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

*Journal of Inequalities and Applications*
**volume 2015**, Article number: 86 (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.

## 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

holds for all \((x,z)\in H\times C\). See [10–12] 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*

### Proof

Since \(P_{C}\) is firmly nonexpansive, we have

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

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*. □

## References

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**100**, 489-490 (1987)Aoyama, K, Iemoto, S, Kohsaka, F, Takahashi, W: Fixed point and ergodic theorems for

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Takahashi, W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)

## 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.

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Kohsaka, F. Ray’s theorem revisited: a fixed point free firmly nonexpansive mapping in Hilbert spaces.
*J Inequal Appl* **2015**, 86 (2015). https://doi.org/10.1186/s13660-015-0606-7

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