or ${\ell }_p$) spaces with $1<p<\mathrm{\infty }$ are q-uniformly smooth and uniformly convex.

• (5) Every ${\ell }_p$ spaces with $1<p<\mathrm{\infty }$ have weakly sequentially continuous generalized duality mappings, but $L_p$ spaces ($1<p<\mathrm{\infty }$, $p\ne 2$) do not have weakly sequentially continuous generalized duality mappings.

Lemma 4.1 Let $X:=L_p$ (or ${\ell }_p$) with $1<p\le 2$. Let C be a nonempty, closed, and convex subset of X. Let $F:C\to X$ be a κ-Lipschitzian and η-strongly accretive operator with constants $\kappa ,\eta >0$. Let $0<\mu <{(\frac{p\eta }{D_p{\kappa }^p})}^{\frac{1}{p-1}}$ and $\tau =\mu (\eta -\frac{D_p{\mu }^{p-1}{\kappa }^p}{p})$. Then for $t\in (0,min\{1,\frac{1}{p\tau }\})$, the mapping $S:C\to X$ defined by $S:=I-t\mu F$ is a contraction with constant $1-t\tau$.

Lemma 4.2 Let $X:=L_p$ (or ${\ell }_p$) with $2\le p<\mathrm{\infty }$. Let C be a nonempty, closed, and convex subset of X. Let $F:C\to X$ be a κ-Lipschitzian and η-strongly accretive operator with constants $\kappa ,\eta >0$. Let $0<\mu <\frac{2\eta }{(p-1){\kappa }^2}$ and $\tau =\mu (\eta -\frac{(p-1)\mu {\kappa }^2}{2})$. Then for $t\in (0,min\{1,\frac{1}{2\tau }\})$, the mapping $S:C\to X$ defined by $S:=I-t\mu F$ is a contraction with constant $1-t\tau$.

Lemma 4.3 Let $X:=H$ be a real Hilbert space. Let C be a nonempty, closed, and convex subset of X. Let $F:C\to X$ be a κ-Lipschitzian and η-strongly accretive operator with constants $\kappa ,\eta >0$. Let $0<\mu <\frac{2\eta }{{\kappa }^2}$ and $\tau =\mu (\eta -\frac{\mu {\kappa }^2}{2})$. Then for $t\in (0,min\{1,\frac{1}{2\tau }\})$, the mapping $S:C\to X$ defined by $S:=I-t\mu F$ is a contraction with constant $1-t\tau$.

4.1 Implicit iteration schemes

Theorem 4.4 Let C be a nonempty, closed, and convex subset of an ${\ell }_p$ space for $1<p\le 2$. Let $Q_C$, F, G, V, and T be the same as in Theorem  3.4. Assume that $0<\mu <{(\frac{p\eta }{D_p{\kappa }^p})}^{\frac{1}{p-1}}$ and $0\le \gamma L<\tau$, where $\tau =\mu (\eta -\frac{D_p{\mu }^{p-1}{\kappa }^p}{p})$. For $\sigma \in (\frac{L_{\delta ,\lambda }}{\tau -\gamma L},min\{1,\frac{1}{p\tau },\frac{1+L_{\delta ,\lambda }}{\tau -\gamma L}\})$ and $t\in (0,1)$, the sequence $\{x_t\}$ defined by (3.2) converges strongly to $x^{\ast }\in Fix(T)$ as $t\to 0$, where $x^{\ast }$ is the unique solution of the variational inequality (3.3).

Theorem 4.5 Let C be a nonempty, closed, and convex subset of an ${\ell }_p$ space for $2\le p<\mathrm{\infty }$. Let $Q_C$, F, G, V, and T be the same as in Theorem  3.4. Assume that $0<\mu <\frac{2\eta }{(p-1){\kappa }^2}$ and $0\le \gamma L<\tau$, where $\tau =\mu (\eta -\frac{(p-1)\mu {\kappa }^2}{2})$. For $\sigma \in (\frac{L_{\delta ,\lambda }}{\tau -\gamma L},min\{1,\frac{1}{2\tau },\frac{1+L_{\delta ,\lambda }}{\tau -\gamma L}\})$ and $t\in (0,1)$, the sequence $\{x_t\}$ defined by (3.2) converges strongly to $x^{\ast }\in Fix(T)$ as $t\to 0$, where $x^{\ast }$ is the unique solution of the variational inequality (3.3).

Remark 4.6 If the spaces $L_p$ has a weakly sequentially continuous generalized duality mappings, then we obtain Theorems 4.4 and 4.5 hold for $L_p$ spaces with $1<p<\mathrm{\infty }$, $p\ne 2$.

4.2 Explicit iteration schemes

Theorem 4.7 Let C be a nonempty, closed, and convex subset of an ${\ell }_p$ space for $1<p\le 2$. Let $Q_C$, F, G, V, and $W_n$ be the same as in Theorem  3.5. Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $(0,1)$ which satisfy the conditions (C1) and (C2) in Theorem  3.5 and $\{{\theta }_{n,k}\}$ satisfies (H1)-(H3). Then the sequence $\{x_n\}$ defined by (3.14) converges strongly to $x^{\ast }\in \mathcal{F}$ as $n\to \mathrm{\infty }$, where $x^{\ast }$ is the unique solution of the variational inequality (3.15).

Theorem 4.8 Let C be a nonempty, closed, and convex subset of an ${\ell }_p$ space for $2\le p<\mathrm{\infty }$. Let $Q_C$, F, G, V, and $W_n$ be the same as in Theorem  3.5. Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are sequences in $(0,1)$ which satisfy the conditions (C1) and (C2) in Theorem  3.5 and $\{{\theta }_{n,k}\}$ satisfies (H1)-(H3). Then the sequence $\{x_n\}$ defined by (3.14) converges strongly to $x^{\ast }\in \mathcal{F}$ as $n\to \mathrm{\infty }$, where $x^{\ast }$ is the unique solution of the variational inequality (3.15).

Remark 4.9 If the spaces $L_p$ has a weakly sequentially continuous generalized duality mappings, then we obtain Theorems 4.7 and 4.8 hold for $L_p$ spaces with $1<p<\mathrm{\infty }$, $p\ne 2$.

5 Numerical examples

In this section, we give a simple example and some numerical experiment result to explain the convergence of the sequence (3.14) as follows:

Example 5.1 Let $X=\mathbb{R}$ and $C=[0,\frac{1}{2}]$. Let $q=2$ and $j_q=I$. We define a mapping $Q_C$ as follows:

$$Q_{C}x=\{\begin{array}{cc}\frac{x}{|x|},\hfill & x\in (-\mathrm{\infty },0)\cup (\frac{1}{2},\mathrm{\infty }),\hfill \\ x,\hfill & x\in [0,\frac{1}{2}].\hfill \end{array}$$

In terms of Theorem 3.5, set $\sigma =\mu =\gamma =1$ and ${\alpha }_n=\frac{1}{n}$. Then we see that ${\alpha }_n=\frac{1}{n}$ satisfies (C1) and (C2) with ${\sigma }_n=\frac{1}{n^2}$. Moreover, we define the mappings F, G, and V as follows:

$$Fx=\frac{1}{3}(x^2+2x),Gx=x\text{and}Vx=x^2.$$

It is easy to observe that F is 1-Lipschitzian and $\frac{2}{3}$-strongly accretive, G is 1-strongly accretive and λ-strictly pseudo-contraction for $\lambda >0$ and V is 1-Lipschitzian. For each $n\in \mathbb{N}$, set $S_n=I$. We show that $W_n=I$. Since $T_{n,k}={\theta }_{n,k}{S}_k+(1-{\theta }_{n,k})I$, where $S_k$ is a ${\lambda }_k$-strictly pseudo-contractive mapping and $\{{\theta }_{n,k}\}$ satisfies (H1)-(H3). It is observe that $T_{n,k}$ is a nonexpansive mapping. From (2.5), we have

$$\begin{array}{c}{W}_1=U_{1,1}=t_1{T}_{1,1}{U}_{1,2}+(1-t_1)I,\hfill \\ W_2=U_{2,1}=t_1{T}_{2,1}{U}_{2,2}+(1-t_1)I\hfill \\ \phantom{W_2}=t_1{T}_{2,1}(t_2{T}_{2,2}{U}_{2,3}+(1-t_2)I)+(1-t_1)I\hfill \\ \phantom{W_2}=t_1{t}_2{T}_{2,1}{T}_{2,2}{U}_{2,3}+t_1(1-t_2)T_{2,1}+(1-t_1)I,\hfill \\ W_3=U_{3,1}=t_1{T}_{3,1}{U}_{3,2}+(1-t_1)I\hfill \\ \phantom{W_3}=t_1{T}_{3,1}(t_2{T}_{3,2}{U}_{3,3}+(1-t_2)I)+(1-t_1)I\hfill \\ \phantom{W_3}=t_1{t}_2{T}_{3,1}{T}_{3,2}{U}_{3,3}+t_1(1-t_2)T_{3,1}+(1-t_1)I\hfill \\ \phantom{W_3}=t_1{t}_2{T}_{3,1}{T}_{3,2}(t_3{T}_{3,3}{U}_{3,4}+(1-t_3)I)+t_1(1-t_2)T_{3,1}+(1-t_1)I\hfill \\ \phantom{W_3}=t_1{t}_2{t}_3{T}_{3,1}{T}_{3,2}{T}_{3,3}+t_1{t}_2(1-t_3)T_{3,1}{T}_{3,2}+t_1(1-t_2)T_{3,1}+(1-t_1)I\hfill \end{array}$$

and we compute (2.5) in a similar way to above, we obtain

$$\begin{array}{rcl}{W}_n& =& U_{n,1}\\ =& t_1{t}_2\cdot \cdot \cdot t_n{T}_{n,1}{T}_{n,2}\cdot \cdot \cdot T_{n,n}+t_1{t}_2\cdot \cdot \cdot t_{n-1}(1-t_n)T_{n,1}{T}_{n,2}\cdot \cdot \cdot T_{n,n-1}\\ +t_1{t}_2\cdot \cdot \cdot t_{n-2}(1-t_{n-1})T_{n,1}{T}_{n,2}\cdot \cdot \cdot T_{n,n-2}+\cdots +t_1(1-t_2)T_{n,1}+(1-t_1)I.\end{array}$$

Since $S_n=I$ and $t_n=\alpha$, for all $n\in \mathbb{N}$, we have

$$W_n=[{\alpha }^n+{\alpha }^{n-1}(1-\alpha )+\cdots +\alpha (1-\alpha )+(1-\alpha )]=I.$$

Under the above assumptions, (3.14) is simplified as follows:

$$\{\begin{array}{c}{x}_1\in C:=[0,\frac{1}{2}],\hfill \\ x_{n+1}=(1-\frac{2}{3n})x_n+\frac{2}{3n}{x}_n^2.\hfill \end{array}$$

(5.1)

Since the assumptions of Theorem 3.5 are satisfied in Example 5.1, the sequence (5.1) converges to $x^{\ast }=0$, which is the unique fixed point of $S_n$.

Next, we show the numerical results by using MATLAB 7.11.0. We presented numerical comparisons for two cases of iteration process with different initial values, which show the convergence of the sequence (5.1).

When we choose $x_1=0.05$ and $x_1=0.1$, we see that the iteration process of sequence $\{x_n\}$ converges to $x^{\ast }=0$ at $n=8\text{,}615$ and $n=28\text{,}946$, respectively, as shown in Table 1 and Figures 1 and 2.

Figure 1

The iteration process with initial value ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0.05}$ .

Figure 2

The iteration process with initial value ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0.1}$ .

Table 1 The value of sequence $\mathbf{\{}{\mathit{x}}_{\mathit{n}}\mathbf{\}}$ with iteration values ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0.05}$ and ${\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0.1}$

From the figures, we can see that $\{x_n\}$ is a monotone decreasing sequence and converges to 0, but an iterative process with initial value $x_1=0.05$ is converges faster than an iterative process with initial value $x_1=0.1$.

Remark 5.2 Note that Lemma 3.1 and Lemma 3.2 play an important role in the proof of Theorems 3.4 and 3.5. These are proved in the framework of the more general q-uniformly smooth Banach space.

Remark 5.3 Our main result extends the main result of Ceng et al. [28] in the following respects:

• (1) An iterative process (1.10) is to extend to a general iterative process defined over the set of fixed points of an infinite family of strict pseudo-contractions in a more general q-uniformly smooth Banach space.

• (2) The self contraction mapping $f:H\to H$ in [[28], Theorem 3.2] is extended to the case of a nonself Lipschitzian mapping $V:C\to X$ on a nonempty, closed, and convex subset C of a real q-uniformly smooth Banach space X.

• (3) The control condition (C3) in [[28], Theorem 3.2] is removed by weaker than control condition $|{\alpha }_{n+1}-{\alpha }_n|\le \circ ({\alpha }_n)+{\sigma }_n$ with ${\sum }_{n=1}^{\mathrm{\infty }}{\sigma }_n<\mathrm{\infty }$.

Furthermore, our method is extended to develop a new iterative method and method of proof is very different from that in Ceng et al. [28] because our method involves the sunny nonexpansive retraction and the infinite family of strict pseudo-contractions.