Two Generic Convergence Results for Infinite Products of Generalized Nonexpansive Mappings

Abstract

In our 2014 work with M. Gabour, we introduced a metric space of generalized nonexpansive self-mappings of bounded and closed subsets of a Banach space and studied, using the Baire category approach, the asymptotic behavior of iterates of a generic operator belonging to this class. In the definition of a generalized nonexpansive mapping the norm is replaced by a general function which can be symmetric as a particular case. In this paper, we prove the convergence of infinite products of generalized nonexpansive self-mappings to a common fixed point in a generic setting.

1. Introduction

Nonexpansive mappings (in other words, those Lipschitz mappings the Lipschitz constant of which is equal to one) have been studied intensively in recent decades. See, for example, refs. [1,2,3,4,5,6,7,8,9,10] and the references cited therein. This research activity has its origins in Banach’s classical theorem [11] regarding the existence of a unique fixed point for strict contractions (that is, those mappings the Lipschitz constant of which is strictly less than one). It also concerns the asymptotic behavior of both exact and inexact iterates of a nonexpansive mapping, and their possible convergence. These theoretical investigations have found significant and diverse applications in the engineering, medical and natural sciences [9,10,12,13,14,15]. In this vein, we introduced and studied in [16] a certain class of nonlinear operators, which we now proceed to describe.

Suppose that $K\subset X$ is a bounded, closed and convex set in a Banach space $(X,\parallel ·\parallel )$, and that a continuous function $f:X\to [0,\infty )$ satisfies $f\left(0\right)=0$,

$$sup\left\{\right|f\left(z\right)|:z\in K-K\}<\infty ,$$

and possesses the following properties:

(P1) for every $ϵ>0$, there exists $\delta >0$ such that if $z_1,z_2\in K$ satisfy $f(z_1-z_2)\le \delta$, then the inequality $\parallel z_1-z_2\parallel \le ϵ$ is true;

(P2) for every $\gamma \in (0,1)$, there exists $\varphi \left(\gamma \right)\in (0,1)$ for which

$$f\left(\gamma (z_1-z_2)\right)\le \varphi \left(\gamma \right)f(z_1-z_2) \mathrm{for} \mathrm{each} \mathrm{pair} \mathrm{of} \mathrm{points} z_1,z_2\in K;$$

(P3) the function $({\xi }_1,{\xi }_2)\mapsto f({\xi }_1-{\xi }_2)$, ${\xi }_1,{\xi }_2\in K$, is uniformly continuous on $K\times K$.

Set

$$\mathrm{diam}\left(K\right):=sup\{\parallel z_1-z_2\parallel :z_1,z_2\in K\}.$$

Denote by $\mathcal{B}$ the collection of all continuous operators $A:K\to K$ which satisfy

$$f(A{z}_1-A{z}_2)\le f(z_1-z_2) \mathrm{for} \mathrm{every} \mathrm{pair} \mathrm{of} \mathrm{points} z_1,z_2\in K.$$

For each pair of operators $T,S\in \mathcal{B}$, define

$$d(T,S):=sup\{\parallel Tz-Sz\parallel :z\in K\}.$$

Clearly, $(\mathcal{B},d)$ is a complete metric space.

Using the Baire category approach, we showed in [16] that a typical (generic) mapping in the space $\mathcal{B}$ has a unique fixed point which attracts all its iterates. It should be mentioned that the classical theorem of De Blasi and Myjak [17] regarding nonexpansive mappings is a special case of this result, where the continuous function $f=\parallel ·\parallel$. As a matter of fact, the mappings defined above can be considered generalized nonexpansive mappings with respect to f which can be symmetric as a particular case. Such a viewpoint, where in some problems of functional analysis the norm is replaced by a general function, was employed, for instance, in [18,19] in the study of generalized best approximation problems. At this point, it is worth noting that generalized nonexpansive mappings were also studied in [20,21]. In the present paper, we continue our studies of genericity in nonlinear analysis and establish the convergence of infinite products of generalized nonexpansive self-mappings to a common fixed point in a generic setting. In this connection, we remark in passing that infinite products of operators and studies of their convergence properties arise naturally in many areas of mathematics and its applications such as approximation theory, computed tomography, image recovery, optimization theory and population biology.

2. Main Results

We continue to use the definitions, notations and assumptions introduced in the Introduction. In particular, we suppose that $f:X\to [0,\infty )$ is a continuous function for which the equality $f\left(0\right)=0$ is true and $f(K-K)$ is a bounded set, and which has properties (P1)–(P3).

Denote by $\mathcal{A}$ the collection of all continuous operators $A:K\to K$ and by $\mathcal{M}$ the collection of all sequences ${\left\{A_t\right\}}_{t=1}^{\infty }\subset \mathcal{A}$. We denote the identity operator by $A^0$ for every mapping $A\in \mathcal{A}$.

The set $\mathcal{M}$ is equipped with the metric $\rho :\mathcal{M}\times \mathcal{M}\to [0,\infty )$ which is defined by

$$\rho ({\left\{T_t\right\}}_{t=1}^{\infty },{\left\{S_t\right\}}_{t=1}^{\infty })=sup\{\parallel T_{t}z-S_{t}z\parallel :z\in K,t=1,2,\dots \},$$

$${\left\{T_t\right\}}_{t=1}^{\infty },{\left\{S_t\right\}}_{t=1}^{\infty }\in \mathcal{M}.$$

(1)

It is not difficult to see that $(\mathcal{M},\rho )$ is a complete metric space. We use the topology which is induced in $\mathcal{M}$ by $\rho$.

Denote by ${\mathcal{M}}_{ne}$ the collection of all sequences ${\left\{A_t\right\}}_{t=1}^{\infty }\in \mathcal{M}$ such that for every integer $t\ge 1$, we have

$$f(A_t{z}_1-A_t{z}_2)\le f(z_1-z_2),z_1,z_2\in K.$$

(2)

In view of property (P3), ${\mathcal{M}}_{ne}$ is a closed subset of $\mathcal{M}$. In the sequel the topological subspace ${\mathcal{M}}_{ne}\subset \mathcal{M}$ is equipped with the relative topology.

A collection E of mappings $A:K\to K$ is called uniformly equicontinuous (ue, for short) if for every positive number $ϵ$, there is a positive number $\delta$ such that $\parallel T{z}_1-T{z}_2\parallel \le ϵ$ for every operator $T\in E$ and every pair of points $z_1,z_2\in K$ which satisfies $\parallel z_1-z_2\parallel \le \delta$. Define

$${\mathcal{M}}_{ue}:=\{{\left\{A_t\right\}}_{t=1}^{\infty }\in \mathcal{M}:{\left\{A_t\right\}}_{t=1}^{\infty } \mathrm{is} \mathrm{a} \mathit{ue} \mathrm{set}\}.$$

(3)

It is clear that ${\mathcal{M}}_{ue}$ is a closed subset of the metric space $\mathcal{M}$. We equip ${\mathcal{M}}_{ue}\subset \mathcal{M}$ with the relative topology. Properties (P1)–(P3) imply that ${\mathcal{M}}_{ne}\subset {\mathcal{M}}_{ue}$.

Denote by ${\mathcal{M}}_{ne}^{*}$ the collection of all ${\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ne}$ satisfying

$${\cap }_{t=1}^{\infty }\{x\in K:A_{t}x=x\}\ne \varnothing$$

and denote by ${\overline{\mathcal{M}}}_{ne}^{*}$ the closure of ${\mathcal{M}}_{ne}^{*}$ in the complete metric space ${\mathcal{M}}_{ne}$. We equip the topological subspace ${\overline{\mathcal{M}}}_{ne}^{*}\subset {\mathcal{M}}_{ne}$ with the relative topology.

Denote by ${\mathcal{M}}_{ue}^{*}$ the collection of all sequences of operators ${\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ue}$ for which there is a point $\widehat{x}\in K$ such that for every natural number t,

$$A_t\widehat{x}=\widehat{x}$$

and

$$f(A_t\xi -\widehat{x})\le f(\xi -\widehat{x}) \mathrm{for} \mathrm{every} \xi \in K,$$

(4)

and denote by ${\overline{\mathcal{M}}}_{ue}^{*}$ its closure in the complete metric space ${\mathcal{M}}_{ue}$. We equip ${\overline{\mathcal{M}}}_{ue}^{*}\subset {\mathcal{M}}_{ue}$ with the relative topology.

In this paper, we prove the following two generic convergence results. Note that in the case where $f=\parallel ·\parallel$ they are proved in [8].

Theorem 1.

There is a set $\mathcal{F}\subset {\overline{\mathcal{M}}}_{ne}^{*}$, which is a countable intersection of open and everywhere dense subsets of the space ${\overline{\mathcal{M}}}_{ne}^{*}$, such that for every sequence of operators ${\left\{B_t\right\}}_{t=1}^{\infty }\in \mathcal{F}$, there is a point $x_{*}\in K$ for which the following two assertions are true.

1. $B_t{x}_{*}=x_{*}$ for all positive integers t.

2. For every positive number ϵ, there is an open neighborhood $\mathcal{U}$ of ${\left\{B_t\right\}}_{t=1}^{\infty }$ in the topological space ${\overline{\mathcal{M}}}_{ne}^{*}$ and an integer $N\ge 1$ so that for every sequence of mappings ${\left\{C_t\right\}}_{t=1}^{\infty }\in \mathcal{U}$, every natural number $T\ge N$, every mapping $r:\{1,\dots ,T\}$ and every point $x\in K$, we have

$$\parallel C_{r\left(T\right)}·\cdots ·C_{r\left(1\right)}x-x_{*}\parallel \le ϵ.$$

Theorem 2.

There is a set $\mathcal{F}\subset {\overline{\mathcal{M}}}_{ue}^{*}$, which is a countable intersection of open and everywhere dense subsets of the topological space ${\overline{\mathcal{M}}}_{ue}^{*}$, such that for every sequence of operators ${\left\{B_t\right\}}_{t=1}^{\infty }\in \mathcal{F}$, there is a point $x_{*}\in K$ for which the following two assertions are true.

1. $B_t{x}_{*}=x_{*}$ for all natural numbers t and

$$f(B_t\xi -x_{*})\le f(\xi -x_{*}),\xi \in K,t=1,2,\dots .$$

2. For every positive number ϵ, there is an open neighborhood $\mathcal{U}$ of the sequence of operators ${\left\{B_t\right\}}_{t=1}^{\infty }$ in the topological space ${\overline{\mathcal{M}}}_{ue}^{*}$ and an integer $N\ge 1$ so that for every sequence of operators ${\left\{C_t\right\}}_{t=1}^{\infty }\in \mathcal{U}$, every natural number $T\ge N$, every mapping $r:\{1,\dots ,T\}$ and every point $x\in K$, we have

$$\parallel C_{r\left(T\right)}·\cdots ·C_{r\left(1\right)}x-x_{*}\parallel \le ϵ.$$

3. An Auxiliary Result

For every bounded mapping $T:K\to X$ define

$$\parallel T\parallel :=sup\{\parallel T\xi \parallel :\xi \in K\}.$$

In our study, we will use the next result (Lemma 6.9 of [8]).

Lemma 1.

Assume that F is a nonempty uniformly equicontinuous collection of mappings $T:K\to K$, $N\ge 1$ is an integer and $ϵ>0$. Then there is $\delta >0$ such that for every sequence of mappings ${\left\{T_t\right\}}_{t=1}^N\subset F$, every sequence of mappings ${\left\{S_t\right\}}_{t=1}^N$, where the (not necessarily continuous) mappings $S_t:K\to K$, $t=1,\dots N$, satisfy

$$\parallel S_t-T_t\parallel \le \delta ,t=1,\dots N,$$

and every point $\xi \in K$, the following inequality is true:

$$\parallel T_N·\cdots ·T_1\xi -S_N·\cdots ·S_1\xi \parallel \le ϵ.$$

4. Proofs of Theorems

We prove Theorems 1 and 2 simultaneously.

For every sequence of mappings $\mathbf{A}={\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ue}^{*}$, there is $x\left(\mathbf{A}\right)\in K$ for which

$$A_{t}x\left(\mathbf{A}\right)=x\left(\mathbf{A}\right),t=1,2,\dots ,$$

(5)

$$f(A_{t}y-x\left(\mathbf{A}\right))\le f(y-x\left(\mathbf{A}\right)),y\in K,t=1,2,\dots .$$

(6)

Let $\mathbf{A}={\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ue}^{*}$ and $\gamma \in (0,1)$. For all natural numbers $t,$ define a mapping $A_{t\gamma }:K\to K$ by

$$A_{t\gamma }x=(1-\gamma )A_{t}x+\gamma x\left(\mathbf{A}\right),x\in K.$$

(7)

Evidently, ${\left\{A_{t\gamma }\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ue}$, if ${\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ne}$, then ${\left\{A_{t\gamma }\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ne}$, and

$$A_{t\gamma }x\left(\mathbf{A}\right)=x\left(\mathbf{A}\right),t=1,2,\dots .$$

(8)

By property (P2), (6) and (7), for every point $y\in K$ and every integer $t\ge 1$,

$$f(A_{t\gamma }y-x\left(\mathbf{A}\right))=f\left((1-\gamma )(A_{t}y-x\left(\mathbf{A}\right))\right)$$

$$\le \varphi (1-\gamma )f(A_{t}y-x\left(\mathbf{A}\right))\le \varphi (1-\gamma )f(y-x\left(\mathbf{A}\right)).$$

(9)

Thus ${\left\{A_{t\gamma }\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ue}^{*}$, and if ${\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ne}$, then ${\left\{A_{t\gamma }\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ne}^{*}$. Clearly, the collection

$$\{{\left\{A_{t\gamma }\right\}}_{t=1}^{\infty }:{\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ue}^{*},\gamma \in (0,1)\}$$

is an everywhere dense subset of the topological space ${\overline{\mathcal{M}}}_{ue}^{*}$ and

$$\{{\left\{A_{t\gamma }\right\}}_{t=1}^{\infty }:{\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ne}^{*},\gamma \in (0,1)\}$$

is an everywhere dense subset of the topological space ${\overline{\mathcal{M}}}_{ne}^{*}$.

In view of property (P3) for every positive integer q, there is

$${\delta }_q\in (0,{\left(4q\right)}^{-1})$$

such that the following property is true:

(a) if the points $z_1,z_2,{\xi }_1,{\xi }_2\in K$ satisfy $\parallel z_i-{\xi }_i\parallel \le 2{\delta }_q$, $i=1,2$, then

$$|f(z_1-z_2)-f({\xi }_1-{\xi }_2)|\le {\left(4q\right)}^{-1}.$$

Let ${\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ue}^{*}$, $\gamma \in (0,1)$ be given and let q be a natural number. In view of property (P1) there is

$${\lambda }_q\in (0,{\delta }_q)$$

for which the following implication holds:

$$\mathrm{if} \mathrm{the} \mathrm{points} z_1,z_2\in K \mathrm{and} f(z_1-z_2)\le 2{\lambda }_q, \mathrm{then} \parallel z_1-z_2\parallel \le {\delta }_q.$$

(10)

Fix a natural number $n(\gamma ,q)$ for which

$$\varphi {(1-\gamma )}^{n(\gamma ,q)}sup\{f(z-z^{\prime }):z,z^{\prime }\in K\}<4^{-1}{\lambda }_q.$$

(11)

By (9) and (11), the following property holds:

(b) For every natural number $T\ge n(\gamma ,q)$, every mapping $r:\{1,\dots ,T\}\to \{1,2,\dots \}$ and every point $x\in K$, we have

$$f(A_{r\left(T\right)\gamma }\cdots A_{r\left(1\right)\gamma }x-x\left(\mathbf{A}\right))\le \varphi {(1-\gamma )}^{T}f(x-x\left(\mathbf{A}\right))\le 4^{-1}{\lambda }_q.$$

Property (b) and Equation (10) imply that the following property is true:

(c) For each integer $T\ge n(\gamma ,q)$, each mapping $r:\{1,\dots ,T\}\to \{1,2,\dots \}$ and each point $x\in K$, we have

$$\parallel A_{r\left(T\right)\gamma }\cdots A_{r\left(1\right)\gamma }x-x\left(\mathbf{A}\right)\parallel \le {\delta }_q.$$

Lemma 1 implies that there is an open neighborhood

$$\mathcal{U}({\left\{A_t\right\}}_{t=1}^{\infty },\gamma ,q)$$

of ${\left\{A_{t\gamma }\right\}}_{t=1}^{\infty }$ in the topological space ${\overline{\mathcal{M}}}_{ue}^{*}$ (respectively, in ${\overline{\mathcal{M}}}_{ne}^{*}$ if ${\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ne}^{*}$) such that the following property is true:

(d) For each ${\left\{C_t\right\}}_{t=1}^{\infty }\in \mathcal{U}({\left\{A_t\right\}}_{t=1}^{\infty },\gamma ,q)$, each $x\in K$ and each

$$r:\{1,\dots ,n(\gamma ,q\left)\right\}\to \{1,2,\dots \},$$

we have

$$\rho ({\left\{C_t\right\}}_{t=1}^{\infty },{\left\{A_{t\gamma }\right\}}_{t=1}^{\infty })\le {\delta }_q$$

(12)

and

$$\parallel A_{r\left(n\right(\gamma ,q\left)\right)\gamma }\cdots A_{r\left(1\right)\gamma }x-C_{r\left(n\right(\gamma ,q\left)\right)}\cdots C_{r\left(1\right)}x\parallel \le {\delta }_q.$$

(13)

Assume that

$${\left\{C_t\right\}}_{t=1}^{\infty }\in \mathcal{U}({\left\{A_t\right\}}_{t=1}^{\infty },\gamma ,q),r:\{1,\dots ,n(\gamma ,q)\}\to \{1,2,\dots \}.$$

(14)

Property (d) and (14) imply (13) for each $x\in K$. Property (c) and inequality (13) imply that for each $x\in K$,

$$\parallel C_{r\left(n\right(\gamma ,q\left)\right)}\cdots C_{r\left(1\right)}x-x\left(\mathbf{A}\right)\parallel \le 2{\delta }_q.$$

(15)

Hence the next property is true:

(e) For every point $x\in K$, every sequence of mappings

$${\left\{C_t\right\}}_{t=1}^{\infty }\in \mathcal{U}({\left\{A_t\right\}}_{t=1}^{\infty },\gamma ,q)$$

and every mapping

$$r:\{1,\dots ,n(\gamma ,q\left)\right\}\to \{1,2,\dots \},$$

inequality (15) is true.

Set

$$\mathcal{F}={\cap }_{q=1}^{\infty }\cup \{\mathcal{U}({\left\{A_t\right\}}_{t=1}^{\infty },\gamma ,i):{\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ue}^{*},\gamma \in (0,1),i=q,q+1,\dots \}$$

(16)

(respectively,

$$\mathcal{F}={\cap }_{q=1}^{\infty }\cup \{\mathcal{U}({\left\{A_t\right\}}_{t=1}^{\infty },\gamma ,i):{\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ne}^{*},\gamma \in (0,1),i=q,q+1,\dots \}).$$

Evidently, $\mathcal{F}$ is a countable intersection of open and everywhere dense subsets of the topological space ${\overline{\mathcal{M}}}_{ue}^{*}$ (respectively, ${\overline{\mathcal{M}}}_{ne}^{*}$).

Suppose that

$${\left\{B_t\right\}}_{t=1}^{\infty }\in \mathcal{F},ϵ>0.$$

(17)

Fix an integer $q\ge 1$ such that

$$q^{-1}<ϵ.$$

(18)

By (16) and (17), there exist $\mathbf{A}={\left\{A_t\right\}}_{t=1}^{\infty }\in {\mathcal{M}}_{ue}^{*}$ (respectively, ${\mathcal{M}}_{ne}^{*}$), a positive number $\gamma <1$ and a natural number $i\ge q$ for which

$${\left\{B_t\right\}}_{t=1}^{\infty }\in \mathcal{U}({\left\{A_t\right\}}_{t=1}^{\infty },\gamma ,i).$$

(19)

It follows from (15), (18), (19) and property (e) that for every natural number t, every natural number $p\ge n(\gamma ,i)$ and every point $x\in K$, we have

$$\parallel B_t^{p}x-x\left(\mathbf{A}\right)\parallel \le 2{\delta }_i<{\left(2i\right)}^{-1}\le {\left(2q\right)}^{-1}<ϵ.$$

(20)

Since $ϵ$ is an arbitrary positive number, the inequality above implies that for every point $x\in K$ and every natural number t, there exists the limit

$$\underset{p\to \infty }{lim}{B}_t^{p}x.$$

When combined with inequality (20), the above relation implies that for each $x\in K$ and each natural number t,

$$\parallel \underset{p\to \infty }{lim}{B}_t^{p}x-x\left(\mathbf{A}\right)\parallel \le 2{\delta }_i<{\left(2i\right)}^{-1}<ϵ.$$

(21)

Since $ϵ$ is an arbitrary positive number, the above inequality implies that there exists a point $x_{*}\in K$ for which

$$\underset{p\to \infty }{lim}{B}_t^{p}x=x_{*},x\in K,t=1,2,\dots ,$$

(22)

$$B_t{x}_{*}=x_{*},t=1,2,\dots ,$$

(23)

and

$$\parallel x\left(\mathbf{A}\right)-x_{*}\parallel \le 2{\delta }_i<ϵ.$$

(24)

Property (e), (15), (18) and (24) imply that for every point $x\in K$, every sequence of mappings ${\left\{C_t\right\}}_{t=1}^{\infty }\in \mathcal{U}({\left\{A_t\right\}}_{t=1}^{\infty },\gamma ,i)$, every natural number $T\ge n(\gamma ,i)$ and every mapping $r:\{1,\dots ,T\}\to \{1,2,\dots \}$, we have

$$\parallel C_{r\left(T\right)}\cdots C_{r\left(1\right)}x-x_{*}\parallel \le \parallel x_{*}-x\left(\mathbf{A}\right)\parallel +\parallel C_{r\left(T\right)}\cdots C_{r\left(1\right)}x-x\left(\mathbf{A}\right)\parallel$$

$$\le 4{\delta }_i<i^{-1}\le q^{-1}<ϵ.$$

Assume now that $x\in K$ and that t is a positive integer. Property (a), (4.8) and (4.15) imply that

$$\parallel B_{t}x-A_{t\gamma }x\parallel \le {\delta }_i.$$

(25)

In view of property (a), (24) and (25),

$$|f(B_{t}x-x_{*})-f(A_{t\gamma x}-x\left(\mathbf{A}\right))|\le {\left(4i\right)}^{-1}.$$

When combined with (8), (9), (18), (24) and property (a), the inequality above implies that

$$f(B_{t}x-x_{*})\le {\left(4i\right)}^{-1}+f(A_{t\gamma }x-x\left(\mathbf{A}\right))\le ϵ/2+f(x-x\left(\mathbf{A}\right))\le f(x-x_{*})+ϵ$$

and

$$f(B_{t}x-x_{*})\le f(x-x_{*})+ϵ.$$

Since $ϵ$ is an arbitrary positive number, we conclude that

$$f(B_{t}x-x_{*})\le f(x-x_{*}).$$

This completes the proofs of both Theorems 1 and 2.