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Nonconvex hybrid algorithm for a family of countable quasiLipschitz mappings and application
Fixed Point Theory and Applications volume 2015, Article number: 214 (2015)
Abstract
The purpose of this article is to establish a kind of nonconvex hybrid iteration algorithms and to prove relevant strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasiLipschitz mappings in Hilbert spaces. Meanwhile, the main result is applied to get the common fixed points of finite family of quasiasymptotically nonexpansive mappings. It is worth pointing out that a nonconvex 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 quasiLipschitz mappings. The results presented in this article are interesting extensions of some current results.
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 [1–8]. 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 [9–24] 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, MartinezYanes and Xu [27] proposed a modification of the Ishikawa iteration method for nonexpansive mapping in a Hilbert space. MartinezYanes 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 quasiasymptotically 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 \(\langle zy,x z\rangle\geq0\) for every \(y\in C\). Recall that \(T:C\rightarrow C\) is nonexpansive if \(\TxTy\\leq \xy\\) 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 quasiLipschitz, if the following conditions hold:

(1)
the fixed point set \(F(T)\) is nonempty;

(2)
\(\Txp\\leq L\xp\\) for all \(x \in C\), \(p \in F(T)\),
where \(1\leq L<+\infty\) is a constant. T is said to be quasinonexpansive, 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 nonconvex hybrid iteration algorithms and to prove relevant strong convergence theorems of common fixed points for a uniformly closed asymptotically family of countable quasiLipschitz mappings in Hilbert spaces. Meanwhile, the main result was applied to get the common fixed points of finite family of quasiasymptotically nonexpansive mappings. It is worth pointing out that a nonconvex 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 quasiLipschitz mappings. The results presented in this article are interesting extensions of some current results.
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
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
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 \(z=tx+(1t)y\) for any \(t \in(0,1)\), we have
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
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
converges strongly to \(P_{F}x_{0}\), where \(\overline{\operatorname{co}}\, C_{n}\) denotes the closed convex closure of \(C_{n}\) for all \(n\geq1\), \(A=\{z \in H: \zP_{F}x_{0}\\leq1\}\).
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
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
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
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
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
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
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
From the definition of \(D_{n}\), we have
This implies that \(D_{n}\) is closed and convex, for all \(n\geq0\). Next, we show that
In fact, for any \(z \in C_{n} \), we have
From
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 \(x_{n} \in \overline{\operatorname{co}}\, C_{n1}\), we know that
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
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
as \(n\rightarrow\infty\). Since \(\alpha_{n} \in(a,1]\subset[0,1]\), from the above limit we have
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
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
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
Take \(1+(L_{0}1)\alpha_{0}=\sqrt{\frac{5}{2}}\), we have
It is easy to show that \(z_{1}=(1,3), z_{2}=(1,3) \in C_{0}\). But
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 quasinonexpansive mapping from C into itself. Assume that \(\alpha_{n} \in(a,1]\) holds for some \(a \in(0,1)\). Then \(\{x_{n}\}\) generated by
converges strongly to \(P_{F(T)}x_{0}\), where \(A=\{z \in H: \zP_{F}x_{0}\ \leq1\}\).
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 quasinonexpansive 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
converges strongly to \(P_{F(T)}x_{0}\), where \(A=\{z \in H: \zP_{F}x_{0}\ \leq1\}\).
Application to family of quasiasymptotically nonexpansive mappings
In this section, we will apply the above result to study the following finite family of asymptotically quasinonexpansive mappings \(\{T_{n}\}_{n=0}^{N1}\). Let
where F denotes the common fixed point set of \(\{T_{n}\}_{n=0}^{N1}\), \(\lim_{j\rightarrow\infty}k_{i,j}=1\) for all \(0\leq i \leq N1\). The finite family of asymptotically quasinonexpansive mappings \(\{T_{n}\}_{n=0}^{N1}\) is said to be uniformly LLipschitz, if
for all \(i=0,1,2,\ldots,N1\), \(j\geq1\), where \(L\geq1\).
Theorem 3.1
Let C be a closed convex subset of a Hilbert space H, and let \(\{T_{n}\}_{n=0}^{N1} : C\rightarrow C\) be a uniformly LLipschitz finite family of asymptotically quasinonexpansive 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
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 \(n\geq0\), \(A=\{z \in H: \zP_{F}x_{0}\\leq1\} \).
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
as \(n\rightarrow\infty\). Observe that
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
as \(k\rightarrow\infty\). That is, \(p \in F=\bigcap_{n=0}^{N1}F(T_{n})\). 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
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
On the other hand, for any \(p\in \bigcap_{n=0}^{\infty}F(T_{i(n)}^{j(n)})\), let \(n=0,1,2, \ldots, N1\), we obtain
which implies that
Hence
□
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 LLipschitz asymptotically quasinonexpansive 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
converges strongly to \(P_{F(T)}x_{0}\), where \(\overline{\operatorname{co}}\, C_{n}\) denotes the closed convex closure of \(C_{n}\) for all \(n\geq1\), \(A=\{z \in H: \zP_{F}x_{0}\\leq1\}\).
Proof
Take \(T_{n}\equiv T\) in Theorem 3.1, we obtain Corollary 3.2. □
Since a nonexpansive mapping must be a Lipschitz asymptotically quasinonexpansive mapping, from Corollary 3.2, we can obtain Corollary 2.8.
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 nonzero 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:
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
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 nonconvex 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.
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Acknowledgements
This project is supported by major project of Hebei North University under grant (No. ZD201304).
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Guan, J., Tang, Y., Ma, P. et al. Nonconvex hybrid algorithm for a family of countable quasiLipschitz mappings and application. Fixed Point Theory Appl 2015, 214 (2015). https://doi.org/10.1186/s1366301504574
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DOI: https://doi.org/10.1186/s1366301504574
MSC
 47H05
 47H09
 47H10
Keywords
 nonexpansive mapping
 hybrid algorithm
 Cauchy sequence
 closed quasinonexpansive