Open Access

Non-convex hybrid algorithm for a family of countable quasi-Lipschitz mappings and application

  • Jinyu Guan1,
  • Yanxia Tang1,
  • Pengcheng Ma1,
  • Yongchun Xu1Email author and
  • Yongfu Su2
Fixed Point Theory and Applications20152015:214

https://doi.org/10.1186/s13663-015-0457-4

Received: 28 July 2015

Accepted: 5 November 2015

Published: 21 November 2015

Abstract

The purpose of this article is to establish a kind of non-convex hybrid iteration algorithms and to prove relevant strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in Hilbert spaces. Meanwhile, the main result is applied to get the common fixed points of finite family of quasi-asymptotically nonexpansive mappings. It is worth pointing out that a non-convex hybrid iteration algorithm is first presented in this article, a new technique is applied in our process of proof. Finally, an example is given which is a uniformly closed asymptotically family of countable quasi-Lipschitz mappings. The results presented in this article are interesting extensions of some current results.

Keywords

nonexpansive mappinghybrid algorithmCauchy sequenceclosed quasi-nonexpansive

MSC

47H0547H0947H10

1 Introduction

Construction of fixed points of nonexpansive mappings (and asymptotically nonexpansive mappings) is an important subject in the theory of nonexpansive mappings and finds application in a number of applied areas. Recently, a great deal of literature on iteration algorithms for approximating fixed points of nonexpansive mappings has been published since one has a variety of applications in inverse problem, image recovery, and signal processing; see [18]. Mann’s iteration process [1] is often used to approximate a fixed point of the operators, but it has only weak convergence (see [3] for an example). However, strong convergence is often much more desirable than weak convergence in many problems that arise in infinite dimensional spaces (see [7] and references therein). So, attempts have been made to modify Mann’s iteration process so that strong convergence is guaranteed (see [924] and references therein).

In 2003, Nakajo and Takahashi [25] proposed a modification of Mann iteration method for a single nonexpansive mapping in a Hilbert space. In 2006, Kim and Xu [26] proposed a modification of Mann iteration method for asymptotically nonexpansive mapping T in a Hilbert space. They also proposed a modification of the Mann iteration method for asymptotically nonexpansive semigroup in a Hilbert space. In 2006, Martinez-Yanes and Xu [27] proposed a modification of the Ishikawa iteration method for nonexpansive mapping in a Hilbert space. Martinez-Yanes and Xu [27] proposed also a modification of the Halpern iteration method for nonexpansive mapping in a Hilbert space. In 2008, Su and Qin [28] proposed first a monotone hybrid iteration method for nonexpansive mapping in a Hilbert space. In 2015, Dong and Lu [29] proposed a new iteration method for nonexpansive mapping in a Hilbert space. In 2015, Liu et al. [30] proposed a new iteration method for a finite family of quasi-asymptotically pseudocontractive mappings in a Hilbert spaces.

Throughout this paper, let H be a real Hilbert space with inner product \(\langle\cdot,\cdot\rangle\) and norm \(\|\cdot\|\). We write \(x_{n} \rightarrow x\) to indicate that the sequence \(\{x_{n}\}\) converges strongly to x. We write \(x_{n} \rightharpoonup x\) to indicate that the sequence \(\{x_{n}\}\) converges weakly to x. Let C be a nonempty, closed, and convex subset of H, we denote by \(P_{C}(\cdot)\) the metric projection onto C. It is well known that \(z = P_{C}(x)\) is equivalent to that \(z\in C\) and \(\langle z-y,x- z\rangle\geq0\) for every \(y\in C\). Recall that \(T:C\rightarrow C\) is nonexpansive if \(\|Tx-Ty\|\leq \|x-y\|\) for all \(x,y \in C\). A point \(x\in C\) is a fixed point of T provided \(Tx=x\). Denote by \(F(T)\) the set of fixed points of T, that is, \(F(T)=\{x\in C:Tx=x\}\). It is well known that \(F(T)\) is closed and convex. A mapping \(T: C\rightarrow C\) is said to be quasi-Lipschitz, if the following conditions hold:
  1. (1)

    the fixed point set \(F(T)\) is nonempty;

     
  2. (2)

    \(\|Tx-p\|\leq L\|x-p\|\) for all \(x \in C\), \(p \in F(T)\),

     
where \(1\leq L<+\infty\) is a constant. T is said to be quasi-nonexpansive, if \(L=1\).

Recall that a mapping \(T:C\rightarrow C\) is said to be closed if \(x_{n}\rightarrow x\) and \(\|Tx_{n}-x_{n}\|\rightarrow0\) as \(n\rightarrow\infty\) implies \(Tx=x\). A mapping \(T:C\rightarrow C\) is said to be weak closed if \(x_{n}\rightharpoonup x\) and \(\|Tx_{n}-x_{n}\|\rightarrow0\) as \(n\rightarrow\infty\) implies \(Tx=x\). It is obvious that a weak closed mapping must be a closed mapping, the inverse is not true.

Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let \(\{T_{n}\}\) be sequence of mappings from C into itself with a nonempty common fixed point set F. \(\{T_{n}\}\) is said to be uniformly closed if for any convergent sequence \(\{z_{n}\} \subset C\) such that \(\|T_{n}z_{n}-z_{n}\|\rightarrow0\) as \(n\rightarrow\infty\), the limit of \(\{z_{n}\}\) belongs to F.

The purpose of this article is to establish a kind of non-convex hybrid iteration algorithms and to prove relevant strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasi-Lipschitz mappings in Hilbert spaces. Meanwhile, the main result was applied to get the common fixed points of finite family of quasi-asymptotically nonexpansive mappings. It is worth pointing out that a non-convex hybrid iteration algorithm was first presented in this article, a new technique has been applied in our process of proof. Finally, an example has been given which is a uniformly closed asymptotically family of countable quasi-Lipschitz mappings. The results presented in this article are interesting extensions of some current results.

2 Main results

The following lemma is well known and is useful for our conclusions.

Lemma 2.1

Let C be a nonempty, closed, and convex subset of real Hilbert space H. Given \(x \in H\) and \(z\in C\). Then \(z = P_{C}x\) if and only if we have the relation
$$\langle x-z, z-y \rangle\geq0 $$
for all \(y \in C\).

Definition 2.2

Let H be a Hilbert space, let C be a closed convex subset of E, and let \(\{T_{n}\}\) be a family of countable quasi-\(L_{n}\)-Lipschitz mappings from C into itself, \(\{T_{n}\}\) is said to be asymptotically, if \(\lim_{n\rightarrow\infty}L_{n}=1\).

Lemma 2.3

Let H be a Hilbert space, let C be a closed convex subset of E, and let \(\{T_{n}\}\) be a uniformly closed asymptotically family of countable quasi-\(L_{n}\)-Lipschitz mappings from C into itself. Then the common fixed point set F is closed and convex.

Proof

Let \(p_{n} \in F\) and \(p_{n}\rightarrow p\) as \(n\rightarrow\infty\), we have
$$\|T_{n}p_{n}-p_{n}\|=0\rightarrow0, \quad p_{n}\rightarrow p $$
as \(n\rightarrow\infty\). Since \(\{T_{n}\}\) is uniformly closed, we know that \(p \in F\), therefore F is closed. Next we show that F is also convex. For any \(x,y \in F\), let \(z=tx+(1-t)y\) for any \(t \in(0,1)\), we have
$$\begin{aligned} \Vert T_{n}z-z\Vert ^{2} =& \langle T_{n}z-z,T_{n}z-z\rangle \\ =& \Vert T_{n}z\Vert ^{2}-2 \langle T_{n}z,z \rangle+ \Vert z\Vert ^{2} \\ =& \Vert T_{n}z\Vert ^{2}-2 \bigl\langle T_{n}z, tx+(1-t)y\bigr\rangle + \Vert z\Vert ^{2} \\ =& \Vert T_{n}z\Vert ^{2}-2 t\langle T_{n}z, x \rangle+2(1-t)\langle T_{n}z, y\rangle + \Vert z\Vert ^{2} \\ =&t \Vert T_{n}z-x\Vert ^{2}+(1-t)\Vert T_{n}z-y\Vert ^{2}-t\Vert x\Vert ^{2}-(1-t) \Vert y\Vert ^{2}+\Vert z\Vert ^{2} \\ \leq& t L_{n}^{2}\Vert z-x\Vert ^{2}+(1-t)L_{n}^{2} \Vert z-y\Vert ^{2}-t\Vert x\Vert ^{2}-(1-t)\Vert y \Vert ^{2}+\Vert z\Vert ^{2} \\ =& t \Vert z-x\Vert ^{2}+(1-t)\Vert z-y\Vert ^{2}-t\Vert x\Vert ^{2}-(1-t)\Vert y\Vert ^{2}+\Vert z\Vert ^{2} \\ &{} +t\bigl(L_{n}^{2}-1\bigr)\Vert z-x\Vert ^{2}+(1-t) \bigl(L_{n}^{2}-1\bigr)\Vert y-x \Vert ^{2} \\ =& \Vert z\Vert ^{2}- 2\langle z, z\rangle+\Vert z\Vert ^{2} \\ &{}+t\bigl(L_{n}^{2}-1\bigr)\Vert z-x\Vert ^{2}+(1-t) \bigl(L_{n}^{2}-1\bigr)\Vert y-x \Vert ^{2}\rightarrow0 \end{aligned}$$
as \(n\rightarrow\infty\). Since \(z\rightarrow z\), and \(\{T_{n}\}\) is uniformly closed, \(z\in F\). Therefore F is convex. This completes the proof. □

The following conclusion is well known.

Lemma 2.4

Let C be a closed convex subset of a Hilbert space H, for any given \(x_{0} \in H\), we have
$$p=P_{C}x_{0}\quad \Leftrightarrow\quad \langle p-z, x_{0}-p\rangle\geq0, \quad \forall z \in C. $$

Theorem 2.5

Let C be a closed convex subset of a Hilbert space H, and let \(\{T_{n}\} : C\rightarrow C\) be a uniformly closed asymptotically family of countable quasi-\(L_{n}\)-Lipschitz mappings from C into itself. Assume that \(\alpha_{n} \in(a,1]\) holds for some \(a \in(0,1)\). Then \(\{x_{n}\}\) generated by
$$\textstyle\begin{cases} x_{0} \in C=Q_{0} \quad \textit{chosen arbitrarily}, \\ y_{n} =(1-\alpha_{n})x_{n}+ \alpha_{n}T_{n}x_{n},\quad n\geq0, \\ C_{n}=\{z\in C: \|y_{n}-z\|\leq(1+(L_{n}-1)\alpha_{n}) \|x_{n}-z\|\}\cap A,\quad n\geq0 , \\ Q_{n}=\{ z \in Q_{n-1}: \langle x_{n}-z, x_{0}-x_{n}\rangle\geq0 \}, \quad n\geq1, \\ x_{n+1}=P_{\overline{\operatorname{co}}\, C_{n}\cap Q_{n}}x_{0}, \end{cases} $$
converges strongly to \(P_{F}x_{0}\), where \(\overline{\operatorname{co}}\, C_{n}\) denotes the closed convex closure of \(C_{n}\) for all \(n\geq1\), \(A=\{z \in H: \|z-P_{F}x_{0}\|\leq1\}\).

Proof

We split the proof into seven steps.

Step 1. It is obvious that \(\overline{\operatorname{co}}\, C_{n}\), \(Q_{n}\) are closed and convex for all \(n\geq0\). Next, we show that \(F\cap A\subset\overline{\operatorname{co}}\, C_{n}\) for all \(n\geq0\). Indeed, for each \(p\in F\cap A\), we have
$$\begin{aligned} \Vert y_{n}-p\Vert & =\bigl\Vert (1-\alpha_{n})x_{n}+ \alpha_{n}T_{n}x_{n}-p\bigr\Vert \\ &=\bigl\Vert \alpha_{n}(x_{n}-p)+(1-\alpha_{n}) (T_{n}x_{n}-p)\bigr\Vert \\ & \leq(1-\alpha_{n})\Vert x_{n}-p\Vert + \alpha_{n}L_{n}\Vert x_{n}-p\Vert \\ & = \bigl(1+(L_{n}-1)\alpha_{n}\bigr) \Vert x_{n}-z\Vert \end{aligned}$$
and \(p\in A\), so \(p\in C_{n}\) which implies that \(F\cap A\subset C_{n}\) for all \(n\geq0\). Therefore, \(F\cap A\subset\overline{\operatorname{co}}\, C_{n}\) for all \(n\geq0\).
Step 2. We show that \(F\cap A \subset \overline{\operatorname{co}}\, C_{n}\cap Q_{n} \) for all \(n\geq0\). It suffices to show that \(F\cap A \subset Q_{n}\), for all \(n\geq0\). We prove this by mathematical induction. For \(n=0\), we have \(F\cap A \subset C= Q_{0}\). Assume that \(F\cap A \subset Q_{n}\). Since \(x_{n+1}\) is the projection of \(x_{0}\) onto \(\overline{\operatorname{co}}\, C_{n}\cap Q_{n}\), from Lemma 2.1, we have
$$\langle x_{n+1}-z,x_{n+1}-x_{0}\rangle\leq0, \quad \forall z\in\overline {\operatorname{co}}\, C_{n}\cap Q_{n} $$
as \(F\cap A \subset\overline{\operatorname{co}}\, C_{n}\cap Q_{n} \), the last inequality holds, in particular, for all \(z\in F\cap A\). This together with the definition of \(Q_{n+1}\) implies that \(F\cap A\subset Q_{n+1}\). Hence the \(F\cap A\subset\overline{\operatorname{co}}\, C_{n}\cap Q_{n}\) holds for all \(n\geq0\).
Step 3. We prove \(\{x_{n}\}\) is bounded. Since F is a nonempty, closed, and convex subset of C, there exists a unique element \(z_{0}\in F\) such that \(z_{0}=P_{F}x_{0}\). From \(x_{n+1}=P_{\overline{\operatorname{co}}\, C_{n}\cap Q_{n}}x_{0}\), we have
$$\|x_{n+1}-x_{0}\|\leq\|z-x_{0}\| $$
for every \(z\in\overline{\operatorname{co}}\, C_{n}\cap Q_{n}\). As \(z_{0}\in F \cap A \subset\overline{\operatorname{co}}\, C_{n}\cap Q_{n}\), we get
$$\|x_{n+1}-x_{0}\|\leq\|z_{0}-x_{0}\| $$
for each \(n\geq0\). This implies that \(\{x_{n}\}\) is bounded.
Step 4. We show that \(\{x_{n}\}\) converges strongly to a point of C (we show that \(\{x_{n}\}\) is a Cauchy sequence). As \(x_{n+1}=P_{\overline{\operatorname{co}}\, C_{n}\cap Q_{n}}x_{0}\subset Q_{n}\) and \(x_{n}=P_{Q_{n}}x_{0}\) (Lemma 2.4), we have
$$\|x_{n+1}-x_{0}\|\geq\|x_{n}-x_{0}\| $$
for every \(n\geq0\), which together with the boundedness of \(\|x_{n}-x_{0}\|\) implies that there exists the limit of \(\|x_{n} -x_{0}\|\). On the other hand, from \(x_{n+m}\in Q_{n}\), we have \(\langle x_{n}-x_{n+m}, x_{n}-x_{0}\rangle\leq0\) and hence
$$\begin{aligned} \begin{aligned} \|x_{n+m}-x_{n}\|^{2}&=\bigl\Vert (x_{n+m}-x_{0})-(x_{n}-x_{0})\bigr\Vert ^{2} \\ &\leq\|x_{n+m}-x_{0}\|^{2}-\|x_{n}-x_{0} \|^{2}-2\langle x_{n+m}-x_{n}, x_{n}-x_{0} \rangle \\ &\leq\|x_{n+m}-x_{0}\|^{2}-\|x_{n}-x_{0} \|^{2}\rightarrow0 ,\quad n\rightarrow\infty \end{aligned} \end{aligned}$$
for any \(m\geq1\). Therefore \(\{x_{n}\}\) is a Cauchy sequence in C, then there exists a point \(q\in C\) such that \(\lim_{n\rightarrow\infty}x_{n}=q\).
Step 5. We show that \(y_{n}\rightarrow q\), as \(n\rightarrow\infty\). Let
$$D_{n}=\bigl\{ z \in C: \|y_{n}-z\|^{2}\leq \|x_{n}-z\|^{2}+4(L_{n}-1) (L_{n}+1) \bigr\} . $$
From the definition of \(D_{n}\), we have
$$\begin{aligned} D_{n} =&\bigl\{ z\in C: \langle y_{n}-z,y_{n}-z \rangle\leq\langle x_{n}-z,x_{n}-z \rangle +(L_{n}-1) (L_{n}+1)2 \bigr\} \\ =&\bigl\{ z\in C: \|y_{n}\|^{2}-2 \langle y_{n},z \rangle+\|z\|^{2} \leq\|x_{n}\|^{2}-2\langle x_{n}, z\rangle+\|z \|^{2}+4(L_{n}-1) (L_{n}+1) \bigr\} \\ =&\bigl\{ z\in C: 2 \langle x_{n}- y_{n},z\rangle \leq\|x_{n}\|^{2}-\|y_{n}\|^{2}+4(L_{n}-1) (L_{n}+1) \bigr\} . \end{aligned}$$
This implies that \(D_{n}\) is closed and convex, for all \(n\geq0\). Next, we show that
$$C_{n}\subset D_{n}, \quad n\geq0. $$
In fact, for any \(z \in C_{n} \), we have
$$\begin{aligned} \|y_{n}-z\|^{2}&\leq\bigl(1+(L_{n}-1) \alpha_{n}\bigr)^{2} \|x_{n}-z\|^{2} \\ &=\|x_{n}-z\|^{2}+\bigl[2(L_{n}-1) \alpha_{n}+(L_{n}-1)^{2}\alpha_{n}^{2} \bigr]\|x_{n}-z\|^{2} \\ &\leq\|x_{n}-z\|^{2}+\bigl[2(L_{n}-1)+(L_{n}-1)^{2} \bigr]\|x_{n}-z\|^{2} \\ &= \|x_{n}-z\|^{2}+(L_{n}-1) (L_{n}+1) \|x_{n}-z\|^{2}. \end{aligned}$$
From
$$C_{n}=\bigl\{ z\in C: \|y_{n}-z\|\leq\bigl(1+(L_{n}-1) \alpha_{n}\bigr) \|x_{n}-z\|\bigr\} \cap A,\quad n\geq0, $$
we have \(C_{n}\subset A\), \(n\geq0\). Since A is convex, we also have \(\overline{\operatorname{co}}\, C_{n}\subset A\), \(n\geq0\). Consider \(x_{n} \in \overline{\operatorname{co}}\, C_{n-1}\), we know that
$$\begin{aligned} \|y_{n}-z\|^{2} & \leq\|x_{n}-z \|^{2}+(L_{n}-1) (L_{n}+1)\|x_{n}-z \|^{2} \\ & \leq\|x_{n}-z\|^{2}+4(L_{n}-1) (L_{n}+1). \end{aligned}$$
This implies that \(z \in D_{n}\) and hence \(C_{n}\subset D_{n}\), \(n\geq0\). Since \(D_{n}\) is convex, we have \(\overline{\operatorname{co}} (C_{n})\subset D_{n}\), \(n\geq0\). Therefore
$$\|y_{n}-x_{n+1}\|^{2} \leq\|x_{n}-x_{n+1} \|^{2}+4(L_{n}-1) (L_{n}+1)\rightarrow0 $$
as \(n\rightarrow\infty\). That is, \(y_{n}\rightarrow q\) as \(n\rightarrow\infty\).
Step 6. We show that \(q\in F\). From the definition of \(y_{n}\), we have
$$\alpha_{n}\|T_{n}x_{n}-x_{n}\|= \|y_{n}-x_{n}\|\rightarrow0 $$
as \(n\rightarrow\infty\). Since \(\alpha_{n} \in(a,1]\subset[0,1]\), from the above limit we have
$$\lim_{n\rightarrow\infty}\|T_{n}x_{n}-x_{n}\|= 0. $$
Since \(\{T_{n}\}\) is uniformly closed and \(x_{n}\rightarrow q\), we have \(q\in F\).
Step 7. We claim that \(q=z_{0}=P_{F}x_{0}\), if not, we have that \(\|x_{0}-p\|>\|x_{0}-z_{0}\|\). There must exist a positive integer N, if \(n>N\) then \(\|x_{0}-x_{n}\|>\|x_{0}-z_{0}\|\), which leads to
$$ \|z_{0}-x_{0}\|^{2}=\|z_{0}-x_{n}+x_{n}-x_{0} \|^{2} =\|z_{0}-x_{n}\|^{2}+\|x_{n}-x_{0} \|^{2}+2\langle z_{0}-x_{n}, x_{n}-x_{0} \rangle. $$
It follows that \(\langle z_{0}-x_{n}, x_{n}-x_{0}\rangle<0\) which implies that \(z_{0} \, \overline{\in}\, Q_{n}\), so that \(z_{0}\, \overline{\in}\, F\), this is a contradiction. This completes the proof. □

Next, we give an example of \(C_{n}\) not involving a convex subset.

Example 2.6

Let \(H=R^{2}\), \(T_{n}: R^{2}\rightarrow R^{2}\) be a sequence of mappings defined by
$$T_{n}: (t_{1},t_{2})\mapsto\biggl(t_{1}, \frac{1}{8}t_{2}\biggr), \quad \forall (t_{1},t_{2}) \in R^{2}, \forall n\geq0. $$
It is obvious that \(\{T_{n}\}\) is a uniformly closed asymptotically family of countable quasi-\(L_{n}\)-Lipschitz mappings with the common fixed point set \(F=\{(t_{1},0): t_{1} \in(-\infty,+\infty)\}\). Take \(x_{0}=(4,0)\), \(\alpha_{0}=\frac{6}{7}\), we have
$$y_{0}=\frac{1}{7}x_{0}+\frac{6}{7}T_{0}x_{0}= \biggl(4\times\frac{1}{7}+\frac {4}{8}\times\frac{6}{7}, 0 \biggr)=(1,0). $$
Take \(1+(L_{0}-1)\alpha_{0}=\sqrt{\frac{5}{2}}\), we have
$$C_{0}=\biggl\{ z \in R^{2}: \|y_{0}-z\|\leq\sqrt{ \frac{5}{2}}\|x_{0}-z\| \biggr\} . $$
It is easy to show that \(z_{1}=(1,3), z_{2}=(-1,3) \in C_{0}\). But
$$z'=\frac{1}{2}z_{1}+\frac{1}{2}z_{2}=(0,3) \, \overline{\in}\, C_{0}, $$
since \(\|y_{0}-z'\|=2\), \(\|x_{0}-z'\|=1\). Therefore \(C_{0}\) is not convex.

Corollary 2.7

Let C be a closed convex subset of a Hilbert space H, and let \(T : C\rightarrow C\) be a closed quasi-nonexpansive mapping from C into itself. Assume that \(\alpha_{n} \in(a,1]\) holds for some \(a \in(0,1)\). Then \(\{x_{n}\}\) generated by
$$\textstyle\begin{cases} x_{0} \in C=Q_{0} \quad \textit{chosen arbitrarily}, \\ y_{n} =(1-\alpha_{n})x_{n}+ \alpha_{n}Tx_{n}, \quad n\geq0, \\ C_{n}=\{z\in C: \|y_{n}-z\|\leq \|x_{n}-z\|\}\cap A,\quad n\geq0 , \\ Q_{n}=\{ z \in Q_{n-1}: \langle x_{n}-z, x_{0}-x_{n}\rangle\geq0 \}, \quad n\geq1, \\ x_{n+1}=P_{ C_{n}\cap Q_{n}}x_{0}, \end{cases} $$
converges strongly to \(P_{F(T)}x_{0}\), where \(A=\{z \in H: \|z-P_{F}x_{0}\| \leq1\}\).

Proof

Take \(T_{n}\equiv T\), \(L_{n}\equiv1\) in Theorem 2.5, in this case, \(C_{n}\) is closed and convex, for all \(n\geq0\), by using Theorem 2.5, we obtain Corollary 2.7. □

Since a nonexpansive mapping must be a closed quasi-nonexpansive mapping, from Corollary 2.7, we obtain the following result.

Corollary 2.8

Let C be a closed convex subset of a Hilbert space H, and let \(T : C\rightarrow C\) be a nonexpansive mapping from C into itself. Assume that \(\alpha_{n} \in(a,1]\) holds for some \(a \in(0,1)\). Then \(\{x_{n}\}\) generated by
$$\textstyle\begin{cases} x_{0} \in C=Q_{0} \quad \textit{chosen arbitrarily}, \\ y_{n} =(1-\alpha_{n})x_{n}+ \alpha_{n}Tx_{n}, \quad n\geq0, \\ C_{n}=\{z\in C: \|y_{n}-z\|\leq \|x_{n}-z\|\}\cap A, \quad n\geq0 , \\ Q_{n}=\{ z \in Q_{n-1}: \langle x_{n}-z, x_{0}-x_{n}\rangle\geq0 \}, \quad n\geq1, \\ x_{n+1}=P_{ C_{n}\cap Q_{n}}x_{0}, \end{cases} $$
converges strongly to \(P_{F(T)}x_{0}\), where \(A=\{z \in H: \|z-P_{F}x_{0}\| \leq1\}\).

3 Application to family of quasi-asymptotically nonexpansive mappings

In this section, we will apply the above result to study the following finite family of asymptotically quasi-nonexpansive mappings \(\{T_{n}\}_{n=0}^{N-1}\). Let
$$\bigl\Vert T_{i}^{j}x-p\bigr\Vert \leq k_{i,j} \|x-p\|, \quad \forall x \in C, p \in F, $$
where F denotes the common fixed point set of \(\{T_{n}\}_{n=0}^{N-1}\), \(\lim_{j\rightarrow\infty}k_{i,j}=1\) for all \(0\leq i \leq N-1\). The finite family of asymptotically quasi-nonexpansive mappings \(\{T_{n}\}_{n=0}^{N-1}\) is said to be uniformly L-Lipschitz, if
$$\bigl\Vert T_{i}^{j}x-T_{i}^{j}y \bigr\Vert \leq L\|x-y\|, \quad \forall x, y \in C $$
for all \(i=0,1,2,\ldots,N-1\), \(j\geq1\), where \(L\geq1\).

Theorem 3.1

Let C be a closed convex subset of a Hilbert space H, and let \(\{T_{n}\}_{n=0}^{N-1} : C\rightarrow C\) be a uniformly L-Lipschitz finite family of asymptotically quasi-nonexpansive mappings with nonempty common fixed point set F. Assume that \(\alpha_{n} \in(a,1]\) holds for some \(a \in(0,1)\). Then \(\{x_{n}\}\) generated by
$$\textstyle\begin{cases} x_{0} \in C=Q_{0} \quad \textit{chosen arbitrarily}, \\ y_{n} =(1-\alpha_{n})x_{n}+ \alpha_{n}T_{i(n)}^{j(n)}x_{n}, \quad n\geq0, \\ C_{n}=\{z\in C: \|y_{n}-z\|\leq(1+(k_{i(n),j(n)}-1)\alpha_{n}) \|x_{n}-z\|\}\cap A, \quad n\geq 0 , \\ Q_{n}=\{ z \in Q_{n-1}: \langle x_{n}-z, x_{0}-x_{n}\rangle\geq0 \}, \quad n\geq1, \\ x_{n+1}=P_{\overline{\operatorname{co}}\, C_{n}\cap Q_{n}}x_{0}, \end{cases} $$
converges strongly to \(P_{F}x_{0}\), where \(\overline{\operatorname{co}}\, C_{n}\) denotes the closed convex closure of \(C_{n}\) for all \(n\geq1\), \(n=(j(n)-1)N+i(n)\) for all \(n\geq0\), \(A=\{z \in H: \|z-P_{F}x_{0}\|\leq1\} \).

Proof

It is sufficient to prove the following two conclusions.

Conclusion 1

\(\{T_{i(n)}^{j(n)}\}_{n=0}^{\infty}\) is a uniformly closed asymptotically family of countable quasi-\(L_{n}\)-Lipschitz mappings from C into itself.

Conclusion 2

\(F=\bigcap_{n=0}^{N}F(T_{n})=\bigcap_{n=0}^{\infty}F(T_{i(n)}^{j(n)})\), where \(F(T)\) denotes the fixed point set of the mapping T.

Proof of Conclusion 1

Let
$$\bigl\Vert T_{i(n)}^{j(n)}x_{n}-x_{n}\bigr\Vert \rightarrow0, \quad x_{n}\rightarrow p $$
as \(n\rightarrow\infty\). Observe that
$$\begin{aligned} \Vert T_{i(n)}x_{n}-x_{n}\Vert \leq&\bigl\Vert T_{i(n)}^{j(n)}x_{n}-x_{n}\bigr\Vert + \bigl\Vert T_{i(n)}^{j(n)}x_{n}-T_{i(n)}x_{n} \bigr\Vert \\ \leq&\bigl\Vert T_{i(n)}^{j(n)}x_{n}-x_{n} \bigr\Vert + L\bigl\Vert T_{i(n)}^{j(n)-1}x_{n}-x_{n} \bigr\Vert \\ \leq&\bigl\Vert T_{i(n)}^{j(n)}x_{n}-x_{n} \bigr\Vert + L\bigl\Vert T_{i(n)}^{j(n-N)}x_{n}-T_{i(n)}^{j(n-N)}x_{n-N} \bigr\Vert \\ &{} + L\bigl\Vert T_{i(n-N)}^{j(n-N)}x_{n-N}-x_{n-N} \bigr\Vert +L\Vert x_{n-N}-x_{n}\Vert \\ \leq& \bigl\Vert T_{i(n)}^{j(n)}x_{n}-x_{n} \bigr\Vert +\bigl(L+L^{2}\bigr)\Vert x_{n-N}-x_{n} \Vert \\ &{} +L\bigl\Vert T_{i(n-N)}^{j(n-N)}x_{n-N}-x_{n-N} \bigr\Vert \end{aligned}$$
from which it turns out that \(\|T_{i(n)}x_{n}-x_{n}\|\rightarrow0\) as \(n\rightarrow\infty\). This implies there exists subsequence \(\{n_{k}\}\subset\{x_{n}\}\) such that
$$\Vert T_{i}x_{n_{k}}-x_{n_{k}}\Vert \rightarrow0, \quad i=0,1,2, \ldots, N-1 $$
as \(k\rightarrow\infty\). That is, \(p \in F=\bigcap_{n=0}^{N-1}F(T_{n})\). Therefore, \(p \in \bigcap_{n=0}^{\infty}F(T_{i(n)}^{j(n)})\), hence \(\{T_{i(n)}^{j(n)}\}\) is uniformly closed. On the other hand, we have
$$\bigl\Vert T_{i(n)}^{j(n)}x-p\bigr\Vert \leq k_{i(n),j(n)} \|x-p\|, \quad \forall x \in C, p\in\bigcap_{n=0}^{\infty}F \bigl(T_{i(n)}^{j(n)}\bigr), $$
and \(\lim_{n\rightarrow\infty}k_{i(n),j(n)}=1\). So, \(\{T_{i(n)}^{j(n)}\}\) is a uniformly closed asymptotically family of countable quasi-\(L_{n}\)-Lipschitz mappings from C into itself with \(L_{n}=k_{i(n),j(n)}\). □

Proof of Conclusion 2

It is obvious that
$$\bigcap_{n=0}^{N-1}F(T_{n})\subset \bigcap_{n=0}^{\infty}F\bigl(T_{i(n)}^{j(n)} \bigr). $$
On the other hand, for any \(p\in \bigcap_{n=0}^{\infty}F(T_{i(n)}^{j(n)})\), let \(n=0,1,2, \ldots, N-1\), we obtain
$$p\in F(T_{0}), \qquad p\in F(T_{1}),\qquad p\in F(T_{2}),\qquad \ldots,\qquad p\in F(T_{n-1}), $$
which implies that
$$\bigcap_{n=0}^{N-1}F(T_{n})\supset \bigcap_{n=0}^{\infty}F\bigl(T_{i(n)}^{j(n)} \bigr). $$
Hence
$$\bigcap_{n=0}^{N-1}F(T_{n})= \bigcap_{n=0}^{\infty}F\bigl(T_{i(n)}^{j(n)} \bigr). $$
 □

By using Theorem 2.5, the iterative sequence \(\{x_{n}\}\) converges strongly to \(P_{\bigcap_{n=0}^{\infty}F(T_{i(n)}^{j(n)})}x_{0}=P_{F}x_{0}\). This completes the proof of Theorem 3.1.  □

Corollary 3.2

Let C be a closed convex subset of a Hilbert space H, and let \(T : C\rightarrow C \) be a L-Lipschitz asymptotically quasi-nonexpansive mappings with nonempty fixed point set F. Assume that \(\alpha_{n} \in(a,1]\) holds for some \(a \in(0,1)\). Then \(\{x_{n}\}\) generated by
$$\textstyle\begin{cases} x_{0} \in C=Q_{0} \quad \textit{chosen arbitrarily}, \\ y_{n} =(1-\alpha_{n})x_{n}+ \alpha_{n}T^{n}x_{n},\quad n\geq0, \\ C_{n}=\{z\in C: \|y_{n}-z\|\leq(1+(k_{n}-1)\alpha_{n}) \|x_{n}-z\|\}\cap A, \quad n\geq0 , \\ Q_{n}=\{ z \in Q_{n-1}: \langle x_{n}-z, x_{0}-x_{n}\rangle\geq0 \}, \quad n\geq1, \\ x_{n+1}=P_{\overline{\operatorname{co}}\, C_{n}\cap Q_{n}}x_{0}, \end{cases} $$
converges strongly to \(P_{F(T)}x_{0}\), where \(\overline{\operatorname{co}}\, C_{n}\) denotes the closed convex closure of \(C_{n}\) for all \(n\geq1\), \(A=\{z \in H: \|z-P_{F}x_{0}\|\leq1\}\).

Proof

Take \(T_{n}\equiv T\) in Theorem 3.1, we obtain Corollary 3.2. □

Since a nonexpansive mapping must be a Lipschitz asymptotically quasi-nonexpansive mapping, from Corollary 3.2, we can obtain Corollary 2.8.

4 Example

Conclusion 4.1

Let H be a Hilbert space, \(\{x_{n}\}_{n=1}^{\infty}\subset H\) be a sequence such that it converges weakly to a non-zero element \(x_{0}\) and \(\|x_{i}-x_{j}\|\geq 1\) for any \(i\neq j\). Define a sequence of mappings \(T_{n}: H\rightarrow H\) as follows:
$$T_{n}(x)= \textstyle\begin{cases} L_{n}x_{n} & \textit{if } x=x_{n}\ (\exists n\geq 1) , \\ -x & \textit{if } x\neq x_{n}\ (\forall n\geq1), \end{cases} $$
where \(\{L_{n}\}_{n=1}^{\infty}\) is a sequence of number such that \(L_{n}>1\), \(\lim_{n\rightarrow\infty}L_{n}=1\). Then \(\{T_{n}\}\) is a uniformly closed asymptotically family of countable quasi-\(L_{n}\)-Lipschitz mappings with the common fixed point set \(F=\{0\}\).

Proof

It is obvious that \(\{T_{n}\}\) has a unique common fixed point 0. Next, we prove that \(\{T_{n}\}\) is uniformly closed. In fact, for any strong convergent sequence \(\{z_{n}\}\subset E\) such that \(z_{n}\rightarrow z_{0}\) and \(\|z_{n}-T_{n}z_{n}\|\rightarrow0\) as \(n\rightarrow\infty\), there exists sufficiently large natural number N such that \(z_{n}\neq x_{m}\), for any \(n, m >N\). Then \(T_{n}z_{n}=-z_{n}\) for \(n>N\), it follows from \(\|z_{n}-T_{n}z_{n}\|\rightarrow0\) that \(2z_{n}\rightarrow 0\) and hence \(z_{0} \in F\). Finally, from the definition of \(\{T_{n}\}\), we have
$$\|T_{n}x-0\|= \|T_{n}x\|\leq\|L_{n}x \|=L_{n}\|x-0\|, \quad \forall x \in H, $$
so that \(\{T_{n}\}\) is a uniformly closed asymptotically family of countable quasi-\(L_{n}\)-Lipschitz mappings. □

Remark

In the result of Liu et al. [30], the boundedness of C was assumed and the hybrid iterative process was complex. In our hybrid iterative process, \(C_{n}\) was constructed as a non-convex set can makes it more simple, meanwhile, the boundedness of C can be removed. Of course, a new technique has been applied in our process of proof.

Declarations

Acknowledgements

This project is supported by major project of Hebei North University under grant (No. ZD201304).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Hebei North University, Zhangjiakou, China
(2)
Department of Mathematics, Tianjin Polytechnic University, Tianjin, China

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© Guan et al. 2015