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

## 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  showed the following fixed point theorem for nonexpansive mappings in Hilbert spaces.

### Theorem 1.1

(Browder’s theorem )

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  showed that the converse of Browder’s theorem holds.

### Theorem 1.2

(Ray’s theorem )

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  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. , 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 )

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 . 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, )

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  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. □

## References

1. Browder, FE: Fixed-point theorems for noncompact mappings in Hilbert space. Proc. Natl. Acad. Sci. USA 53, 1272-1276 (1965)

2. Ray, WO: The fixed point property and unbounded sets in Hilbert space. Trans. Am. Math. Soc. 258, 531-537 (1980)

3. Sine, R: On the converse of the nonexpansive map fixed point theorem for Hilbert space. Proc. Am. Math. Soc. 100, 489-490 (1987)

4. Aoyama, K, Iemoto, S, Kohsaka, F, Takahashi, W: Fixed point and ergodic theorems for λ-hybrid mappings in Hilbert spaces. J. Nonlinear Convex Anal. 11, 335-343 (2010)

5. Takahashi, W, Yao, J-C, Kohsaka, F: The fixed point property and unbounded sets in Banach spaces. Taiwan. J. Math. 14, 733-742 (2010)

6. Kohsaka, F: Existence of fixed points of nonspreading mappings with Bregman distances. In: Nonlinear Mathematics for Uncertainty and Its Applications, pp. 403-410. Springer, Berlin (2011)

7. Brezis, H: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2011)

8. Browder, FE: Convergence theorems for sequences of nonlinear operators in Banach spaces. Math. Z. 100, 201-225 (1967)

9. Bruck, RE Jr: Nonexpansive projections on subsets of Banach spaces. Pac. J. Math. 47, 341-355 (1973)

10. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

11. Goebel, K, Reich, S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984)

12. 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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Correspondence to Fumiaki Kohsaka. 