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Ray’s theorem revisited: a fixed point free firmly nonexpansive mapping in Hilbert spaces

Journal of Inequalities and Applications20152015:86

https://doi.org/10.1186/s13660-015-0606-7

Received: 30 October 2014

Accepted: 23 February 2015

Published: 6 March 2015

Abstract

We give another proof of a strong version of Ray’s theorem ensuring that every unbounded closed convex subset of a Hilbert space admits a fixed point free firmly nonexpansive mapping.

Keywords

  • firmly nonexpansive mapping
  • fixed point
  • Hilbert space
  • Ray’s theorem
  • unbounded set

MSC

  • 47H09
  • 47H10
  • 46C05

1 Ray’s theorem and its strong version

In 1965, Browder [1] showed the following fixed point theorem for nonexpansive mappings in Hilbert spaces.

Theorem 1.1

(Browder’s theorem [1])

Let C be a nonempty closed convex subset of a Hilbert space H. If C is bounded, then every nonexpansive self-mapping on C has a fixed point.

Ray [2] showed that the converse of Browder’s theorem holds.

Theorem 1.2

(Ray’s theorem [2])

Let C be a nonempty closed convex subset of a Hilbert space H. If every nonexpansive self-mapping on C has a fixed point, then C is bounded.

Later, Sine [3] gave a simple proof of Theorem 1.2 by applying a version of the uniform boundedness principle and the convex combination of a sequence of metric projections onto closed and convex sets.

Recently, Aoyama et al. [4], obtained a counterpart of Theorem 1.2 for λ-hybrid mappings in Hilbert spaces by using the following strong version of Ray’s theorem.

Theorem 1.3

(A strong version of Ray’s theorem [4])

Let C be a nonempty closed convex subset of a Hilbert space H. If every firmly nonexpansive self-mapping on C has a fixed point, then C is bounded.

It should be noted that Theorem 1.3 was actually shown by using Theorem 1.2 in [4]. See also [5, 6] on generalizations of Theorem 1.3 for firmly nonexpansive type mappings in Banach spaces.

In this paper, motivated by the papers mentioned above, we give another proof of Theorem 1.3 by using a version of the uniform boundedness principle and a single metric projection onto a closed and convex set. Since every firmly nonexpansive mapping is nonexpansive, Theorem 1.3 immediately implies Theorem 1.2.

2 A fixed point free firmly nonexpansive mapping

Throughout this paper, every linear space is real. The inner product and the induced norm of a Hilbert space H are denoted by \(\langle \cdot , \cdot \rangle\) and \(\Vert \cdot \Vert \), respectively. The dual space of a Banach space X is denoted by \(X^{*}\). The following is a version of the uniform boundedness principle.

Theorem 2.1

(see, for instance, [7])

If C is a nonempty subset of a Banach space X such that \(x^{*}(C)\) is bounded for each \(x^{*}\in X^{*}\), then C is bounded.

Let C be a nonempty closed convex subset of a Hilbert space H. Then a self-mapping T on C is said to be nonexpansive if \(\Vert Tx-Ty\Vert \leq \Vert x-y\Vert \) for all \(x,y\in C\); firmly nonexpansive [8, 9] if \(\Vert Tx-Ty\Vert ^{2} \leq \langle Tx-Ty, x-y \rangle \) for all \(x,y\in C\). The set of all fixed points of T is denoted by \(\mathrm {F}(T)\). The mapping T is said to be fixed point free if \(\mathrm {F}(T)\) is empty. It is well known that for each \(x\in H\), there exists a unique \(z_{x}\in C\) such that \(\Vert z_{x}-x\Vert \leq \Vert y-x\Vert \) for all \(y\in C\). The metric projection \(P_{C}\) of H onto C, which is defined by \(P_{C}x=z_{x}\) for all \(x\in H\), is a firmly nonexpansive mapping of H onto C. This fact directly follows from the fact that the equivalence
$$\begin{aligned} z=P_{C}x \quad\Longleftrightarrow\quad\sup_{y\in C} \langle y-z, x-z \rangle\leq0 \end{aligned}$$
(2.1)
holds for all \((x,z)\in H\times C\). See [1012] for more details on nonexpansive mappings.

We first show the following lemma.

Lemma 2.2

Let C be a nonempty closed convex subset of a Hilbert space H, a be an element of H, and T be the mapping defined by \(Tx=P_{C}(x+a)\) for all \(x\in C\). Then T is a firmly nonexpansive self-mapping on C such that
$$\begin{aligned} \mathrm {F}(T) = \Bigl\{ u\in C: \langle u, a \rangle =\sup _{y\in C} \langle y, a \rangle \Bigr\} . \end{aligned}$$
(2.2)

Proof

Since \(P_{C}\) is firmly nonexpansive, we have
$$\begin{aligned} \Vert Tx-Ty\Vert ^{2} \leq \bigl\langle P_{C}(x+a)-P_{C}(y+a), (x+a)-(y+a) \bigr\rangle = \langle Tx-Ty, x-y \rangle \end{aligned}$$
for all \(x,y\in C\). Thus T is a firmly nonexpansive self-mapping on C. Fix any \(u\in C\). According to (2.1), we know that
$$\begin{aligned} Tu=u \quad\Longleftrightarrow\quad\sup_{y\in C} \bigl\langle y-u, (u+a)-u \bigr\rangle \leq0 \quad\Longleftrightarrow\quad \langle u, a \rangle=\sup _{y\in C} \langle y, a \rangle \end{aligned}$$
and hence (2.2) holds. □

Using Theorem 2.1 and Lemma 2.2, we give another proof of Theorem 1.3.

Proof of Theorem 1.3

If C is unbounded, then Theorem 2.1 implies that \(x^{*}(C)\) is unbounded for some \(x^{*}\in H^{*}\). Since H is a real Hilbert space, we have \(a\in H\) such that \(\sup_{y\in C} \langle y, a \rangle=\infty\). By Lemma 2.2 and the choice of a, the mapping T defined as in Lemma 2.2 is a fixed point free firmly nonexpansive self-mapping on C. □

Declarations

Acknowledgements

The author would like to thank the anonymous referees for carefully reading the original version of the manuscript. The author is supported by Grant-in-Aid for Young Scientists No. 25800094 from the Japan Society for the Promotion of Science.

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

(1)
Department of Computer Science and Intelligent Systems, Oita University, Oita, Japan

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Copyright

© Kohsaka; licensee Springer. 2015

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