On an open question of V. Colao and G. Marino presented in the paper “Krasnoselskii–Mann method for nonself mappings”
 Meifang Guo^{1},
 Xia Li^{1} and
 Yongfu Su^{2}Email authorView ORCID ID profile
Received: 20 February 2016
Accepted: 1 August 2016
Published: 11 August 2016
Abstract
Let H be a Hilbert space and let C be a closed convex nonempty subset of H and \(T : C\rightarrow H\) a nonself nonexpansive mapping. A map \(h : C\rightarrow R\) defined by \(h(x) := \inf \{\lambda \ge 0 : \lambda x+(1\lambda )Tx \in C \}\). Then, for a fixed \(x_0 \in C\) and for \(\begin{aligned}{\alpha _0} = \max \left\{ {\frac{1}{2},h({x_0})} \right\}\end{aligned}\), Krasnoselskii–Mann algorithm is defined by \(x_{n+1}=\alpha _n+(1\alpha _n)Tx_n,\) where \(\alpha _{n+1}=\max \{\alpha _n, h(x_{x_{n+1}})\}\). Recently, Colao and Marino (Fixed Point Theory Appl 2015:39, 2015) have proved both weak and strong convergence theorems when C is a strictly convex set and T is an inward mapping. Meanwhile, they proposed a open question for a countable family of nonself nonexpansive mappings. In this article, authors will give an answer and will prove the further generalized results with the examples to support them.
Keywords
Mathematics Subject Classification
Background
 \((C_1)\) :

\(T:C\rightarrow C\);
 \((C_2)\) :

\(\sum _{n=0}^{\infty }\alpha _n(1\alpha _n)=\infty\).
Many authors are interested in lowering condition \((C_1)\) by allowing T to be nonself at the price of strengthening the requirements on the sequence \(\{\alpha _n\}\) and on the set C.
Recently, Colao and Marino (2015) introduced a new search strategy for the coefficients \(\{\alpha _n\}\) and they have proved that the Krasnoselskii–Mann algorithm (1) is well defined for this particular choice of the sequence \(\{\alpha _n\}\). Also they have proved both weak and strong convergence results for the above algorithm (1) when C is a strictly convex set and T is inward.
Lemma VG
 \((P_1)\) :

for any \(x \in C, h(x) \in [0, 1]\) and \(h(x) = 0\) if and only if \(Tx \in C\);
 \((P_2)\) :

for any \(x\in C\) and any \(\alpha \in [h(x), 1], \alpha x+(1\alpha )Tx \in C;\)
 \((P_3)\) :

if T is an inward mapping, then \(h(x) < 1\) for any \(x \in C\);
 \((P_4)\) :

whenever \(Tx \overline{\in } C, h(x)x+(1h(x))Tx \in \partial C.\)
The following is a main result of Colao and Marino (2015).
Theorem VG
Meanwhile, Colao and Marino presented the following open question.
In this paper, we will give a determinate answer for above open question VG and will give the further generalized results.
The answer for the open question and main result
The following notions will be used in this paper. Of course, these notions have been also presented in the paper of Colao and Marino (2015).
Definition 1
Definition 2
A set \(C \subset H\) is said to be strictly convex if it is convex and with the property that \(x, y \in \partial C\) and \(t \in (0, 1)\) implies that \(tx + (1  t)y \in C^0\). In other words, if the boundary \(\partial C\) does not contain any segment. Where \(\partial C\) is the boundary of C and \(C^0\) is the interior of C.
Definition 3
In order to clearly answer the open question VG, we give the following notions.
Definition 4
Let D, C be two closed and convex nonempty sets in a Hilbert space H and \(D\subset C\). For any sequence \(\{x_n\}\subset C\), if \(\{x_n\}\) converges strongly to an element \(x^* \in \partial C \setminus D, \ x_n\ne x^*\) implies that \(\{x_n\}\) is not Fej\(\acute{e}\)rmonotone with respect to the set \(D \subset C\), we called that, the pair (D, C) satisfies Scondition.
Definition 5
Let \(\{T_n\}\) be sequence of mappings from H into itself with nonempty common fixed point set \(F=\cap _{n=1}^{\infty }F(T_n)\). The \(\{T_n\}\) is said to be uniformly weakly closed if for any convergent sequence \(\{z_n\} \subset C\) such that \(\Vert T_nz_nz_n\Vert \rightarrow 0\) as \(n\rightarrow \infty\), the weak cluster points of \(\{z_n\}\) belong to F.
Lemma 6
The following theorem is main result which is also a answer to the open question of Colao and Marino.
Theorem 7
 1.
if there exist \(a,b \in (0,1)\) such that \(\alpha _n \in [a,b]\) for all \(n\ge 0\), the \(\{x_n\}\) weakly converges to a common fixed point \(p \in F\).
 2.
if \(\sum _{n=0}^{\infty }(1\alpha _n)<+\infty\), and (F, C) satisfies Scondition, the \(\{x_n\}\) converges strongly to a common fixed point \(p \in F\).
Proof
Remark
The proved theorem is a partial answer to the open question that it is not completely satisfactory. In fact the assumption that can not approach to 1, imposes a restriction a priori on \(\alpha _n\). It remains an open question whether the thesis holds without assumptions a priori on.
Definition 8
By using the same method of proof as in Theorem 7, we can prove Theorem 7 is still right for quasinonexpansive mappings. Therefore, we can get the further generalized result as follows.
Theorem 9
 1.
if there exist \(a,b \in (0,1)\) such that \(\alpha _n \in [a,b]\) for all \(n\ge 0\), the \(\{x_n\}\) weakly converges to a common fixed point \(p \in F\).
 2.
if \(\sum _{n=0}^{\infty }(1\alpha _n)<+\infty\), and (F, C) satisfies Scondition, the \(\{x_n\}\) converges strongly to a common fixed point \(p \in F\).
Examples
Theorem 10
(Rockafellar Rockafellar 1970). Let H be a Hilbert space and let A be a monotone operator from H into itself. Then A is maximal if and only if \(R(I + rA)=H\). for all \(r>0\), where I is the identity operator.
Example 11
Let H be a Hilbert space, let A be a maximal monotone operator from H into itself such that \(A^{1}0 \ne \emptyset\), let \(J_r\) be the resolvent of A, where \(r>0\). For \(r_n>0, \ \limsup _{n\rightarrow \infty } r_n>0\), the \(\{J_{r_n}\}_{n=0}^{\infty }\) is an uniformly weakly closed countable family of quasinonexpansive mappings.
Proof
Example 12
Example 13
Conclusions
 1.
if there exist \(a,b \in (0,1)\) such that \(\alpha _n \in [a,b]\) for all \(n\ge 0\), the \(\{x_n\}\) weakly converges to a common fixed point \(p \in F\).
 2.
if \(\sum _{n=0}^{\infty }(1\alpha _n)<+\infty\), and (F, C) satisfies Scondition, the \(\{x_n\}\) converges strongly to a common fixed point \(p \in F\).
Declarations
Authors' contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
This project is supported by the National Natural Science Foundation of China under Grant (11071279).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis 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
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