Iterative approximation for split common fixed point problem involving an asymptotically nonexpansive semigroup and a total asymptotically strict pseudocontraction

Abstract

In this paper, we prove the strong convergence theorem for split feasibility problem involving a uniformly asymptotically regular nonexpansive semigroup and a total asymptotically strict pseudocontractive mapping in Hilbert spaces. Our main results improve and extend some recent results in the literature.

MSC:47H06, 47H09, 47J05, 47J25.

1 Introduction

In this paper, we assume that H is a real Hilbert space with the inner product $〈\cdot ,\cdot 〉$ and the norm $\parallel \cdot \parallel$. Let I denote the identity operator on H. Let C and Q be nonempty, closed and convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively. The split feasibility problem (SFP) is to find a point

$$x\in C\text{ such that }Ax\in Q,$$

(1.1)

where $A:H_1\to H_2$ is a bounded linear operator. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [1] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. The SFP attracts the attention of many authors due to its application in signal processing. Various algorithms have been invented to solve it (see, for example, [3–11] and references therein).

Note that the split feasibility problem (1.1) can be formulated as a fixed point equation by using the fact

$$P_C(I-\gamma A^{\ast }(I-P_Q)A)x^{\ast }=x^{\ast };$$

(1.2)

that is, $x^{\ast }$ solves SFP (1.1) if and only if $x^{\ast }$ solves fixed point equation (1.2) (see [8] for details). This implies that we can use fixed point algorithms (see [12–14]) to solve SFP. A popular algorithm that solves SFP (1.1) is due to Byrne’s CQ algorithm [2] which is found to be a gradient-projection method (GPM) in convex minimization. Subsequently, Byrne [3] applied KM iteration to the CQ algorithm, and Zhao and Yang [15] applied KM iteration to the perturbed CQ algorithm to solve the SFP. It is well known that the CQ algorithm and the KM algorithm for a split feasibility problem do not necessarily converge strongly in the infinite-dimensional Hilbert spaces.

Now let us recall the definitions of some operators that will be used in this paper.

Let $T:H\to H$ be a mapping. A point $x\in H$ is said to be a fixed point of T provided that $Tx=x$, and denote by $F(T)$ the fixed point set of T.

Definition 1.1 The mapping $T:H\to H$ is said to be

• (a) nonexpansive if

  $$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\mathrm{\forall }x,y\in H;$$

• (b) strictly pseudocontractive if there exists a constant $k\in [0,1)$ such that

  $${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+k{\parallel (x-y)-(Tx-Ty)\parallel }^2,\mathrm{\forall }x,y\in H;$$

• (c) $(k,\{{\mu }_n\},\{{\xi }_n\},\varphi )$-total asymptotically strict pseudocontractive if there exists a constant $k\in [0,1)$ and sequences $\{{\mu }_n\}\subset [0,\mathrm{\infty })$, $\{{\xi }_n\}\subset [0,\mathrm{\infty })$ with ${\mu }_n\to 0$ and ${\xi }_n\to 0$ as $n\to \mathrm{\infty }$, and a continuous and strictly increasing function $\varphi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $\varphi (0)=0$ such that for all $n\ge 1$, $x,y\in H$,

  $${\parallel T^{n}x-T^{n}y\parallel }^2\le {\parallel x-y\parallel }^2+k{\parallel (x-y)-(Tx-Ty)\parallel }^2+{\mu }_n\varphi (\parallel x-y\parallel )+{\xi }_n,\mathrm{\forall }x,y\in H.$$

One parameter family $\mathrm{\Gamma }:=\{S(t):0\le t<\mathrm{\infty }\}$ is said to be a (continuous) Lipschitzian semigroup on a real Hilbert space H if the following conditions are satisfied:

• (1) $S(0)x=x$ for all $x\in H$;

• (2) $S(s+t)=S(s)S(t)$ for all $s,t\ge 0$;

• (3) for each $t>0$, there exists a bounded measurable function $L_t:(0,\mathrm{\infty })\to [0,\mathrm{\infty })$ such that

  $$\parallel S(t)x-S(t)y\parallel \le L_t\parallel x-y\parallel ,x,y\in H;$$

• (4) for each $x\in H$, the mapping $S(\cdot )x$ from $[0,\mathrm{\infty })$ into H is continuous.

A Lipschitzian semigroup Γ is called nonexpansive (or contractive) if $L_t=1$ for all $t>0$ and asymptotically nonexpansive if ${lim\hspace{0.17em}sup}_{t\to \mathrm{\infty }}{L}_t\le 1$, respectively. Let $F(\mathrm{\Gamma })$ denote the common fixed point set of the semigroup Γ, i.e., $F(\mathrm{\Gamma }):=\{x\in K:S(t)x=x,\mathrm{\forall }t>0\}$.

Let H be a real Hilbert space, $\mathrm{\Gamma }:=\{S(t):0\le t<\mathrm{\infty }\}$ be a continuous operator semigroup on H. Then Γ is said to be uniformly asymptotically regular (in short, u.a.r.) on H if for all $h\ge 0$ and any bounded subset C of H,

$$\underset{t\to \mathrm{\infty }}{lim}\underset{x\in C}{sup}\parallel S(h)(S(t)x)-S(t)x\parallel =0.$$

The nonexpansive semigroup $\{{\sigma }_t:t>0\}$ defined by the following lemma is an example of u.a.r. operator semigroup. Other examples of u.a.r. operator semigroup can be found in Examples 17, 18 of [16].

Lemma 1.2 (See Lemma 2.7 of [17])

Let D be a bounded closed convex subset of H, and $\mathrm{\Gamma }:=\{S(t):t>0\}$ be a nonexpansive semigroup on H such that $F(\mathrm{\Gamma })$ is nonempty. For each $h>0$, set ${\sigma }_t(x)=\frac{1}{t}{\int }_0^{t}S(s)xds$, then

$$\underset{t\to \mathrm{\infty }}{lim}\underset{x\in D}{sup}\parallel S(h)({\sigma }_{t}x)-{\sigma }_{t}x\parallel =0.$$

Example 1.3 (See [18])

The set $\{{\sigma }_t:t>0\}$ defined by Lemma 1.2 is an u.a.r. nonexpansive semigroup.

Several authors have proved several convergence theorems using several iterative schemes for fixed points of nonexpansive semigroups in the literature. See, for example, [16–22] and the references contained therein.

In this paper, we shall focus our attention on the following split common fixed point problem (SCFP):

$$\text{find }x\in C\text{ such that }Ax\in Q,$$

(1.3)

where $A:H_1\to H_2$ is a bounded linear operator, $\{S(t):t\ge 0\}$ is a uniformly asymptotically regular nonexpansive semigroup on $H_1$ and T is a uniformly L-Lipschitzian continuous and $(k,\{{\mu }_n\},\{{\xi }_n\},\varphi )$-total asymptotically strict pseudocontractive mapping with nonempty fixed point sets $C:={\bigcap }_{t\ge 0}F(S(t))$ and $Q:=F(T)$, and denote the solution set of the two-operator SCFP by

$$\mathrm{\Omega }:=\{y\in C:Ay\in Q\}=C\cap A^{-1}(Q).$$

(1.4)

Recall that ${\bigcap }_{t\ge 0}F(S(t))$ and $F(T)$ are closed and convex subsets of $H_1$ and $H_2$, respectively. If $\mathrm{\Omega }\ne \mathrm{\varnothing }$, we have that Ω is a closed and convex subset of $H_1$. The split common fixed point problem (SCFP) is a generalization of the split feasibility problem (SFP) and the convex feasibility problem (CFP) (see [2, 23]).

In order to solve (1.3), Censor and Segal [23] proposed and proved, in finite-dimensional spaces, the convergence of the following algorithm:

$$x_{n+1}=S(x_n+\gamma A^t(T-I)A{x}_n),n\ge 1,$$

(1.5)

where $\gamma \in (0,\frac{2}{\lambda })$, with λ being the largest eigenvalue of the matrix $A^{t}A$ ($A^t$ stands for matrix transposition) and S and T are quasi-nonexpansive operators.

In 2011, Moudafi [5] introduced the following relaxed algorithm:

$$x_{n+1}=(1-{\alpha }_n)y_n+{\alpha }_{n}S{y}_n,n\ge 1,$$

(1.6)

where $y_n=x_n+\gamma A^{\ast }(T-I)A{x}_n$, $\beta \in (0,1)$, ${\alpha }_n\in (0,1)$, and $\gamma \in (0,\frac{1}{\lambda \beta })$, with λ being the spectral radius of the operator $A^{\ast }A$. Moudafi proved weak convergence result of algorithm (1.6) in Hilbert spaces where S and T are quasi-nonexpansive operators. We observe that strong convergence result can be obtained in the results of Moudafi [5] if a compactness-type condition like demicompactness is imposed on the operator S. Furthermore, we can also obtain a strong convergence result by suitably modifying algorithm (1.6).

Recently, Zhao and He [24] introduced the following viscosity approximation algorithm

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

(1.7)

where $f:H_1\to H_1$ is a contraction of modulus $\rho >0$, $w_n\in (0,\frac{1}{2})$, $\gamma \in (0,\frac{1}{\lambda })$, with λ being the spectral radius of the operator $A^{\ast }A$, and they proved strong convergence results concerning (1.3) for quasi-nonexpansive operators S and T in real Hilbert spaces. Inspired by the work of Zhao and He [24], Moudafi [6] quite recently revisited the viscosity-type approximation method (1.7) above introduced in [24]. First, he proposed a simple proof of the strong convergence of the iterative sequence $\{x_n\}$ defined by (1.7) based on attracting operator properties and then proposed a modification of this algorithm (1.7) and proved its strong convergence (see Theorem 2.1 of [6]).

Very recently Chang et al. [25] proved the following convergence theorem for multiple-set split feasibility problem (MSSFP) (1.3) for a family of multi-valued quasi-nonexpansive mappings and a total asymptotically pseudocontractive mapping in infinitely dimensional Hilbert spaces.

Theorem 1.4 Let $H_1$ and $H_2$ be two real Hilbert spaces, $A:H_1\to H_2$ be a bounded linear operator and $A^{\ast }:H_2\to H_1$ be the adjoint of A. Let ${\{S_i\}}_{i=1}^{\mathrm{\infty }}:H_1\to CB(H_1)$ be a family of multi-valued quasi-nonexpansive mappings and for each $i\ge 1$, $S_i$ is demiclosed at  0. Let $T:H_2\to H_2$ be a uniformly L-Lipschitzian continuous and $(k,\{{\mu }_n\},\{{\xi }_n\},\varphi )$-total asymptotically strict pseudocontractive mapping satisfying ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{\xi }_n<\mathrm{\infty }$. Suppose that there exist constants $M_0>0$, $M_1>0$ such that $\varphi (\lambda )\le M_0{\lambda }^2$, $\mathrm{\forall }\lambda >M_1$. Let $C:={\bigcap }_{i=1}^{\mathrm{\infty }}F(S_i)\ne \mathrm{\varnothing }$ and $Q:=F(T)$. Assume that for each $p\in C$, $S_{i}p=\{p\}$ for each $i\ge 1$. Let $\{x_n\}$ be the sequence generated by

$$\{\begin{array}{l}{x}_1\in H_1\mathit{\text{chosen arbitrarily}},\\ x_{n+1}={\alpha }_{0,n}{y}_n+{\sum }_{i=1}^{\mathrm{\infty }}{\alpha }_{i,n}{w}_{i,n},w_{i,n}\in S_i{y}_n,\\ y_n=x_n+\gamma A^{\ast }(T^n-I)A{x}_n,n\ge 1,\end{array}$$

(1.8)

where $\{{\alpha }_{i,n}\}\subset (0,1)$ and $\gamma >0$ satisfy the following conditions:

• (a) ${\sum }_{i=0}^{\mathrm{\infty }}{\alpha }_{i,n}=1$ for each $n\ge 1$;

• (b) for each $i\ge 1$, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{\alpha }_{0,n}{\alpha }_{i,n}>0$;

• (c) $\gamma \in (0,\frac{1-k}{{\parallel A\parallel }^2})$.

If Ω (the set of solutions of multiple-set split feasibility problem (1.3)) is nonempty, then both ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ and ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$ converge weakly to some point $x\in \mathrm{\Omega }$. In addition, if there exists a positive integer m such that $S_m$ is semi-compact, then both ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ and ${\{y_n\}}_{n=1}^{\mathrm{\infty }}$ converge strongly to $x\in \mathrm{\Omega }$.

We observe on Theorem 1.4 that:

• (1) Theorem 1.4 gives a weak convergence result for multiple-set split feasibility problem (1.3) for a family of multi-valued quasi-nonexpansive mappings and a total asymptotically pseudocontractive mapping in infinitely dimensional Hilbert spaces. In order to get strong convergence, Chang et al. [25] imposed a compactness-type condition (semi-compactness) on the mappings ${\{S_i\}}_{i=1}^{\mathrm{\infty }}$. This compactness condition appears strong as only few mappings are semi-compact.

• (2) It is an interesting problem to extend the results of Theorem 1.4 to the nonexpansive semigroup case so that strong convergence is obtained. In order to obtain a strong convergence result in Theorem 1.4 without compactness-type condition for the nonexpansive semigroup case, a modification of (1.8) is necessary. This modification could be an implicit iterative scheme or an explicit iterative scheme. In the implicit iterative scheme, the computation of the next iteration $x_{n+1}$ involves solving a nonlinear equation at every step of the iteration, a task which may pose the same difficulty level as the initial problem. Therefore, in order to get a strong convergence result for the split common fixed point problem for a nonexpansive semigroup case and a total asymptotically pseudocontractive mapping in infinitely dimensional Hilbert spaces without compactness-type condition, a modification of (1.8), which is an explicit iterative scheme, is necessary. This leads to the following natural question.

Question Can we modify the iterative scheme (1.8) so that strong convergence is guaranteed for a split common fixed point problem involving a uniformly asymptotically regular nonexpansive semigroup and a total asymptotically pseudocontractive mapping in infinitely dimensional Hilbert spaces without any compactness-type condition assumed?

Our interest in this paper is to answer the above question. We thus modify the iterative scheme (1.8) and prove a strong convergence result for the split common fixed point problem for a uniformly asymptotically regular nonexpansive semigroup and a total asymptotically pseudocontractive mapping in infinitely dimensional Hilbert spaces without any further compactness-type condition assumed. Our results improve the corresponding results of Chang et al. [25] and many recent and important results that the results of Chang et al. [25] improved and extended like Censor et al. [26, 27], Yang [10], Moudafi [28], Xu [9], Censor and Segal [23], Masad and Reich [29] and others.

2 Preliminaries

We first recall some definitions, notations and conclusions which will be needed in proving our main results.

• $x_n\to x$ means that $x_n\to x$ strongly;

• $x_n\rightharpoonup x$ means that $x_n\to x$ weakly.

Next, we state the following well-known lemmas which will be used in the sequel.

Lemma 2.1 Let H be a real Hilbert space. Then the following well-known results hold:

• (i) ${\parallel x+y\parallel }^2={\parallel x\parallel }^2+2〈x,y〉+{\parallel y\parallel }^2$, $\mathrm{\forall }x,y\in H$;

• (ii) ${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2〈y,x+y〉$, $\mathrm{\forall }x,y\in H$;

• (iii) ${\parallel \lambda x+(1-\lambda )y\parallel }^2=\lambda {\parallel x\parallel }^2+(1-\lambda ){\parallel y\parallel }^2-\lambda (1-\lambda )\parallel x-y\parallel$, $\mathrm{\forall }x,y\in H$, $\mathrm{\forall }\lambda \in [0,1]$.

Lemma 2.2 (Chang et al. [25])

Let $T:H\to H$ be a uniformly L-Lipschitzian continuous and $(k,\{{\mu }_n\},\{{\xi }_n\},\varphi )$-total asymptotically strict pseudocontractive mapping, then $I-T$ is demiclosed at origin.

Lemma 2.3 (Alber et al. [30])

Let $\{{\lambda }_n\}$ and $\{{\gamma }_n\}$ be nonnegative, $\{{\alpha }_n\}$ be positive real numbers such that

$${\lambda }_{n+1}\le {\lambda }_n-{\alpha }_n{\lambda }_n+{\gamma }_n,n\ge 1.$$

Let for all $n>1$,

$$\frac{{\gamma }_n}{{\alpha }_n}\le c_1\mathit{\text{and}}{\alpha }_n\le \alpha .$$

Then ${\lambda }_n\le max\{{\lambda }_1,K_{\ast }\}$, where $K_{\ast }=(1+\alpha )c_1$.

Lemma 2.4 (Xu [31])

Let $\{a_n\}$ be a sequence of nonnegative real numbers satisfying the following relation:

$$a_{n+1}\le (1-{\alpha }_n)a_n+{\alpha }_n{\sigma }_n+{\gamma }_n,n\ge 0,$$

where

• (i) $\{a_n\}\subset [0,1]$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (ii) ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\sigma }_n\le 0$;

• (iii) ${\gamma }_n\ge 0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\gamma }_n<\mathrm{\infty }$.

Then $a_n\to 0$ as $n\to \mathrm{\infty }$.

3 Main results

For solving the split common fixed point problem (1.3), we assume that the following conditions are satisfied:

• (1) $H_1$ and $H_2$ are two real Hilbert spaces, $A:H_1\to H_2$ is a bounded linear operator and $A^{\ast }:H_2\to H_1$ is the adjoint of A;

• (2) $\{S(t):t\ge 0\}$ is a uniformly asymptotically regular nonexpansive semigroup on $H_1$;

• (3) $T:H_2\to H_2$ is a uniformly L-Lipschitzian continuous and $(k,\{{\mu }_n\},\{{\xi }_n\},\varphi )$-total asymptotically strict pseudocontractive mapping satisfying the following conditions:

• (i) ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$; ${\sum }_{n=1}^{\mathrm{\infty }}{\xi }_n<\mathrm{\infty }$;

• (ii) $\{{\alpha }_n\}$ is a real sequence in $(0,1)$ such that ${\mu }_n=o({\alpha }_n)$, ${\xi }_n=o({\alpha }_n)$, ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (iii) there exist constants $M_0>0$, $M_1>0$ such that $\varphi (\lambda )\le M_0{\lambda }^2$, $\mathrm{\forall }\lambda >M_1$;

• (iv) $C:={\bigcap }_{t\ge 0}F(S(t))\ne \mathrm{\varnothing }$, $Q:=F(T)\ne \mathrm{\varnothing }$ and $\mathrm{\Omega }\ne \mathrm{\varnothing }$.

In this section, we introduce the following algorithm and prove its strong convergence for solving split common fixed point problem (1.3).

Theorem 3.1 Let $H_1$, $H_2$, A, $A^{\ast }$, $\{S(t):t\ge 0\}$, T, C, Q, k, $\{{\mu }_n\}$, $\{{\xi }_n\}$, ϕ and L satisfy the above conditions (i)-(iv). Let $\{x_n\}$ be the sequence generated by $x_1\in H_1$,

$$\{\begin{array}{l}{u}_n=(1-{\alpha }_n)x_n,\\ y_n=u_n+\gamma A^{\ast }(T^n-I)A{u}_n,\\ x_{n+1}={\beta }_n{y}_n+(1-{\beta }_n)S(t_n)y_n,n\ge 1,\end{array}$$

(3.1)

where $t_n\to \mathrm{\infty }$ and $\{{\beta }_n\}\subset (0,1)$ and $\gamma >0$ satisfy the following conditions:

• (a) $0<ϵ\le {\beta }_n\le b<1$;

• (b) $\gamma \in (0,\frac{1-k}{{\parallel A\parallel }^2})$.

If Ω is nonempty, then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to an element of Ω.

Proof Since ϕ is continuous, it follows that ϕ attains maximum (say M) in $[0,M_1]$ and by our assumption, $\varphi (\lambda )\le M_0{\lambda }^2$, $\mathrm{\forall }\lambda >M_1$. In either case, we have that

$$\varphi (\lambda )\le M+M_0{\lambda }^2,\mathrm{\forall }\lambda \in [0,\mathrm{\infty }).$$

Let $x^{\ast }\in \mathrm{\Omega }$. Then, by the convexity of ${\parallel \cdot \parallel }^2$, we obtain

$$\begin{array}{rcl}{\parallel u_n-x^{\ast }\parallel }^2& =& {\parallel (1-{\alpha }_n)x_n-x^{\ast }\parallel }^2={\parallel (1-{\alpha }_n)(x_n-x^{\ast })+{\alpha }_n(-x^{\ast })\parallel }^2\\ \le & (1-{\alpha }_n){\parallel x_n-x^{\ast }\parallel }^2+{\alpha }_n{\parallel x^{\ast }\parallel }^2.\end{array}$$

(3.2)

From (3.1) and Lemma 2.1(i), we obtain that

$$\begin{array}{rcl}{\parallel y_n-x^{\ast }\parallel }^2& =& {\parallel u_n-x^{\ast }+\gamma A^{\ast }(T^n-I)A{u}_n\parallel }^2\\ =& {\parallel u_n-x^{\ast }\parallel }^2+2\gamma 〈u_n-x^{\ast },A^{\ast }(T^n-I)A{u}_n〉+{\gamma }^2{\parallel A^{\ast }(T^n-I)A{u}_n\parallel }^2.\end{array}$$

(3.3)

Since

$$\begin{array}{rcl}{\gamma }^2{\parallel A^{\ast }(T^n-I)A{u}_n\parallel }^2& =& {\gamma }^2〈A^{\ast }(T^n-I)A{u}_n,A^{\ast }(T^n-I)A{u}_n〉\\ =& {\gamma }^2〈A{A}^{\ast }(T^n-I)A{u}_n,(T^n-I)A{u}_n〉\\ \le & {\gamma }^2{\parallel A\parallel }^2{\parallel (T^n-I)A{u}_n\parallel }^2,\end{array}$$

(3.4)

$A{x}^{\ast }\in Q=F(T)$ and T is a total asymptotically strict pseudocontractive mapping, then we obtain

$$\begin{array}{c}〈u_n-x^{\ast },A^{\ast }(T^n-I)A{u}_n〉\hfill \\ =〈A(u_n-x^{\ast }),(T^n-I)A{u}_n〉\hfill \\ =〈A(u_n-x^{\ast })+(T^n-I)A{u}_n-(T^n-I)A{u}_n,(T^n-I)A{u}_n〉\hfill \\ =〈T^{n}A{u}_n-A{x}^{\ast },(T^n-I)A{u}_n〉-{\parallel (T^n-I)A{u}_n\parallel }^2\hfill \\ =\frac{1}{2}[{\parallel T^{n}A{u}_n-A{x}^{\ast }\parallel }^2+{\parallel (T^n-I)A{u}_n\parallel }^2-{\parallel A{u}_n-A{x}^{\ast }\parallel }^2]\hfill \\ -{\parallel (T^n-I)A{u}_n\parallel }^2\hfill \\ \le \frac{1}{2}[{\parallel A{u}_n-A{x}^{\ast }\parallel }^2+k{\parallel (T-I)A{x}_n\parallel }^2+{\mu }_n\varphi \left(\parallel A{u}_n-A{x}^{\ast }\parallel \right)+{\xi }_n]\hfill \\ +\frac{1}{2}[{\parallel (T^n-I)A{u}_n\parallel }^2-{\parallel A{u}_n-A{x}^{\ast }\parallel }^2]\hfill \\ -{\parallel (T^n-I)A{u}_n\parallel }^2\hfill \\ \le \frac{k-1}{2}{\parallel (T^n-I)A{u}_n\parallel }^2+\frac{{\mu }_n}{2}(M+M_0{\parallel A{u}_n-A{x}^{\ast }\parallel }^2)+\frac{{\xi }_n}{2}.\hfill \end{array}$$

(3.5)

Substituting (3.5) and (3.4) into (3.3), we have

$$\begin{array}{rl}{\parallel y_n-x^{\ast }\parallel }^2\le & {\parallel u_n-x^{\ast }\parallel }^2-\gamma (1-k-\gamma {\parallel A\parallel }^2){\parallel (T^n-I)A{u}_n\parallel }^2\\ +{\mu }_n\gamma (M+M_0{\parallel A{u}_n-A{x}^{\ast }\parallel }^2)+\gamma {\xi }_n.\end{array}$$

(3.6)

Putting (3.6) and (3.2) into (3.1), we obtain

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^2& =& {\parallel {\beta }_n(y_n-x^{\ast })+(1-{\beta }_n)(S(t_n)y_n-x^{\ast })\parallel }^2\\ \le & {\beta }_n{\parallel y_n-x^{\ast }\parallel }^2+(1-{\beta }_n){\parallel S(t_n)y_n-x^{\ast }\parallel }^2\\ \le & {\beta }_n{\parallel y_n-x^{\ast }\parallel }^2+(1-{\beta }_n){\parallel y_n-x^{\ast }\parallel }^2={\parallel y_n-x^{\ast }\parallel }^2\\ \le & {\parallel u_n-x^{\ast }\parallel }^2-\gamma (1-k-\gamma {\parallel A\parallel }^2){\parallel (T^n-I)A{u}_n\parallel }^2\\ +{\mu }_n\gamma (M+M_0{\parallel A{u}_n-A{x}^{\ast }\parallel }^2)+\gamma {\xi }_n\\ \le & (1-{\alpha }_n){\parallel x_n-x^{\ast }\parallel }^2+{\alpha }_n{\parallel x^{\ast }\parallel }^2-\gamma (1-k-\gamma {\parallel A\parallel }^2){\parallel (T^n-I)A{u}_n\parallel }^2\\ +{\mu }_n\gamma (M+M_0{\parallel A\parallel }^2((1-{\alpha }_n){\parallel x_n-x^{\ast }\parallel }^2+{\alpha }_n{\parallel x^{\ast }\parallel }^2))+\gamma {\xi }_n\\ =& {\parallel x_n-x^{\ast }\parallel }^2-({\alpha }_n-{\mu }_n\gamma M_0{\parallel A\parallel }^2(1-{\alpha }_n)){\parallel x_n-x^{\ast }\parallel }^2+{\alpha }_n{\parallel x^{\ast }\parallel }^2\\ +{\mu }_n\gamma M+\gamma {\alpha }_n{M}_0{\parallel A\parallel }^2{\parallel x^{\ast }\parallel }^2+\gamma {\xi }_n\\ -\gamma (1-k-\gamma {\parallel A\parallel }^2){\parallel (T^n-I)A{u}_n\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-({\alpha }_n-{\mu }_n\gamma M_0{\parallel A\parallel }^2(1-{\alpha }_n)){\parallel x_n-x^{\ast }\parallel }^2+{\alpha }_n{\parallel x^{\ast }\parallel }^2\\ +{\mu }_n\gamma M+\gamma {\alpha }_n{M}_0{\parallel A\parallel }^2{\parallel x^{\ast }\parallel }^2+\gamma {\xi }_n\\ =& {\parallel x_n-x^{\ast }\parallel }^2-({\alpha }_n-{\mu }_n\gamma M_0{\parallel A\parallel }^2(1-{\alpha }_n)){\parallel x_n-x^{\ast }\parallel }^2+{\sigma }_n,\end{array}$$

(3.7)

where ${\sigma }_n={\alpha }_n{\parallel x^{\ast }\parallel }^2{\mu }_n\gamma M+\gamma {\alpha }_n{M}_0{\parallel A\parallel }^2{\parallel x^{\ast }\parallel }^2+\gamma {\xi }_n$. Since ${\mu }_n=o({\alpha }_n)$ and ${\xi }_n=o({\alpha }_n)$, we may assume without loss of generality that there exist constants $k_0\in (0,1)$ and $M_2>0$ such that for all $n\ge 1$,

$$\frac{{\mu }_n}{{\alpha }_n}\le \frac{1-k_0}{M_0\gamma {\parallel A\parallel }^2(1-{\alpha }_n)}\text{and}\frac{{\sigma }_n}{{\alpha }_n}\le M_2.$$

Thus, we obtain

$${\parallel x_{n+1}-x^{\ast }\parallel }^2\le {\parallel x_n-x^{\ast }\parallel }^2-{\alpha }_n{k}_0{\parallel x_n-x^{\ast }\parallel }^2+{\sigma }_n.$$

By Lemma 2.3, we have that

$${\parallel x_n-x^{\ast }\parallel }^2\le max\{{\parallel x_1-x^{\ast }\parallel }^2,(1+k_0)M_2\}.$$

Therefore, $\{x_n\}$ is bounded. Furthermore, the sequences $\{y_n\}$ and $\{u_n\}$ are bounded.

The rest of the proof will be divided into two parts.

Case 1. Suppose that there exists $n_0\in \mathbb{N}$ such that ${\{\parallel x_n-x^{\ast }\parallel \}}_{n=n_0}^{\mathrm{\infty }}$ is nonincreasing. Then ${\{\parallel x_n-x^{\ast }\parallel \}}_{n=1}^{\mathrm{\infty }}$ converges and

$${\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_{n+1}-x^{\ast }\parallel }^2\to 0,n\to \mathrm{\infty }.$$

From (3.6), we have that

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^2& \le & {\parallel y_n-x^{\ast }\parallel }^2\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-\gamma (1-k-\gamma {\parallel A\parallel }^2){\parallel (T^n-I)A{x}_n\parallel }^2\\ +{\mu }_n\gamma (M+M_0{\parallel A{x}_n-A{x}^{\ast }\parallel }^2)+\gamma {\xi }_n\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-\gamma (1-k-\gamma {\parallel A\parallel }^2){\parallel (T^n-I)A{x}_n\parallel }^2\\ +{\mu }_n\gamma (M+M_0{\parallel A{x}_n-A{x}^{\ast }\parallel }^2)+\gamma {\xi }_n+{\alpha }_n{\parallel x^{\ast }\parallel }^2.\end{array}$$

This implies that

$$\begin{array}{rcl}\gamma (1-k-\gamma {\parallel A\parallel }^2){\parallel (T^n-I)A{x}_n\parallel }^2& \le & {\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_{n+1}-x^{\ast }\parallel }^2\\ +{\mu }_n\gamma (M+M_0{\parallel A{x}_n-A{x}^{\ast }\parallel }^2)+\gamma {\xi }_n+{\alpha }_n{\parallel x^{\ast }\parallel }^2\end{array}$$

and

$$\gamma (1-k-\gamma {\parallel A\parallel }^2){\parallel (T^n-I)A{x}_n\parallel }^2\to 0,n\to \mathrm{\infty }.$$

Hence, we obtain

$$\parallel (T^n-I)A{x}_n\parallel \to 0,n\to \mathrm{\infty }.$$

(3.8)

Also, we observe that

$$\parallel y_n-x_n\parallel =\gamma A^{\ast }\parallel (T^n-I)A{x}_n\parallel \to 0,n\to \mathrm{\infty }.$$

Using (3.6) and Lemma 2.1(iii) in (3.1), we have

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^2& =& {\parallel (1-{\beta }_n)(y_n-x^{\ast })+{\beta }_n(S(t_n)y_n-x^{\ast })\parallel }^2\\ \le & (1-{\beta }_n){\parallel y_n-x^{\ast }\parallel }^2+{\beta }_n{\parallel S(t_n)y_n-x^{\ast }\parallel }^2-(1-{\beta }_n){\beta }_n\parallel y_n-S(t_n)y_n\parallel \\ =& {\parallel y_n-x^{\ast }\parallel }^2-(1-{\beta }_n){\beta }_n\parallel y_n-S(t_n)y_n\parallel \\ \le & {\parallel x_n-x^{\ast }\parallel }^2-\gamma (1-k-\gamma {\parallel A\parallel }^2){\parallel (T^n-I)A{x}_n\parallel }^2\\ +{\mu }_n\gamma (M+M_0{\parallel A{x}_n-A{x}^{\ast }\parallel }^2)+\gamma {\xi }_n-(1-{\beta }_n){\beta }_n\parallel y_n-S(t_n)y_n\parallel .\end{array}$$

This implies from (3.2) and condition (b) that

$$\begin{array}{rcl}(1-{\beta }_n){\beta }_n\parallel y_n-S(t_n)y_n\parallel & \le & {\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_{n+1}-x^{\ast }\parallel }^2\\ +{\mu }_n\gamma (M+M_0{\parallel A{x}_n-A{x}^{\ast }\parallel }^2)+\gamma {\xi }_n\\ \le & {\parallel x_n-x^{\ast }\parallel }^2-{\parallel x_{n+1}-x^{\ast }\parallel }^2+{\alpha }_n{\parallel x^{\ast }\parallel }^2\\ +{\mu }_n\gamma (M+M_0{\parallel A{x}_n-A{x}^{\ast }\parallel }^2)+\gamma {\xi }_n.\end{array}$$

From condition (a) we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_n-S(t_n)y_n\parallel =0.$$

(3.9)

Hence, for any $t\ge 0$,

$$\begin{array}{rl}\parallel S(t)y_n-y_n\parallel & \le \parallel S(t)y_n-S(t)S(t_n)y_n\parallel +\parallel S(t)S(t_n)y_n-S(t_n)y_n\parallel +\parallel S(t_n)y_n-y_n\parallel \\ \le \parallel S(t)S(t_n)y_n-S(t_n)y_n\parallel +2\parallel S(t_n)y_n-y_n\parallel \to 0,n\to \mathrm{\infty }.\end{array}$$

(3.10)

We obtain from (3.1) that

$$\begin{array}{rcl}{\parallel x_{n+1}-x_n\parallel }^2& =& {\parallel {\beta }_n(y_n-x_n)+(1-{\beta }_n)(S(t_n)y_n-x_n)\parallel }^2\\ \le & {\beta }_n{\parallel y_n-x_n\parallel }^2+(1-{\beta }_n){\parallel S(t_n)y_n-x_n\parallel }^2\\ \le & {\beta }_n{\parallel y_n-x_n\parallel }^2+(1-{\beta }_n){(\parallel S(t_n)y_n-y_n\parallel +\parallel y_n-x_n\parallel )}^2.\end{array}$$

Since ${lim}_{n\to \mathrm{\infty }}\parallel y_n-x_n\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel y_n-S(t_n)y_n\parallel =0$, we have

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

Consequently,

$$\begin{array}{rcl}\parallel u_{n+1}-u_n\parallel & =& \parallel (1-{\alpha }_{n+1})x_{n+1}-(1-{\alpha }_n)x_n\parallel \\ \le & |{\alpha }_{n+1}-{\alpha }_n|\parallel x_{n+1}\parallel +(1-{\alpha }_n)\parallel x_{n+1}-x_n\parallel \to 0,n\to \mathrm{\infty }.\end{array}$$

(3.11)

Using the fact that T is uniformly L-Lipschitzian, we have

$$\begin{array}{rcl}\parallel TA{u}_n-A{u}_n\parallel & \le & \parallel TA{u}_n-T^{n+1}A{u}_n\parallel +\parallel T^{n+1}A{u}_n-T^{n+1}A{u}_{n+1}\parallel \\ +\parallel T^{n+1}A{u}_{n+1}-A{u}_{n+1}\parallel +\parallel A{u}_{n+1}-A{u}_n\parallel \\ \le & L\parallel A{u}_n-T^{n}A{u}_n\parallel +(L+1)\parallel A{u}_{n+1}-A{u}_n\parallel +\parallel T^{n+1}A{u}_{n+1}-A{u}_{n+1}\parallel \\ \le & L\parallel A{u}_n-T^{n}A{u}_n\parallel +(L+1)\parallel A\parallel \parallel u_{n+1}-u_n\parallel +\parallel T^{n+1}A{u}_{n+1}-A{u}_{n+1}\parallel .\end{array}$$

From (3.8) and (3.11), we obtain

$$\parallel (T-I)A{u}_n\parallel \to 0,n\to \mathrm{\infty }.$$

(3.12)

Since $\{x_n\}$ is bounded, there exists $\{x_{n_j}\}$ of $\{x_n\}$ such that $x_{n_j}\rightharpoonup z\in H_1$. Using the fact that $x_{n_j}\rightharpoonup z\in H_1$ and $\parallel y_n-x_n\parallel \to 0$, $n\to \mathrm{\infty }$, we have that $y_{n_j}\rightharpoonup z\in H_1$. Similarly, $u_{n_j}\rightharpoonup z\in H_1$ since $\parallel u_n-x_n\parallel \to 0$, $n\to \mathrm{\infty }$.

We next show that $z\in {\bigcap }_{t\ge 0}F(S(t))=C$. Assume the contrary that $z\ne S(t)z$ for some $t\ge 0$. Then, by Opial’s condition, we obtain from (3.10) that

$$\begin{array}{rcl}\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel y_{n_j}-z\parallel & <& \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel y_{n_j}-S(t)z\parallel \\ \le & \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}(\parallel y_{n_j}-S(t)y_{n_j}\parallel +\parallel S(t)y_{n_j}-S(t)z\parallel )\\ \le & \parallel y_{n_j}-z\parallel .\end{array}$$

This is a contradiction. Hence, $z\in {\bigcap }_{t\ge 0}F(S(t))=C$. On the other hand, since A is a linear bounded operator, it follows from $u_{n_j}\rightharpoonup z\in H_1$ that $A{u}_{n_j}\rightharpoonup Az\in H_2$. Hence, from (3.12), we have that

$$\parallel TA{u}_{n_j}-A{u}_{n_j}\parallel =\parallel TA{u}_{n_j}-A{u}_{n_j}\parallel \to 0,j\to \mathrm{\infty }.$$

Since T is demiclosed at zero, we have that $Az\in F(T)=Q$. Hence $z\in \mathrm{\Omega }$.

Next, we prove that $\{x_n\}$ converges strongly to z. From (3.6) and Lemma 2.1(ii), we have

$$\begin{array}{rcl}{\parallel x_{n+1}-z\parallel }^2& \le & {\parallel y_n-z\parallel }^2\\ \le & {\parallel u_n-z\parallel }^2-\gamma (1-k-\gamma {\parallel A\parallel }^2){\parallel (T^n-I)A{u}_n\parallel }^2\\ +{\mu }_n\gamma (M+M_0{\parallel A{u}_n-Az\parallel }^2)+\gamma {\xi }_n\\ \le & {\parallel u_n-z\parallel }^2+{\mu }_n\gamma (M+M_0{\parallel A{u}_n-Az\parallel }^2)+\gamma {\xi }_n\\ \le & {\parallel u_n-z\parallel }^2+M^{\ast }{\mu }_n+\gamma {\xi }_n\\ =& {\parallel (1-{\alpha }_n)(x_n-z)-{\alpha }_{n}z\parallel }^2+M^{\ast }{\mu }_n+\gamma {\xi }_n\\ \le & {(1-{\alpha }_n)}^2{\parallel x_n-z\parallel }^2-2{\alpha }_n〈u_n-z,z〉+M^{\ast }{\mu }_n+\gamma {\xi }_n\\ \le & (1-{\alpha }_n){\parallel x_n-z\parallel }^2-2{\alpha }_n〈u_n-z,z〉+M^{\ast }{\mu }_n+\gamma {\xi }_n,\end{array}$$

(3.13)

where $M^{\ast }>\gamma {sup}_{n\ge 1}(M+M_0{\parallel A{u}_n-Az\parallel }^2)>0$. It is clear that $-2〈u_n-z,z〉\to 0$, $n\to \mathrm{\infty }$ and ${\sum }_{n=1}^{\mathrm{\infty }}{M}^{\ast }{\mu }_n<\mathrm{\infty }$; ${\sum }_{n=1}^{\mathrm{\infty }}\gamma {\xi }_n<\mathrm{\infty }$. Now, using Lemma 2.4 in (3.13), we have $\parallel x_n-z\parallel \to 0$. So $x_n\to z$ as $n\to \mathrm{\infty }$.

Case 2. Assume that $\{\parallel x_n-x^{\ast }\parallel \}$ is not a monotonically decreasing sequence. Set ${\mathrm{\Gamma }}_n={\parallel x_n-x^{\ast }\parallel }^2$ and let $\tau :\mathbb{N}\to \mathbb{N}$ be a mapping for all $n\ge n_0$ (for some $n_0$ large enough) by

$$\tau (n):=max\{k\in \mathbb{N}:k\ge n,{\mathrm{\Gamma }}_k\le {\mathrm{\Gamma }}_{k+1}\}.$$

Clearly, τ is a nondecreasing sequence such that $\tau (n)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$ and

$${\mathrm{\Gamma }}_{\tau (n)}\le {\mathrm{\Gamma }}_{\tau (n)+1},\mathrm{\forall }n\ge n_0.$$

From (3.9), it is easy to see that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel y_{\tau (n)}-S(t_{\tau (n)})y_{\tau (n)}\parallel =0.$$

Furthermore, we can show that

$$\parallel (T-I)A{u}_{\tau (n)}\parallel \to 0,n\to \mathrm{\infty }.$$

By a similar argument as above in Case 1, we conclude immediately that $x_{\tau (n)}$, $y_{\tau (n)}$ and $u_{\tau (n)}$ weakly converge to z as $\tau (n)\to \mathrm{\infty }$. At the same time, from (3.13), we note that for all $n\ge n_0$,

$$\begin{array}{rcl}0& \le & {\parallel x_{\tau (n)+1}-z\parallel }^2-{\parallel x_{\tau (n)}-z\parallel }^2\\ \le & {\alpha }_{\tau (n)}[-2〈u_{\tau (n)}-z,z〉-{\parallel x_{\tau (n)}-z\parallel }^2]+M^{\ast }{\mu }_{\tau (n)}+\gamma {\xi }_{\tau (n)},\end{array}$$

which gives

$${\parallel x_{\tau (n)}-z\parallel }^2\le -2〈u_{\tau (n)}-z,z〉+M^{\ast }{\mu }_{\tau (n)}+\gamma {\xi }_{\tau (n)}.$$

Hence, we deduce that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_{\tau (n)}-z\parallel =0.$$

Therefore,

$$\underset{n\to \mathrm{\infty }}{lim}{\mathrm{\Gamma }}_{\tau (n)}=\underset{n\to \mathrm{\infty }}{lim}{\mathrm{\Gamma }}_{\tau (n)+1}=0.$$

Furthermore, for $n\ge n_0$, it is easy to see that ${\mathrm{\Gamma }}_{\tau (n)}\le {\mathrm{\Gamma }}_{\tau (n)+1}$ if $n\ne \tau (n)$ (that is $\tau (n)<n$) because ${\mathrm{\Gamma }}_j\ge {\mathrm{\Gamma }}_{j+1}$ for $\tau (n)+1\le j\le n$. As a consequence, we obtain for all $n\ge n_0$,

$$0\le {\mathrm{\Gamma }}_n\le max\{{\mathrm{\Gamma }}_{\tau (n)},{\mathrm{\Gamma }}_{\tau (n)+1}\}={\mathrm{\Gamma }}_{\tau (n)+1}.$$

This shows that $lim{\mathrm{\Gamma }}_n=0$ and hence $\{x_n\}$ converges strongly to z. This completes the proof. □

Based on Lemma 1.2 and Example 1.3, we can deduce the following corollary from Theorem 3.1.

Corollary 3.2 Let $H_1$ and $H_2$ be two real Hilbert spaces, $A:H_1\to H_2$ be a bounded linear operator and $A^{\ast }:H_2\to H_1$ be the adjoint of A. Let $\mathrm{\Im }:=\{S(t):0\le t<\mathrm{\infty }\}$ be a one-parameter nonexpansive semigroup on $H_1$. Let $T:H_2\to H_2$ be a uniformly L-Lipschitzian continuous and $(k,\{{\mu }_n\},\{{\xi }_n\},\varphi )$-total asymptotically strict pseudocontractive mapping satisfying the following conditions:

• (i) ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$; ${\sum }_{n=1}^{\mathrm{\infty }}{\xi }_n<\mathrm{\infty }$;

• (ii) $\{{\alpha }_n\}$ is a real sequence in $(0,1)$ such that ${\mu }_n=o({\alpha }_n)$, ${\xi }_n=o({\alpha }_n)$, ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$; ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (iii) there exist constants $M_0>0$, $M_1>0$ such that $\varphi (\lambda )\le M_0{\lambda }^2$, $\mathrm{\forall }\lambda >M_1$.

Let $C:={\bigcap }_{t\ge 0}F(S(t))\ne \mathrm{\varnothing }$, $Q:=F(T)\ne \mathrm{\varnothing }$ and $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be the sequence generated by $x_1\in H_1$,

$$\{\begin{array}{l}{u}_n=(1-{\alpha }_n)x_n,\\ y_n=u_n+\gamma A^{\ast }(T^n-I)A{u}_n,\\ x_{n+1}={\beta }_n{y}_n+(1-{\beta }_n)(\frac{1}{t_n}{\int }_0^{t_n}T(u)du)y_n,n\ge 1,\end{array}$$

(3.14)

where $\{{\beta }_n\}\subset (0,1)$ and $\gamma >0$ satisfy the following conditions:

• (a) $0<ϵ\le {\beta }_n\le b<1$;

• (b) $\gamma \in (0,\frac{1-k}{{\parallel A\parallel }^2})$.

If Ω is nonempty, then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to an element of Ω.

A strong mean convergence theorem for nonexpansive mappings was first established for odd mappings by Baillon [32] and it was later generalized to that of nonlinear semigroups by Reich [33]. It follows from the above proof that Theorem 3.1 is valid for nonexpansive mappings. Thus, we also have the following mean ergodic theorem for nonexpansive mappings in Hilbert spaces.

Corollary 3.3 Let $H_1$ and $H_2$ be two real Hilbert spaces, $A:H_1\to H_2$ be a bounded linear operator and $A^{\ast }:H_2\to H_1$ be the adjoint of A. Let S be a nonexpansive mapping on  $H_1$. Let $T:H_2\to H_2$ be a uniformly L-Lipschitzian continuous and $(k,\{{\mu }_n\},\{{\xi }_n\},\varphi )$-total asymptotically strict pseudocontractive mapping satisfying the following conditions:

• (i) ${\sum }_{n=1}^{\mathrm{\infty }}{\mu }_n<\mathrm{\infty }$; ${\sum }_{n=1}^{\mathrm{\infty }}{\xi }_n<\mathrm{\infty }$;

• (ii) $\{{\alpha }_n\}$ is a real sequence in $(0,1)$ such that ${\mu }_n=o({\alpha }_n)$, ${\xi }_n=o({\alpha }_n)$, ${lim}_{n\to \mathrm{\infty }}{\alpha }_n=0$, ${\sum }_{n=1}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }$;

• (iii) there exist constants $M_0>0$, $M_1>0$ such that $\varphi (\lambda )\le M_0{\lambda }^2$, $\mathrm{\forall }\lambda >M_1$.

Let $C:={\bigcap }_{t\ge 0}F(S(t))\ne \mathrm{\varnothing }$, $Q:=F(T)\ne \mathrm{\varnothing }$ and $\mathrm{\Omega }\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be the sequence generated by $x_1\in H_1$,

$$\{\begin{array}{l}{u}_n=(1-{\alpha }_n)x_n,\\ y_n=u_n+\gamma A^{\ast }(T^n-I)A{u}_n,\\ x_{n+1}={\beta }_n{y}_n+(1-{\beta }_n)(\frac{1}{n+1}{\sum }_{j=0}^n{S}^j{y}_n),n\ge 1,\end{array}$$

(3.15)

where $\{{\beta }_n\}\subset (0,1)$ and $\gamma >0$ satisfy the following conditions:

• (a) $0<ϵ\le {\beta }_n\le b<1$;

• (b) $\gamma \in (0,\frac{1-k}{{\parallel A\parallel }^2})$.

If Ω is nonempty, then the sequence ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ converges strongly to an element of Ω.