Keywords

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let X be a complete metric space. According to Baire’s theorem, the intersection of every countable collection of open dense subsets of X is dense in X. This rather simple, yet powerful result has found many applications. In particular, given a property which elements of X may have, it is of interest to determine whether this property is generic, that is, whether the set of elements which do enjoy this property contains a countable intersection of open dense sets. Such an approach, when a certain property is investigated for the whole space X and not just for a single point in X, has already been successfully applied in many areas of Analysis. In this chapter we discuss several recent results in metric fixed point theory which exhibit these generic phenomena.

1.1 Hyperbolic Spaces

It turns out that the class of hyperbolic spaces is a natural setting for our generic results. In this section we briefly review this concept.

Let (X,ρ) be a metric space and let R 1 denote the real line. We say that a mapping c:R 1→X is a metric embedding of R 1 into X if

$$\rho\bigl(c(s),c(t)\bigr)=|s-t| $$

for all real s and t. The image of R 1 under a metric embedding will be called a metric line. The image of a real interval [a,b]={t∈R 1:a≤t≤b} under such a mapping will be called a metric segment.

Assume that (X,ρ) contains a family M of metric lines such that for each pair of distinct points x and y in X, there is a unique metric line in M which passes through x and y. This metric line determines a unique metric segment joining x and y. We denote this segment by [x,y]. For each 0≤t≤1, there is a unique point z in [x,y] such that

$$\rho(x,z)=t\rho(x,y) \quad\mbox{and}\quad \rho(z,y)=(1-t)\rho(x,y). $$

This point will be denoted by (1−t)x⊕ty.

We will say that X, or more precisely (X,ρ,M), is a hyperbolic space if

$$\rho\biggl({1 \over2}x \oplus{1 \over2}y, {1 \over2}x \oplus{1\over2}z\biggr)\le {1 \over2}\rho(y,z) $$

for all x, y and z in X.

An equivalent requirement is that

$$\rho\biggl({1 \over2}x \oplus{1 \over2}y, {1 \over2}w \oplus{1 \over2}z\biggr)\le {1 \over2} \bigl(\rho(x,w)+ \rho(y,z)\bigr) $$

for all x, y, z and w in X. A set K⊂X is called ρ-convex if [x,y]⊂K for all x and y in K.

It is clear that all normed linear spaces are hyperbolic. A discussion of more examples of hyperbolic spaces and in, particular, of the Hilbert ball can be found, for instance, in [66, 68, 81, 124].

In the sequel we will repeatedly use the following fact (cf. pp. 77 and 104 of [68] and [124]): If (X,ρ,M) is a hyperbolic space, then

$$ \rho\bigl((1-t)x \oplus tz, (1-t)y \oplus tw\bigr) \le(1-t) \rho(x,y)+t\rho(z,w) $$
(1.1)

for all x, y, z and w in X and 0≤t≤1.

1.2 Successive Approximations

Let (X,ρ,M) be a complete hyperbolic space and let K be a closed ρ-convex subset of X. Denote by \(\mathcal{A}\) the set of all operators A:K→K such that

$$\rho(Ax,Ay) \le\rho(x,y) \quad\mbox{for all } x,y \in K. $$

In other words, the set \(\mathcal{A}\) consists of all the nonexpansive self-mappings of K.

Fix some θ∈K and for each s>0, set

$$B(\theta,s)=B(s)=\bigl\{ x \in K: \rho(x,\theta) \le s\bigr\} . $$

For the set \(\mathcal{A}\) we consider the uniformity determined by the following base:

$$E(n,\varepsilon )=\bigl\{ (A,B) \in\mathcal{A} \times\mathcal{A}: \rho (Ax,Bx) \le \varepsilon , x \in B(n)\bigr\} , $$

where ε>0 and n is a natural number. Clearly the space \(\mathcal{A}\) with this uniformity is metrizable and complete. We equip the space \(\mathcal{A}\) with the topology induced by this uniformity.

A mapping A:K→K is called regular if there exists a necessarily unique x A ∈K such that

$$\lim_{n \to\infty}A^nx=x_A \quad\mbox{for all } x \in K. $$

A mapping A:K→K is called super-regular if there exists a necessarily unique x A ∈K such that for each s>0,

$$A^nx \to x_A \quad\mbox{as } n \to\infty, \mbox{ uniformly on } B(s). $$

Denote by I the identity operator. For each pair of operators A,B:K→K and each t∈[0,1], define an operator tA⊕(1−t)B by

$$\bigl(tA \oplus(1-t)B\bigr) (x)=tAx\oplus(1-t)Bx,\quad x \in K. $$

Note that if A and B belong to \(\mathcal{A}\), then so does tA⊕(1−t)B.

In Chap. 2 we establish generic existence and uniqueness of a fixed point for a generic mapping, convergence of iterates of a generic nonexpansive mapping, stability of the fixed point under small perturbations of a mapping and many other results. Among these results are the following two theorems obtained in [132].

The first result shows that in addition to (locally uniform) power convergence, super-regular mappings also provide stability, while the second result shows that most mappings in \(\mathcal{A}\) are, in fact, super-regular. This is an improvement of the classical result of De Blasi and Myjak [49] who established power convergence (to a unique fixed point) for a generic nonexpansive self-mapping of a bounded closed convex subset of a Banach space.

Theorem 1.1

Let A:K→K be super-regular and let ε, s be positive numbers. Then there exist a neighborhood U of A in \(\mathcal{A}\) and an integer n 0≥2 such that for each B∈U, each x∈B(s) and each integer n≥n 0, we have ρ(x A ,B n x)≤ε.

Theorem 1.2

There exists a set \(\mathcal{F}_{0} \subset \mathcal{A}\) which is a countable intersection of open everywhere dense sets in \(\mathcal{A}\) such that each \(A \in\mathcal{F}_{0}\) is super-regular.

1.3 Contractive Mappings

In Chap. 3 we consider the class of contractive mappings which we now define.

Let K be a bounded, closed and convex subset of a Banach space (X,∥⋅∥).

Denote by \(\mathcal{A}\) the set of all operators A:K→K such that

$$\Vert Ax-Ay\Vert \le\Vert x-y\Vert \quad\mbox{for all } x,y \in K. $$

Set

$$d(K)=\sup\bigl\{ \Vert x-y\Vert : x,y \in K\bigr\} . $$

We equip the set \(\mathcal{A}\) with the metric h(⋅,⋅) defined by

$$h(A,B)=\sup\bigl\{ \Vert Ax-Bx\Vert : x \in K\bigr\} ,\quad A, B \in \mathcal{A}. $$

Clearly, the metric space \((\mathcal{A},h)\) is complete.

We say that a mapping \(A \in \mathcal{A}\) is contractive if there exists a decreasing function ϕ A:[0,d(K)]→[0,1] such that

$$\phi^A(t)<1 \quad\mbox{for all } t \in\bigl(0,d(K)\bigr] $$

and

$$\Vert Ax-Ay\Vert \le\phi^A\bigl(\Vert x-y\Vert \bigr)\Vert x-y\Vert \quad \mbox{for all } x,y \in K. $$

The notion of a contractive mapping, as well as its modifications and applications, were studied by many authors. See, for example, [114, 116] and the references mentioned there. We now quote a convergence result which is valid in all complete metric spaces [114].

Theorem 1.3

Assume that \(A \in \mathcal{A}\) is contractive. Then there exists a unique x A ∈K such that A n x→x A as n→∞, uniformly on K.

In Chap. 3 we show that most of the mappings in \(\mathcal{A}\) (in the sense of Baire’s categories) are, in fact, contractive and prove the following result obtained in [131].

Theorem 1.4

There exists a set \(\mathcal{F}\) which is a countable intersection of open everywhere dense sets in \(\mathcal{A}\) such that each \(A \in \mathcal{F}\) is contractive.

Note that at least in Hilbert space the set of strict contractions is only of the first Baire category in \(\mathcal{A}\) [13, 49].

In Chap. 3 we continue with a discussion of nonexpansive mappings which are contractive with respect to a given subset of their domain. We now define this class of mappings.

Let K be a closed (not necessarily bounded) ρ-convex subset of the complete hyperbolic space (X,ρ,M). Denote by \(\mathcal{A}\) the set of all nonexpansive self-mappings of K.

For each x∈K and each subset E⊂K, let ρ(x,E)=inf{ρ(x,y):y∈E}. For each x∈K and each r>0, set

$$B(x,r)=\bigl\{ y \in K: \rho(x,y) \le r\bigr\} . $$

Fix θ∈K. We equip the set \(\mathcal{A}\) with the same uniformity and topology as in the previous section.

Let F be a nonempty, closed and ρ-convex subset of K. Denote by \(\mathcal{A}^{(F)}\) the set of all \(A \in\mathcal{A}\) such that Ax=x for all x∈F. Clearly, \(\mathcal{A}^{(F)}\) is a closed subset of \(\mathcal{A}\). We consider the topological subspace \(\mathcal{A}^{(F)} \subset\mathcal{A}\) with the relative topology.

An operator \(A \in\mathcal{A}^{(F)}\) is said to be contractive with respect to F if for any natural number n, there exists a decreasing function \(\phi^{A}_{n} : [0,\infty) \to[0,1]\) such that

$$\phi^A_n(t)<1 \quad\mbox{for all } t>0 $$

and

$$\rho(Ax,F) \le\phi^A_n\bigl(\rho(x,F)\bigr)\rho(x,F) \quad \mbox{for all } x \in B(\theta,n). $$

Clearly, this definition does not depend on our choice of θ∈K.

The following result, which was obtained in [131], shows that the iterates of an operator in \(\mathcal{A}^{(F)}\) converge to a retraction of K onto F.

Theorem 1.5

Let \(A \in\mathcal{A}^{(F)}\) be contractive with respect to F. Then there exists \(B \in\mathcal {A}^{(F)}\) such that B(K)=F and A n x→Bx as n→∞, uniformly on B(θ,m) for any natural number m.

Finally, we present the following theorem of [131] which shows that if \(\mathcal{A}^{(F)}\) contains a retraction, then almost all the mappings in \(\mathcal{A}^{(F)}\) are contractive with respect to F.

Theorem 1.6

Assume that there exists

$$Q \in\mathcal{A}^{(F)}\quad \textit{such that}\quad Q(K)=F. $$

Then there exists a set \(\mathcal{F} \subset\mathcal{A}^{(F)}\) which is a countable intersection of open everywhere dense sets in \(\mathcal{A}^{(F)}\) such that each \(B \in\mathcal{F}\) is contractive with respect to F.

1.4 Infinite Products

In Chap. 6 we present several results concerning the asymptotic behavior of (random) infinite products of generic sequences of nonexpansive, as well as uniformly continuous, operators on closed and convex subsets of a complete hyperbolic space.

Let (X,∥⋅∥) be a Banach space and let K be a nonempty, bounded, closed and convex subset of X with the topology induced by the norm ∥⋅∥.

Denote by \(\mathcal{A}\) the set of all sequences \(\{A_{t}\}_{t=1}^{\infty}\), where each A t :K→K is a continuous operator, t=1,2,… . Such a sequence will occasionally be denoted by a boldface A.

For the set \(\mathcal{A}\) we consider the metric \(\rho_{s}: \mathcal{A} \times\mathcal{A} \to[0,\infty)\) defined by

$$\begin{aligned} &\rho_s\bigl(\{A_t\}_{t=1}^{\infty}, \{B_t\}_{t=1}^{\infty}\bigr)= \sup\bigl\{ \Vert A_tx-B_tx\Vert : x \in K, t=1,2,\ldots\bigr\} , \\ &\quad\{A_t\}_{t=1}^{\infty}, \{B_t \}_{t=1}^{\infty} \in\mathcal{A}. \end{aligned}$$

It is easy to see that the metric space \((\mathcal{A},\rho_{s})\) is complete. The topology generated in \(\mathcal{A}\) by the metric ρ s will be called the strong topology.

In addition to this topology on \(\mathcal{A}\), we will also consider the uniformity determined by the base

$$\begin{aligned} E(N,\varepsilon ) =&\bigl\{ \bigl(\{A_t\}_{t=1}^{\infty}, \{B_t\}_{t=1}^{\infty}\bigr) \in \mathcal{A} \times \mathcal{A}: \\ &\Vert A_tx-B_tx\Vert \le \varepsilon , t=1,\ldots, N, x \in K\bigr\} , \end{aligned}$$

where N is a natural number and ε>0. It is easy to see that the space \(\mathcal{A}\) with this uniformity is metrizable (by a metric \(\rho_{w}: \mathcal{A} \times\mathcal{A} \to[0,\infty)\)) and complete. The topology generated by ρ w will be called the weak topology.

Define

$$\mathcal{A}_{ne}=\bigl\{ \{A_t\}_{t=1}^{\infty} \in\mathcal{A}: A_t \mbox{ is nonexpansive for } t=1,2,\ldots\bigr\} . $$

Clearly, \(\mathcal{A}_{ne}\) is a closed subset of \(\mathcal{A}\) in the weak topology. We will consider the topological subspace \(\mathcal{A}_{ne} \subset \mathcal{A}\) with both the weak and strong relative topologies.

In Theorem 2.1 of [129] we showed that for a generic sequence \(\{C_{t}\}_{t=1}^{\infty}\) in the space \(\mathcal{A}_{ne}\) with the weak topology,

$$\Vert C_T\cdot\cdots\cdot C_1x-C_T\cdot \cdots\cdot C_1y\Vert \to0 \quad\mbox{as } T \to\infty, $$

uniformly for all x,y∈K. (Such results are usually called weak ergodic theorems in the population biology literature; see [43, 107].)

Here is the precise formulation of this weak ergodic theorem.

Theorem 1.7

There exists a set \(\mathcal{F} \subset \mathcal{A}_{ne}\), which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets of \(\mathcal{A}_{ne}\), such that for each \(\{B_{t}\}_{t=1}^{\infty} \in \mathcal{F}\) and each ε>0, there exist a neighborhood U of \(\{B_{t}\}_{t=1}^{\infty}\) in \(\mathcal{A}_{ne}\) with the weak topology and a natural number N such that:

For each \(\{C_{t}\}_{t=1}^{\infty} \in U\), each x,y∈K, and each integer T≥N,

$$\Vert C_T\cdot\cdots\cdot C_1x-C_T\cdot \cdots\cdot C_1y\Vert \le \varepsilon . $$

Note that in Chap. 6 we also prove a random version of this theorem.

We will say that a set E of operators A:K→K is uniformly equicontinuous (ue) if for any ε>0, there exists δ>0 such that ∥Ax−Ay∥≤ε for all A∈E and all x,y∈K satisfying ∥x−y∥≤δ.

Define

$$\mathcal{A}_{ue}=\bigl\{ \{A_t\}_{t=1}^{\infty} \in\mathcal{A}: \{A_t\}_{t=1}^{\infty} \mbox{ is a (ue) set}\bigr\} . $$

Clearly, \(\mathcal{A}_{ue}\) is a closed subset of \(\mathcal{A}\) in the strong topology.

We will consider the topological subspace \(\mathcal{A}_{ue} \subset \mathcal{A}\) with both the weak and strong relative topologies.

Denote by \(\mathcal{A}_{ne}^{\ast}\) the set of all \(\{A_{t}\}_{t=1}^{\infty} \in\mathcal{A}_{ne}\) which have a common fixed point and denote by \(\bar{\mathcal{A}}_{ne}^{\ast}\) the closure of \(\mathcal{A}_{ne}^{\ast}\) in the strong topology of the space \(\mathcal{A}_{ne}\).

Let \(\mathcal{A}_{ue}^{\ast}\) be the set of all \(\mathbf{A}=\{A_{t}\}_{t=1}^{\infty} \in\mathcal{A}_{ue}\) for which there exists x(A)∈K such that for each integer t≥1,

$$A_tx(\mathbf{A})=x(\mathbf{A}) \qquad\mbox{and}\qquad \bigl\Vert A_ty-x(\mathbf{A})\bigr\Vert \le\bigl\Vert y-x(\mathbf{A})\bigr\Vert \quad \mbox{for all }y \in K, $$

and denote by \(\bar{\mathcal{A}}^{\ast}_{ue}\) the closure of \(\mathcal {A}^{\ast}_{ue}\) in the strong topology of the space \(\mathcal{A}_{ue}\).

We consider the topological subspaces \(\bar{\mathcal{A}}^{\ast}_{ne}\) and \(\bar{\mathcal{A}}^{\ast}_{ue}\) with the relative strong topologies. In Theorem 2.4 of [129] we showed that a generic sequence \(\{C_{t}\}_{t=1}^{\infty}\) in the space \(\bar{\mathcal{A}}^{\ast}_{ue}\) has a unique common fixed point x ∗ and all random products of the operators \(\{C_{t}\}_{t=1}^{\infty}\) converge to x ∗, uniformly for all x∈K. We now quote this theorem.

Theorem 1.8

There exists a set \(\mathcal{F} \subset\bar{\mathcal{A}}_{ue}^{\ast}\), which is a countable intersection of open everywhere dense (in the strong topology) subsets of \(\bar{\mathcal{A}}^{\ast}_{ue}\), such that for each \(\{B_{t}\}_{t=1}^{\infty} \in\mathcal{F}\), there exists x ∗∈K for which the following assertions hold:

  1. 1.

    B t x ∗=x ∗, t=1,2,… , and

    $$\Vert B_ty-x_{\ast}\Vert \le\Vert y-x_{\ast}\Vert, \quad y \in K, t=1,2, \ldots. $$
  2. 2.

    For each ε>0, there exist a neighborhood U of \(\{B_{t}\}_{t=1}^{\infty}\) in \(\bar{\mathcal{A}}_{ue}^{\ast}\) with the strong topology and a natural number N such that for each \(\{C_{t}\}_{t=1}^{\infty} \in U\), each integer T≥N, each mapping r:{1,…,T}→{1,2,…}, and each x∈K,

    $$\Vert C_{r(T)}\cdot\cdots\cdot C_{r(1)}x-x_{\ast}\Vert \le \varepsilon . $$

In [129] we also proved an analog of this theorem for the space \(\bar{\mathcal{A}}_{ne}^{\ast}\).

We remark in passing that one can easily construct an example of a sequence of operators \(\{A_{t}\}_{t=1}^{\infty} \in\mathcal{A}_{ue}^{\ast}\) for which the convergence properties described in the previous theorem do not hold. Namely, they do not hold for the sequence each term of which is the identity operator.

Now assume that F is a nonempty, closed and convex subset of K and that Q:K→F is a nonexpansive operator such that

$$Qx=x,\quad x \in F. $$

Such an operator Q is usually called a nonexpansive retraction of K onto F (see [68]). Denote by \(\mathcal {A}_{ne}^{(F)}\) the set of all \(\{A_{t}\}_{t=1}^{\infty} \in\mathcal {A}_{ne}\) such that

$$A_tx=x,\quad x \in F, t=1,2,\ldots. $$

Clearly, \(\mathcal{A}_{ne}^{(F)}\) is a closed subset of \(\mathcal {A}_{ne}\) in the weak topology. We equip the topological subspace \(\mathcal{A}_{ne}^{(F)} \subset\mathcal{A}_{ne}\) with both the weak and strong relative topologies.

In Theorem 3.1 of [129] we showed that for a generic sequence of operators \(\{B_{t}\}_{t=1}^{\infty}\) in the space \(\mathcal {A}_{ne}^{(F)}\) with the weak topology there exists a nonexpansive retraction P ∗:K→F such that

$$B_t\cdot\cdots\cdot B_1x \to P_{\ast}x \quad\mbox{as } t \to \infty, $$

uniformly for all x∈K. We end this section with the precise statement of this convergence theorem.

Theorem 1.9

There exists a set \(\mathcal{F} \subset \mathcal{A}_{ne}^{(F)}\), which is a countable intersection of open (in the weak topology) everywhere dense (in the strong topology) subsets of \(\mathcal{A}_{ne}^{(F)}\), such that for each \(\{B_{t}\}_{t=1}^{\infty} \in\mathcal{F}\), the following assertions hold:

  1. 1.

    There exists an operator P ∗:K→F such that

    $$\lim_{t \to\infty}B_t\cdot\cdots\cdot B_1x=P_{\ast}x \quad\textit{for each}\ x \in K. $$
  2. 2.

    For each ε>0, there exist a neighborhood U of \(\{B_{t}\}_{t=1}^{\infty}\) in \(\mathcal{A}_{ne}^{(F)}\) with the weak topology and a natural number N such that for each \(\{C_{t}\}_{t=1}^{\infty} \in U\), each integer T≥N, and each x∈K,

    $$\Vert C_T\cdot\cdots\cdot C_1x-P_{\ast}x\Vert \le \varepsilon . $$

Theorem 3.2 of [129] is a random version of this theorem.

1.5 Contractive Set-Valued Mappings

In Chap. 9 we study contractive set-valued mappings.

Assume that (X,∥⋅∥) is a Banach space, K is a nonempty, bounded and closed subset of X and there exists θ∈K such that for each x∈K,

$$t x+ (1-t)\theta\in K, \quad t \in(0,1). $$

We consider the complete metric space K with the metric ∥x−y∥, x,y∈K. Denote by S(K) the set of all nonempty closed subsets of K. For x∈K and D⊂K, set

$$\rho(x,D)=\inf\bigl\{ \Vert x-y\Vert : y \in D\bigr\} , $$

and for each C,D∈S(K), let

$$H(C,D)=\max\Bigl\{ \sup_{x \in C}\rho(x,D), \sup _{y \in D}\rho(y,C)\Bigr\} . $$

We equip the set S(K) with the Hausdorff metric H(⋅,⋅). It is well known that the metric space (S(K),H) is complete.

Denote by \(\mathcal{A}\) the set of all nonexpansive operators T:S(K)→S(K). For the set \(\mathcal{A}\) we consider the metric \(\rho_{\mathcal{A}}\) defined by

$$\rho_\mathcal{A}(T_1,T_2)=\sup\bigl\{ H \bigl(T_1(D),T_2(D)\bigr): D \in S(K)\bigr\} ,\quad T_1,T_2 \in\mathcal{A}. $$

Denote by \(\mathcal{N}\) the set of all mappings T:K→S(K) such that

$$H\bigl(T(x),T(y)\bigr) \le\Vert x-y\Vert, \quad x,y \in K. $$

Set

$$d(K)=\sup\bigl\{ \Vert x-y\Vert : x,y \in K\bigr\} . $$

A mapping \(T \in \mathcal{N}\) is called contractive if there exists a decreasing function ϕ:[0,d(K)]→[0,1] such that

$$\phi(t)<1 \quad\mbox{for all } t \in\bigl(0,d(K)\bigr] $$

and

$$H\bigl(T(x),T(y)\bigr) \le\phi\bigl(\Vert x-y\Vert \bigr)\Vert x-y\Vert \quad\mbox{for all } x,y \in K. $$

Assume that \(T \in \mathcal{N}\). For each D∈S(K), denote by \(\tilde{T}(D)\) the closure of the set ⋃{T(x):x∈D} in the norm topology.

It was shown in [144] that for any \(T \in \mathcal{N}\), the mapping \(\tilde{T}\) belongs to \(\mathcal{A}\) and moreover, the mapping \(\tilde{T}\) is contractive if and only if the mapping T is contractive.

We equip the set \(\mathcal{N}\) with the metric \(\rho_{\mathcal{N}}\) defined by

$$\rho_\mathcal{N}(T_1,T_2)=\sup\bigl\{ H \bigl(T_1(x),T_2(x)\bigr): x \in K\bigr\} ,\quad T_1,T_2 \in \mathcal{N}. $$

It is not difficult to verify that the metric space \((\mathcal{N},\rho_{\mathcal{N}})\) is complete.

For each \(T \in \mathcal{N}\) set \(P(T)=\tilde{T}\). It is easy to see that for each \(T_{1},T_{2} \in \mathcal{N}\),

$$\rho_\mathcal{A}\bigl(P(T_1),P(T_2)\bigr)= \rho_\mathcal{N}(T_1,T_2). $$

Denote

$$\mathcal{B}=\bigl\{ P(T): T \in \mathcal{N}\bigr\} . $$

Clearly, the metric spaces \((\mathcal{B}, \rho_{\mathcal{A}})\) and \((\mathcal{N}, \rho_{\mathcal{N}})\) are isometric.

In [144] we obtained the following results.

Theorem 1.10

Assume that the operator \(T \in \mathcal{N}\) is contractive. Then there exists a unique set A T ∈S(K) such that \(\tilde{T}(A_{T})=A_{T}\) and \((\tilde{T})^{n}(B) \to A_{T}\) as n→∞, uniformly for all B∈S(K).

Theorem 1.11

There exists a set \(\mathcal{F}\), which is a countable intersection of open and everywhere dense subsets of \((\mathcal{N},\rho_{\mathcal{N}})\), such that each \(T \in \mathcal{F}\) is contractive.

1.6 Nonexpansive Set-Valued Mappings

Let (X,∥⋅∥) be a Banach space and denote by S co (X) the set of all nonempty, closed and convex subsets of X. For x∈X and D⊂X, set

$$\rho(x,D)=\inf\bigl\{ \Vert x-y\Vert : y \in D\bigr\} , $$

and for each C,D∈S co (X), let

$$H(C,D)=\max\Bigl\{ \sup_{x \in C}\rho(x,D), \sup _{y \in D}\rho(y,C)\Bigr\} . $$

The interior of a subset D⊂X will be denoted by \(\operatorname{int}(D)\). For each x∈X and each r>0, set B(x,r)={y∈X:∥y−x∥≤r}. For the set S co (X) we consider the uniformity determined by the following base:

$$\mathcal{G}(n)=\bigl\{ (C,D) \in S_{co}(X) \times S_{co}(X): H(C,D) \le n^{-1}\bigr\} , $$

n=1,2,… . It is well known that the space S co (X) with this uniformity is metrizable and complete. We endow the set S co (X) with the topology induced by this uniformity.

Assume now that K is a nonempty, closed and convex subset of X, and denote by S co (K) the set of all D∈S co (X) such that D⊂K. Clearly, S co (K) is a closed subset of S co (X). We equip the topological subspace S co (K)⊂S co (X) with its relative topology.

Denote by \(\mathcal{N}_{co}\) the set of all mappings T:K→S co (K) such that T(x) is bounded for all x∈K and

$$H\bigl(T(x),T(y)\bigr) \le\Vert x-y\Vert, \quad x,y \in K. $$

In other words, the set \(\mathcal{N}_{co}\) consists of those nonexpansive set-valued self-mappings of K which have nonempty, bounded, closed and convex point images.

Fix θ∈K. For the set \(\mathcal{N}_{co}\) we consider the uniformity determined by the following base:

$$\begin{aligned} \mathcal{E}(n) =&\bigl\{ (T_1,T_2) \in \mathcal{N}_{co} \times\mathcal{N}_{co}: H \bigl(T_1(x),T_2(x)\bigr) \le n^{-1} \\ &\mbox{for all } x \in K \mbox{ satisfying } \Vert x-\theta\Vert \le n\bigr\} , \quad n=1,2,\ldots. \end{aligned}$$

It is not difficult to verify that the space \(\mathcal{N}_{co}\) with this uniformity is metrizable and complete.

The following result is well known [45, 102]; see also [116].

Theorem 1.12

Assume that T:K→S(K), γ∈(0,1), and

$$H\bigl(T(x),T(y)\bigr) \le\gamma\Vert x-y\Vert, \quad x,y \in K. $$

Then there exists x T ∈K such that x T ∈T(x T ).

The existence of fixed points for set-valued mappings which are merely nonexpansive is more delicate and was studied by several authors. See, for example, [67, 94, 119] and the references therein. We now state a result established in [145] which shows that if \(\operatorname{int}(K)\) is nonempty, then a generic nonexpansive mapping does have a fixed point. This result will be proved in Chap. 9.

Theorem 1.13

Assume that \(\operatorname{int}(K) \neq \emptyset\). Then there exists an open everywhere dense set \(\mathcal{F} \subset \mathcal{N}_{co}\) with the following property: for each \(\widehat{S} \in\mathcal{F}\), there exist \(\bar{x} \in K\) and a neighborhood \(\mathcal{U}\) of \(\widehat{S}\) in \(\mathcal{N}_{co}\) such that \(\bar{x} \in S(\bar{x})\) for each \(S \in\mathcal{U}\).

1.7 Porosity

In this section we present a refinement of the classical result obtained by De Blasi and Myjak [49]. This refinement involves the notion of porosity which we now recall [51, 123, 180, 182].

Let (Y,d) be a complete metric space. We denote by B(y,r) the closed ball of center y∈Y and radius r>0. A subset E⊂Y is called porous (with respect to the metric d) if there exist α∈(0,1) and r 0>0 such that for each r∈(0,r 0] and each y∈Y, there exists z∈Y for which

$$B(z,\alpha r) \subset B(y,r) \setminus E. $$

A subset of the space Y is called σ-porous (with respect to d) if it is a countable union of porous subsets of Y.

Remark 1.14

It is known that in the above definition of porosity, the point y can be assumed to belong to E.

Since porous sets are nowhere dense, all σ-porous sets are of the first Baire category. If Y is a finite-dimensional Euclidean space, then σ-porous sets are of Lebesgue measure 0. In fact, the class of σ-porous sets in such a space is much smaller than the class of sets which have Lebesgue measure 0 and are of the Baire first category. Also, every Banach space contains a set of the first Baire category which is not σ-porous.

To point out the difference between porous and nowhere dense sets, note that if E⊂Y is nowhere dense, y∈Y and r>0, then there is a point z∈Y and a number s>0 such that B(z,s)⊂B(y,r)∖E. If, however, E is also porous, then for small enough r we can choose s=αr, where α∈(0,1) is a constant which depends only on E.

Let (X,ρ,M) be a complete hyperbolic space and K⊂X a nonempty, bounded, closed and ρ-convex set. Once again we denote by \(\mathcal{A}\) the set of all nonexpansive self-mappings of K. For each \(A, B \in\mathcal{A}\) we again define

$$ h(A,B)=\sup\bigl\{ \rho(Ax,Bx): x \in K\bigr\} . $$
(1.2)

It is easy to verify that \((\mathcal{A},h)\) is a complete metric space.

The following result was established in [142].

Theorem 1.15

There exists a set \(\mathcal{F} \subset \mathcal{A}\) such that the complement \(\mathcal{A} \setminus\mathcal{F}\) is σ-porous in \((\mathcal{A}, h)\) and for each \(A \in\mathcal {F}\) the following property holds:

There exists a unique x A ∈K for which Ax A =x A and A n x→x A as n→∞, uniformly on K.

Proof

Set

$$ d(K)=\sup\bigl\{ \rho(x,y): x,y \in K\bigr\} . $$
(1.3)

Fix θ∈K. For each integer n≥1, denote by \(\mathcal {A}_{n}\) the set of all \(A \in\mathcal{A}\) which have the following property:

(C1):

There exists a natural number p(A) such that

$$ \rho\bigl(A^{p(A)}x,A^{p(A)}y\bigr) \le1/n \quad \mbox{for all } x,y \in K. $$
(1.4)

Let n≥1 be an integer. We will show that \(\mathcal{A}\setminus \mathcal{A}_{n}\) is porous in \((\mathcal{A},h)\). To this end, let

$$ \alpha=\bigl(d(K)+1\bigr)^{-1}(8n)^{-1}. $$
(1.5)

Assume that \(A \in\mathcal{A}\) and r∈(0,1]. Set

$$ \gamma=2^{-1}r\bigl(d(K)+1\bigr)^{-1} $$
(1.6)

and define \(A_{\gamma} \in\mathcal{A}\) by

$$ A_{\gamma}x=(1-\gamma)Ax \oplus\gamma\theta,\quad x \in K. $$
(1.7)

It is easy to see that

$$ \rho(A_{\gamma}x,A_{\gamma}y) \le(1-\gamma)\rho(x,y), \quad x,y \in K, $$
(1.8)

and

$$ h(A,A_{\gamma}) \le\gamma d(K). $$
(1.9)

Choose a natural number p for which

$$ p>r^{-1}\bigl(d(K)+1\bigr)^24n+1. $$
(1.10)

Let \(B \in\mathcal{A}\) satisfy

$$ h(A_{\gamma},B) \le\alpha r, $$
(1.11)

and let x,y∈K. We will show that ρ(B p x,B p y)≤1/n. (We use the convention that C 0=I, the identity operator.)

Assume the contrary. Then for i=0,…,p,

$$ \rho\bigl(B^ix,B^iy\bigr) >1/n. $$
(1.12)

It follows from (1.11), (1.2), (1.8) and (1.12) that for i=0,…,p−1,

$$\begin{aligned} \rho\bigl(B^{i+1}x,B^{i+1}y\bigr) \le&\rho \bigl(B^{i+1}x,A_{\gamma}B^ix\bigr) + \rho \bigl(A_{\gamma}B^ix,A_{\gamma}B^iy\bigr)+\rho \bigl(A_{\gamma}B^iy,B^{i+1}y\bigr) \\ \le& \alpha r+\rho\bigl(A_{\gamma}B^ix,A_{\gamma}B^iy \bigr)+\alpha r \\ \le& 2\alpha r+(1-\gamma)\rho\bigl(B^ix,B^iy\bigr) \le \rho\bigl(B^ix,B^iy\bigr)+2\alpha r-\gamma/n \end{aligned}$$

and

$$\rho\bigl(B^ix,B^iy\bigr)-\rho\bigl(B^{i+1}x,B^{i+1}y \bigr) \ge\gamma/n -2\alpha r. $$

When combined with (1.3), (1.6) and (1.5), this latter inequality implies that

$$\begin{aligned} d(K) \ge&\rho(x,y)-\rho\bigl(B^px,B^py\bigr) \\ =& \sum_{i=0}^{p-1}\bigl[\rho \bigl(B^ix,B^iy\bigr)-\rho\bigl(B^{i+1}x,B^{i+1}y \bigr)\bigr] \ge p(\gamma/n-2\alpha r) \\ \ge& p\bigl[r\bigl(d(K)+1\bigr)^{-1}(2n)^{-1}-2r \bigl(d(K)+1\bigr)^{-1}(8n)^{-1}\bigr] \\ \ge& pr \bigl(d(K)+1\bigr)^{-1}(4n)^{-1} \end{aligned}$$

and

$$p \le r^{-1}d(K) \bigl(d(K)+1\bigr)4n, $$

a contradiction (see (1.10)). Thus ρ(B p x,B p y)≤1/n for all x,y∈K. This means that

$$ \bigl\{ B \in\mathcal{A}: h(A_{\gamma},B) \le\alpha r\bigr\} \subset\mathcal{A}_n. $$
(1.13)

It now follows from (1.9), (1.6) and (1.5) that

$$\begin{aligned} \bigl\{ B \in\mathcal{A}: h(A_{\gamma},B) \le\alpha r\bigr\} \subset& \bigl\{ B \in\mathcal{A}: h(A,B) \le\alpha r +\gamma d(K)\bigr\} \\ \subset&\bigl\{ B \in\mathcal{A}: h(A,B) \le r\bigr\} . \end{aligned}$$

In view of (1.13) this inclusion implies that \(\mathcal{A} \setminus\mathcal{A}_{n}\) is porous in \((\mathcal{A},h)\). Define \(\mathcal{F}=\bigcap_{n=1}^{\infty} \mathcal{A}_{n}\). Then \(\mathcal{A} \setminus\mathcal{F}\) is σ-porous in \((\mathcal{A},h)\).

Let \(A \in \mathcal{F}\). It follows from property (C1) that for each integer n≥1, there exists a natural number s such that ρ(A i x,A j y)≤1/n for all x,y∈K and all integers i,j≥s. Since n is an arbitrary natural number, we conclude that for each x∈K, \(\{A^{i}x\}_{i=1}^{\infty}\) is a Cauchy sequence which converges to a point x ∗∈K satisfying Ax ∗=x ∗ and moreover, A i x→x ∗ as i→∞, uniformly on K. This completes the proof of Theorem 1.15. □

1.8 Examples

Most of the results obtained in this book are generic existence theorems. Usually, we study a certain property for a class of problems which is identified with a complete metric space and it is shown that for a typical (generic) element of this space the corresponding problem has a unique solution. Of course, such results are of interest only if there is a problem which does not possess the desired property. It should be mentioned that such problems do exist. Let us consider, for instance, the space of mappings discussed in Sect. 1.2. By Theorem 1.2, a typical element of this space is super-regular. It is easy to see that the identity operator is not super-regular. If our metric space is a Banach space, then any translation is not super-regular. Of course both of these mappings are not contractive too. In the book we also consider other examples which are more interesting and complicated.

In Sect. 3.4 we construct a contractive mapping A:[0,1]→[0,1] such that none of its powers is a strict contraction. Section 3.5 contains an example of a mapping A:[0,1]→[0,1] such that

$$ \begin{array}{c} |Ax-Ay| \le|x-y| \quad\mbox{for all } x,y \in[0,1], \\ A^n x \to0 \quad\mbox{as } n \to\infty, \mbox{ uniformly on } [0,1], \end{array} $$

and for each integer m≥0, the power A m is not contractive. In Sect. 3.6 we construct a nonexpansive mapping with nonuniformly convergent powers.

In Sect. 2.24 we construct an example of an operator T on a complete metric space such that all of its orbits converge to its unique fixed point and for any nonsummable sequence of errors and any initial point, there exists a divergent inexact orbit with a convergent subsequence. In Sect. 2.26 we construct an example of an operator T on a certain complete metric space X (a bounded, closed and convex subset of a Banach space) such that all of its orbits converge to its unique fixed point, and for any nonsummable sequence of errors and any initial point, there exists an inexact orbit which does not converge to any compact set.