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On Some Boundedness and Convergence Properties of a Class of Switching Maps in Probabilistic Metric Spaces with Applications to Switched Dynamic Systems
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.
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 , 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. Section 2 introduces and 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. On and Classes of Self-Mappings in Probabilistic Metric Spaces
Let us define a probabilistic distance , where is a nonempty abstract set represented by for each , where is a set of distribution functions. A distribution function is a mapping which is nondecreasing and left-continuous with and .
The ordered pair is a probabilistic metric (PM) space if for any and all the following conditions hold :The triplet is a Menger space where is a PM-space and is a triangular norm which satisfies the inequality , , .
Note that for and for if since is nondecreasing and left-continuous. Note also that every metric space can be realized as a PM-space by taking being defined by for all [1–4]. In the following, is the space of all mappings which are left-continuous and nondecreasing with and . The space is partially ordered by the usual pointwise ordering of functions; namely, if and only if , , and its maximal element is the distribution .
Definition 1. Let be a PM-space. A mapping is said to be of -class for some function if
Definition 2. Let be PM-space. A mapping is said to be of -class for some functions ifwhere the functions satisfy , .
Note that if is of -class, then it is also of -class. Note also that is nonexpansive if it is of -class with and, in particular, a probabilistic strict contraction if it is of -class with . Also, if is of -class with , then it is nonexpansive (probabilistic strictly contractive) [1–4]. If is of -class with , then it is expansive [1–4]. If there is some with such thatthen is expansive (even if is not of -class for some subject to ).
The following technical result follows.
Lemma 3. The following properties hold:(i)Let be a PM-space and let be a mapping of -class. Consider the sequences and built by , , with , for some given . Then, where , , .(ii)If is of -class, then , .(iii)If is a mapping of either -class or -class with for the given , then , .(iv)If is a mapping of -class with , , , and are in (i.e., is the closure of , i.e., the extended nonnegative real semiline) for the given , then If is a mapping of -class, then , .
Proof. It follows recursively from (3) with , , , with , for the given thatProperty (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 ; then, , , and the conditions that is nondecreasing in the argument and lead from (8) to the existence of the limit , . Property (iv) is proved closely to Property (iii) by first getting (7a) and (7b) directly from the definitions of , , , and , .
The subsequent example illustrates that Lemma 3 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 space with being defined by for all for the distribution function defined by:for some left-continuous nondecreasing functions withAssume also that is everywhere lower-semicontinuous and is everywhere upper-semicontinuous. Then,with , , and , . Assume following Lemma 3(iv) that and with , , . Note that , , , and are allowed to be dependent on . Then, if is a mapping of -class so that (7a) and (7b) of Lemma 3 hold, one gets for any given from (7a) and (7b). Thus, one gets for any given the following:(a)If for some given , then, since as well, one gets and if since , then and if since the above superior and inferior limits equalize unity.(b)If , then If, furthermore, , then . If, in addition, , then , , and , ; then , , 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 leading to that is, , , , or, in other words, for any distance from a given to a given , .(c)If , then .(d)Now, assume that is a mapping of -class with and then ; that is, the mapping is expansive. Then, if implying that (and also since ), one concludes from (13b) that since and . The constraints (13a) still hold for each such that but the sequence diverges to while and .
Example 5. Assume that , , independent of . Then, is of -class and nonexpansive but also probabilistic noncontractive. If , then one gets from (13a) and (13b)Assume instead that for some sequence , . Then, if , one has for any thatwhich simplifies as , , if :
From Lemma 3(iii), one gets directly the subsequent result.
Proposition 6. Let be a PM-space and let be a mapping of either -class or -class and there is a strictly increasing sequence of nonnegative integers fulfilling such that, for some given , , ; then , .
Proof. It follows from Lemma 3 that if , , then , , , , , and then .
Note that Proposition 6 includes as a particular case that of probabilistic strict contractions which are then mappings of -class with .
Definition 7 (see ). Let be a PM-space and a nonempty subset of . The probabilistic diameter of is a mapping defined by .
Definition 8 (see [2, 4]). Let be a PM-space and a nonempty subset of . The nonempty set is said to be probabilistically bounded if , that is, if the supremum of its probabilistic diameter .
We can define the set unboundedness as the concept opposite to Definition 8 as follows.
Definition 9 (see [2, 4]). Let be a PM-space and a nonempty subset of . The nonempty set is said to be probabilistically unbounded if , that is, if .
The boundedness and unboundedness of sequences can be easily defined as supported by Definitions 8 and 9 as follows.
Definition 10 (see [2, 4]). Let be a PM-space. The sequence is probabilistically bounded if .
Definition 11 (see ). Let be a PM-space. Then, the sequence is(1)probabilistically convergent to a point , denoted by , if for every and there exists some such that(2)Cauchy if for every and there exists some such thatA PM-space is complete if every Cauchy sequence is probabilistically convergent.
Proposition 12. Let be a PM-space. Then,(1) for some if and only if the following limit exists: ;(2) is a Cauchy sequence if and only if , .
Proof. If , then there exists some such that , , for every and . Thus, since , , then, by taking , one gets .
Conversely, if , then for every and there exists some such that , ; thus . Assume that this is not true. Thus, there is some subsequence such that for some and while for any since is nondecreasing in the argument and one gets the following contradiction for some :Proposition 12(1) has been proved. The proof of Proposition 12(2) is very close and it is omitted.
Proposition 13. Let be a PM-space. Then, the sequence is probabilistically bounded if and only if , where with for some , that is, if and only if for some .
Proof. If is bounded, then the result is direct for any sequence . Assume that is not bounded and proceed by contradiction by assuming that is probabilistically bounded and for some . On the other hand, since is probabilistically bounded, then, for all , there is such that , . Since , there is such that , , and some . This implies from the contrapositive equivalent logic proposition to the third property of (1) of being a PM-space that either , and then the sequence is not probabilistically bounded, (a contradiction), or , , for any given , and then for some fixed . Now, assume that, for all such that , . Thus, the subset of , , is unbounded since its probabilistic diameter is less than one; that is, , , and then the sequence is probabilistically unbounded, again a contradiction. It has been proved that if is bounded, then for some , where is the complementary to in . It remains to prove that if for some , then is probabilistically bounded. Since , then , , for some and all . It follows from the third property of (1) that , , . Thus, is probabilistically bounded.
3. Switched Maps Defined by and 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 . Based on the above elementary idea, this section relies on switching maps built with a prefixed number of either -class or -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 -class, or -class, self-mappings are characterized by constants instead of functions in Definitions 1 and 2.
Let be PM-space and let be a set of (primary) self-mappings of -class for some constants , for . A switching map from to with respect to the switching law generates a sequencefor each given for some and we informally can say that the th primary self-mapping is “active” at the th value (or sample) of the sequence . See also [23–27]. In other words, the switching map on is defined by one of the self-mappings () for each and it has associated piecewise constant functions such that , for each and each . The set of switching samples of a sequence is a (proper or improper) subset of , so-called the switching set, defined by . Note that a switching set is a strictly ordered set for the standard strict ordering relation “”. Since , , are self-mappings of with constants , , thenfor any so that one has recursively from (3) for a sequence generated by , , for any given where , , , isand is the number of times that the th self-mapping for some is “active” in the interval for each . If , , are self-mappings of -class, then one has instead of (25)
Theorem 14. Let be a PM-space and let be self-mappings of -class for some constants for . Then, the following properties hold:(i)Assume that there is (at least) a self-mapping for some which is a probabilistic strict contraction. Then, there are infinitely many switching laws such that their associate switching maps are probabilistic strict contractions.(ii)Under the conditions of the above proposition, there are infinitely many switching laws such that their associate switching maps are probabilistic strict contractions and, furthermore, they consist of infinitely many alternate active switching maps of the form or with .(iii)If, in addition, is complete, then any sequence , under any switching law fulfilling either Property (i) or Property (ii) for any given initial point is Cauchy and probabilistically convergent.
Proof. It follows from (27) that Property (i) is fulfilled for sequences generated as for any by any of the infinitely many switching maps built under switching laws which fulfilsince there is a finite nonnegative integer depending on the subsequence which is a terminal switching point such that for for any such a sequence . Thus, one gets from (27) thatsince is nondecreasing and left-continuous with and then the sequence built as , is a Cauchy sequence and is a probabilistic strict contraction. Property (i) has been proved.
Property (ii) follows with alternate (probabilistic strict contraction versus remaining self-mapping) switching laws defined by a switching set fulfilling the fact that if, for any , for has active (perhaps nonprobabilistic strict contractions) self-mappings for , then is defined such that is active on ; that is, for with being defined with large enough such thatwhich lead toso that we get again (29) and a similar conclusion. Property (iii) is obvious from the fact that is complete and is Cauchy under switching laws fulfilling either Property (i) or Property (ii).
The following result is a direct consequence of Theorem 14 since mappings of -class are also of -class.
Corollary 15. Let be a PM-space and let be self-mappings of -class for some constants , , for . Then, Theorem 14 still holds.
Theorem 16. Let be a complete Menger space with and let be self-mappings of -class for some constants for all with at least being a probabilistic strict contraction for some . Let be a switching law and let be a sequence generated as , , for any given such that their associate switching map is defined by for and all and some finite . Then, which is the unique fixed point of the strict contraction .
Proof. Since is a strict probabilistic contraction, , , , from (29) since for . Then is probabilistically convergent to which is a fixed point of the probabilistic strict contraction as proved by contradiction. Assume that this is false so that and then since is Cauchy, , which is nondecreasing and left-continuous, and is a probabilistic strict contraction, one gets, for any given and and all and some ,for some , , which implies that but since can be chosen arbitrarily, it suffices to take to get a contradiction. Then, which is proved to be unique again by contradiction. Assume that this is not the case so that there exist and , , which are fixed points of the probabilistic strict contraction . Thus, one gets the contradictionso that . Since for for some finite , we can write and for to get that generated by , , for any given arbitrary is probabilistically convergent to .
The above result is a direct consequence of Theorem 14 which is also valid if is of -class. However, note that, under the alternate switching laws in Theorem 14(ii), the limit points of sequences generated through the switching maps are, in general, dependent on the initial points of the sequences and on the switching law.
The following result generalizes Theorem 16 without assuming any special contractive condition on at least one of , , with the only condition on the operators being that all of them are either of -class or -class.
Theorem 17. Let be a PM-space, let be self-mappings of -class for some constants , , , and let be a switching mapping associated with a switching law which generates a sequence as , , for some given . Let be the set of switching points, that is, for any given , , provided that , if and only if . Then, the following properties hold:(i) for all , , , , where where is the number of times that the th self-mapping for each is “active” in the interval for some .(ii)If is upper-semicontinuous at for some given , where , then has the following property: If is lower-semicontinuous at for some given , where , then has the following property:(iii)If are of -class for some constants , , then has the following property: And all