Krasnoselskii-Mann method for non-self mappings
- Vittorio Colao^{1} and
- Giuseppe Marino^{1, 2}Email author
https://doi.org/10.1186/s13663-015-0287-4
© Colao and Marino; licensee Springer. 2015
Received: 4 December 2014
Accepted: 3 March 2015
Published: 13 March 2015
Abstract
Let H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If \(T:C\to H\) is a non-self and non-expansive mapping, we can define a map \(h:C\to\mathbb{R}\) by \(h(x):=\inf\{\lambda\geq 0:\lambda x+(1-\lambda)Tx\in C\}\). Then, for a fixed \(x_{0}\in C\) and for \(\alpha_{0}:=\max\{1/2, h(x_{0})\}\), we define the Krasnoselskii-Mann algorithm \(x_{n+1}=\alpha _{n}x_{n}+(1-\alpha_{n})Tx_{n}\), where \(\alpha_{n+1}=\max\{\alpha_{n},h(x_{n+1})\}\). We will prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping.
Keywords
1 Introduction
Let C be a closed, convex and nonempty subset of a Hilbert space H and let \(T:C\to H\) be a non-expansive mapping such that the fixed point set \(\operatorname{Fix}(T):=\{x\in C:Tx=x\}\) is not empty.
- (C1)
T is a self-mapping, i.e., \(T:C\to C\) and
- (C2)
\(\{\alpha_{n}\}\) is such that \(\sum_{n}\alpha _{n}(1-\alpha_{n})=+\infty\).
Historically, the inward condition and its generalizations were widely used to prove convergence results for both implicit [8–11] and explicit (see, e.g., [1, 12–14]) 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 \(P:H\to C\) at each step.
We point out that 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.
We stress that the main difference between the classical Krasnoselskii-Mann and our algorithm is that the choice of the coefficient \(\alpha_{n}\) is not made a priori in the latter, but it is constructed step to step and determined by the values of the map T and the geometry of the set C.
2 Main result
We will make use of the following.
Definition 1
We refer to [15] for a comprehensive survey on the properties of the inward mappings.
Definition 2
Definition 3
Lemma 1
- (P1)
for any \(x\in C\), \(h(x)\in[0,1]\) and \(h(x)=0\) if and only if \(Tx\in C\);
- (P2)
for any \(x\in C\) and any \(\alpha\in[h(x),1]\), \(\alpha x+(1-\alpha)Tx\in C\);
- (P3)
if T is an inward mapping, then \(h(x)<1\) for any \(x\in C\);
- (P4)
whenever \(Tx\notin C\), \(h(x)x+(1-h(x))Tx\in\partial C\).
Proof
Our main result is the following.
Theorem 1
- 1.
C is strictly convex and
- 2.
T is a non-expansive mapping, which satisfies the inward condition (2) and such that \(\operatorname{Fix}(T)\neq\emptyset\),
Proof
Remark 1
Following the same line of proof, it can be easily seen that the same results hold true if the starting coefficient \(\alpha_{0}=\max\{ \frac{1}{2},h(x_{0})\}\) is substituted by \(\alpha_{0}=\max\{b,h(x_{0})\}\), where \(b\in(0,1)\) is a fixed and arbitrary value. In the statement of Theorem 1, the value \(b=\frac{1}{2}\) was taken to ease the notation.
We also note that the value \(h(x_{n})\) can be replaced, in practice, by \(h_{n}=1-\frac{1}{2^{j_{n}}}\), where \(j_{n}:=\min\{j\in\mathbb {N}:(1-\frac{1}{2^{j}})x_{n}+\frac{1}{2^{j}}Tx_{n}\in C\}\).
Remark 2
As it follows from the proof, the condition \(\sum_{n}(1-\alpha _{n})<\infty\) provides a localization result for the fixed point \(x^{*}\) as a side result. Indeed, in this case, it holds that \(x^{*}=v_{1}=v_{2}\) belongs to the boundary ∂C of the set C.
Remark 3
We illustrate the statement of our results with a brief example.
Example 1
We conclude the paper by including few question that appear to be still open to the best of our knowledge.
Question 1
It has been proved that the Krasnoselskii-Mann algorithm converges for general classes of mappings (see, e.g., [20] and [21]). By maintaining the same assumption on the set C and the inward condition of the involved map, it appears to be natural to ask for which classes of mappings the same result of Theorem 1 still holds.
Question 2
We refer to [22] and [23] for two examples regarding the classical Krasnoselskii-Mann algorithm.
Question 3
On the other hand, we observe that the strict convexity of the set C does appear to be unusual for results regarding the convergence of Krasnoselskii-Mann iterations. We do not know if our result can hold for a convex and closed set C, even at the price of strengthening the requirements on the map T.
Declarations
Acknowledgements
This project was funded by Ministero dell’Istruzione, dell’Universitá e della Ricerca (MIUR).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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