- Research
- Open Access
- Published:
Non-convex hybrid algorithm for a family of countable quasi-Lipschitz 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 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.
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 [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, 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)
the fixed point set \(F(T)\) is nonempty;
-
(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
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 \(z=tx+(1-t)y\) 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 \(n\geq1\), \(A=\{z \in H: \|z-P_{F}x_{0}\|\leq1\}\).
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
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 \(x_{n} \in \overline{\operatorname{co}}\, C_{n-1}\), 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 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
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
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
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
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
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
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, \(p \in F=\bigcap_{n=0}^{N-1}F(T_{n})\). 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
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 \(n=0,1,2, \ldots, N-1\), 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 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
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:
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 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.
References
Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506-510 (1953)
Halpern, B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957-961 (1967)
Genel, A, Lindenstrass, J: An example concerning fixed points. Isr. J. Math. 22, 81-86 (1975)
Youla, D: Mathematical theory of image restoration by the method of convex projection. In: Stark, H (ed.) Image Recovery: Theory and Applications, pp. 29-77. Academic Press, Orlando (1987)
Moudafi, A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46-55 (2000)
Xu, HK: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279-291 (2004)
Bauschke, HH, Combettes, PL: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26(2), 248-264 (2001)
Podilchuk, CI, Mammone, RJ: Image recovery by convex projections using a least-squares constraint. J. Opt. Soc. Am. 7(3), 517-521 (1990)
Matsushita, S-Y, Takahashi, W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory 134, 257-266 (2005)
Alber, YI: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 15-50. Dekker, New York (1996)
Alber, YI, Reich, S: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panam. Math. J. 4, 39-54 (1994)
Kamimura, S, Takahashi, W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938-945 (2002)
Cioranescu, I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic, Dordrecht (1990)
Takahashi, W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
Butnariu, D, Reich, S, Zaslavski, AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. J. Appl. Anal. 7, 151-174 (2001)
Butnariu, D, Reich, S, Zaslavski, AJ: Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numer. Funct. Anal. Optim. 24, 489-508 (2003)
Censor, Y, Reich, S: Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization. Optimization 37, 323-339 (1996)
Rockafellar, RT: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75-88 (1970)
Takahashi, W: Convex Analysis and Approximation Fixed Points. Yokohama Publishers, Yokohama (2000) (in Japanese)
Kohsaka, F, Takahashi, W: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstr. Appl. Anal. 2004, 239-249 (2004)
Ohsawa, S, Takahashi, W: Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces. Arch. Math. 81, 439-445 (2003)
Reich, S: Constructive techniques for accretive and monotone operators. In: Applied Nonlinear Analysis (Proceedings of the Third International Conference, University of Texas, Arlington, TX, 1978), pp. 335-345. Academic Press, New York (1979)
Reich, S: A weak convergence theorem for the alternating method with Bregman distance. In: Kartsatos, AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 313-318. Dekker, New York (1996)
Solodov, MV, Svaiter, BF: Forcing strong convergence of proximal point iterations in Hilbert space. Math. Program. 87, 189-202 (2000)
Nakajo, K, Takahashi, W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 279, 372-379 (2003)
Kim, T-H, Xu, H-K: Strong convergence of modified Mann iterations for asymptotically mappings and semigroups. Nonlinear Anal. 64, 1140-1152 (2006)
Martinez-Yanes, C, Xu, H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal. 64, 2400-2411 (2006)
Su, Y, Qin, X: Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators. Nonlinear Anal. 68, 3657-3664 (2008)
Dong, Q, Lu, Y: A new hybrid algorithm for a nonexpansive mapping. Fixed Point Theory Appl. 2015, 37 (2015)
Liu, Y, Zheng, L, Wang, P, Zhou, H: Three kinds of new hybrid projection methods for a finite family of quasi-asymptotically pseudocontractive mappings in Hilbert spaces. Fixed Point Theory Appl. 2015, 118 (2015)
Acknowledgements
This project is supported by major project of Hebei North University under grant (No. ZD201304).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Rights and permissions
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.
About this article
Cite this article
Guan, J., Tang, Y., Ma, P. et al. Non-convex hybrid algorithm for a family of countable quasi-Lipschitz mappings and application. Fixed Point Theory Appl 2015, 214 (2015). https://doi.org/10.1186/s13663-015-0457-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13663-015-0457-4
MSC
- 47H05
- 47H09
- 47H10
Keywords
- nonexpansive mapping
- hybrid algorithm
- Cauchy sequence
- closed quasi-nonexpansive