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Iterative approximation for split common fixed point problem involving an asymptotically nonexpansive semigroup and a total asymptotically strict pseudocontraction
Fixed Point Theory and Applications volume 2014, Article number: 131 (2014)
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 \u3008\cdot ,\cdot \u3009 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
where A:{H}_{1}\to {H}_{2} is a bounded linear operator. The SFP in finitedimensional 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
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 gradientprojection 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 infinitedimensional 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 TxTy\parallel \le \parallel xy\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in H; 
(b)
strictly pseudocontractive if there exists a constant k\in [0,1) such that
{\parallel TxTy\parallel}^{2}\le {\parallel xy\parallel}^{2}+k{\parallel (xy)(TxTy)\parallel}^{2},\phantom{\rule{1em}{0ex}}\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 xy\parallel}^{2}+k{\parallel (xy)(TxTy)\parallel}^{2}+{\mu}_{n}\varphi (\parallel xy\parallel )+{\xi}_{n},\phantom{\rule{1em}{0ex}}\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)xS(t)y\parallel \le {L}_{t}\parallel xy\parallel ,\phantom{\rule{1em}{0ex}}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,
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)x\phantom{\rule{0.2em}{0ex}}ds, then
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):
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 LLipschitzian 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 twooperator SCFP by
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 finitedimensional spaces, the convergence of the following algorithm:
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 quasinonexpansive operators.
In 2011, Moudafi [5] introduced the following relaxed algorithm:
where {y}_{n}={x}_{n}+\gamma {A}^{\ast}(TI)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 quasinonexpansive operators. We observe that strong convergence result can be obtained in the results of Moudafi [5] if a compactnesstype 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
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 quasinonexpansive operators S and T in real Hilbert spaces. Inspired by the work of Zhao and He [24], Moudafi [6] quite recently revisited the viscositytype 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 multipleset split feasibility problem (MSSFP) (1.3) for a family of multivalued quasinonexpansive 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 multivalued quasinonexpansive mappings and for each i\ge 1, {S}_{i} is demiclosed at 0. Let T:{H}_{2}\to {H}_{2} be a uniformly LLipschitzian 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
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{1k}{{\parallel A\parallel}^{2}}).
If Ω (the set of solutions of multipleset 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 semicompact, 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 multipleset split feasibility problem (1.3) for a family of multivalued quasinonexpansive mappings and a total asymptotically pseudocontractive mapping in infinitely dimensional Hilbert spaces. In order to get strong convergence, Chang et al. [25] imposed a compactnesstype condition (semicompactness) on the mappings {\{{S}_{i}\}}_{i=1}^{\mathrm{\infty}}. This compactness condition appears strong as only few mappings are semicompact.

(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 compactnesstype 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 compactnesstype 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 compactnesstype 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 compactnesstype 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 wellknown lemmas which will be used in the sequel.
Lemma 2.1 Let H be a real Hilbert space. Then the following wellknown results hold:

(i)
{\parallel x+y\parallel}^{2}={\parallel x\parallel}^{2}+2\u3008x,y\u3009+{\parallel y\parallel}^{2}, \mathrm{\forall}x,y\in H;

(ii)
{\parallel x+y\parallel}^{2}\le {\parallel x\parallel}^{2}+2\u3008y,x+y\u3009, \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 xy\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 LLipschitzian continuous and (k,\{{\mu}_{n}\},\{{\xi}_{n}\},\varphi )total asymptotically strict pseudocontractive mapping, then IT 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
Let for all n>1,
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:
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 LLipschitzian 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},
where {t}_{n}\to \mathrm{\infty} and \{{\beta}_{n}\}\subset (0,1) and \gamma >0 satisfy the following conditions:

(a)
0<\u03f5\le {\beta}_{n}\le b<1;

(b)
\gamma \in (0,\frac{1k}{{\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
Let {x}^{\ast}\in \mathrm{\Omega}. Then, by the convexity of {\parallel \cdot \parallel}^{2}, we obtain
From (3.1) and Lemma 2.1(i), we obtain that
Since
A{x}^{\ast}\in Q=F(T) and T is a total asymptotically strict pseudocontractive mapping, then we obtain
Substituting (3.5) and (3.4) into (3.3), we have
Putting (3.6) and (3.2) into (3.1), we obtain
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,
Thus, we obtain
By Lemma 2.3, we have that
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
From (3.6), we have that
This implies that
and
Hence, we obtain
Also, we observe that
Using (3.6) and Lemma 2.1(iii) in (3.1), we have
This implies from (3.2) and condition (b) that
From condition (a) we have
Hence, for any t\ge 0,
We obtain from (3.1) that
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
Consequently,
Using the fact that T is uniformly LLipschitzian, we have
From (3.8) and (3.11), we obtain
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
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
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
where {M}^{\ast}>\gamma {sup}_{n\ge 1}(M+{M}_{0}{\parallel A{u}_{n}Az\parallel}^{2})>0. It is clear that 2\u3008{u}_{n}z,z\u3009\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
Clearly, τ is a nondecreasing sequence such that \tau (n)\to \mathrm{\infty} as n\to \mathrm{\infty} and
From (3.9), it is easy to see that
Furthermore, we can show that
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},
which gives
Hence, we deduce that
Therefore,
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},
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 oneparameter nonexpansive semigroup on {H}_{1}. Let T:{H}_{2}\to {H}_{2} be a uniformly LLipschitzian 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},
where \{{\beta}_{n}\}\subset (0,1) and \gamma >0 satisfy the following conditions:

(a)
0<\u03f5\le {\beta}_{n}\le b<1;

(b)
\gamma \in (0,\frac{1k}{{\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 LLipschitzian 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},
where \{{\beta}_{n}\}\subset (0,1) and \gamma >0 satisfy the following conditions:

(a)
0<\u03f5\le {\beta}_{n}\le b<1;

(b)
\gamma \in (0,\frac{1k}{{\parallel A\parallel}^{2}}).
If Ω is nonempty, then the sequence {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} converges strongly to an element of Ω.
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Acknowledgements
This research was supported by National Research Council of Thailand (NRCT) and University of Phayao under Grant R020057216003.
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Cholamjiak, P., Shehu, Y. Iterative approximation for split common fixed point problem involving an asymptotically nonexpansive semigroup and a total asymptotically strict pseudocontraction. Fixed Point Theory Appl 2014, 131 (2014). https://doi.org/10.1186/168718122014131
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DOI: https://doi.org/10.1186/168718122014131
Keywords
 total asymptotically strict pseudocontractive mapping
 nonexpansive semigroup
 split common fixedpoint problems
 strong convergence
 Hilbert spaces