Open Access

Some Fixed Point Properties of Self-Maps Constructed by Switched Sets of Primary Self-Maps on Normed Linear Spaces

Fixed Point Theory and Applications20102010:438614

DOI: 10.1155/2010/438614

Received: 15 September 2009

Accepted: 25 March 2010

Published: 4 May 2010

Abstract

This paper is devoted to the investigation of the existence of fixed points in a normed linear space endowed with a norm for self-maps from to which are constructed from a given class of so-called primary self- maps being also from to . The construction of the self-maps of interest is performed via a so-called switching rule which is a piecewise-constant map from a set to some finite subset of the positive integers or a sequence map which domain in some discrete subset of .

1. Introduction

This paper is devoted to the investigation of the existence of fixed points in a normed linear space with norm for self-maps from to which are constructed from a given class of so-called primary self-maps from to . The construction of the maps of interest is performed via a so-called switching rule which is a piecewise-constant map from a set to some finite subset of the positive integers. The potential discontinuity points of such a self-map in a discrete subset are the so-called switching points at which a new primary self-map in a class is activated to construct the self-map of interest, each of those self-maps depends also on some given switching rule .

In particular, , for all where are two consecutive elements in the sequence ST generated by the switching rule such that , for all , for all provided that there is no .

The class of primary self-maps used to generate the self-map from to might contain itself, in the most general case, a class of contractive primary self-maps from to , a class of large contractive self-maps from to , a class of nonexpansive self-maps from to , as well as a class of expansive self-maps from to X. The problem is easily extendable to the case when the switching rule is a discrete sequence of domain in a discrete set of and of codomain again being the set of nonnegative integers. The study is of particular interest for its potential application to the study of eventual fixed points in the state-trajectory solution of either continuous-time or discrete-time switched dynamic systems [16], constructed from a fixed set of primary parameterizations.

It is well known that contractive self-maps in normed linear spaces and in metric spaces possess a fixed point which is unique in Banach spaces and in complete metric spaces , [710]. Under additional boundedness-type conditions, a large contractive self-map from to which generates uniformly bounded iterates for any number of iterations still possesses a unique fixed point in a complete metric space (or in a Banach space [6]. Some nonexpansive self-maps as well as certain expansive self-maps also possess fixed points (see, e.g., [11, 12]). On the other hand, pseudocontractive self-maps and semicontinuous compact maps in Banach spaces can also possess fixed points [13, 14]. Those features motivate in this paper the choice of the given class of primary self-maps for this investigation. It is also taken into account as motivation that unforced linear time-invariant dynamic systems are exponentially stable to the origin if the matrix of dynamics is a stability matrix. In the case that such a matrix has some pair of complex conjugate eigenvalues then the solution is bounded and the solution trajectory may oscillate, and if there is some eigenvalue with positive real part (i.e., within the instability region), then the state trajectory solution is unbounded. The two first situations can be discussed using the fixed pointy formalism, [1, 2, 6]. Thus, it is of interest to have some extended formalism to investigate time-varying switched dynamic systems obtained under switched linear primary parameterizations not all of them being necessarily stable and then associated with asymptotic contractive self-maps. It is also of interest the basic investigation on the existence of fixed points of discontinuous self-maps which are identical to some self-maps in a prescribed class over each connected subset of being generated from a switching rule of switching points in the sequence . The switching rule which governs the definition of the self-map from to from the primary class of self-maps from to is shown to be crucial for to possess a fixed point. is assumed to be a normed linear space which is not necessarily either a Banach space or assumed to have some uniform structure [15].

Three examples of the formalism are provided two of them being referred to the use of arbitrary primary self-maps on while the third one refers to linear time-varying dynamic systems subject to simultaneous parameterization switching and impulsive controls.

2. Notation

where .

where .

, respectively, are the sets of -real, respectively, -complex vector functions of domain of class , that is, times -real, respectively, -complex continuously differentiable everywhere in its definition domain .

, respectively, are the sets of -real or, respectively, -complex vector functions of domain of class with its th time derivative being necessarily everywhere piecewise-continuous in . Thus, , respectively, are the sets of all -real, respectively, -complex vector functions being piecewise—continuous everywhere in .

If is a first-class discontinuity point of , then , simplifying the customary notation , denotes the lower limit of at and denotes the right limit of at .

denotes the Lebesgue measure of a subset of .

is the discrete measure of the sequence (A being Lebesgue-measurable) defined via the Kronecker-delta defined by if and , otherwise.

The symbols , and mean logic conjunction, disjunction, and negation, respectively, and is a subset of the positive integer numbers.

To establish the general framework for the formulation, consider a set which is a proper or improper subset of (it could also be a subset of ) and being a linear normed space endowed with the norm .

Consider also a class of so-called primary (i.e., auxiliary) self-maps , for all to be used to build the class of maps under study through Fixed Point Theory which is defined by
(2.1)
The class of primary self-maps of , which generate the class , is defined by for some proper or improper subset of , for all . In many applications ; for all . However, the possibility of taking different subsets of remains open within this formulation, for instance, for cases when the various parameterizations (or some of them) are not defined or, simply, not allowed to switch arbitrarily but each with its own switching restrictions. The map ( ) is the so-called switching rule being an integer-valued switching function if is a subset of of nonzero measure and a nonnegative integer-valued switching sequence of real domain if is a subset of . Each element of for which the initial condition is fixed is also axiomatically considered a switching point for any arbitrary switching rule. That intuitively means that a switching rule involves a switch . The discrete switching sequence of switching points , indexed for and which is generated by the switching rule [35], is defined as follows,
(2.2)
is the number of switching points which is either finite or infinity numerable. (the so-called Switching Property of the switching rule ): the switching sequence ST is defined according to the -Property:
(2.3)
Note that each switching-dependent integer is some integer in the set which defines the configuration within the set of configurations which remains active within the interval for two consecutive switching points so that for any three consecutive ,
(2.4)

The discrete switching sequence may also be viewed as a discrete strictly ordered set of real or integer nonnegative numbers of first element (i.e., first switching point) in the sense that such that . Also, such that ; for all . Note that the switching sequence is a strictly ordered sequence of real numbers which depends on the switching rule . If the switching rule is a piecewise continuous function, that is, is a countable union of real intervals then its discontinuity point happen at points of ST since the function changes to another primary function being distinct from the previous one. It is being supposed through the manuscript that (being either or ) and .

The above framework is useful for examples of composed functions, multi-parameterizations of dynamic systems, and so forth involving mappings with some kind of switching. In particular, it is useful to investigate the stability and asymptotic stability of certain dynamic systems which switched parameterizations. Examples of problems situations adjusting to the above description are [35].
  1. (a)
    , a subset of , is the state space of a continuous-time linear time-varying unforced dynamic system described for by
    (2.5)
     

is a real matrix function of piecewise constant entries whose image is , where , for all with if .

Then, where is a fundamental matrix of (2.5), for all on the interval , for all and is a piecewise constant function taking values in the integer set which changes value at each so that , for all . The unique state trajectory-solution for each given initial conditions is with being differentiable everywhere in with having first-class discontinuities in ST.

The eventual discontinuity points of the piecewise continuous switching function , that is,
(2.6)
are the discontinuity points of the state trajectory time-derivative since they generate discontinuities in at least one entry of , for all .
  1. (b)
    , a subset of , is the state space of a discrete-time linear time-varying unforced dynamic system described for by
    (2.7)
     

is a real matrix sequence whose image is , where , for all .

The changes of value of the discrete switching function at a sample imply changes of values in at least one entry of

The class of primary self-maps of is a union of disjoint sets, of respective disjoint indexing sets and whose sum of respective cardinals equalizes that is
(2.8)
with at least one of them being nonempty. Note from (2.1) that any function in is constructed by taking a function in its primary class for each interval ; for all and some . In this way, , with , is identical to some , for some within each real interval for each pair with the switching rule being defined in such a way that and . Thus, ; for all . The sets and considered in this section of the manuscript are defined in the sequel.
  1. (1)

    The class of strictly contractive primary self-maps from to is defined as follows.

     

Definition 2.1.

belongs to the class of strictly contractive primary self-maps from to if for any , for all , the following inequality holds:
(2.9)

where for some real constants where denotes the value in of at .

Note from Definition 2.1 that for each in it exists a function for some real constant , for all . It is assumed that is the first element in ST. The terminology "strictly contractive self-map" is used for the standard contractions referred to in the Banach contraction principle [6, 16] as a counterpart of the alternative terminology used for large contractions [6], here introduced below. Since strict contractions are also large contractions, since contractive self-maps are also nonexpansive ones, and since the sets and are investigated as corresponding unions of disjoint components, the class of large contractions (resp., that of nonexpansive self-maps) are characterized as members of a set which excludes the strict contractions (resp., as members of a class which excludes any contractive self-map).

Remark 2.2.

Note that the dependence of the functions on the switching rule is a generalization which often occurs in practical cases. For instance, if a dynamic system (2.5) changes its parameterization at time from a stability matrix to another one then ; some so that , for all what leads to
(2.10)
for some sufficiently small real constant if . Since is also a fundamental matrix of a time-invariant linear dynamic system (i.e., the active system parameterization starting at time ) for initial vector state , then it is of exponential order so that for for some real constant and some real constants (being norm-dependent) and ; .
  1. (2)

    The class ofnonstrictly contractive large-contractive primary self-maps from to is now defined as follows.

     

Definition 2.3.

belongs to the class of nonstrictly contractive large-contractive primary self-maps from to if it fulfils the joint condition for any , for all , where Conditions and are defined as follows.

(a) satisfies Condition
(2.11)

(b) satisfies Condition for all such that

(2.12)

where for some real constant .

Definition 2.3 applies to all in with for some real constant , for all .
  1. (3)

    The class of noncontractive nonexpansive primary self-maps from to is defined as follows.

     

Definition 2.4.

belongs to the class of noncontractive nonexpansive primary self-maps from to if for any , , the following inequality holds:
(2.13)
Note that the above inequality is fulfilled by any , for all .
  1. (4)

    The class of expansive upper-bounded primary self-maps from to satisfying a global Lipschitz condition and an additional bounding property is now defined as follows.

     

Definition 2.5.

belongs to the class of expansive upper-bounded primary self-maps from to satisfying a global Lipschitz condition and an additional bounding property if for any , the following inequalities hold:
(2.14)

where and for some real constants and .

The above upper-bounding condition has been assumed to facilitate the subsequent exposition. Note that there exists and for some finite real constants and for any in , for all and that . Note also that this requirement is not very restrictive since it is fulfilled, for instance, by compact self-maps from to , also by bounded self-maps from to and, even, by unbounded piecewise-continuous maps of positive exponential order. That means that any state-trajectory solution generated from bounded initial conditions in globally exponentially stable continuous-time linear dynamic systems fulfil such a property for finite time in . Many nonlinear dynamic systems whose state-trajectory solutions do not exhibit finite escape times also possess this property.
  1. (5)

    The class of neither nonexpansive nor expansive primary self-maps from to satisfying a global Lipschitz condition is defined in the sequel. This class includes, for instance, primary self-maps which are expansive and globally Lipschitzian and nonexpansive over alternate subsets of of finite measure and primary functions which are, for instance, asymptotically nonexpansive while being expansive for proper (then finite) subsets .

     

Definition 2.6.

belongs to the class of neither nonexpansive nor expansive primary self-maps from to satisfying a global Lipschitz condition if it satisfies the following inequality:
(2.15)

where is uniformly upper-bounded by some finite real constant .

There exist being uniformly upper-bounded by some finite real constants , for each in , for all . Note that

Since , and all the self-mappings from to in all the component subsets satisfy a global Lipschitz condition, the following result is direct via recursion for each pair of consecutive elements of from the definition of the class of primary functions for as a union of disjoint classes and Definition 2.1 and Definitions 2.3–2.6.

Lemma 2.7.

, being generated from a class of primary self-maps of in by any switching rule , possesses the two following properties:
  1. (i)
    (2.16)
     
  1. (ii)
    (2.17)
     

Proof.

Any function in is constructed for each interval by taking a function in its primary class , for all and some . In this way, , with , is identical to some , for some within each real interval for each pair with the switching rule being defined in such a way that and . Thus, , for all . The primary class consists of a disjoint union of classes defined in Definition 2.1 and Definitions 2.3–2.6 which all have upper-bounding functions of the form of the first inequality in (2.16). The function can be directly expanded for any and , provided that some next consecutive exists, and for any , otherwise (i.e., if switching ends, i.e., has finite cardinal, and the last switching point is ) via recursion from the preceding interswitching intervals ; , for all .

Remark 2.8.

The last logic proposition for the validity of Property (ii) of Lemma 2.7 means that if and, otherwise, that is, if so that is the last element in ST (with the physical sense that the switching process generated by the switch rule stops in finite time), then .

Lemma 2.7(ii) leads to the following direct consequent result.

Lemma 2.9.

Assume that is Lebesgue-mesasurable with . Then, the self-map , being generated from a class of primary self-maps from to , which satisfies the given assumptions, from the switching rule , is strictly contractive if there exists some finite with such that
(2.18)
with for all , . The condition (2.18) implies
(2.19)
where , for all , , and also
(2.20)

where , for all , .

The self-map is still strictly contractive if (2.18) holds by replacing for some finite even if .

Proof.

It follows that if (2.18) holds, then proceeding inductively
(2.21)
so that one gets
(2.22)
so that , being generated from a classof primary self-maps from to from the switching rule , is strictly contractive from Banach's contraction principle. Equations (2.19) and (2.20) are a direct consequence of the fact that (2.18) implies directly
(2.23)

The second part of the result is obvious since the finite interval of may be removed from the discussion by still keeping the strict contraction property.

To discuss some practical situations that guarantee the fulfilment of the condition (2.18), let us define the following subsets of , as a union of disjoint components associated with some proper or improper subset of the class of primary functions being active to build each on some nonempty subset of the subset of :
(2.24)
with
(2.25)
being such that
(2.26)
(2.27)
where for any such that for any proper or improper subset of and
(2.28)

Note that depends on , but not on the , since a particular switching rule can remove some primary self-maps from generating a particular self-map from to . Note also that the above decompositions are also extendable "mutatis-mutandis" to any subsets and .

The interpretation of is the number of strictly contractive primary self-maps from to ; that is, members of being active on any of the subsets of finite measure of . Note that the sets and are, respectively, the set of switching points used to build from the primary class of functions on by the switching rule and which restrict its image (since its domain is restricted) to some which is the subset of active primary functions for some nonempty subset of . The interpretations of the disjoint decompositions of, in general, nonconnected subsets, of the sets and ST in (2.28) are related to. Note that for any given , one has by construction
(2.29)
(2.30)

where the sets are connected disjoint subsets of since and are consecutive switching points under the switching rule , if (i.e., infinity numerable if generates infinitely many switching points), and with and , otherwise. However, the subsets and of are not necessarily connected for any given switching rule since it can make a particular primary function at disjoint subsets of active to build

Note that the set also includes any subset being obtained by replacing in (2.30) with any other of the disjoint components of . Note also that Condition (2.18) needs the presence of a strictly contractive self-map as a member of the primary functions for the given switching rule as it is discussed in the subsequent result.

Lemma 2.10.

A necessary condition for the strict contractive condition (2.18) to hold is that switching rule which generates has at least one self-map being a member of the primary self-maps .

Proof.

Proceed by contradiction by assuming that the generalized condition obtained from (2.18) for any finite holds with . If then and then (2.18) does not hold and, instead, we have,
(2.31)
so that
(2.32)
and, then, either
(2.33)
provided that it exists, or if the above limit does not exist, then is not strictly contractive. A second possibility is and so that . Then, for any such a
(2.34)

Now, if then the same contradiction as above follows. Otherwise, if then either fails, for all , or it holds for some but there is no such that (2.18) holds since it has been assumed that satisfies .

Remark 2.11.

Concerning Lemma 2.10, note that fixed points can still exist for even if . A such a situation can happen, for instance, if a self-map of is built with a switching rule involving above primary functions in a class . It is well known that a large contraction self-map in a Banach space can possess fixed points (see, e.g., [6]). However, Lemma 2.10 proves that if , the strict contraction condition (2.18) does not hold for any

The mains result of this section follows.

Theorem 2.12.

Assume that is Lebesgue-mesasurable with and consider a switching rule which generates , with defined by the class of primary self-maps from to satisfying the given assumptions. The following properties hold.

(i) has a fixed point under Lemma 2.9 if and

(ii) has a fixed point if and and, furthermore, such that the boundedness condition
(2.35)

holds in the case that is not Lipschitzian.

(iii) has a fixed point if and and, furthermore, such that the boundedness condition
(2.36)

holds in the case that is not Lipschitzian; then has a fixed point.

Proof.
  1. (i)
    Proceed by contradiction by taking into account the set inclusion properties (2.30) and by assuming that the extended form of (2.18) in Lemma 2.9 to any replacement (finite) does not hold. Since and , there exists a finite such that for ; for all and for (some finite ), it follows by complete induction that
    (2.37)
     
for some since , for all which is a contradiction. Then, Lemma 2.9 holds with for the valid replacement so that is strictly contractive and has a fixed point.
  1. (ii)
    Since is a large contraction, the real sequence below is bounded monotonically strictly decreasing for any pair since
    (2.38)
     
Thus, for any prefixed and such that
(2.39)
provided that is sufficiently close to zero satisfying where is the Lipschitz constant of the self-map from to , provided to be Lipschitzian, and any given . If as , then the self-map from to has a fixed point in and the result is proven. Furthermore, the error sequence which maps in is a Cauchy sequence and has a zero fixed point in . Also, is a Cauchy sequence with a limit in which is a fixed point of . Otherwise, assume that such that is not a Cauchy sequence or, if so, it does not converge to zero while satisfying ; for all Then, such that
(2.40)
Then, there is a finite sufficiently large such that the above result is a contradiction for all . Then, the self-map the Lipschitzian self-map from to has a fixed point in . Now, if is not Lipschitzian, but it satisfies the given boundedness alternative condition, then
(2.41)

which is again a contradiction for for all such that concluding that has a fixed point.(iii)Since then either

(1) and then the proof follows from Property (i), or

(2) and then the proof follows from Property (ii), or

(3) and then the proof also follows from Property (ii).

Remark 2.13 (An interpretation of Theorem 2.12).

Theorem 2.12 extends the Banach contraction principle of strictly contractive maps and the fixed point properties of large contractions to the case when the self-map is defined via switching-based combination of contractive primary self-maps as follows. If , then the self-map from to is built with a set of strictly contractive self-maps from to on a subset of its domain of infinity Lebesgue measure. If, furthermore, , then there is a finite (e.g., a finite time instant if ) such that all the primary self-maps used to build the self-map are strictly contractive for all . A conclusion is that the self-map from to is strictly contractive so that it has a fixed point. A close reasoning leads to the conclusion that the self-map from to is a large contraction if and or if and . In all those cases, the subset of the domain of where each primary self-map is activated by the switching rule are not necessarily connected. In the case when (Theorem 2.12(iii)) the joint subset of the domain of where the primary self-maps building are either strictly contractive or large contractive has infinite measure, what leads to the same conclusions about the existence of fixed points as in the two former cases, although it is not necessarily connected. A counterpart of Theorem 2.12 can be formulated for the case when is discrete countable sequence (say, e.g., ). In this case, the finite Lebesgue measures referred to in Theorem 2.12 are replaced by the cardinals of finite subsequences of and the infinity Lebesgue measures are replaced by (i.e., sequences of countable infinity many nonnegative integers). The usefulness of the extended results ofTheorem 2.12relies on its use on the stability properties of switched dynamic systems with asymptotic convergence of their state trajectory solutions to a fixed point. They also rely, to a more basic level, on the fixed point properties of maps which are not necessarily Lipschitz-continuous but being built with Lipschitz-continuous functions through a switching process.

Remark 2.14.

In the case that the existence of a fixed point is not guaranteed under the given conditions. Some "ad hoc" conditions for the existence of fixed points are given in the next section.

Remark 2.15.

If a fixed point exists for a particular self-map built with a class of primary self-maps switched according to a rule under the sufficiency-type conditions of Theorem 2.12, then such a fixed point is unique in the Banach space . The result is directly extendable to complete metric spaces what allows to consider a parallel formulation for the case that the domain of , that is , is an infinite sequence on nonnegative real numbers. If the formalism is applied on a compact metric space, then it is not required for large contractions the fulfilment of the boundedness condition of Theorem 2.12(ii)-(iii) from Edelstein fixed point theorem [8] which can be proven using the Meir-Keeler theorem [9] as observed in [10].

The following property from to being a fixed point space relative to a set of maps [16] is obvious under the conditions which guarantee the existence of at least a fixed point in for any in the class .

Assertion 2.16.

The Banach space is a fixed point space relative to the class of self-maps which satisfies any of the properties of Theorem 2.12.

Example 2.17.

Consider and a set of primary self-maps all having the structure
(2.42)
for some given (being piecewise continuous and uniformly bounded on ); , for all . Note that the primary self-maps depend on a switching rule which is a piecewise constant real function defined as , for all where is the switching sequence. The class consists of the piecewise continuous self-maps f on built as follows:
(2.43)

Note that , for all , for all

If , for all then
(2.44)
Theorem 2.12 is applied as follows. If all the primary self-maps are strictly contractive for the switching rule , that is, , for all that is, a sufficiently fast switching cadence is used, it follows that the real self-map possesses a fixed point from Banach contraction principle for any switching rule. It is not difficult to see that the property also holds if the primary self-maps are large contractions or there are mixed large and strict contractions used by the switching rule to build the self-map Each particular fixed point may depend on the switching rule and is a fixed point space for the class of self-maps built in such a way. The property may be generalized by using also primary nonexpansive maps with associated having unity absolute upper-bound provided that the switching rule involves nonexpansive self-maps being always used on subsets of of finite Lebesgue measure or, otherwise, combined with contractive primary self-maps both (alternately) run on sets of infinity Lebesgue measure. If has also nonexpansive self maps, then a fixed point does not exist for all the class of constructed self-maps so that is not a fixed point space. Finally, if expansive primary self-maps are also considered, then a fixed point still exists for switching rules satisfying a condition like that of Lemma 2.9 for some finite real according to the following constraint:
(2.45)
for some real constant , where
(2.46)

The above results are directly extendable to the linear space endowed with any Euclidean norm.

Example 2.18.

If the replacements , are performed in (2.42) to define the class of primary self-maps, that is, a sufficiently slow switching cadence is used, then, the residence interval taken for the next switch after the switch at makes to strictly decrease the function . As a result, the existence of a fixed point is guaranteed by any switching rule involving at least a primary self-map generating or sufficiently large residence intervals compared with the times where the remaining primary self-maps are used to generate .

3. Some Extensions

It is now assumed that the class of primary self-maps from to still satisfies weaker assumptions than the given ones in the previous section as follows.

The real constants are not necessarily upper-bounds for the primary self-maps from to in . Instead, the class is redefined so that the upper-bounding functions , for all , for all are assumed to be nonnegative and uniformly upper-bounded by finite constants (possibly exceeding or being equal to unity within some subsets of their definition domains) on . Furthermore, they are assumed to be asymptotically strictly contractive (i.e., taking asymptotically values being less than unity) in the precise sense that for some . Note that this condition implies that for any given with arbitrary such that , the upper-bounding function associated with satisfies the limiting upper-bounding condition .

Note that the condition is fulfilled if is uniformly bounded in and it is also monotone strictly decreasing on some . In this case, there is also a subinterval in which , for all . This condition is important in practice since exponentially stable dynamics such as those in (2.5) systems fulfil it. Thus, it is possible to construct switching rules which respect a sufficiently large minimum residence time interval at least at one of their stable parameterizations to guarantee the existence of a fixed point and the exponential stability to the origin in the dynamic system is unforced as a result.

The class is assumed to be the set of asymptotically large contractive primary self-maps from to in the sense that Condition for any primary self-map from to in the class is replaced with its asymptotic counterpart:
(3.1)

so that fulfils the strict upper-bounding condition in some subset of of infinite Lebesgue measure. The Condition is left unaltered.

The class of noncontractive nonexpansive self-maps from to is defined to fulfil a similar condition to the above one by using instead under nonstrict inequality.

The remaining classes of primary functions are assumed to be as those given in Section 2.

Although three of the subsets of primary self-maps are redefined under weaker conditions, the notation of Section 2 is kept for them in order to facilitate the exposition. The subsequent notation , for , stands for switching points of the rule acting on the primary class of self-maps from to being within the subset If the arguments and are omitted, then is understood to be within any subset of .

The main result of this section extends Theorem 2.12 as follows.

Theorem 3.1.

Under the assumptions in this section, assume also that all the self-maps of the class in are also uniformly bounded, that is Lebesgue-measurable with and consider a switching rule which generates , with defined by the set of primary self-maps from to satisfying all the above assumptions. The following properties hold.

(i) has a fixed point if , and a minimum finite residence interval being sufficiently large compared to is respected at any , for all before the next switching in the following precise sense:
(3.2)

or there is a finite number of switches with the last switching point being to a primary self-map in .

(ii) has a fixed point if and and, furthermore, such that the boundedness condition
(3.3)

holds in the case that is not Lipschitzian, then has a fixed point and a minimum finite residence interval being sufficiently large compared to is respected at any , for all .

(iii) has a fixed point if and and, furthermore, such that the boundedness condition
(3.4)

holds in the case that is not Lipschitzian, then has a fixed point providing that a minimum residence interval is respected for at least one of the asymptotically strictly contractive or asymptotically large contractive self-maps in .

Proof.
  1. (i)

    Assume , . Since is uniformly bounded piecewise-continuous since all the functions in are also uniformly bounded, the corresponding rate over-bounding functions are also uniformly bounded. Then, since , then the following situations can occur.

     
The last switching occurs at a finite point in with switching of the self-map from to to an asymptotically contractive primary self-map. Also, is the left boundary of a connected interval of being of infinity Lebesgue measure. Formally: such that for some , that is, there is no switching point being larger than the largest switching point in ST under the switching rule . Then,
(3.5)

provided that for any given positive real constant such that is sufficiently large but finite since . Thus, the self-map has a fixed point.

There is no last switching point but after a finite switching points all the switching points exceeding some sufficiently large finite one involve switches to asymptotically strictly contractive primary self-maps from to .

Then, such that and for some . First, assume that and generates the next switching point under a primary self-map in with , and
(3.6)
This situation can occur of a simply connected subinterval of Using a parallel reasoning to that of case ( involving complete induction, one gets that since
(3.7)

with provided that if for some with being sufficiently large but finite. It is again concluded that the self-map has a fixed point.

There is no last switching point but, after a finite switching point, the sequence of all the switching points exceeding some sufficiently large finite one contains an infinite sequence of switching points to primary self-maps from to which are not asymptotically contractive. This case cannot occur since then contradicting the given assumptions.

Property (i) has been fully proven.

Properties (ii)-(iii) are proven in a similar way to their stronger parallel properties in Theorem 2.12 by using the upper-bounding limiting property of (2.18) for the extended class of primary self-maps. The detailed proof is omitted.

Corollary 3.2.

Theorem 3.1(i) is fulfilled for any switching rule such that the minimum residence intervals referred to are respected in only one of the asymptotically strictly contractive primary self-maps. Theorem 3.1(ii) is extendable to the fulfilment of a sufficiently large residence interval by one of the asymptotically large contractive primary self-maps. Theorem 3.1(iii) is extendable to the fulfilment of the above property by either one of the asymptotically strictly contractive or one of the asymptotically large contractive primary self-maps.

Theorem 3.1 addresses the case when the subset of , where is defined via not asymptotically (strict or large) contractive primary self-maps in , has a finite Lebesgue measure; that is, switches in-between primary self-maps can involve no contractive self-maps over finite intervals. It is furthermore interesting to make that assumption more powerful by considering that is the countable union of infinitely many connected subsets of finite Lebesgue measure whose boundaries are each pair of consecutive switching points. Sets formed by unions of some finite number of those subsets for consecutive switching points are assumed to contain at least one asymptotically either strict or large contractive primary self-map generating . The subsequent result extends Theorem 3.1 to the case when the conditions of Lemma 2.9 are modified to their asymptotic versions. It is admitted that the sets of primary self-maps which are not contractive may be asymptotically compensated by the contractive ones, so that the built by the switching rule is asymptotically contractive in some sense to guarantee the existence of a fixed point. Its proof follows directly by combining a directly extended Lemma 2.9 to its asymptotic version with Theorem 3.1 since any is uniformly bounded on its definition domain.

Corollary 3.3.

Assume that all the self-maps of the class in are also uniformly bounded, that is Lebesgue-measurable with , and consider a switching rule which generates , with defined by the class of primary self-maps from to satisfying either
(3.8)
for some sufficiently large or the Condition of Section 2 for large contractions, together with the (asymptotic) modified condition
(3.9)

Then, has a fixed point.

Remark 3.4.

The results of Sections 2 and 3 are extendable directly to the discrete case for the sets and by replacing Lebesgue measures with discrete ones.

Example 3.5.

Fixed point theory is a useful tool to investigate the stability of dynamic systems including standard linear continuous-time or discrete systems and time-delay systems [2, 6] as well as hybrid dynamic systems including coupled continuous-time and discrete-time subsystems [1]. Now, it is discussed the case of a delay-free continuous-time system under a switching rule operating among a given set of parameterizations and subject to controlled and uncontrolled impulses. Consider the linear dynamic unforced time-varying system:
(3.10)
where and , which are not required to be disjoint, are the real sequences of impulsive time instants where feedback control impulses and open-loop (i.e., feedback-free) control impulses occur, respectively, with the control Dirac distributions being and , respectively, of respective piecewise-constant function matrices of dynamics and control matrix functions and being run by a switching rule , for all , for all where ST is the strictly ordered sequence of switching time instants. The real impulsive amplitude sequences and of elements in are assumed to be uniformly bounded and can be finite or infinite. An empty or nonempty sequence of impulsive time instants can occur as follows for some :
(3.11)

either within any simply connected time interval , where are two consecutive switching points, or within any interval if the switching rule generates a finite sequence ST of switching time instants of last element . The following impulsive constraints are assumed.

(1) such that for any two consecutive impulsive time instants , if any, within with being any two consecutive switching time instants. Also, such that for any two consecutive switching time instants. The interpretation is that there is no accumulation point either of switching time instants or of impulsive time instants.

(2)If then and if , then

(3)If then there is no and

(4)If and then there is no

(5)If is of finite Lebesgue measure, then (i.e., there is at most a finite number of impulsive instants in any finite time interval within two consecutive switching instants).

(6)If there is a finite number of switches generated by the switching law , so that a finite is the last one, with being of infinite Lebesgue measure then (i.e., an infinite or finite number of impulsive instants can occur within an infinite time interval).

(7)If then is of infinity Lebesgue measure so that .

The unique state trajectory solution of this dynamic system satisfies
(3.12)
where and is the th identity matrix. One gets by applying the above relations recursively for the sequence of tie instants
(3.13)
so that
(3.14)
with
(3.15)
where and is the numerical radius (or 2-matrix measure with respect to the spectral -norm) of and the fundamental matrix function is upper-bounded as follows for any and some real constants and , for all . Then,
(3.16)
If is the last switching time instant, then and
(3.17)
and furthermore,
(3.18)
Also, one gets for any finite , any , for all
(3.19)
where is defined directly from the above expression, provided that there is at least one switching time instant within , where
(3.20)
The case that there is no impulse but one switch within is included in the above formula by removing the norms since the involved are zero. The case of no switch-no impulse occurring in is also particular case of the above formula (3.19) resulting to be for
(3.21)
It follows directly from recursion in (3.21) that for any switching rule if there exists a such that
(3.22)
so that the state-trajectory solution possesses a fixed point exists since
(3.23)

It turns out that if there is at least one stability matrix , then there are always switching rules which lead to a state-trajectory solution possessing a fixed point. Since , since , for any sufficiently large residence interval such that (i.e., for a sufficiently large time interval free of switches and impulses previous to the next switch after each switch to the stability matrix has happened), then the associated map for this switching is asymptotically strictly contractive. They can occur also switches to nonexpansive ( ) or expansive ( ) generated by the switching rule but a fixed point always exists for such a rule if for some , there is a dominance of the switching intervals associated with so that . In the presence of impulses, the result is still valid by increasing, if necessary, the residence interval before to the next switch after switches to the parameterizing matrix have happened. It is possible to achieve a constant for some since the norm upper-bounding real function of time is monotone strictly decreasing related to the residence interval .

Remark 3.6.

It is important to point out that it is obvious that the generalization of the given formalism to switching rules is direct; that is the codomain of coincides with the image of so that infinitely many distinct primary self-maps are used to construct . This implies necessarily that the switching rule generates infinitely many switches so that the discrete measure of ST is infinity.

Declarations

Acknowledgments

The author is grateful to the Spanish Ministry of Education by its partial support of this work through Grant DPI2009-07197. He is also grateful to the Basque Government by its support through Grants GIC07143-IT-269-07 and SAIOTEK S-PE08UN15. He is also grateful to the reviewers for their interesting comments which have been very useful to him in order to improve the previous version of the manuscript.

Authors’ Affiliations

(1)
Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco

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© M. De la Sen. 2010

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