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.
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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.
For a real sequence \(\{\alpha_{n}\}\subset(0,1)\), we will consider the iterations
If T is a self-mapping, the iterative scheme above has been studied in an impressive amount of papers (see [1] and the references therein) in the last decades and it is often called ‘segmenting Mann’ [2–4] or ‘Krasnoselskii-Mann’ (e.g., [5, 6]) iteration.
A general result on algorithm (1) is due to Reich [7] and states that the sequence \(\{x_{n}\}\) weakly converges to a fixed point of the operator T under the following assumptions:
-
(C1)
T is a self-mapping, i.e., \(T:C\to C\) and
-
(C2)
\(\{\alpha_{n}\}\) is such that \(\sum_{n}\alpha _{n}(1-\alpha_{n})=+\infty\).
In this paper, we are interested in lowering condition (C1) by allowing T to be non-self at the price of strengthening the requirements on the sequence \(\{\alpha_{n}\}\) and on the set C. Indeed, we will assume that C is a strictly convex set and that the non-expansive map \(T:C\to H\) is inward.
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.
As an example, in [12], the following algorithm is studied:
where \(T:C\to H\) satisfies the weakly inward condition, f is a contraction and \(P:H\to C\) is a non-expansive retraction.
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.
To overcome the necessity of using an auxiliary mapping P, for an inward and non-expansive mapping \(T:C\to H\), we will introduce a new search strategy for the coefficients \(\{\alpha_{n}\}\) and we will prove that the Krasnoselskii-Mann algorithm
is well defined for this particular choice of the sequence \(\{\alpha _{n}\}\). Also we will prove both weak and strong convergence results for the above algorithm when C is a strictly convex set.
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
A map \(T:C\to H\) is said to be inward (or to satisfy the inward condition) if, for any \(x\in C\), it holds
We refer to [15] 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
In other words, if the boundary ∂C does not contain any segment.
Definition 3
A sequence \(\{y_{n}\}\subset C\) is Fejér-monotone with respect to a set \(D\subset C\) if, for any element \(y\in D\),
For a closed and convex set C and a map \(T:C\to H\), we define a mapping \(h:C\to\mathbb{R}\) as
Note that the above quantity is a minimum since C is closed. In the following lemma, we group the properties of the function defined above.
Lemma 1
Let C be a nonempty, closed and convex set, let \(T:C\to H\) be a mapping and define \(h:C\to\mathbb{R}\) as in (3). Then the following properties hold:
-
(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
Properties (P1) and (P2) follow directly from the definition of h. To prove (P3), observe that (2) implies
for some \(c\geq1\). As a consequence,
In order to verify (P4), we first note that \(h(x)>0\) by property (P1) and that \(h(x)x+(1-h(x))Tx\in C\). Let \(\{\eta_{n}\}\subset(0,h(x))\) be a sequence of real numbers converging to \(h(x)\) and note that, by the definition of h, it holds
for any \(n\in\mathbb{N}\). Since \(\eta_{n}\to h(x)\) and
it follows that \(z_{n}\to h(x)x+(1-h(x))Tx\in C\), so that this last must belong to ∂C. □
Our main result is the following.
Theorem 1
Let C be a convex, closed and nonempty subset of a Hilbert space H and let \(T:C\to H\) be a mapping. Then the algorithm
is well defined.
If we further assume that
-
1.
C is strictly convex and
-
2.
T is a non-expansive mapping, which satisfies the inward condition (2) and such that \(\operatorname{Fix}(T)\neq\emptyset\),
then \(\{x_{n}\}\) weakly converges to a point \(p\in \operatorname{Fix}(T)\). Moreover, if \(\sum_{n=0}^{\infty}(1-\alpha_{n})<\infty\), then the convergence is strong.
Proof
To prove that the algorithm is well defined, it is sufficient to note that \(\alpha_{n}\in[h(x_{n}),1]\) for any \(n\in\mathbb{N}\); then, by recalling property (P2) from Lemma 1, it immediately follows that
Assume now that T satisfies the inward condition. In this case, by property (P3) of the previous lemma, we obtain that the non-decreasing sequence \(\{\alpha_{n}\}\) is contained in \([\frac{1}{2},1)\). Also, since T is non-expansive and with at least one fixed point, it follows by standard arguments that \(\{x_{n}\}\) is Fejér-monotone with respect to \(\operatorname{Fix}(T)\) and, as a consequence, both \(\{x_{n}\}\) and \(\{Tx_{n}\}\) are bounded.
Firstly, assume that \(\sum_{n=0}^{\infty}(1-\alpha_{n})=\infty\). Then, since \(\alpha_{n}\geq\frac{1}{2}\), we derive that \(\sum_{n=0}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty\) and from Lemma 2 of [16] we obtain that
This fact, together with the Fejér-monotonicity of \(\{x_{n}\}\) proves that the sequence weakly converges in \(\operatorname{Fix}(T)\) (see [17], Proposition 2.1).
Suppose that
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 \(x_{n}\to x^{*}\in C\).
Note that T satisfies the inward condition. Then, by applying properties (P2) and (P3) from Lemma 1, we obtain that \(h(x^{*})<1\) and that for any \(\mu\in(h(x^{*}),1)\) it holds
On the other hand, we observe that since \(\lim_{n\to\infty}\alpha_{n}=1\) by (5) and since \(\alpha_{n}=\max\{\alpha _{n-1}, h(x_{n})\}\) holds, it follows that we can choose a sub-sequence \(\{x_{n_{k}}\}\) with the property that \(\{h(x_{n_{k}})\}\) is non-decreasing and \(h(x_{n_{k}})\to1\). In particular, for any \(\mu<1\),
eventually holds.
Choose \(\mu_{1},\mu_{2}\in(h(x^{*}),1)\) with \(\mu_{1}\neq\mu_{2}\) and set \(v_{1}:=\mu_{1}x^{*}+(1-\mu_{1})Tx^{*}\) and \(v_{2}:=\mu _{2}x^{*}+(1-\mu_{2})Tx^{*}\). Then, whenever \(\mu\in[\mu_{1},\mu_{2}]\), by (6) we have that \(v:=\mu x^{*}+(1-\mu)Tx^{*}\in C\). Moreover,
since \(x_{n}\to x^{*}\). This last, together with (7), implies that \(v\in\partial C\) and \([v_{1},v_{2}]\subset\partial C\), since μ is arbitrary.
By the strict convexity of C, we derive that
and \(x^{*}=Tx^{*}\) must necessarily hold, i.e., \(\{x_{n}\}\) strongly converges to a fixed point of T. □
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
In [18], for a closed and convex set C, the map
was introduced and used in conjunction with an iterative scheme to approximate a fixed point of minimum norm (see also [19]). Indeed, in the above mentioned paper, it is proved that the iterative scheme
strongly converges under the assumptions that \(\{\alpha_{n}\}\) is a sequence in \((0,1)\) such that \(\lim_{n}\frac{\alpha_{n}}{(1-\lambda_{n})}=0\) and that \(\sum_{n}(1-\lambda_{n})\alpha_{n}=\infty\). We point out that the mentioned conditions appear to be difficult to be checked as they involve the geometry of the set C.
We illustrate the statement of our results with a brief example.
Example 1
Let \(H=l^{2}(\mathbb{R})\) and let \(C:=B_{1}\cap B_{2}\), where \(B_{1}:=\{ (t_{i})_{i\in\mathbb{N}}:(t_{1}-49.995)^{2}+\sum_{i=2}^{\infty }t_{i}^{2}\leq(50.005)^{2}\}\) and \(B_{2}:=\{(t_{i})_{i\in\mathbb{N}}:\sum_{i=1}^{\infty}t_{i}^{2}\leq 1\}\). Then C is a nonempty, closed and strictly convex subset of H. Let \(T:C\to H\) be the map defined by \(T(t_{1},t_{2},\ldots,t_{i},\ldots ):=(-t_{1},t_{2},\ldots,t_{i},\ldots)\), then T is a non-expansive inward map with \(\operatorname{Fix}(T)=\{(0,t_{2},\ldots ,t_{i},\ldots):\sum_{i=2}^{\infty}t_{i}^{2}\leq1\}\). If we use the algorithm
then, by the natural symmetry of the problem, we obtain the constant sequence
If we use the algorithm
then \(\{x_{n}\}\) still converges in \(\operatorname{Fix}(T)\), but \(\{x_{n}\}\cap \operatorname{Fix}(T)=\emptyset\) whenever \(t_{i}\neq0\).
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
Under which assumptions can algorithm (4) 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?
We refer to [22] and [23] for two examples regarding the classical Krasnoselskii-Mann algorithm.
Question 3
In the classical literature, it has been proved that the inward condition can be often dropped in favor of a weaker condition. For example, a mapping \(T:C\to X\) is said to be weakly inward (or to satisfy the weakly inward condition) if
Does Theorem 1 hold even for weakly inward mappings?
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.
References
Chidume, C: Geometric Properties of Banach Spaces and Nonlinear Iterations. Lecture Notes in Mathematics, vol. 1965. Springer, Berlin (2009)
Mann, WR: Mean value methods in iteration. Proc. Am. Math. Soc. 4(3), 506-510 (1953)
Groetsch, CW: A note on segmenting Mann iterates. J. Math. Anal. Appl. 40(2), 369-372 (1972)
Hicks, TL, Kubicek, JD: On the Mann iteration process in a Hilbert space. J. Math. Anal. Appl. 59(3), 498-504 (1977)
Edelstein, M, O’Brien, RC: Nonexpansive mappings, asymptotic regularity and successive approximations. J. Lond. Math. Soc. 2(3), 547-554 (1978)
Hillam, BP: A generalization of Krasnoselski’s theorem on the real line. Math. Mag. 48(3), 167-168 (1975)
Reich, S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67(2), 274-276 (1979)
Xu, H-K, Yin, X-M: Strong convergence theorems for nonexpansive nonself-mappings. Nonlinear Anal. 24(2), 223-228 (1995)
Xu, H-K: Approximating curves of nonexpansive nonself-mappings in Banach spaces. C. R. Acad. Sci. Paris Sér. I Math. 325(2), 151-156 (1997)
Marino, G, Trombetta, G: On approximating fixed points for nonexpansive mappings. Indian J. Math. 34, 91-98 (1992)
Takahashi, W, Kim, G-E: Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces. Nonlinear Anal. 32(3), 447-454 (1998)
Song, Y, Chen, R: Viscosity approximation methods for nonexpansive nonself-mappings. J. Math. Anal. Appl. 321(1), 316-326 (2006)
Song, YS, Cho, YJ: Averaged iterates for non-expansive nonself mappings in Banach spaces. J. Comput. Anal. Appl. 11, 451-460 (2009)
Zhou, H, Wang, P: Viscosity approximation methods for nonexpansive nonself-mappings without boundary conditions. Fixed Point Theory Appl. 2014, 61 (2014)
Kirk, W, Sims, B: Handbook of Metric Fixed Point Theory. Springer, Berlin (2001)
Ishikawa, S: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. Am. Math. Soc. 59(1), 65-71 (1976)
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)
He, S, Zhu, W: A modified Mann iteration by boundary point method for finding minimum-norm fixed point of nonexpansive mappings. Abstr. Appl. Anal. 2013, Article ID 768595 (2013)
He, S, Yang, C: Boundary point algorithms for minimum norm fixed points of nonexpansive mappings. Fixed Point Theory Appl. 2014, 56 (2014)
Schu, J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 158(2), 407-413 (1991)
Marino, G, Xu, H-K: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329(1), 336-346 (2007)
Bauschke, HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 202(1), 150-159 (1996)
Suzuki, T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305(1), 227-239 (2005)
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This project was funded by Ministero dell’Istruzione, dell’Universitá e della Ricerca (MIUR).
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Colao, V., Marino, G. Krasnoselskii-Mann method for non-self mappings. Fixed Point Theory Appl 2015, 39 (2015). https://doi.org/10.1186/s13663-015-0287-4
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DOI: https://doi.org/10.1186/s13663-015-0287-4