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On an open question of V. Colao and G. Marino presented in the paper “Krasnoselskii–Mann method for nonself mappings”
SpringerPlusvolume 5, Article number: 1328 (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.
Background
Let C be a closed, convex and nonempty subset of a Hilbert space H and let \(T : C \rightarrow H\) be a nonexpansive mapping such that the fixed point set \(F(T) := \{x \in C : Tx = x\}\) is nonempty. If T is a selfmapping, for \(x_0\in C\) the Mann iterative scheme
has been studied in an impressive amount of papers (see Chidume 2009 and the references therein) in the last decades and it is often called segmenting Mann (1953), Groetsch (1972), Hicks and Kubicek (1977) or Krasnoselskii–Mann (e.g., Edelstein and OBrien 1978; Hillam 1975) iteration. A general result on algorithm (1) is due to Reich (1979) and states that the sequence \(\{x_n\}\) weakly converges to a fixed point of the operator T under the following assumptions:
 \((C_1)\) :

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

\(\sum _{n=0}^{\infty }\alpha _n(1\alpha _n)=\infty\).
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.
Historically, the inward condition and its generalizations were widely used to prove convergence results for both implicit (Xu and Yin 1995; Xu 1997; Marino and Trombetta 1992; Takahashi and Kim 1998) and explicit (see, e.g., Chidume 2009; Song and Chen 2006; Song and Cho 2009; Zhou and Wang 2014) algorithms. However, we point out that the explicit case was only studied in conjunction with processes involving the calculation of a projection or a retraction \(P : H \rightarrow C\) at each step. As an example, in Song and Chen (2006), the following algorithm is studied:
where \(T : C \rightarrow H\) satisfies the weakly inward condition, f is a contraction and \(P : H \rightarrow C\) is a nonexpansive retraction. However, in many real world applications, the process of calculating P can be a resource consumption task and it may require an approximating algorithm by itself, even in the case when P is the nearest point projection.
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.
For a closed and convex set C and a map \(T : C \rightarrow H\), we define a mapping \(h : C \rightarrow R\) as
Note that the above quantity is a minimum since C is closed. The following lemma is useful which has been proved in Colao and Marino (2015).
Lemma VG
(Colao and Marino 2015) Let C be a nonempty, closed and convex set, let \(T : C \rightarrow H\) be a mapping and define \(h : C \rightarrow R\) as in (2). Then the following properties hold:
 \((P_1)\) :

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
(Colao and Marino 2015) Let C be a convex, closed and nonempty subset of a Hilbert space H and let \(T : C \rightarrow H\) be a mapping. Then the algorithm
is well defined. If we further assume that C is strictly convex and T is a nonexpansive mapping, which satisfies the inward condition (2) and such that \(F(T)\ne \emptyset\). Then \(\{x_n\}\) weakly converges to a point \(p \in F(T)\). Moreover, if \(\sum _{n=0}^{\infty }(1\alpha _n)<+\infty\), then the convergence is strong.
Meanwhile, Colao and Marino presented the following open question.
Open question VG (Colao and Marino 2015) Under which assumptions can algorithm (2) be adapted to produce a converging sequence to a common fixed point for a family of mappings ? In other words, does the algorithm
converge to a common fixed point of the family \(\{T_n\}\), where
and under suitable hypotheses ?
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
A map \(T : C \rightarrow H\) is said to be inward (or to satisfy the inward condition) if, for any \(x \in C\), it holds
We refer to Kirk and Sims (2001) for a comprehensive survey on the properties of the inward mappings.
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
A sequence \(\{y_n\} \subset C\) is Fejérmonotone with respect to a set \(D \subset C\) if, for any element \(y \in D,\)
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
(Reich 1979) Let X be a uniformly convex Banach space, \(\{x_n\}, \{y_n\}\subset X\) be two sequences, if there exists a constant \(d\ge 0\) such that
the \(\lim _{n\rightarrow \infty }\Vert x_ny_n\Vert =0\), where \(t_n \in [a,b]\subset (0,1)\) and a, b are two constants.
The following theorem is main result which is also a answer to the open question of Colao and Marino.
Theorem 7
Let C be a convex, closed and nonempty subset of a Hilbert space H and let \(\{T_n\}_{n=0}^{\infty }: C \rightarrow H\) be a uniformly weakly closed countable family of nonself nonexpansive mappings. Then the algorithm (4) is well defined. Assume that C is strictly convex and each \(T_n\) satisfies the inward condition and such that \(F=\cap _{n=0}^{\infty }F(T_n)\ne \emptyset\). Then the following conclusions hold:

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
(1) for any \(p\in F\) we have
Therefore there exists a constant d such that
and
Moreover
as \(k\rightarrow \infty\). By using Lemma 6, we have
Since \(\{x_{n}\}\) is bounded, there exists a subsequence \(\{x_{n_k}\}\) such that it converges weakly to a element \(x^* \in C.\) Since \(\{T_n\}\) is uniformly weakly closed, it follows \(x^* \in F\). Next we claim that \(\{x_n\}\) converges weakly to this element \(x^*\). If not, there exists a subsequence \(\{x_{m_k}\}\) does not converges weakly to \(x^*\), then there must exist a subsequence \(\{x_{m_{k_i}}\}\) such that it converges weakly to another element \(y^*\ne x^*\) and \(y^* \in F.\) Hilbert space H satisfies Opial’s condition, we have
This is a contradiction. This show that \(\{x_n\}\) converges weakly to this element \(x^*\in F\).
(2) Since
and by the boundedness of \(\{x_n\}\) and \(\{Tx_n\}\), it is promptly obtained that
i.e., \(\{x_n\}\) is a strongly Cauchy sequence and hence \(x_n\rightarrow x^* \in C\). If there exists a natural number N such that \(n>N\) implies \(x_n=x^*\), the conclusion is right. In the other case, note that each \(T_n\) satisfies the inward condition. Then, by applying properties \((P_2)\) and \((P_3)\) from Lemma VG, we obtain that \(h_n(x^*)<1\) for all \(n\ge 0\) and that for any \(\mu _n \in (h_n(x^*),1)\) it holds
On the other hand, we observe that since \(\lim _{n\rightarrow \infty }\alpha _n=1\) and since \(\alpha _{n+1}=\max \{\alpha _n, h_{n+1}(x_{n+1})\}\) holds, it follows that we can choose a subsequence \(\{x_{n_k}\}\) with the property that \(h_{n_k}(x_{n_k})\) is nondecreasing and \(\lim _{k\rightarrow \infty }h_{n_k}(x_{n_k})=1\). Hence
On the other hand
this together with (5) implies \(x^* \in\, \partial \, C\). Since \(\{x_n\}\) is Fej\(\acute{e}\)rmonotone with respect to a set \(F \subset C\), the Scondition implies \(x^* \in F\). This completes the proof. \(\square\)
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
A mapping \(T: C \rightarrow H\) is said to be quasinonexpansive, if the fixed point set F(T) is nonempty and
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
Let C be a convex, closed and nonempty subset of a Hilbert space H and let \(\{T_n\}_{n=0}^{\infty }: C \rightarrow H\) be a uniformly weakly closed countable family of nonself quasinonexpansive mappings. Then the algorithm (4) is well defined. Assume that C is strictly convex and each \(T_n\) satisfies the inward condition and such that \(F=\cap _{n=0}^{\infty }F(T_n)\ne \emptyset\). Then the following conclusions hold:

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
Let A be a multivalued operator from H into itself with domain \(D(A)=\{z\in E: Az\ne \emptyset \}\) and range \(R(A)=\{z\in E: z\in D(A)\}\). An operator A is said to be monotone if
for each \(x_1, x_2 \in D(A)\) and \(y_1\in Ax_1, y_2\in Ax_2\). A monotone operator A is said to be maximal if it’s graph \(G(A)=\{(x, y) : y \in Ax\}\) is not properly contained in the graph of any other monotone operator. We know that if A is a maximal monotone operator, then \(A^{1}0\) is closed and convex. The following result is also wellknown.
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.
Let H be a Hilbert space, and let A be a maximal monotone operator from H into itself. Using Theorem 10, we obtain that for every \(r>0\) and \(x\in H\), there exists a unique \(x_r\) such that
Then we can define a single valued mapping \(J_r :H \rightarrow D(A)\) by \(J_r = (I + rA)^{1}\) and such a \(J_r\) is called the resolvent of A. We know that \(A^{1}0=F(J_r)\) for all \(r > 0\), see Takahashi (2000a, b) for more details.
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
For any \(p\in \bigcap _{n=0}^{\infty }F(J_{r_n})=A^{1}0 \ne \emptyset ,\ w \in H\), from the monotonicity of A, we have
for all \(n\ge 0\). Let \(\{z_n\}\) be a sequence in H such that \(\lim _{n\rightarrow \infty }\Vert z_nJ_{r_n}z_n\Vert =0\). Let q be a weak cluster point of \(\{z_n\}\), then there exists a subsequence \(\{z_{n_k}\}\subset \{z_n\}\) such that \(\{z_{n_k}\}\) converges weakly to q. In this case, we have
It follows from
and the monotonicity of A that
for all \(w\in D(A)\) and \(w^*\in Aw\). Letting \(k\rightarrow \infty\), we have \(\langle wq, w^*\rangle \ge 0\) for all \(w\in D(A)\) and \(w^*\in Aw\). Therefore from the maximality of A, we obtain \(q\in A^{1}0\) and hence \(q \in \cap _{n=1}^{\infty }F(J_{r_n})\). Therefore, \(\{J_{r_n}\}_{n=1}^{\infty }\) is an uniformly weakly closed countable family of quasinonexpansive mappings. This completes the proof. \(\square\)
Example 12
Let \(H=R^2\),
It is obvious that, (D, C) satisfies Scondition.
Example 13
Let \(H=R^2\),
It is obvious that, (D, C) satisfies Scondition.
Conclusions
Let C be a convex, closed and nonempty subset of a Hilbert space H and let \(\{T_n\}_{n=0}^{\infty }: C \rightarrow H\) be a uniformly weakly closed countable family of nonself nonexpansive mappings. Then the algorithm (4) is well defined. Assume that C is strictly convex and each \(T_n\) satisfies the inward condition and such that \(F=\cap _{n=0}^{\infty }F(T_n)\ne \emptyset\). Then the following conclusions hold:

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 \(p \in F\).
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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.
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Keywords
 Hilbert space
 Nonexpansive mapping
 Nonself mapping
 Mann iterative scheme
 Inward condition
 Weak convergence
 Strong convergence
Mathematics Subject Classification
 47H05
 47H09
 47H10