Chapter 1

ROTATIVE MAPPINGS AND MAPPINGS WITH CONSTANT DISPLACEMENT

Wiesława Kaczor

wkaczor@golem.umcs.lublin.pl

Małgorzata Koter-Mórgowska

mkoter@golem.umcs.lublin.pl

Institute of Mathematics Maria Curie-Skłodowska University 20-031 Lublin, Poland

1. Introduction

This part is a continuation of the chapter by K. Goebel, so we adopt all his notations and definitions.

In general, to assure the fixed point property for nonexpansive mappings some assumptions concerning the geometry of the space are added. Another way is to put some additional restrictions on the mapping itself. One of them is the so-called rotativeness. This property seems to be quite natural in the case of nonexpansive mappings. Moreover, it assures the existence of fixed points in the case of k-lipschitzian mappings with k slightly greater than 1.

In what follows mappings with constant displacement will also be discussed.

Let C be a nonempty subset of a Banach space X and T : C → C. We say that T is the mapping with constant displacement if it moves each point in C the same distance, that is, if ∥x − T x∥ = const for each x ∈ C. Recall that the mapping T is said to be nonexpansive if ∥T x − T y∥ ≤ ∥x − y∥ for all x, y ∈ C.

Note that if T is nonexpansive or with constant displacement, then for any x ∈ C and any n ∈ ℕ,

∥x − T^{n}x∥ ≤ n ∥x − T x∥ .  (1)

2. Rotative mappings

Rotativeness is a certain generalization of the well known notion of periodicity. Recall that a mapping T : C → C is said to be n-periodic if T^{n} = Id.

Definition 2.1 A mapping T : C → C is said to be (a, n)-rotative for 0 ≤ a < n, n ≥ 2, if

∥x − T^{n}x∥ ≤ a ∥x − T x∥  (2)

for each x ∈ C.

We will simply say that the mapping is n-rotative if it is (a, n)-rotative with some a < n, and rotative if it is n-rotative for some n ≥ 2. The term “rotative” originates from the fact that all rotations in the euclidean plane satisfy this condition. Clearly, n-periodic mappings are (0, n)-rotative. Note also that all contractions are rotative for all n ≥ 2. Since each nonexpansive mapping satisfies (1), rotativeness of such a mapping is caused either by a certain “turning” of a sequence of successive iterations {x, T x, T^{2}x, ...}, meaning that they do not lie on the same “metrical line”, or by a “shortening” of the distances ∥T^{i}x−T^{i−1}x∥ between consecutive terms of this sequence. In the case of mappings with constant displacement rotativeness is resulted only by a turning of the sequence of successive iterations.

It is worth noting here that if a nonexpansive mapping is (a, n)-rotative, then it is also (m − n + a, m)-rotative for all m ≥ n.

The following theorem shows that the condition of rotativeness is actually quite strong; it assures the existence of fixed points of nonexpansive mappings even without weak compactness, boundedness or another special geometric structure of the set C.

Theorem 2.2 If C is a nonempty, closed and convex subset of a Banach space X, then any nonexpansive rotative mapping T : C → C has a fixed point.

Proof. For x ∈ C and α ∈ [0, 1) consider the mapping T_{x,α} : C → C defined by

T_{x,α}z = (1 − α) x + αT z, z ∈ C.

The mapping T_{x,α} is α-lipschitzian and therefore it is a contraction. By Banach’s Contraction Mapping Principle there exists exactly one fixed point of T_{x,α} in C, i.e. a point F_{α}x ∈ C such that T_{x,α}F_{α}x = F_{α}x. So we can define a mapping F_{α} : C → C whose set of fixed points coincides with the set of fixed points of T and which is also nonexpansive. Indeed, we have

F_{α}x = (1 − α) x + αT F_{α}x, x ∈ C,  (3)

and

∥F_{α}x − F_{α}y∥ = ∥(1 − α) (x − y) + α (T F_{α}x − T F_{α}y)∥
≤ (1 − α) ∥x − y∥ + α ∥F_{α}x − F_{α}y∥ .

From (3) we get

(1 − α) (x − F_{α}x) = α (F_{α}x − T F_{α}x)

for any x ∈ C. Moreover, by iterating F_{α} we obtain

F_{α}^{k}x = (1 − α) F_{α}^{k−1}x + αT F_{α}^{k}x, k = 1, 2, ... .

Now suppose T is (a, n)-rotative. Then

∥F_{α}x − F_{α}^{2}x∥ = ∥F_{α}x − (1 − α) F_{α}x − αT F_{α}^{2}x∥
= α ∥F_{α}x − T F_{α}^{2}x∥
≤ α ∥F_{α}x − T^{n}F_{α}x∥ + α ∥T^{n}F_{α}x − T F_{α}^{2}x∥
≤ αa ∥F_{α}x − T F_{α}x∥ + α ∥T^{n−1}F_{α}x − F_{α}^{2}x∥
= (1 − α) a ∥x − F_{α}x∥ + α ∥T^{n−1}F_{α}x − F_{α}^{2}x∥ .

It then follows that

∥F_{α}x − F_{α}^{2}x∥ ≤ (1 − α) a ∥x − F_{α}x∥ + α ∥T^{n−1}F_{α}x − F_{α}^{2}x∥  (4)

for all x ∈ C. Now, using only the fact that T is nonexpansive, we will use induction on m to establish the following:

α ∥T^{m−1}F_{α}x − F_{α}^{2}x∥ ≤ [(m − 1) − mα + α^{m}] ∥x − F_{α}x∥  (5)
+ α^{m} ∥F_{α}x − F_{α}^{2}x∥ ,

For m = 2 we have

α ∥T F_{α}x − F_{α}^{2}x∥ ≤ α (1 − α) ∥T F_{α}x − F_{α}x∥ + α^{2} ∥T F_{α}x − T F_{α}^{2}x∥
≤ (1 − α)^{2} ∥x − F_{α}x∥ + α^{2} ∥F_{α}x − F_{α}^{2}x∥ .

Assuming (5) to hold for m we will prove it for m + 1.

α ∥T^{m}F_{α}x − F_{α}^{2}x∥ ≤ α (1 − α) ∥T^{m}F_{α}x − F_{α}x∥ + α^{2} ∥T^{m}F_{α}x − T F_{α}^{2}x∥
≤ mα (1 − α) ∥T F_{α}x − F_{α}x∥ + α^{2} ∥T^{m−1}F_{α}x − F_{α}^{2}x∥
= m (1 − α)^{2} ∥x − F_{α}x∥ + α^{2} ∥T^{m−1}F_{α}x − F_{α}^{2}x∥
≤ m (1 − α)^{2} ∥x − F_{α}x∥
+α [(m − 1) − mα + α^{m}] ∥x − F_{α}x∥
+α^{m+1} ∥F_{α}x − F_{α}^{2}x∥
= [m − (m + 1) α + α^{m+1}] ∥x − F_{α}x∥ + α^{m+1} ∥F_{α}x − F_{α}^{2}x∥ .

Finally, using (4) and (5) we conclude

∥F_{α}x − F_{α}^{2}x∥ ≤ (1 − α^{n})^{−1} [(1 − α) a + (n − 1) − nα + α^{n}] ∥x − F_{α}x∥
= [(a + n) (∑_{i=0}^{n−1} α^{i})^{−1} − 1] ∥x − F_{α}x∥  (6)
= g (α) ∥x − F_{α}x∥ , x ∈ C.

Note that g is continuous, decreasing for α ∈ (0, 1] and g(1) = a/n < 1. So there exists α_{0} ∈ (0, 1) such that g(α) < 1 for α ∈ (α_{0}, 1). For such α, the iterates {F_{α}^{n}x} converge for any x ∈ C to a fixed point of T. ■

As we mentioned, in the class of nonexpansive mappings rotativeness appears in a very natural way. Obviously, this condition is independent of nonexpansiveness. Therefore we can also consider k-lipschitzian rotative mappings with k > 1. In this case the condition (2) puts a less natural restriction on the mapping T. However, it assures (provided k is not too large) the fixed point property for closed convex sets in an arbitrary Banach space. Recall that even in a Hilbert space one can construct a self-mapping of the unit ball which is fixed point free and (1 + ε)-lipschitzian with 0 < ε < 1 arbitrarily small (see, e.g., Example 1 in the chapter by K. Goebel). Of course this mapping is not rotative, which can be concluded from the following

Theorem 2.3 If C is a nonempty, closed and convex subset of a Banach space X, then for any n ∈ ℕ and a < n there exists γ > 1 such that any (a, n)-rotative and k-lipschitzian mapping T : C → C has a fixed point provided k < γ.

Proof. Reasoning similar to that in the proof of Theorem 2.2 can be applied. If T is a k-lipschitzian self-mapping of C, then the mapping F_{α} : C → C given by (3) is well defined for α ∈ [0, 1/k) and it is \frac{1−α}{1−αk}-lipschitzian. In this case one can prove by induction that for all m ∈ ℕ,

αk ∥T^{m−1}F_{α}x − F_{α}^{2}x∥ ≤ (1 − α) (\frac{k^{m} − 1}{k − 1} − \frac{1 − α^{m}k^{m}}{1 − αk} ) ∥x − F_{α}x∥  (5′)
+ α^{m}k^{m} ∥F_{α}x − F_{α}^{2}x∥ ,

and then obtain the following counterpart of (6) :

∥F_{α}x − F_{α}^{2}x∥ ≤ \frac{1 − α}{1 − αk} [\frac{a + \frac{k^{n} − 1}{k − 1}}{∑^{n−1} ; i=0 (αk)^{i}} − 1] ∥x − F_{α}x∥  (6′)

= g (α, k) ∥x − F_{α}x∥ .

It is easy to see that g(α, k) approaches g(α), defined in (6), as k → 1. Thus, since g(α) < 1 for α sufficiently close to 1, there exists α ∈ (0, \frac{1}{k}) such that g(α, k) < 1 if k is sufficiently close to 1 (say, if 1 < k < γ) . For such α and k the iterates {F_{α}^{n}x} converge for any x ∈ C to a fixed point of F_{α}, which is also the fixed point of T. ■

Clearly, γ which appears in the proof of the above theorem depends on a, n and the space in which the set C is contained. Thus it is convenient to define the function γ_{n}^{X}(a) as follows

γ_{n}^{X} (a) = inf {k : there is a closed and convex set C ⊂ X and

a fixed point free k-lipschitzian (a, n) -rotative selfmapping of C} .

Now we can reformulate Theorem 2.3 in following way

For any Banach space X, n ≥ 2 and a < n, we have γ_{n}^{X} (a) > 1.

In general, precise values of γ_{n}^{X}(a) are not known. However, in some cases we can give certain estimates.

For an arbitrary positive integer n, only estimates of the value of the function γ_{n}^{X} at 0 are known. Namely,

γ_{n}^{X} (0) ≥ \{ \begin{matrix}2 \\ ^{n−1} & \sqrt{\frac{1}{n−2} (−1 + \sqrt{n (n − 1) − \frac{1}{n−1}})} \end{matrix} \begin{matrix}for n = 2, \\ for n > 2. \end{matrix}  (∗)

Some better results can be obtained for n = 2.

Theorem 2.4 In an arbitrary Banach space X,

γ_{2}^{X} (a) ≥ max \{\frac{1}{2} (2 − a + \sqrt{(2 − a)^{2} + a^{2}}) ,  (7)

\frac{1}{8} (a^{2} + 4 + \sqrt{(a^{2} + 4)^{2} − 64a + 64})\} .

Proof. Let C be a closed and convex subset of X. Suppose T : C → C is k-lipschitzian and (a, 2)-rotative. For arbitrarily chosen α ∈ (0, 1) and x ∈ C, put

z = (1 − α) x + αT x,
u = (1 − α) T^{2}x + αT x.

Then

∥z − T z∥ ≤ ∥z − u∥ + ∥u − T z∥
≤ (1 − α) ∥x − T^{2}x∥ + (1 − α) ∥T^{2}x − T z∥ + α ∥T x − T z∥
≤ (1 − α) a ∥x − T x∥ + (1 − α) k ∥T x − z∥ + αk ∥x − z∥
≤ [(1 − α) a + (1 − α)^{2} k + α^{2}k] ∥x − T x∥
= h_{1} (α) ∥x − T x∥ .

On the other hand,

∥u − T u∥ = ∥(1 − α) T^{2}x + αT x − αT u − (1 − α) T u∥
≤ (1 − α) k ∥T x − u∥ + αk ∥x − u∥
= (1 − α)^{2} k ∥T x − T^{2}x∥ + αk ∥αx + (1 − α) x − (1 − α) T^{2}x − αT x∥
≤ (1 − α)^{2} k^{2} ∥x − T x∥ + α^{2}k ∥x − T x∥ + α (1 − α) k ∥x − T^{2}x∥
≤ [(1 − α)^{2} k^{2} + α^{2}k + α (1 − α) ka] ∥x − T x∥
= h_{2} (α) ∥x − T x∥ .

An easy calculation shows that h_{1} attains its minimum value \frac{4k^{2}+4ak−a^{2}}{8k} at α_{1} = \frac{a+2k}{4k}, while h_{2} attains its minimum value \frac{k(4k−a^{2})}{4(1+k−a)} at α_{2} = \frac{2k−a}{2(1+k−a)}. Setting

\begin{matrix}x_{0}^{1} = x_{0}^{2} = x, \\ x_{n+1}^{1} = (1 − α_{1}) x_{n} + α_{1}T x_{n}, \\ x_{n+1}^{2} = (1 − α_{2}) T^{2}x_{n} + α_{2}T x_{n}, \end{matrix}  n = 0, 1, ...,

we see that the sequences {x_{n}^{i}}, i = 1, 2, satisfy

∥T x_{n+1}^{i} − x_{n+1}^{i}∥ ≤ h_{i} (α_{i}) ∥T x_{n}^{i} − x_{n}^{i}∥

and if

k < max \{\frac{2 − a + \sqrt{(2 − a)^{2} + a^{2}}}{2} , \frac{a^{2} + 4 + \sqrt{(a^{2} + 4)^{2} − 64a + 64}}{8}\} ,

then either h_{1}(α_{1}) < 1 or h_{2}(α_{2}) < 1, which implies that at least one of the sequences {x_{n}^{i}}, i = 1, 2, is convergent to a fixed point of T. ■

It is interesting that the first term in (7) gives a better evaluation for a ∈ [0, 2(\sqrt{2}−1)], while the second for a ∈ [2(\sqrt{2} − 1), 2). In particular, for a = 0 we get γ_{2}^{X}(0) ≥ 2. This evaluation is not the best possible. For example in the case of Hilbert space H one can prove that γ_{2}^{H}(0) ≥ \sqrt{5} ≈ 2.24, which follows from the inequality

γ_{2}^{H} (a) ≥ \sqrt{\frac{5}{a^{2} + 1}}.  (8)

The above evaluation can be obtained using arguments similar to that in the proof of Theorem 2.4 and additionally applying the fact that for all u, v ∈ H and α ∈ [0, 1],

∥(1 − α) u + αv∥^{2} = (1 − α) ∥u∥^{2} + α ∥v∥^{2} − α (1 − α) ∥u − v∥^{2} .

The estimate (8) is also not sharp as the following example shows.

Example 2.5 We will show that in the case of Hilbert space

γ_{2}^{H} (0) ≥ \sqrt{π^{2} − 3} ≈ 2.62 .

To this aim let T : CarrowC, where C ⊂ H is nonempty, closed and convex, be a k-lipschitzian involution. We will prove that if k < \sqrt{π^{2} − 3}, then there is z ∈ [x, T x] such that ∥T z − z∥ ≤ (1 − ε)∥x − T x∥ with some ε ∈ (0, 1). So for x ∈ C and t ∈ (0, 1) put z(t) = (1 − t)x + tT x and suppose, contrary to our claim, that

∥T z (t) − z (t)∥ > (1 − ε) ∥x − T x∥

for all t, ε ∈ (0, 1). Let g(t) = T z(t) − z(t). Then the curve g : t ↦ g(t) is lipschitzian, g(0) = −g(1) and ∥g(t)∥ > (1 − ε)∥g(0)∥ = (1 − ε)∥x − T x∥. Using the fact that the length of a curve laying on S and joining antipodal points exceeds the so-called girth of the sphere, which is π in Hilbert space, we get that the length of g is not less than π(1 − ε)∥x − T x∥. Let 0 = t_{0} < t_{1} < ... < t_{n} = 1, t_{i} − t_{i−1} = \frac{1}{n}. Then for sufficiently large n the length of the piecewise linear curve ⋃_{i=1}^{n}[g(t_{i−1}), g(t_{i})] is also not less than π(1 − ε)∥x − T x∥. It then follows that

π^{2} (1 − ε)^{2} ∥x − T x∥^{2} ≤ (∑_{i=1}^{n} ∥g (t_{i}) − g (t_{i−1})∥)^{2}

≤ n ∑_{i=1}^{n} ∥g (t_{i}) − g (t_{i−1})∥^{2}

= n ∑_{i=1}^{n} ∥T z (t_{i}) − T z (t_{i−1}) − [z (t_{i}) − z (t_{i−1})]∥^{2}

= n ∑_{i=1}^{n}[∥T z (t_{i}) − T z (t_{i−1})∥^{2} + ∥z (t_{i}) − z (t_{i−1})∥^{2}

−2 (T z (t_{i}) − T z (t_{i−1}) , z (t_{i}) − z (t_{i−1}))]

≤ n[(k^{2} + 1) ∑_{i=1}^{n} ∥z (t_{i}) − z (t_{i−1})∥^{2}

−2 ∑_{i=1}^{n} (T z (t_{i}) − T z (t_{i−1}) , z (t_{i}) − z (t_{i−1}))]

≤ n (k^{2} + 1) ∑_{i=1}^{n} \frac{1}{n^{2}} ∥x − T x∥^{2}

−2 (∑_{i=1}^{n} [T z (t_{i}) − T z (t_{i−1})] , T x − x)

= (k^{2} + 3) ∥x − T x∥^{2} .

Hence π(1 − ε) ≤ \sqrt{k^{2} + 3}. Since ε can be arbitrarily chosen, we get the desired result.

It turns out, however, that the above estimate is also not sharp. Namely, if T is a k-lipschitzian involution which maps the whole Hilbert space H into itself, then it has a fixed point provided k < \frac{1}{3}(π +\sqrt{π^{2} − 4}) ≈ 2.78. To see this one can apply arguments similar to that of the preceding example to the curve t ↦ g(t) = F_{α}z(t) − z(t), where z(t) = (1 − t)x + ty, F_{α} : H → H is defined by (3) and y ∈ H is such that F_{α}y = T F_{α}x.

We now give two examples providing upper bounds for γ_{2}^{X}(a).

Example 2.6 Let R : B → S be a k-lipschitzian retraction of the unit ball B onto the unit sphere S. For ε ∈ (0, 1) put

T x = \{ \begin{matrix}−R (\frac{x}{ε}) \\ −\frac{x}{∥x∥} \end{matrix} \begin{matrix}for ∥x∥ ≤ ε, \\ for ε < ∥x∥ ≤ 1. \end{matrix}

Clearly, T maps B into S and is \frac{k}{ε}-lipschitzian. Moreover, since T^{2} = −T, we see that T is fixed point free. It is also easy to show that for x ∈ B,

∥x − T^{2}x∥ ≤ \frac{1 + ε}{1 − ε} ∥x − T x∥ .

So if ε < \frac{1}{3}, then T is (\frac{1+ε}{1−ε}, 2)-rotative. Therefore in any Banach space X,

γ_{2}^{X} (a) ≤ \frac{k_{0} (X) (a + 1)}{a − 1} for a ∈ (1, 2) ,  (9)

where k_{0}(X) is the minimal Lipschitz constant of the retraction of the unit ball onto the unit sphere in X.

In some particular cases better evaluations can be obtained.

Example 2.7 Let X = C[0, 1] and let C ⊂ X be defined as follows

C = {x : x is nondecreasing and x (0) = 0, x (1) = 1} .

For k > 1 set

T x (t) = k max \{x (t) − (1 − \frac{1}{k} ) , 0\} .

It is easy to see that T : C → C is k-lipschitzian and moves each point the same constant distance 1 − \frac{1}{k}. Moreover,

∥x − T^{2}x∥ = 1 − \frac{1}{k^{2}} = (1 + \frac{1}{k} ) ∥x − T x∥ .

So T is (1 + \frac{1}{k}, 2)-rotative. Therefore for 1 < a < 2,

γ_{2}^{C[0,1]} (a) ≤ \frac{1}{a − 1}.  (10)

It follows from Example 5 in the chapter by K. Goebel that (10) holds also for L^{1}(0, 1).

The above results can be illustrated pictorially in Fig. 1.

Figure 1 Set

γ′ (a) = \frac{2 − a + \sqrt{(2 − a)^{2} + a^{2}}}{2} , γ′′ (a) = \frac{a^{2} + 4 + \sqrt{(a^{2} + 4)^{2} − 64a + 64}}{8}

and

D_{1} = {(a, k) ∈ [0, 2) × [0, ∞) : k < max [γ′ (a) , γ′′ (a)]} ,
D_{2} = \{(a, k) ∈ [0, 2) × [0, ∞) : k > \frac{k_{0} (X) (a + 1)}{a − 1}\} ,
D_{3} = \{(a, k) ∈ [0, 2) × [0, ∞) : k > \frac{1}{a − 1}\} ,
D_{4} = ([0, 2) × [0, ∞)) \ (D_{1} ∪ D_{3}) .

It follows from (7), (9) and (10) that the graph of the function γ_{2}^{X} always lies in the region D_{4} ∪ D_{3} and for some spaces X in D_{4}. The “lower” boundary of D_{4} can be

slightly moved up in the case of Hilbert space (see (8)).

3. Firmly lipschitzian mappings

In this section we show that the estimate (10) is sharp for a certain subclass of the family of k-lipschitzian mappings.

Definition 3.1 A mapping T : C → C is said to be firmly k-lipschitzian if for each t ∈ [0, 1] and for any x, y ∈ C,

∥T x − T y∥ ≤ ∥k (1 − t) (x − y) + t (T x − T y)∥ .  (11)

Of course each firmly k-lipschitzian mapping is k-lipschitzian. We will now consider

firmly k-lipschitzian and (a, 2)-rotative mappings. Observe first that for such a mapping T, putting in (11) t = \frac{k}{k+1} and y = T x, we get

∥T x − T^{2}x∥ ≤ \frac{k}{k + 1} ∥x − T^{2}x∥ ≤ \frac{ka}{k + 1} ∥x − T x∥ .  (12)

Next, note that

a ∥x − T x∥ ≥ ∥x − T^{2}x∥ ≥ ∥x − T x∥ − ∥T x − T^{2}x∥

≥ (1 − \frac{ka}{k + 1}) ∥x − T x∥ .

Consequently, if T is firmly k-lipschitzian and (a, 2)-rotative, then

a ≥ \frac{k + 1}{2k + 1}.

On the other hand, it follows from (12) that if k < \frac{1}{a−1}, then T has a fixed point. So if we define an analogue of γ_{n}^{X} by setting

γ̃_{n}^{X} (a) = inf {k : there is a closed and convex set C ⊂ X and a fixed

point free firmly k-lipschitzian (a, n) -rotative selfmapping of C}

we get

γ̃_{2}^{X} (a) ≥ \frac{1}{a − 1}.

Moreover, since the mapping defined in Example 2.7 is firmly k-lipschitzian and (1 + \frac{1}{k}, 2)-rotative,

γ̃_{2}^{C[0,1]} (a) = \frac{1}{a − 1}.

The same equality holds for L^{1}(0, 1) (see Example 5 in the chapter by K. Goebel). We can summarize the results of this section pictorially in Fig. 2.

Figure 2

E_{1} = \{(a, k) ∈ [0, 2) × [0, ∞) : k < \frac{1 − a}{2a − 1}\} ,
E_{2} = \{(a, k) : \frac{1 − a}{2a − 1} ≤ k < \frac{1}{a − 1}\} ,
E_{3} = \{(a, k) : k ≥ \frac{1}{a − 1}\} .

The graph of the function γ̃_{2}^{X} always lies in the region E_{3} and for some spaces X it

coincides with the lower boundary of E_{3}.

The shapes of E_{i} are a bit different if we deal with a Hilbert space. In this case we have the following

Proposition 3.2 The mapping T : C → C, C ⊂ H, is firmly k-lipschitzian if and only if for any x, y ∈ C,

∥T x − T y∥^{2} ≤ k (x − y, T x − T y) .  (13)

Proof. Assume that (11) holds. Then

∥T x − T y∥^{2} ≤ k^{2} (1 − t)^{2} ∥x − y∥^{2} + 2kt (1 − t) (x − y, T x − T y)
+t^{2} ∥T x − T y∥^{2}

and consequently,

∥T x − T y∥^{2} ≤ \frac{k^{2} (1 − t)}{1 + t} ∥x − y∥^{2} + \frac{2kt}{1 + t} (x − y, T x − T y) , t ∈ [0, 1).

Upon passage to the limit as t → 1_{−} we get (13). Conversely, if (13) holds, then

∥k (1 − t) (x − y) + t (T x − T y)∥^{2}
= k^{2} (1 − t)^{2} ∥x − y∥^{2} + 2kt (1 − t) (x − y, T x − T y) + t^{2} ∥T x − T y∥^{2}
≥ (1 − t)^{2} ∥T x − T y∥^{2} + 2t (1 − t) ∥T x − T y∥^{2} + t^{2} ∥T x − T y∥^{2}
= ∥T x − T y∥^{2} ,

which completes the proof. ■

Note that (13) implies that T is monotone, i.e. (x − y, T x − T y) ≥ 0. Consequently, if T is also (a, 2)-rotative, then

a^{2} ∥T x − x∥^{2} ≥ ∥T^{2}x − x∥^{2}
= ∥T^{2}x − T x∥^{2} + 2 (T^{2}x − T x, T x − x) + ∥T x − x∥^{2} ≥ ∥T x − x∥^{2} ,

which gives a ≥ 1. In other words, in a Hilbert space there are no firmly lipschitzian and (a, 2)-rotative mappings with a < 1.

Proposition 3.3

γ̃_{2}^{H} (a) ≥ \frac{2}{a^{2} − 2}.

Proof. Let T : C → C, C − closed and convex subset of H, be firmly k-lipschitzian and (a, 2)-rotative. Then

a^{2} ∥T x − x∥^{2} ≥ ∥T^{2}x − x∥^{2}
= ∥T^{2}x − T x∥^{2} + 2 (T^{2}x − T x, T x − x) + ∥T x − x∥^{2}
≥ ∥T x − x∥^{2} + (\frac{2}{k} + 1) ∥T^{2}x − T x∥^{2} ,

which yields

∥T^{2}x − T x∥^{2} ≤ k \frac{a^{2} − 1}{k + 2} ∥T x − x∥^{2} .

Hence if a ≤ \sqrt{2} or if a > \sqrt{2} and k < \frac{2}{a^{2}−2}, then T has a fixed point. ■

It follows from the above considerations that the region E_{2} in Fig. 2 in the case of Hilbert space is equal to \{(a, k) ∈ [1, 2) × [0, ∞) : k < \frac{2}{a^{2}−2}\}. The sets E_{1} and E_{3} are changed respectively. However, we do not know whether the lower boundary of E_{3}

coincides with the graph of γ̃_{2}^{H}.

4. Mappings with constant displacement

The mapping T considered in Example 2.7 is k-lipschitzian, rotative and moves each point of C the same constant distance (for all x ∈ C, ∥x − T x∥ = 1 − \frac{1}{k}). We will show later that for any noncompact set C and any k > 1 there is a k-lipschitzian mapping

T : C → C satisfying ∥x − T x∥ = const > 0. So it seems to be natural to discuss k-lipschitzian mappings with constant displacement. It follows from (1) that if

d (T) = inf_{x∈C} ∥x − T x∥ ,

then for any mapping T with constant displacement we have d(T^{n}) ≤ nd(T). As we mentioned in Section 2, rotativeness of such a mapping is resulted by a turning of the

sequence of its successive iterations. We can introduce an analogue of γ_{n}^{X} for mappings with constant displacement by setting

γ̄_{n}^{X} (a) = inf {k : there is a closed and convex set C ⊂ X and

a k-lipschitzian (a, n) -rotative selfmapping of C with constant positive displacement} .

Of course, γ_{n}^{X}(a) ≤ γ̄_{n}^{X}(a). Moreover, for an arbitrary Banach space X,
γ̄_{2}^{X} (a) ≥ \frac{4}{a + 2}.  (14)

To see this, it suffices to take u = \frac{1}{2}(T x + T^{2}x) and to verify that if ∥x − T x∥ = d(T) for all x ∈ C, then

d (T) = ∥u − T u∥ ≤ \frac{1}{4}k ∥x − T x∥ + \frac{1}{4}k ∥x − T^{2}x∥ + \frac{1}{4}k ∥T x − T^{2}x∥
= \frac{1}{4}k ∥x − T^{2}x∥ + \frac{1}{2}kd (T) .

Thus

d (T^{2}) ≥ 2 (\frac{2}{k} − 1) d (T) ,  (15)

which implies that for (a, 2)-rotative mapping T it must be either d(T) = 0 or k ≥ \frac{4}{a+2}.

Note that (14) gives a better evaluation of γ̄_{2}^{X}(a) than the general estimate (7). On the other hand, it follows from Example 2.7 that γ̄_{2}^{C[0,1]}(a) ≤ \frac{1}{a−1}, 1 < a < 2. Let us

give another example of mappings with constant displacement.

Example 4.1 Assume R : B → S is a k-lipschitzian retraction. For l ≥ k, define T by setting

T x = x − \frac{Rx}{l}.

Then T is a (1 + \frac{k}{l})-lipschitzian mapping of B onto B(0, 1 − \frac{1}{l}). Clearly,

∥x − T x∥ = \frac{1}{l}.

Considering mappings with constant displacement we can formulate some equivalents to Schauder’s Fixed Point Theorem. They might appear weaker than the Schauder

Theorem itself but they are still its equivalents.

Theorem 4.2 Let C be a nonempty, convex and compact subset of a Banach space X. If a continuous self-mapping T of C moves each point the same constant distance d, then d = 0.

Proof. Clearly, this result follows immediately from Schauder’s Theorem. For the reverse implication, suppose that a continuous mapping T_{1} : C → C is fixed point free. By compactness of C, min_{x∈C} ∥x − T_{1}x∥ = d > 0 and therefore T defined by

T x = x + d\frac{x − T_{1}x}{∥x − T_{1}x∥}, x ∈ C,

is the mapping with constant positive displacement d. ■

Recall that lipschitzian mappings form a dense subset in the set of all continuous

self-mappings of a compact set C. Consequently, Schauder’s Theorem has an equivalent formulation: If C is a nonempty, convex and compact subset of a Banach

space X, then every lipschitzian mapping T : C → C has a fixed point. Similar approximation argument allows us to formulate an analogue of Theorem 4.2.

Theorem 4.3 If C is noncompact and convex, then for each k > 1 there is a klipschitzian mapping T : C → C which moves all points the same constant distance.

Proof. It is known that for any noncompact set C and any k > 1 there is a klipschitzian mapping T_{1} : C → C such that d_{1} = inf{∥x − T_{1}x∥ : x ∈ C} > 0. Therefore

T_{2}x = x + d_{1} \frac{x − T_{1}x}{∥x − T_{1}x∥}, x ∈ C,

is a lipschitzian mapping with constant displacement. Next, setting T = (1−λ)Id+λT_{2} with an appropriate λ, we get a k-lipschitzian mapping such that ∥x − T x∥ = λd_{1}. ■

It was shown at the beginning of this section that mappings with constant displacement behave in a special way. Namely, we have (15), which in turn leads to

the following Proposition 4.4 Given ε ∈ (0, 2], if C is noncompact and convex, then there is a continuous mapping T : C → C such that d(T) = inf_{x∈C} ∥x − T x∥ > 0 and

d(T^{2}) ≥ (2 − ε)d(T).

As such a mapping T we can take any k-lipschitzian mapping with k sufficiently close

to 1, which displaces each point by the same fixed distance. This consideration leads to a subsequent equivalent to Schauder’s Theorem.

Theorem 4.5 Given ε ∈ (0, 2], if C is compact and convex, then for any continuous mapping T : C → C there is x ∈ C such that

∥x − T^{2}x∥ ≤ (2 − ε) ∥x − T x∥ .

In particular, for ε = 2 we have the following, seemingly weaker, equivalent to Schauder’s Theorem.

Theorem 4.6 If C is compact and convex, then any continuous mapping T : C → C has a periodic point with period 2, i.e. T^{2} has a fixed point.

Using Theorem 2.3 we can modify Proposition 4.4 in the following way: Given ε ∈ (0, n], n ≥ 2, if C is noncompact, then there is a continuous mapping T : C → C such that d(T) > 0 and d(T^{n}) ≥ (n − ε)d(T). Therefore we get another equivalent

formulation of Schauder’s Theorem:

Theorem 4.7 Given ε > 0, if C is compact and convex, and T : C → C is a continuous mapping, mapping,continuousthen there exists x ∈ C such that ∥x−T^{n}x∥ ≤ (n − ε)∥x − T x∥.

Theorem 4.8 If C is compact and convex, then any continuous mapping T : C → C has a periodic point with period n, i.e. T^{n} has a fixed point.

Finally, let us add another theorem to our list of equivalent formulations of Schauder’s Theorem.

Theorem 4.9 Given M > 0, if C is compact and convex, then for any continuous mappings T_{1}, T_{2} : C → C there exists x ∈ C such that

∥x − T_{1}x∥ ≤ M ∥x − T_{2}x∥ .

Proof. Clearly, Schauder’s Theorem implies our statement. For the converse, suppose that Schauder’s Theorem does not hold. Then by Theorem 4.7 there is a continuous mapping T_{2} such that ∥x−T_{2}^{n}x∥ > (n−ε)∥x−T_{2}x∥ for all x. Now we can take T_{1} = T_{2}^{n}.

■

5. Notes and remarks

The notion of rotativeness was introduced by K. Goebel and M. Koter in [7]. Basic

results about rotative mappings, which we present in Section 2, originate from [7], [8] and [15]. The estimate (8) and a part of (7) are due to T. Komorowski [14]. In

[11] one can also find some evaluations of γ_{2}^{l^{p}}(a) and γ_{2}^{L^{p}}(a). In the book by K. Goebel and W. A. Kirk [6] one chapter is devoted to rotative mappings. Moreover, in the survey [9] not only the current state of knowledge concerning such mappings is

presented, but also several interesting open problems are posed. It is worth noting here that A. T. Plant and S. Reich [18] extended the notion of

rotativeness onto nonlinear semigroups. If S is a semigroup generated by −A, then S is said to be rotative if for some p > 0 and a < p, ∥x − S(p)x∥ ≤ a∥Ax∥ for all x in the domain of A. It is shown in [18] that if C is a closed subset of Banach space and a

nonexpansive mapping T : C → C is such that A = Id − T satisfies the so-called range condition, then the semigroup S generated by −A is rotative if and only if T is.

The fixed point result for nonexpansive rotative semigroups is also established. As we mentioned, rotative mappings include periodic mappings and in particular involutions. First fixed point theorems for involutions are due to K. Goebel and E. Złotkiewicz [3], [10]. It is shown that if T : C → C, where C is a closed and convex

subset of a Banach space, is a k-lipschitzian involution with k < 2, then T has at least one fixed point. The proof of this fact is a straightforward verification that starting from any x ∈ C, the sequence of iterates \{F^{n}x\} for F = \frac{1}{2}(I + T) always converges to a fixed point of T. W. A. Kirk [13] extended this result by proving that the same is

true if T is n-periodic and such that ∥T^{i}x − T^{i}y∥ ≤ k∥x − y∥ for x, y ∈ C, i = 1, ..., n − 1, where n^{−2}[(n − 1)(n − 2)k^{2} + 2(n − 1)k] < 1. This result

leads to the estimate (∗).

The notion of firmly lipschitzian mappings was introduced in [15] and is analogous to that of firmly nonexpansive mappings originating from Bruck’s paper [2]. Basic results of Section 3 are taken from [15]. Certain properties of firmly lipschitzian mappings and some facts regarding their asymptotic behavior can be found in [12]. Mappings with constant displacement are studied in [5]. This paper contains also a number of other (not presented here) equivalent formulations of Schauder’s Fixed

Point Theorem [19]. Other basic facts used here come from [1] and [17] (existence of a lipschitzian retraction of the unit ball in an infinite dimensional Banach space onto its unit

sphere), [20] (girth of sphere) and [16] (existence of a lipschitzian self-mapping of a noncompact set with positive minimal displacement).

References

[1] Benyamini Y., Sternfeld Y. Spheres in infinite dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), 439-445.

[2] Bruck R. E. Nonexpansive projections on subsets of Banach spaces, Pacific J. Math 48 (1973), 341-357.

[3] Goebel K. Convexity of balls and fixed point theorems for mappings with a nonexpansive square, Compostio Math. 22 (1970), 269-274.

[4] Goebel K. On the minimal displacement problem, Pacific J. Math. 48 (1973), 151-163.

[5] Goebel K., Kaczor W. Remarks on failure of Schauder’s Theorem in noncompact settings, Ann. Univ. Mariae Curie-Skłodowska 51 (1997), 99-108.

[6] Goebel K., Kirk W. A. Topics in metric fixed point theory, Cambridge University Press, 1990 [7] Goebel K., Koter M. A remark on nonexpansive mappings, Canadian Math. Bull. 24, (1981)

113-115.

[8] Goebel K., Koter M. Fixed points of rotative lipschitzian mappings, Rend. Sem. Mat. Fis. Milano 51 (1981), 145-156.

[9] Goebel K., Koter-Mórgowska M. Rotative mappings in metric fixed point theory, Proc. NACA98, World Scientific (1999), 150-156.

[10] Goebel K., Złotkiewicz E. Some fixed point theorems in Banach spaces, Colloquium Math. 23 (1971), 103-106.

[11] Górnicki J. Remarks on fixed points of rotative Lipschitzian mappings, Comment. Math. Univ. Carolinae 40 (1999), 495-510.

[12] Kaczor W., Koter-Mórgowska M. Firmly lipschitzian mappings, Ann. Univ. Mariae Curie-Skłodowska 50 (1996), 77-85.

[13] Kirk W. A. A fixed point theorem for mappings with a nonexpansive iterate, Proc. Amer. Math. Soc. 29 (1971), 294-298.

REFERENCES 15

[14] Komorowski T. Selected topics on lipschitzian mappings, (in Polish) Thesis, Univ. Maria Curie-Skłodowska, 1987.

[15] Koter M. Fixed points of lipschitzian 2-rotative mappings, Boll. Un. Mat. Ital. Ser. VI, 5 (1986), 321-339.

[16] Lin P. K., Sternfeld Y. Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (1985), 633-639.

[17] Nowak B. On the Lipschitz retraction of the unit ball in infinite dimensional Banach spaces onto boundary, Bull. Polish Acad. Sci. 27 (1979), 861-864.

[18] Plant A. T., Reich S. Nonlinear rotative semigroups, Proc. Japan Acad. 58 (1982), 398-401.

[19] Schauder J. Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171-180.

[20] Schäffer J. J. Geometry of spheres in normed spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc., New York-Basel, 1976.

Index

Antipodal points, 6 Banach’s Contraction Mapping Principle, 2 Boundary

lower, 8–10 Boundedness, 2 Class

of nonexpansive mappings, 3 Condition

range, 12 Curve

lipschitzian, 6 piecewise linear, 6

Fixed point property for closed convex sets, 3 for nonexpansive mappings, 1

Function

continuous

decreasing, 3 Graph, 8–10 Hilbert space, 3, 5–6, 8–10 Involution

k-lipschitzian, 6, 12 Involutions, 12 Iterates, 3–4, 12 Iterations, 10 Mapping

(1 + \frac{1}{k} ,2)-rotative, 7

k-lipschitzian, 8 (1 + \frac{k}{l})-lipschitzian, 11

Mapping

(a,2)-rotative firmly k-lipschitzian, 9 monotone, 9

(a,n)-rotative, 1–2

k-lipschitzian, 3 (m − n + a,m)-rotative, 2 k-lipschitzian, 4, 7

with constant displacement, 11 n-periodic, 13 n-rotative, 2 α-lipschitzian, 2 continuous, 11–12 contraction, 2 firmly k-lipschitzian, 8–9 monotone, 9 nonexpansive, 2, 12 rotative, 2

k-lipschitzian, 10

nonexpansive, 2 with constant displacement, 10

Mappings

(a,2)-rotative

k-lipschitzian, 8 k-lipschitzian, 1

with constant displacement, 10 n-periodic, 2 firmly lipschitzian, 13 firmly nonexpansive, 13 lipschitzian, 11 nonexpansive, 1 periodic, 12 rotative, 12

k-lipschitzian, 3 with constant displacement, 1–2, 10–11, 13

Metrical line, 2

Minimal

Lipschitz constant, 7 Minimum, 5 Nonexpansiveness, 3 Periodic point, 12 Periodicity, 1 Retraction

k-lipschitzian, 7, 10 lipschitzian, 13

Rotative, 2

Rotativeness, 1–3, 10, 12 Self-mapping

k-lipschitzian (a,n)-rotative

fixed point free, 4 k-lipschitzian, 3

Self-mapping continuous

with constant displacement, 11 Self-mapping

of a noncompact set lipschitzian, 13,

Self-mapping

of the unit ball

fixed point free, 3 Semigroup, 12 Semigroups

nonexpansive rotative, 12

nonlinear, 12 Set

of all continuous self-mappings, 11

of fixed points, 2 Shortening, 2 Sphere

girth of, 6, 13 Subclass

of k-lipschitzian mappings, 8 Theore

Schauder’s

equivalent formulation, 12 Theorem

Schauder’s Fixed Point, 11, 13

Schauder’s, 12 equivalent formulation, 11 equivalent to, 11

Turning, 2, 10 Upper bounds, 6 Weak compactness, 2