# On Some Boundedness and Convergence Properties of a Class of Switching Maps in Probabilistic Metric Spaces with Applications to Switched Dynamic Systems

Authors: M. De la Sen; A. Ibeas
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This paper investigates some boundedness and convergence properties of sequences which are generated iteratively through switched mappings defined on probabilistic metric spaces as well as conditions of existence and uniqueness of fixed points. Such switching mappings are built from a set of primary self-mappings selected through switching laws. The switching laws govern the switching process in between primary self-mappings when constructing the switching map. The primary self-mappings are not necessarily contractive but if at least one of them is contractive then there always exist switching maps which exhibit convergence properties and have a unique fixed point. If at least one of the self-mappings is nonexpansive or an appropriate combination given by the switching law is nonexpansive, then sequences are bounded although not convergent, in general. Some illustrative examples are also given.


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## Body

## 1. Introduction

The background literature on fixed point theory and applications and associated convergence properties in metric spaces, Banach spaces, probabilistic metric spaces, Menger spaces, and some fuzzy-type versions is very abundant. See, for instance, [1–19] and the references therein. In particular, the theory focused on probabilistic metric spaces, including their specialization to Menger spaces, is also abundant. See, for instance, [1–4, 15, 16, 20] and the references therein. There are also studies in the graph framework for fixed point theory and problems of stability. See, for instance, [21, 22] and the references therein. On the other hand, fixed point theory has a wide range of applications, for instance, in the study of convergence of iterative schemes [17], in particular, of Mann and Jungck types or their many variants [18, 19], and in that of stability of dynamic systems and that of differential and difference equations. A particular class of real world applications refer to the stability of the so-called switched dynamic systems where a switching law assigns active parameterization for the dynamic system through time (or through an iterative discrete process) [23–27].This paper investigates some boundedness and convergence properties of sequences which are generated through a class of switched mappings defined on probabilistic metric spaces, as well as conditions of existence and uniqueness of fixed points. The above switching mappings are defined via the selection as active of a set of primary self-mappings with the activation process governed by a “so-called” switching law. In this way, such switching laws govern the switching process in between primary self-mappings when constructing the switching map. The primary self-mappings are not necessarily contractive but if at least one of them is contractive then there always exist switching maps which exhibit convergence properties and have a unique fixed point. On the other hand, if at least one of the primary self-mappings is nonexpansive or an appropriate combination given by the switching law is nonexpansive, then sequences are bounded although not convergent, in general. Some illustrative examples are also discussed. Section2 introduces Cρk and Ck classes of primary self-mappings in probabilistic metric spaces as well as associated upper- and lower-bounding constraints of the probability density of the built sequences. The above class allows the characterization of strict contractions as well as nonexpansive or expansive self-mappings in the probabilistic metric spaces. In parallel, some needed definitions are revisited while some preliminary results of convergence of sequences, Cauchy sequences, and boundedness of sequences in probabilistic metric spaces and in Menger spaces are obtained. Section 3 gives formalism in probabilistic metric spaces related to the switched maps defined via the activation of the primary self-mappings through switching laws. The obtained results for switched maps rely on boundedness and convergence of sequences in a probabilistic context.


## 2. OnCρk and Ck Classes of Self-Mappings in Probabilistic Metric Spaces

Let us define a probabilistic distanceF:X×X→ΔF, where X is a nonempty abstract set represented by Fx,y for each x,y∈X×X, where ΔF is a set of distribution functions. A distribution function F∈ΔF is a mapping F:R→R0+ which is nondecreasing and left-continuous with inft∈R⁡Ft=0 and supt∈R⁡Ft=1.The ordered pairX,F is a probabilistic metric (PM) space if for any x,y,z∈X and all t,s∈R+ the following conditions hold [1]:(1)1Fx,yt=Ht⟺x=y,whereH∈ΔFis defined byHt=0,ift≤0,1,ift>0;2Fx,yt=Fy,xt;(3)  ifFx,yt=1,Fy,zs=1thenFx,zt+s=1.The triplet X,F,Δ is a Menger space where X,F is a PM-space and Δ is a triangular norm which satisfies the inequality Fx,zt+s≥ΔFx,yt,Fy,zs, ∀x,y,z∈X, ∀t,s∈R+.Note thatFx,y0=Fx,yt=0 for t≤0 and Fx,yt=Fx,y0+=1 for t>0 if x=y since H∈ΔF is nondecreasing and left-continuous. Note also that every metric space X,d can be realized as a PM-space by taking F:X×X→ΔF being defined by Fx,yt=Ht-dx,y for all x,y∈X [1–4]. In the following, D+ is the space of all mappings F:R→0,1 which are left-continuous and nondecreasing with F0=0 and l-F+∞=1. The space D+ is partially ordered by the usual pointwise ordering of functions; namely, F≤G if and only if Ft≤Gt, ∀t∈R, and its maximal element is the distribution Ht [4].Definition 1.
LetX,F be a PM-space. A mapping T:X→X is said to be of Ck-class for some function k:X×X→R+ if(2)FTx,Tyt≥Fx,yk-1x,yt;∀x,y∈X,∀t∈R+.Definition 2.
LetX,F be PM-space. A mapping T:X→X is said to be of Cρk-class for some functions ρ,k:X×X→R+ if(3)Fx,yρ-1x,yt≥FTx,Tyt≥Fx,yk-1x,yt;∀x,y∈X,∀t∈R+,where the functions ρ,k:X×X→R+ satisfy ρx,y≤kx,y, ∀x,y∈X.Note that ifT:X→X is of Cρk-class, then it is also of Ck-class. Note also that T:X→X is nonexpansive if it is of Ck-class with supx,y∈X⁡kx,y≤1 and, in particular, a probabilistic strict contraction if it is of Ck-class with supx,y∈X⁡kx,y<1. Also, if T:X→X is of Cρk-class with supx,y∈X⁡kx,y≤1(supx,y∈X⁡kx,y<1), then it is nonexpansive (probabilistic strictly contractive) [1–4]. If T:X→X is of Cρk-class with 1<infx,y∈X⁡ρx,y≤infx,y∈X⁡kx,y, then it is expansive [1–4]. If there is some ρ:X×X→R+ with infx,y∈X⁡ρx,y>1 such that(4)Fx,yρ-1x,yt≥FTx,Tyt;∀x,y∈X,∀t∈R+,then T:X→X is expansive (even if T:X→X is not of Cρk-class for some k:X×X→R+ subject to 1<infx,y∈X⁡ρx,y≤infx,y∈X⁡kx,y).The following technical result follows.Lemma 3.
The following properties hold:(i)
LetX,F be a PM-space and let T:X→X be a mapping of Cρk-class. Consider the sequences xn⊆X and yn⊆X built by xn+1=Txn, yn+1=Tyn, ∀n∈Z0+ with x0=x, y0=y for some given x,y∈X. Then,(5)Fx,y∏i=0n-1ρi-1x,yt≥FTnx,Tnyt≥Fx,y∏i=0n-1ki-1x,yt,whereknx,y=kTnx,Tny, ρnx,y=ρTnx,Tny, ∀n∈Z0+.(ii)
IfT:X→X is of Ck-class, then FTnx,Tnyt≥Fx,y∏i=0n-1ki-1x,yt, ∀n∈Z0+.(iii)
IfT:X→X is a mapping of either Ck-class or Cρk-class with limn→∞⁡∏i=0nkix,y=0  for the given x,y∈X, then limn→∞⁡FTnx,Tnyt=1, ∀t∈R+.(iv)
IfT:X→X is a mapping of Cρk-class with βn=∏i=0n-1ρi-1x,y, αn=∏i=0n-1ki-1x,y, ∀n∈Z0+, and(6)α=αx,y=liminfn→∞⁡αn,β=βx,y=limsupn→∞⁡βnare inclR0+=R0+∪+∞ (i.e., clR0+ is the closure of R0+, i.e., the extended nonnegative real semiline) for the given x,y∈X, then(7a)Fx,yβnt≥FTnx,Tnyt≥Fx,yαnt;∀n∈Z0+∀t∈R+,(7b)Fx,yβt≥limsupn→∞⁡FTnx,Tnyt≥liminfn→∞⁡FTnx,Tnyt≥Fx,yαt;∀t∈R+.IfT:X→X is a mapping of Ck-class, then liminfn→∞⁡FTnx,Tnyt≥Fx,yαt, ∀t∈R+.Proof.
It follows recursively from (3) with xn+1=Txn, yn+1=Tyn, ∀n∈Z0+, with x0=x, y0=y for the given x,y∈X that(8)Fx,yρ0-1x,yt≥FTx,Tyt≥Fx,yk0-1x,yt,Fx,yρ0-1x,yρ1-1x,yt≥FTx,Tyρ1-1x,yt≥FT2x,T2yt≥FTx,Tyk1-1x,yt≥Fx,yk0-1x,yk1-1x,yt,⋮Fx,y∏i=0n-1ρi-1x,yt≥FTnx,Tnyt≥Fx,y∏i=0n-1ki-1x,yt;∀x,y∈X,∀t∈R+,∀n∈Z0+.Property (i) has been proved and the proof of Property (ii) follows directly by just using the lower-bounding part of the recursion. Property (iii) follows since limn→∞⁡∏i=0n-1ki-1x,y=+∞; then, limn→∞⁡∏i=0n-1ki-1x,yt=+∞, ∀t∈R+, and the conditions that Fx,yt is nondecreasing in the argument t and supt∈R+⁡Fx,yt=limsupt→∞⁡Fx,yt=1 lead from (8) to the existence of the limit limn→∞⁡FTnx,Tnyt=1, ∀t∈R+. Property (iv) is proved closely to Property (iii) by first getting (7a) and (7b) directly from the definitions of βn, αn, β, and α, ∀n∈Z0+.The subsequent example illustrates that Lemma3 is useful for the characterization of probabilities which can be less than one (i.e., the probabilistic certainty) through lower-bounds and upper-bounds in probabilistic metric spaces.Example 4.
Let us consider the metric spaceX,d with F:X×X→ΔF being defined by Fx,yt=Ht-dx,y for all x,y∈X for the distribution function Hab∈ΔF defined by:(9)Habt=at,ift≤0,bt,ift>0for some left-continuous nondecreasing functions a,b:R0+→0,1 with(10)bx,y,t≥ax,y,t=ax,y,-t;∀t∈R,limt→-∞⁡ax,y,t=0,limt→-∞⁡bx,y,t=1.Assume also that a:R0+→0,1 is everywhere lower-semicontinuous and b:R0+→0,1 is everywhere upper-semicontinuous. Then,(11)Habαt-dx,y=aαt-dx,y,ift≤dx,yα,bαt-dx,y,ift>dx,yα,∀x,y∈Xwith ax,y,0-=ax,y,0+=bx,y,0-=0, limt→-∞⁡ax,y,t=0, and limt→+∞⁡bx,y,t=1, ∀x,y∈X. Assume following Lemma 3(iv) that β=βx,y=limsupn→∞⁡βn and α=αx,y=liminfn→∞⁡αn with βn=βnx,y=∏i=0n-1ρi-1x,y, αn=αnx,y=∏i=0n-1ki-1x,y, ∀n∈Z0+. Note that α,  β,  αn,  and βn are allowed to be dependent on x,y. Then, if T:X→X is a mapping of Cρk-class so that (7a) and (7b) of Lemma 3 hold, one gets for any given x,y∈X(12a)bβnt-dx,y=Fx,yβnt=Habβnt-dx,y≥limsupn→∞⁡FTnx,Tnyt=limsupn→∞⁡Habt-dTnx,Tny≥liminfn→∞⁡FTnx,Tnyt=liminfn→∞⁡Habt-dTnx,Tny≥Fx,yαnt=bαnt-dx,y=Habαnt-dx,y;∀n∈Z0+,(12b)bβt-dx,y=Fx,yβt=Habβt-dx,y≥limsupn→∞⁡FTnx,Tnyt=limsupn→∞⁡Habt-dTnx,Tny≥liminfn→∞⁡FTnx,Tnyt=liminfn→∞⁡Habt-dTnx,Tny≥Fx,yαt=bαt-dx,y=Habαt-dx,yfrom (7a) and (7b). Thus, one gets for any given x,y∈X the following:(a)
Iftn>dx,y/αn for some given n∈Z0+, then, since tn>dx,y/βn as well, one gets(13a)bβntn-dx,y≥FTnx,Tnytn≥bαntn-dx,yand ift>dx,y/α since t>dx,y/β, then(13b)bβt-dx,y≥limsupn→∞⁡FTnx,Tnyt≥liminfn→∞⁡FTnx,Tnyt≥bαt-dx,yand∃limt→+∞⁡limn→∞⁡FTnx,Tnyt=1 if α>0 since the above superior and inferior limits equalize unity.(b)
Ifdx,y/βn<t≤dx,y/αn, then(14)Habαnt-dx,y=aαnt-dx,y≤liminfn→∞⁡FTnx,Tnyt≤limsupn→∞⁡FTnx,Tnyt≤Habβnt-dx,y=bβnt-dx,y.If, furthermore,β=α>0, then ∃limt→+∞⁡limn→∞⁡FTnx,Tnyt=1. If, in addition, α=+∞, then β=+∞, at=0, and bt=1, ∀t∈R+; then ∃limn→∞⁡FTnx,Tnyt=1, ∀t∈R+, which is the basic convergence suitable property in probabilistic metric spaces for probabilistic strictly contractive mappings in the existing literature. Note that this case includes the case under Lemma 3(iii) when limn→∞∏i=0nkix,y=0 leading to(15)limn→∞⁡FTnx,Tnyt=limn→∞⁡Habt-dTnx,Tny≥Fx,y+∞=H+∞=1;that is,limn→∞FTnx,Tnyt=1, ∀x,y∈X, ∀t∈R+, or, in other words, for any distance dx,y from a given x∈X to a given y∈X, limn→∞dTnx,Tny=0.(c)
Ift≤dx,y/βn, then liminfn→∞FTnx,Tnyt≥aβnt-dx,y.(d)
Now, assume thatT:X→X is a mapping of Cρk-class with 1<infx,y∈X⁡ρx,y≤infx,y∈X⁡kx,y and then β=α=0; that is, the mapping is expansive. Then, if t→+∞ implying that t>dx,y/β (and also t>dx,y/β since α=β=0), one concludes from (13b) that(16)inft>dx,y/βn⁡FTnx,Tnyt=1,supt≤dx,y/βn⁡FTnx,Tnyt<1;∀x,y≠x∈X,limn→+∞⁡inft>dx,y/βn⁡FTnx,Tnyt=1,limn→+∞⁡supt≤dx,y/βn⁡FTnx,Tnyt<1;∀x,y≠x∈Xsinceαn→0 and βn→0. The constraints (13a) still hold for each n∈Z0+ such that tn>dx,y/αn but the sequence tn diverges to +∞ while αn→0 and βn→0.Example 5.
Assume thatα=β=αn=βn=1, ∀n∈Z0+, independent of x,y∈X. Then, T:X→X is of C11-class and nonexpansive but also probabilistic noncontractive. If t>dx,y, then one gets from (13a) and (13b)(17)FTnx,Tnyt=Fx,yt=bt-dx,y,limt→+∞⁡limn→∞⁡FTnx,Tnyt=limt→+∞⁡bt-dx,y=limt→+∞⁡Habt=1;∀x,y∈X.Assume instead that αnk=βnk=1 for some sequence nk⊆Z0+, ∀k∈Z0+. Then, if tnk>dx,y, one has for any x,y∈X that(18)FTnx,Tnytnk=btnk-dx,yiftnk>dx,y;bαnt-dx,y≤FTnx,Tnyt≤bβnt-dx,y,∀t∈tnk,tnk+1,∀n∈nk,nk+1ift>dx,yβn,which simplifies as bαnt-dx,y≤FTnx,Tnyt≤bβnt-dx,y, ∀n∈Z0+, if t>dx,y/βn:(19)limt→+∞⁡limk→∞⁡FTnkx,Tnkyt=limk→∞⁡btnk-dx,y=limt→+∞⁡bt-dx,y=limk→∞⁡Habtnk=1.From Lemma3(iii), one gets directly the subsequent result.Proposition 6.
LetX,F be a PM-space and let T:X→X be a mapping of either Ck-class or Cρk-class and there is a strictly increasing sequence of nonnegative integers nk fulfilling limk→∞⁡supnk+1-nk<+∞ such that, for some given x,y∈X, ∏i=nknk+nk+1kix,y<1, ∀k∈Z0+; then limn→∞⁡FTnx,Tnyt=1, ∀t∈R+.Proof.
It follows from Lemma3 that if ∏i=nknk+nk+1kix,y<1, ∀k∈Z0+, then limk→∞∏i=0∑i=0knikix,y=0, limk→∞∏i=0∑i=0kni+nkix,y=0, limk→∞∏i=0∑i=0kni+nki-1x,y=∞, ∀n∈nk,nk+1, ∀k∈Z0+, and then limn→∞∏i=0nki-1x,y=∞.Note that Proposition6 includes as a particular case that of probabilistic strict contractions T:X→X which are then mappings of Ck-class with 0<α-1=supx,y∈Xkx,y<1.Definition 7 (see [2]).
LetX,F be a PM-space and A a nonempty subset of X. The probabilistic diameter of A is a mapping DA:R0+→0,1 defined by DAz=supt<z⁡infx,y∈A⁡Fx,yt.Definition 8 (see [2, 4]).
LetX,F be a PM-space and A a nonempty subset of X. The nonempty set A is said to be probabilistically bounded if supz∈R0+⁡DAz=1, that is, if the supremum of its probabilistic diameter DA∈D+.We can define the set unboundedness as the concept opposite to Definition8 as follows.Definition 9 (see [2, 4]).
LetX,F be a PM-space and A a nonempty subset of X. The nonempty set A is said to be probabilistically unbounded if supz∈R0+⁡DAz<1, that is, if DA∉D+.The boundedness and unboundedness of sequencesxn⊆X can be easily defined as supported by Definitions 8 and 9 as follows.Definition 10 (see [2, 4]).
LetX,F be a PM-space. The sequence xn⊆X is probabilistically bounded if supz∈R0+⁡supt<z⁡infn,m∈Z0+⁡infxn,xm∈X⁡Fxn,xmt=1.Definition 11 (see [1]).
LetX,F be a PM-space. Then, the sequence xn⊆X is(1)
probabilistically convergent to a pointx∈X, denoted by xn→x, if for every ε∈R+ and λ∈0,1 there exists some N=Nε,λ∈Z0+ such that(20)Fxn,xε>1-λ;∀n∈Z0+≥N;(2)
Cauchy if for everyε∈R+ and λ∈0,1 there exists some N=Nε,λ∈Z0+ such that(21)Fxn,xmε>1-λ;∀n,m∈Z0+≥N.A PM-space X,F is complete if every Cauchy sequence is probabilistically convergent.Proposition 12.
LetX,F be a PM-space. Then,(1)
xn→x⊆X for some x∈X if and only if the following limit exists: limε→0+⁡limn→∞⁡Fxn,xε=1;(2)
xn⊆X is a Cauchy sequence if and only if limε→0+⁡limn→∞⁡Fxn,xn+mε=1, ∀m∈Z0+.Proof.
Ifxn→x, then there exists some N=Nε,λ∈Z0+ such that Fxn,xε>1-λ, ∀n∈Z0+≥N, for every ε∈R+ and λ∈0,1. Thus, since 0≤Fxn,xε≤1, ∀n∈Z0+, then, by taking λ→0+, one gets limn→∞⁡Fxn,x0+=limε→0+⁡limn→∞⁡Fxn,xε=1.
Conversely, iflimn→∞⁡Fxn,x0+=1, then for every ε∈R+ and λ∈0,1 there exists some N=Nε,λ∈Z0+ such that Fxn,xε>1-λ, ∀n∈Z0+≥N; thus xn→x. Assume that this is not true. Thus, there is some subsequence xnk⊆xn such that Fxnk,xε≤1-λ for some ε∈R+ and λ∈0,1 while limn→∞⁡Fxn,xε=limn→∞⁡Fxn,x0+=1 for any ε∈R+ since Fxn,xε is nondecreasing in the argument ε and one gets the following contradiction for some λ∈0,1:(22)1-λ≥limk→∞⁡Fxnk,xε=limn→∞⁡Fxn,xε=limn→∞⁡Fxn,x0+=1.Proposition 12(1) has been proved. The proof of Proposition 12(2) is very close and it is omitted.Proposition 13.
LetX,F be a PM-space. Then, the sequence xn⊆X is probabilistically bounded if and only if DaMxn=1, where DaMxn=supz∈R0+⁡Daxnz with Daxnz=supt<z⁡infxn∈X⁡Fa,xnt for some a∈X, that is, if and only if DaMxn∈D+ for some a∈X.Proof.
IfX is bounded, then the result is direct for any sequence S=xn⊆X. Assume that X is not bounded and proceed by contradiction by assuming that S⊆X is probabilistically bounded and DaMS∉D+ for some a∈X. On the other hand, since S⊆X is probabilistically bounded, then, for all xk,xm∈S, there is t0=t0xk,xm∈R0+ such that Fxk,xmt=1, ∀t∈R0+>t0. Since DaMS∉D+, there is xk∈S such that Fa,xkt<1, ∀t∈R0+>t01, and some t01=t01xk,a∈R0+. This implies from the contrapositive equivalent logic proposition to the third property of (1) of X,F being a PM-space that either Fxk,xmt/2<1, and then the sequence S⊆X is not probabilistically bounded, ∀t∈R0+>t01 (a contradiction), or Fa,xmt/2<1, ∀t∈R0+>t01, for any given xm∈S, and then limsupt→+∞⁡Fa,xmt/2<1 for some fixed a,xm∈X. Now, assume that, for all a∈X such that a∉S, limsupt→+∞⁡Fa,xmt/2<1. Thus, the subset Aa=a,xm of X, ∀a∈X, is unbounded since its probabilistic diameter is less than one; that is, DAa∉D+, ∀a∈X, and then the sequence S is probabilistically unbounded, again a contradiction. It has been proved that if S⊂X is bounded, then DaMS∈D+ for some a∈X∩S-, where S- is the complementary to   S in X. It remains to prove that if DaMS∈D+ for some a∈X∩S-, then S⊂X is probabilistically bounded. Since DaMS∈D+, then Fa,xkt=1, ∀t∈R0+>t0, for some t0=t0a,xk∈R0+ and all xk∈S. It follows from the third property of (1) that Fxn,xmt=1, ∀t∈R0+>2t0, ∀xn,xm∈S. Thus, S⊂X is probabilistically bounded.


## 3. Switched Maps Defined byCρk and Ck Classes of Primary Self-Mappings and a Class of Dynamic Systems

Switching processes are a very important tool in some applications of discrete-time and continuous-time dynamic systems. The basic idea is how to switch in-between alternative parameterizations of a system by using either “ad hoc” or even arbitrary switching laws while keeping or improving essential suitable properties like global or asymptotic stability or convergence to the equilibrium points. See [23–26] and some references therein. The formalism can also rely on the definitions of iteration-dependent maps in iterative schemes of Mann or Jungck type or its generalizations so as to get appropriate convergence properties [18, 19, 23]. Note that a switching process in an iterative scheme can be interpreted as the choice under a switching rule of certain primary self-maps from an available collection of them at certain iteration points; that is, the iterative scheme or the solution equation of a dynamic system is being governed by a switching rule [28]. Based on the above elementary idea, this section relies on switching maps built with a prefixed number of either Cρk-class or Ck-class, self-mappings on PM-spaces subject to switching rules which select the new selected mapping and the points at which such new switching occurs. For exposition simplicity, it is assumed that Cρk-class, or Ck-class, self-mappings are characterized by constants instead of functions in Definitions 1 and 2.LetX,F be PM-space and let Ti:X→X be a set of (primary) self-mappings of Cρk-class for some constants ρi≤ki, ki∈R+ for i∈q-=1,2,…,q. A switching map T=Tσnx from Z0+×X to X with respect to the switching law σ:Z0+×X→q- generates a sequence(23)xn+1=Txn=Tσnxnxn=Tixn,∀n∈Z0+,for each given x0∈X for some i=in∈q- and we informally can say that the ith primary self-mapping Ti:X→X is “active” at the nth value (or sample) of the sequence xn [28]. See also [23–27]. In other words, the switching map T=Tσn on X is defined by one of the self-mappings Ti:X→X (i∈q-) for each n∈Z0+ and it has associated piecewise constant functions ρσn,kσn:Z0+×X×X→R+ such that ρ=ρσnx,y∈ρ1,ρ2,…,ρq, k=kσnx,y∈k1,k2,…,kq for each n∈Z0+ and each x,y∈X. The set of switching samples of a sequence xn⊂X is a (proper or improper) subset ZS=ZSxn,σ of Z0+, so-called the switching set, defined by ZS=n∈Z+:Tσnxn≠Tσn-1xn=n0,n1,…,nk,…. Note that a switching set is a strictly ordered set for the standard strict ordering relation “<”. Since Ti:X→X, ∀i∈q-, are self-mappings of Cρk with constants ρi≤ki, ki∈R+, then(24)FTix,xρi-1t≥FTi2x,Tixt≥FTix,xki-1t;∀t∈R+,for any x∈X so that one has recursively from (3) for a sequence xn⊂X generated by xn+1=Tσnxn, ∀n∈Z0+, for any given x0∈X(25)Fx1,x0∏i=0nρσi-1t=Fx1,x0∏i=1qρi-∑j=0k-1γinj,nj+1-γink,nk+lt≥Fxnk+l+1,xnk+lt≥Fx1,x0∏i=0nk+lkσi-1t=Fx1,x0∏i=1qki-∑j=0k-1γinj,nj+1-γink,nk+lt;∀k∈Z0+,∀t∈R+,where l∈Z0+≤nj+1-nj, nk∈ZS, ∀k∈Z0+, is(26)nk=nk-1+∑i=1qγink-1,nk=∑j=0k-1∑i=1qγinj,nj+1and γinj,nj+l∈Z0+ is the number of times that the ith self-mapping Ti:X→X for some i∈q- is “active” in the interval nj,nj+l for each i∈q-. If Ti:X→X, ∀i∈q-, are self-mappings of Ck-class, then one has instead of (25)(27)Fxnk+l+1,xnk+lt≥Fx1,x0∏i=1qki-∑j=0k-1γinj,nj+1-γink,nk+lt;∀k∈Z0+,∀t∈R+.Theorem 14.
LetX,F be a PM-space and let Ti:X→X be self-mappings of Ck-class for some constants ki∈R+ for i∈q-=1,2,…,q. Then, the following properties hold:(i)
Assume that there is (at least) a self-mappingTi:X→X for some i∈q- which is a probabilistic strict contraction. Then, there are infinitely many switching laws σ:Z0+×X→q- such that their associate switching maps T:Z0+×X→X are probabilistic strict contractions.(ii)
Under the conditions of the above proposition, there are infinitely many switching lawsσ:Z0+×X→q- such that their associate switching maps T:Z0+×X→X are probabilistic strict contractions and, furthermore, they consist of infinitely many alternate active switching maps of the form i,j or j,i with j∈q-∖i.(iii)
If, in addition,X,F is complete, then any sequence xn+1=Tσnxn, n∈Z0+ under any switching law σ:Z0+×X→q-  fulfilling either Property (i) or Property (ii) for any given initial point x0∈X is Cauchy and probabilistically convergent.Proof.
It follows from (27) that Property (i) is fulfilled for sequences xn⊂X generated as xn+1=Tσnxn for any x0∈X by any of the infinitely many switching maps T:Z0+×X→X built under switching laws σ:Z0+×X→q- which fulfil(28)limk→∞⁡∑j=0k-1γinj,nj+1=+∞,∑l≠i=1q∑k=0∞γlnk,nk+1<+∞since there is a finite nonnegative integer n∗=n∗σ depending on the subsequence xn:n<n∗ which is a terminal switching point such that σn=i for i≥n∗ for any such a sequence xn. Thus, one gets from (27) that(29)limn→∞⁡Fxn+m,xnt=FTσ0x0,x0+∞=1,∀t∈R+,∀m∈Z+,since F:R→R0+ is nondecreasing and left-continuous with supt∈R⁡Ft=1 and then the sequence xn built as xn+1=Txn=Tσnxn, x0∈X is a Cauchy sequence and T:Z0+×X→X is a probabilistic strict contraction. Property (i) has been proved.
Property (ii) follows with alternate (probabilistic strict contraction versus remaining self-mapping) switching lawsσ:Z0+×X→q- defined by a switching set ZS fulfilling the fact that if, for any k∈Z0+, nk+j∈ZS for j=0,1,…,lk-2 has active (perhaps nonprobabilistic strict contractions) self-mappings Tj:X→X for j≠i∈q-, then nk+lk∈ZS is defined such that Ti:X→X is active on nk+lk-1,nk+lk; that is, xn+1=Tσnxn=Tixn for n∈nk+lk-1,nk+lk with nk+lk being defined with lk large enough such that(30)kink+lk-nk+lk-1<∏j≠i=1qkj∑l=kk+lk-1γjnl,nl+1which lead to(31)limk→∞⁡∑j=0k-1γinj,nj+1=limk→∞⁡∑l≠i=1q∑k=0∞γlnk,nk+1=+∞so that we get again (29) and a similar conclusion. Property (iii) is obvious from the fact that X,F is complete and xn is Cauchy under switching laws fulfilling either Property (i) or Property (ii).The following result is a direct consequence of Theorem14 since mappings of Cρk-class are also of Ck-class.Corollary 15.
LetX,F be a PM-space and let Ti:X→X be self-mappings of Cρk-class for some constants ρi≤ki, ki∈R+, for i∈q-=1,2,…,q. Then, Theorem 14 still holds.Theorem 16.
LetX,F,Δ be a complete Menger space with Δa,b=min⁡a,b and let Tj:X→X be self-mappings of Ck-class for some constants kj∈R+  for all j∈q-=1,2,…,q with at least Ti:X→X being a probabilistic strict contraction for some i∈q-. Let σ:Z0+×X→q- be a switching law and let xn⊂X be a sequence generated as xn+1=Tσnxn, n∈Z0+, for any given x0∈X such that their associate switching map T:Z0+×X→X is defined by Txn=Tσnxn=Tixn for j∈q- and all n≥n∗ and some finite n∗∈Z0+. Then, xn→zi which is the unique fixed point of the strict contraction Ti:X→X.Proof.
SinceTi:X→X is a strict probabilistic contraction, limn→∞⁡Fxn+m,xnt=FTixn∗,xn∗+∞=1, ∀t∈R+, ∀m∈Z+, from (29) since Txn=Tσnxn=Tixn for n≥n∗. Then xn is probabilistically convergent to zi∈X which is a fixed point of the probabilistic strict contraction Ti:X→X as proved by contradiction. Assume that this is false so that zi≠Tizi and then since xn is Cauchy, xn→zi, F:R→R0+ which is nondecreasing and left-continuous, and Ti:X→X is a probabilistic strict contraction, one gets, for any given t∈R+ and λ∈0,1 and all n∈Z0+≥N and some N=Nt,λ∈Z0+,(32)1-λ0t≥Fzi,Tizit≥ΔΔFzi,xnt4,FTixn,xnt4,FTixn,Tizit2≥ΔΔFzi,xnt4,FTixn,xnt4,FTixn,Tzit4≥ΔΔFzi,xnt4,FTixn,xnt4,Fxn,ziki-1t4≥ΔΔFzi,xnt4,FTixn,xnt4,Fxn,zit4>1-λfor some λ0=λ0t>0, ∀t∈R+, which implies that λ∈λ0,1 but since λ∈0,1 can be chosen arbitrarily, it suffices to take λ∈0,λ0 to get a contradiction. Then, zi=Tzi which is proved to be unique again by contradiction. Assume that this is not the case so that there exist zi1=Tnzi1 and zi2=Tnzi2≠zi1, ∀n∈Z0+, which are fixed points of the probabilistic strict contraction Ti:X→X. Thus, one gets the contradiction(33)1>Fzi1,zi20+=Fzi1,zi2t=FTnzi1,Tnzi2t≥Fzi1,zi2ki-nt=Fzi1,zi2+∞=1,∀t∈R+,∀n∈Z0+,so that zi2=zi1=zi=Tizi. Since Txn=Tσnxn=Tixn for n≥n∗ for some finite n∗∈Z0+, we can write xn∗=Tσn∗-1·⋯·Tσ0x0 and xn=Tin-n∗xn∗ for n∈Z0+≥n∗ to get that xn generated by xn+1=Tσnxn, ∀n∈Z0+, for any given arbitrary x0∈X is probabilistically convergent to zi=Tizi.The above result is a direct consequence of Theorem14 which is also valid if Ti:X→X is of Cρk-class. However, note that, under the alternate switching laws in Theorem 14(ii), the limit points of sequences generated through the switching maps T:Z0+×X→X are, in general, dependent on the initial points of the sequences and on the switching law.The following result generalizes Theorem16 without assuming any special contractive condition on at least one of Ti:X→X, ∀i∈q-, with the only condition on the operators being that all of them are either of Cρk-class or Ck-class.Theorem 17.
LetX,F be a PM-space, let Ti:X→X be self-mappings of Cρk-class for some constants ρi≤ki, ki∈R+, ∀i∈q-, and let T:Z0+×X→X be a switching mapping associated with a switching law σ:Z0+×X→q- which generates a sequence xn⊂X as xn+1=Tσnxn, ∀n∈Z0+, for some given x0∈X. Let ZS⊆Z0+ be the set of switching points, that is, for any given nk∈ZS, ∀k∈Z+, provided that nk-1∈ZS, if and only if σnk≠σnk-1=σnk-1. Then, the following properties hold:(i)
(34)FTx0,x0ρ-nt≥Fxn+1,xnt≥FTx0,x0γ-nt;∀t∈R+,for alln∈nk,nk+l∩Z0+, l∈Z0+≤nk+1-nk, nk∈ZS, ∀k∈Z0+, where(35)γ-n=∏i=1qki-∑j=0k-1γinj,nj+1-γink,nk+l,(36)ρ-n=∏i=1qρi-∑j=0k-1γinj,nj+1-γink,nk+l,whereγinj,nj+l∈Z0+ is the number of times that the ith self-mapping Ti:X→X for each i∈q- is “active” in the interval nj,nj+l for some i∈q-.(ii)
IfF:R→0,1 is upper-semicontinuous at ρ-t for some given t∈R+, where ρ-=limsupn→∞⁡ρ-n, then xn has the following property:(37)limsupn→∞⁡Fxn+1,xnt≤FTx0,x0ρ-t.IfF:R→0,1 is lower-semicontinuous at ρ-t for some given t∈R+, where γ_=liminfn→∞⁡γ-n, then xn has the following property:(38)liminfn→∞⁡Fxn+1,xnt≥FTx0,x0γ_t.(iii)
IfTi:X→X are of Ck-class for some constants ki∈R+, ∀i∈q-, then xn has the following property:(39)Fxn+1,xnt≥FTx0,x0γ-nt;∀t∈R+.And alln∈nk,nk+l∩Z0+, l∈Z0+≤nk+1-nk, nk∈ZS, ∀k∈Z0+. If F:R→0,1 is lower-semicontinuous at ρ-t for a given t∈R+, where γ_=liminfn→∞⁡γ-n, then (38) holds.Proof.
Property (i) follows since (34), subject to (35)-(36), is obtained directly from (25)-(26). If F:R→0,1 is upper-semicontinuous at ρ-t, then(40)limsupn→∞⁡Fxn+1,xnt≤limsupρ-n→ρ-⁡FTx0,x0ρ-nt≤FTx0,x0limsupρ-n→ρ-⁡ρ-nt=FTx0,x0ρ-t.In the same way, if F:R→0,1 is lower-semicontinuous at γt, we get liminfn→∞⁡Fxn+1,xnt≥FTx0,x0γ_t=FTx0,x0liminfγ-n→γ⁡γ-nt. This proves Property (ii). Property (iii) is a restriction of Properties (i)-(ii) for the case when Ti:X→X are of Ck-class for some constants ki∈R+, ∀i∈q-.Note that althoughF:R→0,1 is assumed to be everywhere left-continuous in the probabilistic metric framework, this does not mean that it is everywhere lower- and/or upper-semicontinuous. Therefore, some extra related conditions are imposed in Theorem 17(ii)-(iii) allowing obtaining limit upper- and lower-bounds of Fxn+1,xnt as n→∞ via the limit superior and the limit inferior.The following two results related to bounded and unbounded sequences follow from Theorem17.Corollary 18.
LetX,F be a PM-space, let Ti:X→X be self-mappings of Ck-class for some constants ki∈R+, ∀i∈q-, and let T:Z0+×X→X be a switching mapping associated with a switching law σ:Z0+×X→q- which generates a sequence xn⊂X as xn+1=Tσnxn, ∀n∈Z0+, ∀n∈Z0+, for some given x0∈X with switching points in the switching set ZS. Assume also that F:R→0,1 is everywhere lower-semicontinuous with γ_=liminfn→∞⁡γ-n>0 for such a sequence xn. Then, the following properties hold:(i)
xn is probabilistically bounded.(ii)
IfFx0,Tx0t=Ht-dx,Tx, ∀x0∈X, ∀t∈R, then ∃limn→∞⁡Ht-dTn+1x0,Tnx0=1 if t∈dx0,Tx0/γ_,+∞.Proof.
From (39), one concludes (38); that is, liminfn→∞⁡Fxn+1,xnt≥FTx0,x0γ_t, ∀t∈R+. Since γ_>0, then supt∈R0+⁡FTx0,x0γ_t=1 and xn is probabilistically bounded since a nonempty set A=Axnk⊂X which contains all the points of some subsequence xnk⊆xn has the property that DA∈D+. Property (i) has been proved. On the other hand, it follows from (25) and Property (i) that if Fx0,Tx0t=Ht-dx,Tx, ∀x0∈X, ∀t∈R, then(41)limn→∞⁡Ht-dTn+1x0,Tnx0≥Hγ_t-dx0,Tx0=1;t∈dx0,Tx0γ_,+∞and Property (ii) is proved.Since mappings ofCk-class are also of Cρk-class, then Corollary 18 also holds if some of Ti:X→X are self-mappings of Cρk-class for some constants ρi≤ki, ki∈R+, ∀i∈q-.Corollary 19.
LetX,F be a PM-space, let Ti:X→X be self-mappings of Ck-class for some constantski∈R+, ∀i∈q-, and let T:Z0+×X→X be a switching mapping associated with a switching law σ:Z0+×X→q- which generates a sequence xn⊂X as xn+1=Tσnxn, ∀n∈Z0+, ∀n∈Z0+, for some given x0∈X  with switching points in the switching set ZS. Assume also that F:R→0,1 is everywhere upper-semicontinuous with ρ-=limsupn→∞⁡ρ-n=0 for such a sequence xn. Then, the following properties hold:(i)
xn is probabilistically unbounded.(ii)
IfFx0,Tx0t=Ht-dx,Tx, ∀x0∈X, ∀t∈R, then ∃limt→+∞⁡limn→∞⁡Ht-dTn+1x0,Tnx0=1.Proof.
From (39), one concludes (38); that is, limsupn→∞⁡Fxn+1,xnt≤FTx0,x0ρ-t=FTx0,x00=0, ∀t∈R+. Consider a nonempty set A=Axnk⊂X which contains all the points of some subsequence xnk⊆xn. It is obvious that DA∉D+. Then xn is probabilistically unbounded since it has a probabilistically unbounded subsequence. Property (i) has been proved. On the other hand, it follows from (25) and Property (i) that if Fx0,Tx0t=Ht-dx,Tx, ∀x0∈X, ∀t∈R, then(42)limt→+∞⁡limn→∞⁡Ht-dTn+1x0,Tnx0≤limt→+∞⁡Hρ-t-dx0,Tx0=H-dx0,Tx0=H0=0and Property (ii) is proved.Remarks 3.
(1) Note that Corollary18 is fulfilled, in particular, by switched sequences with switching set of finite cardinal with a terminal point of switching to some nonexpansive Cρk (or Ck) self-mapping Ti:X→X; that is, ki∈0,1 for some i∈q-. Note also that Corollary 18 is not fulfilled for terminal switching to an expansive self-mapping.
(2) Note that Corollary19 for Cρk self-mappings can also be applied to self-mappings Ti:X→X, i∈q-, which are only subject to the upper-bounding rule of the probability density function (say self-mappings “of Cρ-class”); that is,(43)FTix0,x0ρi-1t≥FTix0,Ti2x0t;∀t∈R+,while leading to a similar conclusion about probabilistic unboundedness. Note that the result applies, in particular, to switched sequences with switching set of finite cardinal with a terminal point of switching to some expansive Cρk (or Cρ) self-mapping Ti:X→X for some i∈q-; that is, ρi>1 for some i∈q-.
(3) Note that Corollary18 is also fulfilled by switched sequences with switching set of infinite cardinal if γ_>0. Corollary 18 excludes switching laws σ:Z0+×X→q- being built under some expansive self-mapping Ti:X→X for some i∈q- being used so infinitely often such that γ_=0 with either finite terminal switching point or not. This includes, in particular, the case when the switched map is built with a finite terminal switching point to an expansive self-mapping. Then, the probabilistic unboundedness result of Corollary 19 applies.Note that an expansive mapping can give a fixed point which unbounded sequences do not converge to as the next example visualizes.Example 20.
Assume thatX,F is a PM-space with 0∈X and consider a self-map T:X→X such that T0=0. Thus, z=0 is trivially a fixed point of T:X→X. Assume the following cases:(a)
F0,xt≥F0,Txρt, ∀t∈R+, and all x≠0∈X for some real constant ρ>1 so that T:X→X is expansive. Then, F0,Tnxt≤F0,xρ-nt, ∀t∈R+, ∀n∈Z0+, so that limn→∞⁡F0,Tnxt=0, ∀t∈R+, ∀n∈Z0+, and the sequence Tnx is unbounded and does not converge in probability to the fixed point z=0. There is an analogy with the expansive deterministic counterpart examples. For instance, consider a scalar difference equation xn+1=ρxn, ∀n∈Z0+, with ρ>1 and x0≠0. Then, z=0 is a fixed point of the self-mapping on R defining the sequence trajectory solution which is clearly expansive for the metric space R,d with d being any metric, for instance, the Euclidean norm. However, xn→+∞ as n→∞ so that the unique fixed point is an unstable equilibrium point.(b)
F0,Txt≥F0,xk-1t, ∀t∈R+, and all x≠0∈X for some real constant k∈0,1 so that T:X→X is nonexpansive. Then, if furthermore k∈0,1, then T:X→X is strictly contractive and F0,Tnxt≥F0,xk-nt, ∀n∈Z0+, so that limn→∞⁡F0,Tnxt=1 and Tnx→z=0, which is a fixed point, with probability one, is also Cauchy and probabilistically bounded. The deterministic counterpart can be the example xn+1=kxn with x0≠0, ∀n∈Z0+, with k<1 in a metric space R,d, where z=0 is the unique fixed point (and a globally asymptotically stable equilibrium point) of the self-mapping defining the sequence trajectory solution for any initial condition.(c)
In the above case, assumingk=1, then, for any x∈X, FTn+1x,Tnxt≥Fx,Txt, ∀t∈R+, ∀n∈Z0+. If Tx=x, then FTn+1x,Tnxt=1, Tnx=x, ∀n∈Z0+; the mapping T:X→X defining the solution is not expansive but not contractive and any point is a fixed point. A deterministic counterpart can be visualized with the example xn+1=xn, ∀n∈Z0+, for any x0∈X which has infinitely many fixed points which are also (nonasymptotically) stable equilibrium points.(d)
The above discussion can be directly extended to the case of switching maps built under switching laws with finite terminal switching point to an expansive primary self-mapping or for appropriate switching laws with no terminal switching point in the presence of at least one expansive primary self-mapping.A worked numerical example follows.Example 21.
This example aims at numerically illustrating the main results stated and proved in Section3 through Theorem 17 (as a generalization of Theorem 16). For this purpose, consider(44)Fx,yt=Ht-dx,y=0,t≤dx,y,1,t>dx,y,where d(x,y) is the distance induced by the 2-norm, d(x,y)=x-y2. Consider also the switched self-mapping described by the discrete dynamical system xn+1=Tσnxn with Tσn=Aσn+Fσn, Aσn∈A1,A2,A3, and Fσn∈F1,F2,F3. This way, A matrices can be regarded as dynamics matrices while F matrices can be understood as perturbation ones. The dynamics matrices are selected in such a way that in conjunction with the perturbation ones the switched map exhibits different characters (contractive, nonexpansive, and expansive) for each i=1,2,3 so that the effect of switching can be positively noticed. Thus, the dynamics matrices are given by(45)A1=0.10.30.20.10.20.20.10.20.1,A2=0.20.30.40.4250.20.30.10.30.1,A3=1.20.31.40.50.21.30.10.31.1,whose eigenvalues are, respectively, given by(46)specA1=0.741,-0.17+0.153j,-0.17-0.153j,specA2=0.7722,-0.1361+0.1191j,-0.1361-0.1191j,specA3=1.8691,0.7381,-0.1073.The constants characterizing each one of these matrices, interpreted as operators, are calculated from (2) FTx,Ty(t)≥Fx,y(k-1t), which in this particular case takes the form(47)Ht-dAix,Aiy≥Hk-1t-dx,y=Ht-kdx,ykfor each one of the matrices, i=1,2,3. The latter condition is satisfied if d(Aix,Aiy)≤kid(x,y) which results in the considered metric in Aix-Aiy2≤kix-y2. Therefore, for the first matrix, we have(48)A1x-A1y2≤A12x-y2=0.534x-y2<0.54x-y2so that k1=0.54. The 2-norms of the remaining matrices are A22=0.8 and A32=2.528. As it can be seen, A1 and A2 are contractive operators while A3 is expansive. However, the perturbation matrices F will shape the behavior of the operator Tσn=Aσn+Fσn in a different way. To this end, fix ε=0.2 and set the perturbation matrices in such way that Fi2=λmaxFTF≤ε for i=1,2,3. Thus, let them be(49)F1=0.010.02-0.010.02-0.08-0.01-0.010.18-0.01,F2=-0.010.18-0.010.010.010.020.02-0.08-0.01,F3=0.02-0.08-0.01-0.010.18-0.010.010.02-0.01.The perturbation matrices are generated by swapping a file for each subsystem. Therefore, F12=F22=F32=0.1987 near the upper bounding value of ε=0.2. Under these circumstances, the worst-case operator A1+εI is still contractive, A2+εI is nearly nonexpansive and noncontractive, and A3-εI is always expansive providing different dynamics to each of the subsystems. With this setup, we will perform three simulation experiments to illustrate the diversity of dynamical behaviours that can be achieved by modifying the switching law. The initial condition in all experiments is x0=1-12T.Experiment 1.
The switching law never stops and the first subsystem prevails over the other two. The switching law is selected to be periodic, with a period of 8 samples according to the pattern displayed in Figure1. This means that in one period we have 5=γ1>γ2+γ3=2+1; that is, the time interval within which the first subsystem is active is larger than the sum of the intervals corresponding to the other subsystems. Thus, Theorem 17 ensures that the sequence of iterates is probabilistically bounded and converges to the fixed point x=0 as Figure 2 displays for the state components and Figure 3 for the norm of the state. Moreover, Figure 4 shows how Fxn+1,xn(t) converges to the Heaviside function H(t) as the iteration variable n increases.Figure 1
Periodic switching map for the first experiment.Figure 2
Evolution of the state variables for the first experiment.Figure 3
Evolution of the norm of the state for the first experiment.Figure 4
Evolution ofFxn+1,xnt as n increases for the first experiment.Experiment 2.
The switching map converges in finite time to the second subsystem, which is nearly to be nonexpansive and noncontractive. Thus, the switching map is depicted in Figure5. Figures 6 and 7 show, respectively, the evolution of the state variables and their norm as n increases. As Theorem 17 states, the sequence of iterates is bounded and converges to the fixed point because, due to numerical round-off errors, the operator T2 is slightly contractive. However, it converges at a lower rate than in Experiment 1 since the operator T2 is very nearly to be noncontractive and nonexpansive. In this case, Fxn+1,xnt also converges to the Heaviside function H(t) as the iteration variable n increases, as it can be noticed in Figure 8.Figure 5
Switching map for the second experiment.Figure 6
Evolution of the state variables in the second experiment.Figure 7
Evolution of the norm of the state in the second experiment.Figure 8
Evolution ofFxn+1,xnt as n increases for the second experiment.Experiment 3.
The switching law never stops and the third subsystem prevails over the other two. A periodic switching signal is considered again withγ3>γ1+γ2 in a period. Figure 9 gives the switching law. In addition, Figures 10 and 11, respectively, display the evolution of the state variables and their norm as n increases. As we could have expected from Theorem 17, they are not bounded and diverge asymptotically since the expansive operator dominates the period of the switching pattern. Contrarily to the previous experiments, Fxn+1,xnt converges now to the identically null function, since, as n increases, the Heaviside function H(t) is displaced to the right, as it is represented in Figure 12.Figure 9
Switching patterns for the third experiment.Figure 10
Evolution of the state variables for the third experiment.Figure 11
Norm of the state for the third experiment.Figure 12
Evolution ofFxn+1,xnt as n increases for the third experiment.



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*Source: 836283-2015-10-04.xml*