Properties of convergence of a class of iterative processes generated by sequences of self-mappings with applications to switched dynamic systems
© Sen and Ibeas; licensee Springer. 2014
Received: 7 September 2014
Accepted: 27 November 2014
Published: 15 December 2014
This article investigates the convergence properties of iterative processes involving sequences of self-mappings of metric or Banach spaces. Such sequences are built from a set of primary self-mappings which are either expansive or non-expansive self-mappings and some of the non-expansive ones can be contractive including the case of strict contractions. The sequences are built subject to switching laws which select each active self-mapping on a certain activation interval in such a way that essential properties of boundedness and convergence of distances and iterated sequences are guaranteed. Applications to the important problem of stability of dynamic switched systems are also given.
Keywordsexpansive non-expansive contractive and strictly contractive self-mappings switched dynamic systems convergence fixed point stability
The problems of boundedness and convergence of sequences of iterative schemes are very important in numerical analysis and the numerical implementation of discrete schemes. See [1–5] and references therein. In particular,  describes in detail and with rigor the associated problems linked to the theory of fixed points in various types of spaces like metric spaces, complete and compact metric spaces and Banach spaces. This book also contains, discusses and compares results of a number of relevant background references on the subject. In other papers, related problems are focused from a computational point of view including the acceleration of convergence using modified numerical methods like Aitken’s delta-squared methods or Steffensen’s method, [2–5]. On the other hand, there is also a rich background in theory and applications of fixed point theory related to non-expansive, contractive, weakly contractive and strictly contractive mappings as well as related to their counterparts in the framework of common fixed points and coincidence points for several mappings and in the framework of multivalued functions. A (non-exhaustive) list of recent related references is given including new results and a discussion of previous background ones. See, for instance, [1, 6–25], and references therein. Many efforts are also devoted to the formulation of extensions of the above problems to the study of existence and uniqueness of best proximity points in cyclic self-mappings, [8, 13, 14, 16–21], to that of proximal contractions, [13, 14] and to the characterization of approximate fixed and coincidence points, [22, 23]. Direct applications of fixed point theory to the study of the stability of dynamic systems including the property of ultimate boundedness for the trajectory solutions having mixed non-expansive and expansive properties through time or being subject to impulsive controls have been given in  and [24, 25]. Some recent studies of best proximity points of weak ϕ-contractions in ordered metric spaces have been performed in . On the other hand, the existence of best proximity points for 2-cyclic operators in uniformly convex Banach spaces is investigated in . Finally, in , it has been proved that some previous fixed point results and some recently announced best proximity results are equivalent.
This paper is focused on the study of boundedness and convergence of sequences of distances and iterated points and the characterization of fixed points of a class of composite self-maps in metric spaces. Such maps are built with combinations of sets of elementary self-maps which can be expansive or non-expansive and the last ones can be contractive (including the case of strict contractions). The composite maps are defined by switching rules which select some self-map (the ‘active’ self-map) on a certain interval of definition of the running index of the sequence of iterates being built. The above mentioned properties concerning the sequences of iterates being generated from the given initial points are investigated under particular constraints for the switching rule. Note, on the other hand, that the properties of controllability and stability of differential-difference and the various kinds of dynamic systems are of a wide interest in theory and applications [25, 29–52]. See, for instance, related problems associated with continuous-time, discrete-time, digital, and hybrid systems and those involving delayed dynamics [38–42] and [31, 34–37, 44–46], sampled-data systems under constant, non-uniform, and/or multirate sampling and switched systems [34, 36, 45, 46]. In this context, this paper also includes an application of the developed theoretical framework to the stability of (in general) nonlinear switched dynamic systems. The composite self-map generating the trajectory solution sequence from initial conditions is defined with sets of elementary self-maps being associated with distinct active parameterizations which are switched through time by switching rules which guarantee the fulfilment of the suitable properties.
2 Problem statement and main results
The function is the so-called switching law where if is finite and , otherwise, i.e. (i.e. the infinity cardinal of a countable set).
q is the set of distinct parameterizations of the sequence of self-mappings on X in the sense that such a sequence contains a finite or infinite set of distinct self-mappings.
- (3)q is the disjoint union of the sets , where , , , and are, respectively, the indexing sets of strict contractions, contractive self-mappings which are not strict contractions, non-expansive self-mappings which are not contractive, and a class of expansive self-mappings which fulfil a specific expanding condition according to(2.2)
- (4)The composite mapping , defined by(2.3)
The following simple example illustrates how the iterative process can work in a real situation.
Example 2.1 Consider the simple scalar discrete equation (2.1) with with , under the Euclidean metric. The sequence is such that , , where , , and for some given real constants and . It is clear that the self-mapping on R is a strict contraction with a contractive constant , the self-mapping on R is non-expansive with constant , and the self-mapping on R is expansive with constants and . Note that the set of fixed points of reduces to , that is, for is the unique fixed, and equilibrium, point of and while with the whole R being an equilibrium set of . Note also that is a stable equilibrium point of , while it is an unstable equilibrium point of .
The switching law is . If , , it is said that the i th configuration of (2.1) is active in the interval . If , then it is said that is a switching sample of (2.1) since the active configuration becomes modified with such a sample.
If , then as , , and , , as and as , , where .
If the switching law is such that , , for some given real constant then all the sequence of composite self-mappings defined by on X are strict contractions of contractive constant ρ.
If , then as for any given in X.
- (iv)Assume that , for and , and the existence of the limits and uniformly in X as , that is, , , where . Then the self-mapping on X is a strict contraction and, for any given , there is a real constant such that such that one has for all(2.9)(2.10)
and property (iv) has been proven. □
Remark 2.3 Note that the testing sequence of the given switching law in Lemma 2.2 is not necessarily associated with a set of strict contractions although the composite , for , defines a composite strict contraction of non-necessarily strictly contractive self-mappings. The composite sequences for , where , are not necessarily associated with a composite strict contraction. Note also that the testing sequence is not unique for a given switching law since the only requirement is that it be strictly increasing with a maximum prescribed, but arbitrary, separation in-between any two adjacent elements of such a sequence.
Remark 2.4 Lemma 2.2 can be fulfilled, in general, by non-unique sequences of a given switching law in Lemma 2.2 as well as non-unique associated , μ and . A typical case occurs when the switching law consists of strict contractions or converges to a sequence of strict contractions. Those ones can be grouped individually in the limit or this one may be a composite self-mapping of strict limit contractions so that the next result follows.
- (i)There is a (in general, non-unique) decomposition of the composite in a maximum number of strict contractions , , with for the testing sequence , such that so that(2.20)
The decomposition (2.20) in a maximum number of strict contractions is unique if and only if the positive integer numbers , such that are unique. In particular, the decomposition (2.20) is unique if , .
- (iii)Assume that, furthermore, uniformly in X as for some , so that and , . Then the decomposition (2.20) is unique in a maximum number of strict contractions if and only if(2.21a)
since , where . Then with , being unique as well.
Proof Since the switching law is given, any strictly increasing sequence satisfying the constraints , and uniformly in X as for , with and being finite, imply that the composite-self-mapping is a composite self-mapping of at least one and at most strict contractions. So, the maximum number p of strict limit contractions in the composite for the sequence for for a given sequence S exists and satisfies (2.20). Property (i) follows. Uniqueness of the decomposition (2.20) holds if there is a unique , such that and , are strict contractions. In particular, the decomposition (2.16) is trivially unique if ; so that . Property (ii) has been proven.
and . Since (2.21a)-(2.21b) hold, , subject to exists so that is a unique strict contraction on X. Thus, either (2.21b) holds with and the sufficiency part of the property is already proven for with a strict contraction on X for or and (2.21a) holds for some integer . Necessity follows since if and is a strict contraction then for . Uniqueness is trivial since there is a single self-mapping in the decomposition of the composite self-mapping on X.
. The proof of sufficiency is proven by complete induction. Assume that (2.21a)-(2.21b) hold so that , subject to , has to exist so that is a unique strict contraction on X since exists such that exists, being a strict contraction on X. Set and note that and , , so that , then is a unique strict contraction on X for all and some such that the set of positive integers , subject to , is unique. Then the composite is a unique strict contraction, with , , and the positive integer is unique so that the set , subject to , is also unique and , for all and , and then the composite are unique and strictly contractive on X. The proof follows by complete induction, with the existence of a unique positive integer . Then with , being unique. Necessity follows since if (2.20) holds and (2.22) fails for some for a given then the factorization of the composite self-mapping on X, subject to , does not consist of strict contractions. Property (iii) has been proven. □
Note that the decomposition (2.20) of the composite on X is not unique, in general, and so it is not unique, in general, the decomposition (2.23). The decomposition (2.20) is unique if the testing S converges to a finite subsequence of strict contractions so that where p is the maximum number of strict contractions of the decomposition (2.20) and the numbers , are unique. Uniqueness holds in the case that and , .
Note that the assumption uniformly in X as for some , in Theorem 2.5(iii) implies that , . Note also that is a consequence of the fact that (2.20) is a decomposition of strict contraction while some of the limit self-mappings on X reached as a result of the uniform convergence constraint , can be expansive.
If Lemma 2.2(ii) holds then and , with , , are Cauchy sequences, so bounded, and then convergent in X if, in addition, is complete.
If Lemma 2.2(iii) holds with , , for some real then property (i) holds.
If both Lemma 2.2(ii) and Theorem 2.5(i) hold then are Cauchy sequences, so bounded, and then convergent in X if, in addition, is complete for all and , and , where .
Assume that is a compact metric space and that Lemma 2.2(ii) and Theorem 2.5(i) both hold. Then for some and is a strict Picard self-mapping and for any initial , where , . If then the above result holds if is a complete metric space.
then so that , . Since is a strict contraction from Lemma 2.2(ii), it has a unique fixed point from Banach contraction principle since is a compact metric space (i.e. totally bounded and complete, equivalently, if it very family of closed subsets of X with finite intersection property has a nonempty intersection). Then , and for some . As a result, for any initial condition , . If there is no contractive self-mapping on X not being a strict contraction in the switching law, the above holds if is just a complete metric space. □
Note that Theorem 2.6(iv) holds even if for some is not contractive. However, the composite self-mapping is a strict Picard self-mapping and , , . The error estimates and convergence rate are characterized in the subsequent result.
for any given , some real convergent sequence , with , , and some and , where converges .
and as at exponential rate so that (2.27) is proven. Closely analogous proofs to that of (2.27) follow directly for (2.28) and (2.29). □
A discussion of cases of interest concerning the above result follows.
Remark 2.8 (1) Theorem 2.7 refers to the case when with being a composite strictly contractive self-mapping on X of the form (2.20), i.e. possessing p (non-necessarily strictly contractive) fixed configurations which are the limit of the switching law . is a strict Picard self-mapping on X as a result. Also, , satisfying (2.27)-(2.29), and , satisfying (2.27)-(2.29) with and being replaced with 0, for any given , where .
(2) A particular case of interest of Theorem 2.7 is that when , so that , with and some (one of the configurations of the switching law) so that the limit self-mapping on X is a strict contraction, and then a strict Picard self-mapping, and the switching law is such that uniformly in X. Since the testing switching has the property , so that an admissible choice of can be made in (2.27)-(2.29). The interpretation of the presence of the real constant in the error estimates and convergence rate is due to the fact that the sequence of composite self-mappings governed by the switching law to build the iterative scheme for any given is not of the form , while converges to a uniform limit strictly contractive on X with a unique fixed point subject to and and for any given according to (2.27)-(2.29) with the replacement .
. If a fixed point of then (2.39)-(2.41) hold in closed forms under the replacements , and .
provided that the non-expansive self-mapping on X has a fixed point . Then one has so that and is bounded for any initial although convergence to the fixed point is not guaranteed. This is a well-known result from fixed point theory for non-expansive self-mappings and a well-known (non-asymptotic) global stability result in related problems of stability of dynamic systems which can have a global attractor which can be either a fixed point, which is also an equilibrium point which is not asymptotically stable, or a stable limit cycle. If then the above result takes the simpler form .
Then from point-wise convergence, , since z is a fixed point of , and . Taking the limit as in (2.43) yields .
with , and . But if then (since ) if .
3 Numerical examples
This section contains some numerical examples regarding the theoretical results stated in the previous section. Two examples are discussed. The first one considers a scalar time-invariant nonlinear switched system while the second one deals with a linear time-varying switched system.
3.1 Scalar nonlinear switched system
In this case, the trajectory enlarges very much as Figure 4 shows. The time scale in this figure has been reduced to handle the large values of the signal. Hence, a serious problem arises in this example: How can the stability of the switched system be proved under different switching rules? The results stated in Section 2 can be used to solve this question as the following example shows.
3.2 Linear time-varying switched dynamic system
Hence, the stability of the linear time-varying switched system is analyzed by just calculating the product of some constants, easing off the determination of the stability properties of the system.
The authors are very grateful to the Spanish Government for Grant DPI2012-30651 and to the Basque Government and UPV/EHU for Grants IT378-10, SAIOTEK S-PE13UN039 and UFI 2011/07. The authors are also grateful to the referees for their suggestions.
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