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 [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

$$\begin{aligned} \mathrm {F}(T) = \Bigl\{ u\in C: \langle u, a \rangle =\sup _{y\in C} \langle y, a \rangle \Bigr\} . \end{aligned}$$
(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. □