Convergence results for the zero-finding problem and fixed points of nonexpansive semigroups and strict pseudocontractions

Abstract

In this work, we establish strong convergence theorems for solving the fixed point problem of nonexpansive semigroups and strict pseudocontractions, and the zero-finding problem of maximal monotone operators in a Hilbert space. We further apply our result to the convex minimization problem and commutative semigroups.

MSC:47H09, 47H10.

1 Introduction

Let H be a real Hilbert space and K a nonempty, closed, and convex subset of H. Let $T:K\to K$ be a nonlinear mapping. Then T is said to be nonexpansive if $\parallel Tx-Ty\parallel \le \parallel x-y\parallel$ for all $x,y\in K$. The fixed points set of T is denoted by $F(T)$.

In 1953, Mann [21] introduced the following classical iteration for a nonexpansive mapping $T:K\to K$ in a real Hilbert space: $x_1\in K$ and

$$x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n,n\ge 1,$$

(1.1)

where $\{{\alpha }_n\}\subset (0,1)$.

In 1967, Halpern [13] introduced another classical iteration for a nonexpansive mapping $T:K\to K$ in a real Hilbert space: $x_1\in K$ and

$$x_{n+1}={\alpha }_{n}u+(1-{\alpha }_n)T{x}_n,n\ge 1,$$

where $\{{\alpha }_n\}\subset (0,1)$ and $u\in K$ is fixed.

Let $f:K\to K$ be a contraction (i.e., $\parallel f(x)-f(y)\parallel \le \alpha \parallel x-y\parallel$ for all $x,y\in K$ and $\alpha \in [0,1)$). In 2000, Moudafi [25] introduced the viscosity approximation method for a nonexpansive mapping T as follows: $x_1\in K$ and

$$x_{n+1}={\alpha }_{n}f(x_n)+(1-{\alpha }_n)T{x}_n,n\ge 1,$$

(1.2)

where $\{{\alpha }_n\}\subset (0,1)$. It was proved, in a Hilbert space that the sequence $\{x_n\}$ generated by (1.2) strongly converges to a fixed point of T under suitable conditions.

Let A be a strongly positive bounded linear operator on H: that is, there is a constant $\overline{\gamma }$ with property

$$〈Ax,x〉\ge \overline{\gamma }{\parallel x\parallel }^2\mathrm{\forall }x\in H.$$

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:

$$\underset{x\in K}{min}〈Ax,x〉-〈x,b〉,$$

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

Recently, Marino-Xu [22] introduced the following general iterative method for a nonexpansive mapping T in a Hilbert space: $x_1\in H$ and

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

where $\{{\alpha }_n\}\subset (0,1)$, f is a contraction and A is a strongly positive bounded linear operator.

Since then, there have been a number of modified viscosity approximation methods for nonexpansive mappings or nonexpansive semigroups (see, for example, [6, 7, 9, 26, 32, 35, 38, 42, 43]).

Recall that $T:K\to K$ is called a κ-strict pseudocontraction if there exists a constant $0\le \kappa <1$ such that

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

(1.3)

for all $x,y\in K$. It is known that (1.3) is equivalent to the following:

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

for all $x,y\in K$.

The class of strict pseudocontractions was introduced, in 1967, by Browder-Petryshyn [3]. The existence and weak convergence theorems were proved in a real Hilbert space by using Mann iterative algorithm (1.1) with a constant sequence ${\alpha }_n=\alpha$ for all $n\ge 1$. Recently, Marino-Xu [23] and Zhou [44] extended the results of Browder-Petryshyn [3] to Mann’s iteration process (1.1). Since 1967, the study of fixed points for strict pseudocontractions has been investigated by many authors (see, e.g., [1, 28]).

A set-valued mapping $M:H\to 2^H$ is called monotone if for all $x,y\in H$, $f\in M(x)$, and $g\in M(y)$ imply $〈x-y,f-g〉\ge 0$. A monotone mapping M is maximal if its graph $G(M):=\{(f,x)\in H\times H:f\in M(x)\}$ of M is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if for $(x,f)\in H\times H$, $〈x-y,f-g〉\ge 0$ for all $(y,g)\in G(M)$ imply $f\in M(x)$. Let $J_{\lambda }^M={(I+\lambda M)}^{-1}$, $\lambda >0$ be the resolvent of M. It is well known that $J_{\lambda }^M$ is single-valued and $D(J_{\lambda }^M)=H$ for any $\lambda >0$. For each $\lambda >0$, the Yosida approximation of M is defined by $A_{\lambda }=\frac{I-J_{\lambda }^M}{\lambda }$. We know that $(J_{\lambda }^{M}x,A_{\lambda }x)\in G(M)$ for all $\lambda >0$ and $x\in H$.

A fundamental problem of monotone operators is that of finding an element x such that $0\in Mx$. Such a problem is called the zero-finding problem (denoted by $M^{-1}(0)$ the set of solutions) and also includes many concrete examples, such as convex programming and monotone variational inequalities. It is known that if $g:H\to (-\mathrm{\infty },\mathrm{\infty }]$ is a proper lower semicontinuous convex function, then ∂g is maximal monotone and the equation $0\in \partial g(x)$ is reduced to $g(x)=min\{g(y):y\in H\}$ (see [29, 30]).

Initiated by Martinet [24], Rockafellar [30] introduced the following iterative scheme: $x_1\in H$ and

$$x_{n+1}=J_{{\lambda }_n}^M{x}_n,n\ge 1,$$

(1.4)

where $\{{\lambda }_n\}\subset (0,\mathrm{\infty })$ and M is a maximal monotone operator on H. Such an algorithm is called the proximal point algorithm. It was proved that the sequence $\{x_n\}$ generated by (1.4) converges weakly to an element in $M^{-1}(0)$ if ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\lambda }_n>0$.

The convergence of the zero-finding problem of monotone operators has been studied by many authors in several setting (see, for example, [8, 10, 14, 15, 27, 34]).

In this work, motivated by Lau et al. [16–20], Marino-Xu [22], and Saeidi [32], we introduce a new general iterative scheme for solving the fixed- point problem of a nonexpansive semigroup involving a strict pseudocontraction and the zero-finding problem of a maximal monotone operator in the framework of a Hilbert space. Some applications concerning the convex minimization problem and commutative semigroups are also presented.

2 Preliminaries and lemmas

In this section, we state some preliminaries and lemmas which will be used in the sequel.

Let S be a semigroup. We denote by ${\ell }^{\mathrm{\infty }}(S)$ the Banach space of all bounded real-valued functionals on S with supremum norm. For each $s\in S$, we define the left and right translation operators $l(s)$ and $r(s)$ on ${\ell }^{\mathrm{\infty }}(S)$ by

$$(l(s)f)(t)=f(st)\text{and}(r(s)f)(t)=f(ts)$$

for each $t\in S$ and $f\in {\ell }^{\mathrm{\infty }}(S)$, respectively. Let X be a subspace of ${\ell }^{\mathrm{\infty }}(S)$ containing 1. An element μ in the dual space $X^{\ast }$ of X is said to be a mean on X if $\parallel \mu \parallel =\mu (1)=1$. It is well known that μ is a mean on X if and only if

$$\underset{s\in S}{inf}f(s)\le \mu (f)\le \underset{s\in S}{sup}f(s)$$

for each $f\in X$. We often write ${\mu }_t(f(t))$ instead of $\mu (f)$ for $\mu \in X^{\ast }$ and $f\in X$.

Let X be a translation invariant subspace of ${\ell }^{\mathrm{\infty }}(S)$ (i.e., $l(s)X\subset X$ and $r(s)X\subset X$ for each $s\in S$) containing 1. Then a mean μ on X is said to be left invariant (resp. right invariant) if $\mu (l(s)f)=\mu (f)$ (resp. $\mu (r(s)f)=\mu (f)$) for each $s\in S$ and $f\in X$. A mean μ on X is said to be invariant if μ is both left and right invariant [16–18]. S is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. S is a amenable if S is left and right amenable. In this case, ${\ell }^{\mathrm{\infty }}(S)$ also has an invariant mean. It is known that ${\ell }^{\mathrm{\infty }}(S)$ is amenable when S is commutative semigroup or solvable group. However, the free group or semigroup of two generators is not left or right amenable (see [11, 20]). A net $\{{\mu }_{\alpha }\}$ of means on X is said to be left regular [11] if

$$\underset{\alpha }{lim}\parallel l_s^{\ast }{\mu }_{\alpha }-{\mu }_{\alpha }\parallel =0$$

for each $s\in S$, where $l_s^{\ast }$ is the adjoint operator of $l_s$.

Let K be a nonempty, closed, and convex subset of H. A family $\mathcal{S}=\{T(s):s\in S\}$ is called a nonexpansive semigroup on K if for each $s\in S$, the mapping $T(s):K\to K$ is nonexpansive and $T(st)=T(s)T(t)$ for each $s,t\in S$. We denote by $F(\mathcal{S})$ the set of common fixed points of $\mathcal{S}$, i.e.,

$$F(\mathcal{S})=\bigcap _{s\in S}F(T(s))=\bigcap _{s\in S}\{x\in K:T(s)x=x\}.$$

Throughout this article, we denote the open ball of radius r centered at 0 by $B_r$ and also denote the closed and convex hull of $A\subset H$ by $\overline{co}A$. For $\epsilon >0$ and a mapping $T:D\to H$, the set of ε-approximate fixed points of T will be denoted by $F_{\epsilon }(T,D)$, i.e. $F_{\epsilon }(T,D)=\{x\in D:\parallel x-Tx\parallel \le \epsilon \}$.

The following lemmas are important in order to prove our main theorem.

Lemma 2.1 [20, 31, 39]

Let f be a function of a semigroup S into a Banach space E such that the weak closure of $\{f(t):t\in S\}$ is weakly compact and let X be a subspace of ${\ell }^{\mathrm{\infty }}(S)$ containing all the functions $t\mapsto 〈f(t),x^{\ast }〉$ with $x^{\ast }\in E^{\ast }$. Then, for any $\mu \in X^{\ast }$, there exists a unique element $f_{\mu }$ in E such that

$$〈f_{\mu },x^{\ast }〉={\mu }_t〈f(t),x^{\ast }〉$$

for all $x^{\ast }\in E^{\ast }$. Moreover, if μ is a mean on X then

$$\int f(t)d\mu (t)\in \overline{co}\{f(t):t\in S\}.$$

We can write $f_{\mu }$ by $\int f(t)d\mu (t)$.

Lemma 2.2 [20, 31, 39]

Let K be a closed and convex subset of a Hilbert space H, $\mathcal{S}=\{T(s):s\in S\}$ be a nonexpansive semigroup from K into K such that $F(\mathcal{S})\ne \mathrm{\varnothing }$ and X be a subspace of ${\ell }^{\mathrm{\infty }}(S)$ containing 1 and the mapping $t\mapsto 〈T(t)x,y〉$ be an element of X for each $x\in K$ and $y\in H$, and μ be a mean on X.

If we write $T(\mu )x$ instead of $\int T_{t}xd\mu (t)$, then the following hold:

• (i) $T(\mu )$ is a nonexpansive mapping from K into K;

• (ii) $T(\mu )x=x$ for each $x\in F(\mathcal{S})$;

• (iii) $T(\mu )x\in \overline{co}\{T_{t}x:t\in S\}$ for each $x\in K$;

• (iv) if μ is left invariant, then $T(\mu )$ is a nonexpansive retraction from K onto $F(\mathcal{S})$.

Let K be a nonempty, closed, and convex subset of a real Hilbert space H. Then, for any $x\in H$, there exists a unique nearest point in K, denoted by $P_{K}x$, such that

$$\parallel x-P_{K}x\parallel \le \parallel x-y\parallel$$

for all $y\in K$. Such a projection $P_K$ is called the metric projection of H onto K. We also know that for $x\in H$ and $z\in K$, $z=P_{K}x$ if and only if

$$〈x-z,y-z〉\le 0,\mathrm{\forall }y\in K.$$

We know the following subdifferential inequality.

Lemma 2.3 For all $x,y\in H$, there holds the inequality

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,x+y〉.$$

Lemma 2.4 [22]

Let A be a strongly positive bounded linear operator on a Hilbert space H with coefficient $\overline{\gamma }$ and $0<\rho \le {\parallel A\parallel }^{-1}$. Then $\parallel I-\rho A\parallel \le 1-\rho \overline{\gamma }$.

In the sequel, we need the following crucial lemmas.

Lemma 2.5 [41]

Assume $\{a_n\}$ is a sequence of nonnegative real numbers such that

$$a_{n+1}\le (1-{\rho }_n)a_n+{\rho }_n{\delta }_n,n\ge 1,$$

where $\{{\rho }_n\}$ is a sequence in $(0,1)$ and $\{{\delta }_n\}$ is a sequence in $\mathbb{R}$ such that

• (a) ${\sum }_{n=1}^{\mathrm{\infty }}{\rho }_n=\mathrm{\infty }$;

• (b) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\delta }_n\le 0$ or ${\sum }_{n=1}^{\mathrm{\infty }}|{\rho }_n{\delta }_n|<\mathrm{\infty }$.

Then ${lim}_{n\to \mathrm{\infty }}{a}_n=0$.

Lemma 2.6 [36]

Let $\{x_n\}$ and $\{y_n\}$ be bounded sequences in a Banach space E such that

$$x_{n+1}=(1-{\beta }_n)y_n+{\beta }_n{x}_n,\mathrm{\forall }n\ge 1,$$

where $\{{\beta }_n\}$ is a real sequence in $(0,1)$ with $0<{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\beta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\beta }_n<1$. If ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}(\parallel y_{n+1}-y_n\parallel -\parallel x_{n+1}-x_n\parallel )\le 0$, then ${lim}_{n\to \mathrm{\infty }}\parallel y_n-x_n\parallel =0$.

The following crucial results can be found in [1].

Lemma 2.7 [1]

Let K be a nonempty, closed, and convex subset of a real Hilbert space H and let $T:K\to K$ be a κ-strict pseudocontraction such that $F(T)\ne \mathrm{\varnothing }$, then $I-T$ is demiclosed at zero, that is, for all sequence $\{x_n\}\subset K$ with $x_n\rightharpoonup y$ and $\parallel x_n-T{x}_n\parallel \to 0$ it follows that $y=Ty$.

Lemma 2.8 [1]

Let K be a nonempty, closed, and convex subset of a real Hilbert space H and let $T_i:K\to K$ ($i=1,2,\dots ,N$) be a family of ${\kappa }_i$-strict pseudocontractions for some $0\le {\kappa }_i<1$. Assume ${\{{\eta }_i\}}_{i=1}^N$ is a positive sequence such that ${\sum }_{i=1}^N{\eta }_i=1$. Then ${\sum }_{i=1}^N{\eta }_i{T}_i$ is a κ-strict pseudocontraction with $\kappa =max\{{\kappa }_i:1\le i\le N\}$. Moreover, if ${\{T_i\}}_{i=1}^N$ has a common fixed point, then $F({\sum }_{i=1}^N{\eta }_i{T}_i)={\bigcap }_{i=1}^{N}F(T_i)$.

Lemma 2.9 [40]

Let the resolvent $J_{\lambda }^M$ be defined by $J_{\lambda }^M={(I+\lambda M)}^{-1}$, $\lambda >0$. Then the following holds:

$$\parallel J_s^{M}x-J_t^{M}x\parallel \le \left|\frac{s-t}{t}\right|\parallel x-J_t^{M}x\parallel$$

for all $s,t>0$ and $x\in H$.

3 Main result

In this section, we are now ready to prove our main theorem.

Theorem 3.1 Let H be a real Hilbert space and $\mathcal{S}=\{T(t):t\in S\}$ a nonexpansive semigroup on H. Let $M:H\to 2^H$ be a maximal monotone operator and $T:H\to H$ a κ-strict pseudocontraction such that $F:=F(\mathcal{S})\cap M^{-1}(0)\cap F(T)\ne \mathrm{\varnothing }$. Let X be a left invariant subspace of ${\ell }^{\mathrm{\infty }}(S)$ such that $1\in X$, and the function $t\mapsto 〈T(t)x,y〉$ is an element of X for each $x,y\in H$. Let $\{{\mu }_n\}$ be a left regular sequence of means on X such that ${lim}_{n\to \mathrm{\infty }}\parallel {\mu }_{n+1}-{\mu }_n\parallel =0$. Let f be an α-contraction on H and A a strongly positive bounded linear operator with coefficient $\overline{\gamma }$. Let β and γ be real numbers such that $0<\beta <1$ and $0<\gamma <\overline{\gamma }/\alpha$. Let $\{x_n\}$ be generated by $x_1\in H$ and

$$\{\begin{array}{c}{y}_n=J_{{\lambda }_n}^M({\delta }_n{x}_n+(1-{\delta }_n)T{x}_n),\hfill \\ x_{n+1}={\alpha }_n\gamma f(x_n)+\beta x_n+((1-\beta )I-{\alpha }_{n}A)T({\mu }_n)y_n,n\ge 1,\hfill \end{array}$$

where $\{{\alpha }_n\}\subset (0,1)$, $\{{\delta }_n\}\subset (\kappa ,1)$ and $\{{\lambda }_n\}\subset (0,\mathrm{\infty })$ satisfying the conditions:

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

(C2) ${lim}_{n\to \mathrm{\infty }}|{\delta }_{n+1}-{\delta }_n|=0$;

(C3) $\kappa <{lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\delta }_n\le {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\delta }_n<1$;

(C4) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\lambda }_n>0$ and ${lim}_{n\to \mathrm{\infty }}|{\lambda }_{n+1}-{\lambda }_n|=0$.

Then $\{x_n\}$ converges strongly to $p\in F$ which also solves the following variational inequality:

$$〈(\gamma f-A)p,q-p〉\le 0,\mathrm{\forall }q\in F.$$

(3.1)

Proof Since ${\alpha }_n\to 0$, we shall assume that ${\alpha }_n\le (1-\beta ){\parallel A\parallel }^{-1}$ and $1-{\alpha }_n(\overline{\gamma }-\alpha \gamma )>0$. So by Lemma 2.4, we have $\parallel (1-\beta )I-{\alpha }_{n}A\parallel \le 1-\beta -{\alpha }_n\overline{\gamma }$.

First, we show that $\{x_n\}$ is bounded. Let $w\in F$. Put $z_n={\delta }_n{x}_n+(1-{\delta }_n)T{x}_n$ for all $n\in \mathbb{N}$. Then

$$\begin{array}{rcl}{\parallel z_n-w\parallel }^2& =& {\parallel {\delta }_n{x}_n+(1-{\delta }_n)T{x}_n-w\parallel }^2\\ =& {\parallel {\delta }_n(x_n-w)+(1-{\delta }_n)(T{x}_n-w)\parallel }^2\\ =& {\delta }_n{\parallel x_n-w\parallel }^2+(1-{\delta }_n){\parallel T{x}_n-w\parallel }^2-{\delta }_n(1-{\delta }_n){\parallel x_n-T{x}_n\parallel }^2\\ \le & {\delta }_n{\parallel x_n-w\parallel }^2+(1-{\delta }_n){\parallel x_n-w\parallel }^2+(1-{\delta }_n)\kappa {\parallel x_n-T{x}_n\parallel }^2\\ -{\delta }_n(1-{\delta }_n){\parallel x_n-T{x}_n\parallel }^2\\ =& {\parallel x_n-w\parallel }^2+(1-{\delta }_n)(\kappa -{\delta }_n){\parallel x_n-T{x}_n\parallel }^2\\ \le & {\parallel x_n-w\parallel }^2,\end{array}$$

(3.2)

which yields

$$\parallel z_n-w\parallel \le \parallel x_n-w\parallel .$$

Moreover, since $J_{{\lambda }_n}^M$ is firmly nonexpansive,

$$\parallel y_n-w\parallel =\parallel J_{{\lambda }_n}^M{z}_n-w\parallel \le \parallel z_n-w\parallel \le \parallel x_n-w\parallel .$$

(3.3)

From (3.3), we have

$$\begin{array}{rcl}\parallel x_{n+1}-w\parallel & \le & \parallel [(1-\beta )I-{\alpha }_{n}A][T({\mu }_n)y_n-w]\parallel +\parallel {\alpha }_n\gamma [f(x_n)-f(w)]\parallel \\ +\parallel {\alpha }_n[\gamma f(w)-Aw]\parallel +\parallel \beta (x_n-w)\parallel \\ \le & [1-{\alpha }_n(\overline{\gamma }-\alpha \gamma )]\parallel x_n-w\parallel +{\alpha }_n\parallel \gamma f(w)-Aw\parallel \\ \le & max\{\parallel x_n-w\parallel ,\frac{\parallel \gamma f(w)-Aw\parallel }{(\overline{\gamma }-\gamma \alpha )}\}.\end{array}$$

By an induction, we can show that

$$\parallel x_n-w\parallel \le max\{\parallel x_1-w\parallel ,\frac{\parallel \gamma f(w)-Aw\parallel }{(\overline{\gamma }-\gamma \alpha )}\},\mathrm{\forall }n\ge 1.$$

Therefore, $\{x_n\}$ is bounded. So are $\{f(x_n)\}$, $\{y_n\}$, $\{z_n\}$, and $\{T({\mu }_n)y_n\}$.

We next show that

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

Observe that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel T({\mu }_{n+1})y_n-T({\mu }_n)y_n\parallel =0.$$

(3.4)

Indeed,

$$\begin{array}{rcl}\parallel T({\mu }_{n+1})y_n-T({\mu }_n)y_n\parallel & =& \underset{\parallel z\parallel =1}{sup}\left|〈T({\mu }_{n+1})y_n-T({\mu }_n)y_n,z〉\right|\\ =& \underset{\parallel z\parallel =1}{sup}|{({\mu }_{n+1})}_s〈T(s)y_n,z〉-{({\mu }_n)}_s〈T(s)y_n,z〉|\\ \le & \parallel {\mu }_{n+1}-{\mu }_n\parallel \underset{s\in S}{sup}\parallel T(s)y_n\parallel .\end{array}$$

Since $\{y_n\}$ is bounded and ${lim}_{n\to \mathrm{\infty }}\parallel {\mu }_{n+1}-{\mu }_n\parallel =0$, (3.4) holds.

For each $n\in \mathbb{N}$, define $T_{n}x={\delta }_{n}x+(1-{\delta }_n)Tx$. Then $T_n$ is nonexpansive, and hence

$$\begin{array}{rcl}\parallel z_{n+1}-z_n\parallel & =& \parallel T_{n+1}{x}_{n+1}-T_n{x}_n\parallel \\ \le & \parallel T_{n+1}{x}_{n+1}-T_{n+1}{x}_n\parallel +\parallel T_{n+1}{x}_n-T_n{x}_n\parallel \\ \le & \parallel x_{n+1}-x_n\parallel +|{\delta }_{n+1}-{\delta }_n|M_1\end{array}$$

(3.5)

for some big enough constant $M_1>0$.

On the other hand, since $y_n=J_{{\lambda }_n}^M{z}_n$ and $y_{n+1}=J_{{\lambda }_{n+1}}^M{z}_{n+1}$,

$$\begin{array}{rcl}\parallel y_{n+1}-y_n\parallel & =& \parallel J_{{\lambda }_{n+1}}^M{z}_{n+1}-J_{{\lambda }_n}^M{z}_n\parallel \\ \le & \parallel J_{{\lambda }_{n+1}}^M{z}_{n+1}-J_{{\lambda }_n}^M{z}_{n+1}\parallel +\parallel J_{{\lambda }_n}^M{z}_{n+1}-J_{{\lambda }_n}^M{z}_n\parallel \\ \le & \parallel J_{{\lambda }_{n+1}}^M{z}_{n+1}-J_{{\lambda }_n}^M{z}_{n+1}\parallel +\parallel z_{n+1}-z_n\parallel .\end{array}$$

(3.6)

Put $w_n=\frac{x_{n+1}-\beta x_n}{1-\beta }$. Then

$$\begin{array}{rcl}{w}_{n+1}-w_n& =& \frac{1}{1-\beta }[(x_{n+2}-\beta x_{n+1})-(x_{n+1}-\beta x_n)]\\ =& \frac{1}{1-\beta }[{\alpha }_{n+1}(\gamma f(x_{n+1})-AT({\mu }_{n+1})y_{n+1})+(1-\beta )T({\mu }_{n+1})y_{n+1}]\\ -\frac{1}{1-\beta }[{\alpha }_n(\gamma f(x_n)-AT({\mu }_n)y_n)+(1-\beta )T({\mu }_n)y_n]\\ =& \frac{{\alpha }_{n+1}}{1-\beta }(\gamma f(x_{n+1})-AT({\mu }_{n+1})y_{n+1})+T({\mu }_{n+1})y_{n+1}\\ -\frac{{\alpha }_n}{1-\beta }(\gamma f(x_n)-AT({\mu }_n)y_n)-T({\mu }_n)y_n\\ =& \frac{{\alpha }_{n+1}}{1-\beta }(\gamma f(x_{n+1})-AT({\mu }_{n+1})y_{n+1})+(T({\mu }_{n+1})y_{n+1}-T({\mu }_{n+1})y_n)\\ -\frac{{\alpha }_n}{1-\beta }(\gamma f(x_n)-AT({\mu }_n)y_n)-(T({\mu }_n)y_n-T({\mu }_{n+1})y_n)\end{array}$$

which implies

$$\begin{array}{rcl}\parallel w_{n+1}-w_n\parallel & \le & \frac{{\alpha }_{n+1}}{1-\beta }\parallel \gamma f(x_{n+1})-AT({\mu }_{n+1})y_{n+1}\parallel +\parallel y_{n+1}-y_n\parallel \\ +\frac{{\alpha }_n}{1-\beta }\parallel \gamma f(x_n)-AT({\mu }_n)y_n\parallel +\parallel T({\mu }_n)y_n-T({\mu }_{n+1})y_n\parallel .\end{array}$$

(3.7)

Substituting (3.5) and (3.6) into (3.7), we obtain

$$\begin{array}{rcl}\parallel w_{n+1}-w_n\parallel & \le & \frac{{\alpha }_{n+1}}{1-\beta }\parallel \gamma f(x_{n+1})-AT({\mu }_{n+1})y_{n+1}\parallel +\parallel J_{{\lambda }_{n+1}}^M{z}_{n+1}-J_{{\lambda }_n}^M{z}_{n+1}\parallel \\ +\parallel x_{n+1}-x_n\parallel +|{\delta }_{n+1}-{\delta }_n|M_1+\frac{{\alpha }_n}{1-\beta }\parallel \gamma f(x_n)-AT({\mu }_n)y_n\parallel \\ +\parallel T({\mu }_n)y_n-T({\mu }_{n+1})y_n\parallel .\end{array}$$

Using Lemma 2.9, (3.4), (C1), (C2), and (C4), we have

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\parallel w_{n+1}-w_n\parallel -\parallel x_{n+1}-x_n\parallel )\le 0.$$

From Lemma 2.6, we derive

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

It also follows that

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

(3.8)

We next show that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T(t)x_n\parallel =0,\mathrm{\forall }t\in S.$$

Put

$$K=max\{\parallel x_1-w\parallel ,\frac{\parallel \gamma f(w)-Aw\parallel }{(\overline{\gamma }-\gamma \alpha )}\}.$$

Set $D=\{y\in H:\parallel y-w\parallel \le K\}$. Then D is a nonempty bounded closed convex set. Moreover, $\{x_n\}$, $\{y_n\}$, and $\{z_n\}$ are in D. To complete our proof, we follow the proof line as in [2] (see also [19, 20, 33]). Let $\epsilon >0$. From [5], there exists $\delta >0$ such that

$$\overline{co}{F}_{\delta }(T(t);D)+B_{\delta }\subseteq F_{\epsilon }(T(t);D),\mathrm{\forall }t\in S.$$

(3.9)

From Corollary 1.1 in [5], there exists a natural number N such that

$$\parallel \frac{1}{N+1}\sum _{i=0}^{N}T\left(t^{i}s\right)y-T(t)\left(\frac{1}{N+1}\sum _{i=0}^{N}T\left(t^{i}s\right)y\right)\parallel \le \delta ,$$

(3.10)

for all $t,s\in S$ and $y\in D$. Let $t\in S$. Since $\{{\mu }_n\}$ is left regular, there exists $n_0\in \mathbb{N}$ such that

$$\parallel {\mu }_n-l_{t^i}^{\ast }{\mu }_n\parallel \le \frac{\delta }{3(K+\parallel w\parallel )}$$

for all $n\ge n_0$ and $i=1,2,\dots ,N$. So we have for all $n\ge n_0$

(3.11)

Observe, by Lemma 2.2

$$\int \frac{1}{N+1}\sum _{i=0}^{N}T\left(t^{i}s\right)yd{\mu }_n(s)\in \overline{co}\{\frac{1}{N+1}\sum _{i=0}^{N}T{(t)}^i(T(s)y):s\in S\}.$$

(3.12)

Combining (3.10)-(3.12), we derive

$$\begin{array}{rcl}T({\mu }_n)y& =& \int \frac{1}{N+1}\sum _{i=0}^{N}T\left(t^{i}s\right)yd{\mu }_n(s)+(T({\mu }_n)y-\int \frac{1}{N+1}\sum _{i=0}^{N}T\left(t^{i}s\right)yd{\mu }_n(s))\\ \in & \overline{co}\{\frac{1}{N+1}\sum _{i=0}^{N}T{(t)}^i(T(s)y):s\in S\}+B_{\delta /3}\\ \subseteq & \overline{co}{F}_{\delta }(T(t);D)+B_{\delta /3},\end{array}$$

(3.13)

for all $y\in D$ and $n\ge n_0$. Let $t\in S$ and $\epsilon >0$. Then there exists $\delta >0$ which satisfies (3.9). Observe

$$x_{n+1}=T({\mu }_n)y_n+\frac{\beta }{1-\beta }(x_{n+1}-x_n)+\frac{{\alpha }_n}{1-\beta }(\gamma f(x_n)-AT({\mu }_n)y_n).$$

Since $\parallel x_{n+1}-x_n\parallel \to 0$ and ${\alpha }_n\to 0$, there exists $k\in \mathbb{N}$ such that

$$\begin{array}{rcl}{x}_{n+1}& =& T({\mu }_n)y_n+\frac{\beta }{1-\beta }(x_n-x_{n+1})+\frac{{\alpha }_n}{1-\beta }(\gamma f(x_n)-AT({\mu }_n)y_n)\\ \in & \overline{co}{F}_{\delta }(T(t);D)+B_{\delta /3}+B_{\delta /3}+B_{\delta /3}\\ \subseteq & \overline{co}{F}_{\delta }(T(t);D)+B_{\delta }\subseteq F_{\epsilon }(T(t);D),\end{array}$$

for all $n>k$. Hence, ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}\parallel x_n-T(t)x_n\parallel \le \epsilon$. Since $\epsilon >0$ is arbitrary,

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T(t)x_n\parallel =0.$$

(3.14)

We next show that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-z_n\parallel =0.$$

(3.15)

Since $J_{{\lambda }_n}^M$ is firmly nonexpansive and $y_n=J_{{\lambda }_n}^M{z}_n$,

$$\begin{array}{rcl}{\parallel y_n-w\parallel }^2& =& {\parallel J_{{\lambda }_n}^M{z}_n-J_{{\lambda }_n}^{M}w\parallel }^2\\ \le & 〈J_{{\lambda }_n}^M{z}_n-J_{{\lambda }_n}^{M}w,z_n-w〉\\ =& 〈y_n-w,z_n-w〉\\ =& \frac{1}{2}({\parallel y_n-w\parallel }^2+{\parallel z_n-w\parallel }^2-{\parallel z_n-y_n\parallel }^2),\end{array}$$

which implies

$${\parallel y_n-w\parallel }^2\le {\parallel z_n-w\parallel }^2-{\parallel z_n-y_n\parallel }^2.$$

Therefore,

$$\begin{array}{rcl}{\parallel x_{n+1}-w\parallel }^2& =& {\parallel [(1-\beta )(T({\mu }_n)y_n-w)+\beta (x_n-w)]+{\alpha }_n[\gamma f(x_n)-AT({\mu }_n)y_n]\parallel }^2\\ \le & {\parallel (1-\beta )(T({\mu }_n)y_n-w)+\beta (x_n-w)\parallel }^2\\ +2{\alpha }_n〈\gamma f(x_n)-AT({\mu }_n)y_n,x_{n+1}-w〉\\ \le & (1-\beta ){\parallel y_n-w\parallel }^2+\beta {\parallel x_n-w\parallel }^2\\ +2{\alpha }_n〈\gamma f(x_n)-AT({\mu }_n)y_n,x_{n+1}-w〉\\ \le & (1-\beta )({\parallel z_n-w\parallel }^2-{\parallel z_n-y_n\parallel }^2)+\beta {\parallel x_n-w\parallel }^2\\ +2{\alpha }_n〈\gamma f(x_n)-AT({\mu }_n)y_n,x_{n+1}-w〉\\ \le & {\parallel x_n-w\parallel }^2-(1-\beta ){\parallel z_n-y_n\parallel }^2\\ +2{\alpha }_n〈\gamma f(x_n)-AT({\mu }_n)y_n,x_{n+1}-w〉,\end{array}$$

which yields

$$(1-\beta ){\parallel z_n-y_n\parallel }^2\le {\alpha }_n{M}_2+({\parallel x_n-w\parallel }^2-{\parallel x_{n+1}-w\parallel }^2)$$

for some $M_2>0$. Thus, (3.15) holds by (3.8) and ${\alpha }_n\to 0$.

We next show that

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

(3.16)

From (3.2), we have

$${\parallel y_n-w\parallel }^2\le {\parallel z_n-w\parallel }^2\le {\parallel x_n-w\parallel }^2+(1-{\delta }_n)(\kappa -{\delta }_n){\parallel x_n-T{x}_n\parallel }^2.$$

So, we obtain

$$\begin{array}{rcl}{\parallel x_{n+1}-w\parallel }^2& \le & (1-\beta ){\parallel y_n-w\parallel }^2+\beta {\parallel x_n-w\parallel }^2\\ +2{\alpha }_n〈\gamma f(x_n)-AT({\mu }_n)y_n,x_{n+1}-w〉\\ \le & (1-\beta )({\parallel x_n-w\parallel }^2+(1-{\delta }_n)(\kappa -{\delta }_n){\parallel x_n-T{x}_n\parallel }^2)+\beta {\parallel x_n-w\parallel }^2\\ +2{\alpha }_n〈\gamma f(x_n)-AT({\mu }_n)y_n,x_{n+1}-w〉\\ \le & {\parallel x_n-w\parallel }^2+(1-\beta )(1-{\delta }_n)(\kappa -{\delta }_n){\parallel x_n-T{x}_n\parallel }^2+{\alpha }_n{M}_2.\end{array}$$

It follows that

$$(1-\beta )(1-{\delta }_n)({\delta }_n-\kappa ){\parallel x_n-T{x}_n\parallel }^2\le {\alpha }_n{M}_2+{\parallel x_n-w\parallel }^2-{\parallel x_{n+1}-w\parallel }^2.$$

From (C1) and (C3), we conclude that (3.16) holds. Moreover, we get that

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

(3.17)

It is easy to see that $P_F(\gamma f+(I-A))$ is a contraction. So, by Banach’s contraction principle, there exists a unique point p which satisfies the following variational inequality:

$$〈(\gamma f-A)p,q-p〉\le 0,\mathrm{\forall }q\in F.$$

We next show that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(\gamma f-A)p,x_n-p〉\le 0.$$

To this end, we choose a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(\gamma f-A)p,x_n-p〉=\underset{k\to \mathrm{\infty }}{lim}〈(\gamma f-A)p,x_{n_k}-p〉.$$

Since $\{x_n\}$ is bounded and H is reflexive, there exists a point $z\in H$ such that $x_{n_k}\rightharpoonup z$. From (3.15) and (3.17), there exists a corresponding subsequence $\{y_{n_k}\}$ of $\{y_n\}$ (resp. $\{z_{n_k}\}$ of $\{z_n\}$) such that $y_{n_k}\rightharpoonup z$ (resp. $z_{n_k}\rightharpoonup z$).

We next show that $z\in M^{-1}(0)$. Since $y_n=J_{{\lambda }_n}^M{z}_n$,

$$\parallel A_{{\lambda }_n}{z}_n\parallel =\frac{1}{{\lambda }_n}\parallel y_n-z_n\parallel .$$

From (3.15) and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\lambda }_n>0$, we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel A_{{\lambda }_n}{z}_n\parallel =0.$$

(3.18)

Noting that $(z_n,A_{{\lambda }_n}{z}_n)\in G(M)$, by the monotonicity of M, we have

$$〈s-z_n,s^{\ast }-A_{{\lambda }_n}{z}_n〉\ge 0$$

for all $(s,s^{\ast })\in G(M)$. So we obtain

$$〈s-z,s^{\ast }〉\ge 0$$

for all $(s,s^{\ast })\in G(M)$. Hence, $z\in M^{-1}(0)$ by the maximality of M.

On the other hand, from (3.14), we get that $z\in F(\mathcal{S})$ by the demiclosedness of a nonexpansive mapping [4, 12]. Applying Lemma 2.7 to (3.16), we also get that $z\in F(T)$. This shows that $z\in F$, and hence

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}〈(\gamma f-A)p,x_n-p〉=〈(\gamma f-A)p,z-p〉\le 0.$$

(3.19)

We finally show that $x_n\to p$ as $n\to \mathrm{\infty }$. From Lemmas 2.3 and 2.4, we have

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& =& {\parallel [((1-\beta )I-{\alpha }_{n}A)(T({\mu }_n)y_n-p)+\beta (x_n-p)]+{\alpha }_n(\gamma f(x_n)-Ap)\parallel }^2\\ \le & {\parallel ((1-\beta )I-{\alpha }_{n}A)(T({\mu }_n)y_n-p)+\beta (x_n-p)\parallel }^2\\ +2{\alpha }_n〈\gamma f(x_n)-Ap,x_{n+1}-p〉\\ =& {\parallel (1-\beta )\frac{(1-\beta )I-{\alpha }_{n}A}{(1-\beta )}(T({\mu }_n)y_n-p)+\beta (x_n-p)\parallel }^2\\ +2{\alpha }_n\gamma 〈f(x_n)-f(p),x_{n+1}-p〉+2{\alpha }_n〈\gamma f(p)-Ap,x_{n+1}-p〉\\ \le & (1-\beta ){\parallel \frac{(1-\beta )I-{\alpha }_{n}A}{(1-\beta )}(T({\mu }_n)y_n-p)\parallel }^2+\beta {\parallel x_n-p\parallel }^2\\ +2{\alpha }_n\gamma \alpha \parallel x_n-p\parallel \parallel x_{n+1}-p\parallel +2{\alpha }_n〈\gamma f(p)-Ap,x_{n+1}-p〉\\ \le & \frac{{\parallel (1-\beta )I-{\alpha }_{n}A\parallel }^2}{1-\beta }{\parallel T({\mu }_n)y_n-p\parallel }^2+\beta {\parallel x_n-p\parallel }^2\\ +{\alpha }_n\gamma \alpha ({\parallel x_n-p\parallel }^2+{\parallel x_{n+1}-p\parallel }^2)+2{\alpha }_n〈\gamma f(p)-Ap,x_{n+1}-p〉\\ \le & (\frac{{((1-\beta )-\overline{\gamma }{\alpha }_n)}^2}{1-\beta }+\beta +{\alpha }_n\gamma \alpha ){\parallel x_n-p\parallel }^2+{\alpha }_n\gamma \alpha {\parallel x_{n+1}-p\parallel }^2\\ +2{\alpha }_n〈\gamma f(p)-Ap,x_{n+1}-p〉\\ =& (1-(2\overline{\gamma }-\alpha \gamma ){\alpha }_n+\frac{{\overline{\gamma }}^2{\alpha }_n^2}{1-\beta }){\parallel x_n-p\parallel }^2+{\alpha }_n\gamma \alpha {\parallel x_{n+1}-p\parallel }^2\\ +2{\alpha }_n〈\gamma f(p)-Ap,x_{n+1}-p〉.\end{array}$$

It follows that

$$\begin{array}{rcl}{\parallel x_{n+1}-p\parallel }^2& \le & (1-\frac{2{\alpha }_n(\overline{\gamma }-\alpha \gamma )}{1-\alpha \gamma {\alpha }_n}){\parallel x_n-p\parallel }^2\\ +\frac{2{\alpha }_n(\overline{\gamma }-\alpha \gamma )}{1-\alpha \gamma {\alpha }_n}(\frac{1}{\overline{\gamma }-\alpha \gamma }〈\gamma f(p)-Ap,x_{n+1}-p〉\\ +\frac{{\overline{\gamma }}^2{\alpha }_n}{2(1-\beta )(\overline{\gamma }-\alpha \gamma )}{\parallel x_n-p\parallel }^2).\end{array}$$

From (3.19) and (C1), we can apply Lemma 2.5 to conclude that $x_n\to p$ as $n\to \mathrm{\infty }$. This completes the proof. □

From Rockafellar’s theorem [29, 30], we next apply our result to the convex minimization problem in a Hilbert space.

Corollary 3.2 Let H be a real Hilbert space and $\mathcal{S}=\{T(t):t\in S\}$ a nonexpansive semigroup on H. Let $g:H\to (-\mathrm{\infty },\mathrm{\infty }]$ be a proper lower semi-continuous convex function and $T:H\to H$ a κ-strict pseudocontraction such that $F:=F(\mathcal{S})\cap \partial g^{-1}(0)\cap F(T)\ne \mathrm{\varnothing }$. Let X be a left invariant subspace of ${\ell }^{\mathrm{\infty }}(S)$ such that $1\in X$, and the function $t\mapsto 〈T(t)x,y〉$ is an element of X for each $x,y\in H$. Let $\{{\mu }_n\}$ be a left regular sequence of means on X such that ${lim}_{n\to \mathrm{\infty }}\parallel {\mu }_{n+1}-{\mu }_n\parallel =0$. Let f be an α-contraction on H and A a strongly positive bounded linear operator with coefficient $\overline{\gamma }$. Let $\{{\alpha }_n\}$, β, γ, $\{{\delta }_n\}$ and $\{{\lambda }_n\}$ be as in Theorem 3.1. Then the sequence $\{x_n\}$ generated by $x_1\in H$ and

$$\{\begin{array}{c}{z}_n={\delta }_n{x}_n+(1-{\delta }_n)T{x}_n,\hfill \\ y_n={argmin}_{y\in H}\{g(y)+\frac{1}{2{\lambda }_n}{\parallel z_n-y\parallel }^2\},\hfill \\ x_{n+1}={\alpha }_n\gamma f(x_n)+\beta x_n+((1-\beta )I-{\alpha }_{n}A)T({\mu }_n)y_n,n\ge 1,\hfill \end{array}$$

converges strongly to $p\in F$ which also solves the variational inequality (3.1).

Using Lemma 2.8, we next apply our result to a finite family of strict pseudocontractions in a Hilbert space.

Corollary 3.3 Let H be a real Hilbert space and $\mathcal{S}=\{T(t):t\in S\}$ a nonexpansive semigroup on H. Let $M:H\to 2^H$ be a maximal monotone operator and ${\{T_i\}}_{i=1}^N:H\to H$ a family of ${\kappa }_i$-strict pseudocontractions such that $F:=F(\mathcal{S})\cap M^{-1}(0)\cap F(T_1)\cap \cdots \cap F(T_N)\ne \mathrm{\varnothing }$. Let $\kappa =max\{{\kappa }_i:1\le i\le N\}$. Let X be a left invariant subspace of ${\ell }^{\mathrm{\infty }}(S)$ such that $1\in X$, and the function $t\mapsto 〈T(t)x,y〉$ is an element of X for each $x,y\in H$. Let $\{{\mu }_n\}$ be a left regular sequence of means on X such that ${lim}_{n\to \mathrm{\infty }}\parallel {\mu }_{n+1}-{\mu }_n\parallel =0$. Let f be an α-contraction on H and A a strongly positive bounded linear operator with coefficient $\overline{\gamma }$. Let $\{{\alpha }_n\}$, β, γ, $\{{\delta }_n\}$ and $\{{\lambda }_n\}$ be as in Theorem 3.1 and ${\eta }_i\in (0,1)$ with ${\sum }_{i=1}^N{\eta }_i=1$. Then the sequence $\{x_n\}$ generated by $x_1\in H$ and

$$\{\begin{array}{c}{y}_n=J_{{\lambda }_n}^M({\delta }_n{x}_n+(1-{\delta }_n){\sum }_{i=1}^N{\eta }_i{T}_i{x}_n),\hfill \\ x_{n+1}={\alpha }_n\gamma f(x_n)+\beta x_n+((1-\beta )I-{\alpha }_{n}A)T({\mu }_n)y_n,n\ge 1,\hfill \end{array}$$

converges strongly to $p\in F$ which also solves the variational inequality (3.1).

Using the results proved in [37] (see also [19]), we obtain the following corollaries.

Corollary 3.4 Let H be a real Hilbert space. Let $S_1$ and $S_2$ be nonexpansive mappings on H with $S_1{S}_2=S_2{S}_1$. Let $M:H\to 2^H$ be a maximal monotone operator and let $T:H\to H$ be a κ-strict pseudocontraction such that $F:=F(S_1)\cap F(S_2)\cap M^{-1}(0)\cap F(T)\ne \mathrm{\varnothing }$. Let f be an α-contraction on H and A a strongly positive bounded linear operator with coefficient $\overline{\gamma }$. Let $\{{\alpha }_n\}$, β, γ, $\{{\delta }_n\}$, and $\{{\lambda }_n\}$ be as in Theorem 3.1. Then the sequence $\{x_n\}$ generated by $x_1\in H$ and

$$\{\begin{array}{c}{y}_n=J_{{\lambda }_n}^M({\delta }_n{x}_n+(1-{\delta }_n)T{x}_n),\hfill \\ x_{n+1}={\alpha }_n\gamma f(x_n)+\beta x_n+((1-\beta )I-{\alpha }_{n}A)(\frac{1}{n^2}{\sum }_{i=0}^{n-1}{\sum }_{j=0}^{n-1}{S}_1^i{S}_2^j{y}_n),n\ge 1,\hfill \end{array}$$

converges strongly to $p\in F$ which also solves the variational inequality (3.1).

Corollary 3.5 Let H be a real Hilbert space. Let $\mathcal{S}=\{T(t):t\in {\mathbb{R}}_{+}\}$ be a strongly continuous nonexpansive semigroup on H. Let $M:H\to 2^H$ be a maximal monotone operator and $T:H\to H$ a κ-strict pseudocontraction such that $F:=F(\mathcal{S})\cap M^{-1}(0)\cap F(T)\ne \mathrm{\varnothing }$. Let f be an α-contraction on H and A a strongly positive bounded linear operator with coefficient $\overline{\gamma }$. Let $\{{\alpha }_n\}$, β, γ, $\{{\delta }_n\}$, and $\{{\lambda }_n\}$ be as in Theorem 3.1. Then the sequence $\{x_n\}$ generated by $x_1\in H$ and

$$\{\begin{array}{c}{y}_n=J_{{\lambda }_n}^M({\delta }_n{x}_n+(1-{\delta }_n)T{x}_n),\hfill \\ x_{n+1}={\alpha }_n\gamma f(x_n)+\beta x_n+((1-\beta )I-{\alpha }_{n}A)(\frac{1}{t_n}{\int }_0^{t_n}T(s)y_{n}d(s)),n\ge 1,\hfill \end{array}$$

where $\{t_n\}$ is an increasing sequence in $(0,\mathrm{\infty })$ such that ${lim}_{n\to \mathrm{\infty }}{t}_n=\mathrm{\infty }$ and ${lim}_{n\to \mathrm{\infty }}{t}_n/t_{n+1}=1$, converges strongly to $p\in F$ which also solves the variational inequality (3.1).

Corollary 3.6 Let H be a real Hilbert space. Let $\mathcal{S}=\{T(t):t\in {\mathbb{R}}_{+}\}$ be a strongly continuous nonexpansive semigroup on H. Let $M:H\to 2^H$ be a maximal monotone operator and $T:H\to H$ a κ-strict pseudocontraction such that $F:=F(\mathcal{S})\cap M^{-1}(0)\cap F(T)\ne \mathrm{\varnothing }$. Let f be an α-contraction on H and A a strongly positive bounded linear operator with coefficient $\overline{\gamma }$. Let $\{{\alpha }_n\}$, β, γ, $\{{\delta }_n\}$ and $\{{\lambda }_n\}$ be as in Theorem 3.1. Then the sequence $\{x_n\}$ generated by $x_1\in H$ and

$$\{\begin{array}{c}{y}_n=J_{{\lambda }_n}^M({\delta }_n{x}_n+(1-{\delta }_n)T{x}_n),\hfill \\ x_{n+1}={\alpha }_n\gamma f(x_n)+\beta x_n+((1-\beta )I-{\alpha }_{n}A)(a_n{\int }_0^{\mathrm{\infty }}exp(-a_{n}s)T(s)y_{n}d(s)),n\ge 1,\hfill \end{array}$$

where $\{a_n\}$ is a decreasing sequence in $(0,\mathrm{\infty })$ such that ${lim}_{n\to \mathrm{\infty }}{a}_n=0$, converges strongly to $p\in F$ which also solves the variational inequality (3.1).