1. Introduction
There is an abundant and rich background literature available on contractive cyclic self-mappings defined on subsets of complete metric spaces. See, for instance, [
1,
2,
3] and some of the references therein. In [
4], multivalued cyclic self-mappings are investigated while, in [
5], Jungck iterative cyclic iterative processes are focused on. In [
6], some results for cyclic contractions are obtained, which apply for Banach spaces, geodesic metric spaces and metric spaces. In [
7], relatively nonexpansive mappings are investigated on the union of two subsets of a Banach space. If the pair of subsets has a proximal normal structure and a relative nonexpansive mapping,
on
satisfies
and
, where
and
are weakly compact and convex subsets of the Banach space
, then there is some proximal point
such that
. The obtained results are also adapted, in particular, for Banach spaces with strictly convex norms and they are valid for uniformly convex Banach spaces. In [
8], cyclic
-contractions defined on partially ordered orbitally complete metric spaces are considered. Some fixed-point and best-proximity-point theorems are stated and certain links between points of coincidence and common best proximal points are investigated. It is also found that there are conditions under which points of coincidence, common best proximal points and common fixed points are coincident. A study on the best-proximity points for cyclic mappings in multiplicative metric spaces is performed in [
9]. On the other hand, best-proximity results in b-metric spaces endowed with a graph are established in [
10], while, in [
11,
12], some new results for cyclic mappings in complete b-metric-like space are established. Also, some connections on cyclic contractions and contractive self-mappings with Hardy–Rogers self-mappings are investigated in [
13] while proximity results in Busemann convex metric spaces are given in [
14]. In [
15], a so-called BSS-type cyclic zapping is investigated, which is a pair semi-cycle mapping supported with a special contractive rule. On the other hand, some new best proximity and cyclic mapping results are obtained in [
16,
17] concerned with S-cyclic mappings.
It can be pointed out the importance of the concept of approximative compactness for the non-emptiness of the sets of best-proximity points of two nonempty subsets
and
of a metric space. In fact, one of the simplest such conditions is that both best proximity sets are nonempty if
is compact and
is approximatively compact with respect to
, that is, every sequence
such that
as
for some
such that
such have a convergent subsequence [
1,
18,
19].
This paper extends the formalism on cyclic contractions to consider the compositions of cyclic self-mappings with usual strict contractions which are integrated in more general iterative processes in the following way. The subsets of the cyclic disposals are assumed to be nonempty and closed and they can be disjointed or not. The contractive cyclic self-mapping
operates over adjacent nonempty closed subsets of the cyclical disposal in the metric space
as usual, while the strict contraction
operates over each of the individual subsets. The iterative processes which govern the combinations of both self-mappings are built in general under any order of the contributions of each of the mappings and any number of compositions of the self-mapping
with itself. Also, the iterative processes can include any number of compositions in any order of the sets of compositions of both self-mappings
and
with themselves. The subjacent idea of the proposal is to adequate the evolution of sequences in each of the subsets being governed by the self-mapping
or by compositions of the self-mapping
with itself, with switching actions from the current subset to its adjacent one in the cyclic disposal, governed by the self-mapping
. Each switching in-between adjacent subsets of the cyclic disposal can be addressed from the current subset to another subset distinct to the adjacent one if a number of compositions of the cyclic self-mapping
with itself are combined before the next contribution to the iterative process of the contractive self-mapping
, again, with eventual compositions with itself. Some applications of interest, as exhibited in some of the given worked examples, rely on the solution of differential systems of equations in alternative domains which can also have different parameterizations and are subject to mutual switching actions. For instance, each of the subsets of the cyclic disposal can be associated with a particular parameterization of the differential system, while switching actions can govern the activations of other alternative parameterizations in a different set to the current one among those being integrated in the cyclic disposal. This situation may arise in some processes in the real word as, for instance, in atmospheric models, in distillation column towers, in switching processes in financial markets, etc., and these are also relevant in certain control processes [
20,
21,
22,
23].
This paper is organized as follows.
Section 2 gives some preliminary definitions and results on the contractive cyclic self-mapping and the strict contraction to be composed in the iterative processes of the third section. It is basically seen that under the condition that just one of the best proximity sets in the cyclic disposal is a singleton together with the bounded compactness of such a set (or its strict convexity if the metric space is, in addition, a uniformly convex Banach space), this suffices to achieve unique asymptotic cycles formed by a best-proximity point per each adjacent subset. All the sequences with arbitrary initial points in the union of all the subsets converge to such a limit cycle. The results also hold for extended cyclic contractions where the product of the different individual constants (one per transition from each subset to its adjacent one) is less than unity which, in fact, only requires that one of such constants be contractive.
Section 3 is devoted to the convergence of distances of sequences and sequences when the compositions of the cyclic contraction are combined with compositions of the strict contraction in any order of mutual compositions while possibly involving any number of repeated compositions of both maps with themselves. It is assumed that the subsets of the cyclic disposal are nonempty and closed and that they are subsets of a uniformly convex Banach space. In addition, it is assumed that at least one of the subsets of the cyclic disposal is boundedly compact with its best proximity set to the next adjacent subset being a singleton, or alternatively that, at least one of such subsets is strictly convex. It is also assumed that the unique fixed point of the contractive self-mapping
at each subset belongs to the set of best-proximity points to its next adjacent subset in the cyclic disposal.
Section 4 describes some worked examples which illustrate the formal results and, finally, the Conclusions section ends this paper.
2. Preliminary Definitions and Preparatory Results on Cyclic Contractive Self-Mappings and Strict Contractions
Through this paper, and is the logic disjunction “or” of the logic propositions and . Also, and are the respective sets of positive and the non-negative real numbers, and and are the respective sets of positive and the non-negative integer numbers.
A positive matrix (referred to as ) is the one with all its entries being non-negative with at least one positive. is strictly positive (referred to a ) if all its entries are positive. A vector is positive (referred to as ) if all its components are non-negative with at least one being positive. Also, is strictly positive (referred to as ) if all its components are strictly positive.
Consider self-mappings
such that
for
are non-empty closed subsets of a metric space
with
for
and
for
.
for some real constants , such that the distances in-between any adjacent subsets and for and their respective diameters are, respectively, as below:
The distance in-between adjacent subsets is denoted with
D which is zero if such subsets intersect and greater than zero if they do not intersect. More formally,
(if
;
then
), with
then,
for any integer
. Unless otherwise be stated, we will refer to as the “adjacent” subset of
to its next adjacent one
in the cyclic disposal rather than to the preceding one
- (1)
is said to be a -cyclic contractive self-mapping on
- (2)
is:
- (1)
Non-expansive from (2)–(3) for any pair , with , allocated either in the same or in adjacent subsets and for of elements allocated in adjacent subsets;
- (2)
A (strict) contraction from (3) for any pair of elements allocated in the same subset for ;
- (3)
Contractive for any pair allocated in adjacent subsets and for under the constraints of the first equation in (2), that is, if .
Remark 1. 1.1. Note that the constraints for any are clearly needed for the coherency of (2). Note that, if , then the contractive condition (3) applies for
for any if , while this contractive condition applies if , or if
and for if from (2). Also, if , then there exist nonempty closed subsets for each which contain the best approximation point(s) of each subset to its adjacent subset .
1.2. Note that is a -cyclic contraction from (1), [1,2,3], and is a (strict) contraction on
for each
from (3). 1.3. Note that the second condition of (2) implies that if
for then to accomplish with . In order for that constraint to be feasible, it is necessary that a) ; ; b) may be multi-valued at some points in order to accomplish with that constraint in the event that distances
for points of adjacent subsets that exceed the given threshold. That is, if and (or vice versa) for any given but the above constraint is not requested if belong to the same subset for any given .
Remark 2. Note that, since is subject to the constraint ; , this function may be defined by functions ; which are all identical in if such an intersection is non-empty. By virtue of the contraction condition (3), each has a unique fixed point in ; which are all coincident if . Note also that for any and with :so that as and and for some and all .
Since and for all are closed, then there exist nonempty closed subsets for each which contain the best approximation point(s) of each subset
to its adjacent subset . Throughout this paper, it is of interest for the convergence properties the case when
and ; , that is, there is a unique best-proximity point at each
which is also the unique fixed point of . Note that since as with and if
and ; then the best-proximity point
through is jointly the unique fixed point of
and that of ; , then satisfies , andthat is, commutes with at each of the fixed points so that In the case when then since ; .
The next proposed definition leading immediately to an “ad hoc” elementary result refers to the fact that it is only needed a contraction of on one of the transitions from a set to its adjacent one to obtain a cyclic contraction under the whole disposal on the subsets. It is useful if the mapping has different real constants for . It is simply necessary for that purpose that one of contractive constants be small enough to deal with other transitions which can be non-contractive or even expansive.
Definition 1. The -cyclic self-mapping with ; is said to be an extended -cyclic contraction if it satisfies the following condition:
for any , with real constants , any and .
The following result relies on the fact that the existence of a unique convex element in the intersection set
of non-disjoint closed sets ; suffices for the convexity of such an intersection.
Lemma 1. Assume that all the sets for are nonempty, non-disjointed and closed and that there exists some such that is convex. Then, is closed and strictly convex.
Proof. is closed since ; are closed. Also, and is closed and convex then the non-empty closed set is also convex and strictly convex since all its boundary points are extreme since all points in any segment in are necessarily in its interior since it is closed. □
If the subsets intersect, then and have unique fixed points in the intersection set. Formally, one has the subsequent result:
Lemma 2. Assume that all the sets ; of a complete metric space are nonempty, closed and non-disjointed. Then, has a unique fixed point in , and and have an identical unique fixed point in .
Proof. Note that
is nonempty and closed and then
. Also, one has for any given
:
Then,
as
;
and for any given
, there is some
such that for any
and any
such that
, then,
. In particular, since
is Lipschitz-continuous with Lipschitz constant
, one can interchange limit and distance to get:
and
. Also, it is the unique fixed point, that is,
. Otherwise, if
, then, we get the contradiction
and
so that
. All the sequences
converge to this point. Assume that there is some
such that
. Thus,
Then,
. A similar reasoning from (1) with
concludes that
is a Cauchy sequence,
, with
;
. It is proved that
is also the unique fixed point of
. Otherwise, the following contradiction is obtained if
is not a fixed point, that is, if
:
Then, is a fixed point of , which is unique under a similar reasoning that the one used before to prove the uniqueness of the fixed point of since is also a strict contraction if . The proof is complete. The proof is complete. □
Equation (4) of Remark 2 is generalized to an extended -cyclic contraction as follows:
Theorem 1. Assume that all the sets ; of a metric space are nonempty and closed. If is an extended -cyclic contraction, then the following properties hold:
(i) There exists for any and any ; and ; and any given .
(ii) If , then ; and any given ; ; ; , and is a Cauchy sequence, then bounded for being finite, and .
(iii) The sequences
are bounded for any finite
irrespectively of the subsets of the cyclic disposal being disjoint or non-disjoint.
(iv) The properties (i)–(iii) also hold if ;
is an extended -cyclic contraction.
Proof. It follows for
;
that:
so that
and then there exists
for any
and any
.
Property (i) has been proved.
To prove Property (ii), first note that
implies and it is implied by
. Thus, (10) holds with
. This implies for any
and any
that
as
;
, and
for
where
if
and
, otherwise, and then
so that one has for any
:
then, for any
, any
and any
, one gets by taking limits in the above expression as
since
:
and by taking
;
,
in the above expression:
Thus, there is no subsequence
with
being strictly sequence such that there is some
and some
such that
since the following will hold
, a contradiction. Therefore,
is a Cauchy sequence which converges to some
for any given
and any
. Since
, take
to conclude, after permuting the order of the “limit” and “distance” functions, because of the Lipschitz continuity of
, that
Then, so that . Also, . Assume, on the contrary, that . Then,, a contradiction so that and . On the other hand, if then , a contradiction. Then, and it is unique. Assume that then , a contradiction so that .
Property (iii) follows since by taking
in the previous inequality to (10), which leads to:
and then
as
if
is finite. For a fixed
lying in the same subset as
, one has
If
is unbounded for some
then there is some subsequence
with
such that
and then
then, the subsequent contradiction is reached:
Thus, is bounded for all finite . Property (iii) has been proved.
Property (iv) follows directly from the above properties since a -cyclic contraction is also an extended -cyclic contraction. □
Note that, as a consequence of Definition 1 of extended cyclic contractions, Theorem 1 only requires as a necessary condition for its fulfilment that , for some , provided that if . Note also that trivially (1) a -cyclic contraction is also an extended -cyclic contraction with all the contraction constants being identical and less than unity; (2) if is an extended -cyclic contraction, then is a -cyclic contraction with contractive constant . In particular, it is seen that the constraint in the previous inequality to (9) in the proof of Theorem 1 together with ensures an extended cyclic contraction for any built subsequence of elements on the adjacent subsets irrespectively of the feature that some of the individual constants are not less than unity. This condition leads to (9) which is the basis to establish that, for positive integers , sequences of elements built through compositions of the extended cyclic mapping T have similar properties as as those of the sequences of elements of a cyclic self-mapping T. This point of view is also formally reflected in several of the subsequent results.
Lemma 3. Assume that are nonempty and closed subsets of a metric space with ; and that is a -cyclic contraction self-mapping. Assume also that either or is boundedly compact; thus, if each ordered par of subsets in the set has either its first element or its second element boundedly compact if is even and, if is odd, then the same rule applies to the set of pairs and is boundedly compact if is not boundedly compact. Then, there is some finite such that for each .
The above property also holds if is an extended -cyclic contraction self-mapping.
Proof. Since
, then
for any
and
;
so that
;
. If
then
and
is a (strict) contraction so that any sequence
, that is,
is the unique fixed point of
which belongs to any
for
. Since
, the result follows. Now, consider the case
for which the subsets of the cyclic disposal do not intersect. Take any
for any arbitrary
so that
and
. By the hypothesis that
is a
-cyclic contraction self-mapping, it follows that
for any
as
(see Remark 2), so that for any given arbitrary constant
such that
, for a subsequence
for any
, some strictly increasing
and some
. In the same way,
for a subsequence
, for all
and some
. Since either
or
is boundedly compact, the best proximity sets
and
are nonempty. Now, assume that for any
, there exists a real number
such that
so that
Since
as
, then the subsequence of distances
as
. Then, there is
such that
for any
. Thus, if
then, one has
, which implies that
. Since
is arbitrary, it suffices to take
to get a contradiction to
for any
and then there is
such that
;
. The first part of the result has been proved. Now, if
is an extended
-cyclic contraction, then for any integer
, there exists
such that
for some integer
and
as
for any
. Then, from the property
as
for a
-cyclic contraction and these considerations, it follows that
as
;
and any given
since:
and the result follows in the same way as for the case of
-cyclic contractions. The result also holds if
is an extended
-cyclic contraction self-mapping by virtue of (10). □
It can be pointed out that the hypothesis of boundedly compactness of the alternate subsets in the cyclical disposal in Lemma 3 is invoked since, although the distances are finite, meaning , it can happen that the set distance can correspond to points at infinity subject to finite mutual distances since the closed subsets are not assumed to be bounded.
The following result relies on the feature that if all the subsets of the cyclic disposal are nonempty and closed, it is only needed for one of the best-proximity points to be a singleton in order to achieve to convergence of the self-mapping to best-proximity points even if the respective best proximity sets (the mentioned one excepted) are not singletons.
Lemma 4. The following properties hold:
(i) Let
be a metric space and let for be a set of
nonempty closed subsets of
such that . Assume that is a-cyclic contraction and assume also that, for some ,
is boundedly compact and that
is a singleton. Then, the set
of respective fixed points of
from
to
is unique. This set is also the limit set of proximity points of
from each subset to its adjacent one to which all the sequences generated by the cyclic contractive self-mapping converge within each subset.
(ii) Property (i) holds if
is an extended -cyclic contraction self-mapping.
Proof. Since
is a
-cyclic contraction, then
as
for all pair
. Since all the subsets
for
are nonempty and closed and
for any
, then all the subsets
have nonempty subsets of best-proximity points
;
. According to the hypothesis, it exists a subset
, for some
such that
is a singleton. Also,
is trivially nonexpansive and Lipschitz-continuous with unity Lipschitz constant since the constraint
;
leads to:
The continuity allows for the exchange of the orders of limit and distance functions so that
Take
,
; then, one has this result since
,
which follows from the analysis of the three subsequent possible cases for
:
Case a: . Since is Lipschiz, since it is nonexpansive, it follows that .
Case b: and since then for all . Then, is a contradiction and Case b is impossible.
Case c:
does not converge in
. Since the set
is boundedly compact, a subsequence
is convergent in the closed set
to some
for some strictly increasing sequence of nonnegative integers
. Then, the following contradiction follows:
Thus, Case c is also impossible. As a result,
for any
. Now, consider, for any
, the sequence
if
and
if
. Then, one has for
since
,
where
for any
is a fixed point of
in
For sequences generated from initial points
in adjacent subsets, one has that:
so that
and
exists while it is in
. But, since
, then
and
. Also, one has that
and one concludes for both the sequences initialized in
and
in
and the sequences initialized in
that the unique finite best-proximity point
of
is the unique fixed point of
and
is a fixed point of
which is unique, since
is single-valued, and finite since
is finite and
is continuous. Thus, the image of
is a singleton and it is possible to obtain similar conclusions for
for
. That is, although the only guaranteed singleton set of best-proximity points is
, the set
of fixed points of
is unique, which is also the limit set of proximity points of
to which all the sequences generated by the contractive self-mapping converge at each one of the subsets of the cyclic disposal. Property (i) has been proved.
Property (ii) follows directly from Property (i) since a -cyclic contraction is also an extended -cyclic contraction. □
Theorem 2. Let be a uniformly convex Banach space, let ; be nonempty closed strictly convex subsets of with ; , where is the norm-induced metric defined by ; . Let be a -cyclic contraction with contractive constant and take for some arbitrary . Then, the following properties hold:
(i) ; ; are Cauchy sequences, andfor any and any .
(ii) and are unique best-proximity points in each , that is, ; .
(iii) Propositions [(i)–(ii)] also hold if
is an extended -cyclic contraction.
Proof. Note that
if
and
if
. Then, in both cases,
. Property (i) follows from [
1,
2], (see Lemmas 3.7 and 3.8 in [
1]), and Remark 2, for the norm-induced metric
,
;
. The fist part of Property (ii) is also direct from ([
1], Lemmas 3.7 and 3.8). The convergence of the Cauchy sequences of Property (i) to unique best-proximity points in each subset of the cyclic disposal follows from ([
1], Theorem 3.10). Property (iii) follows from Properties (i)–(ii) and Lemma 5(i). □
It is noticed that the part of Theorem 2(i) concerned with the convergence of distances to
for sequences in adjacent subsets holds in any metric space
without invoking the uniform convexity of
and the involved sequences are bounded, see Equation (4). Also, the convexity of the sets of the cyclical disposal is invoked in the above theorem for the uniqueness of all the best-proximity points at each subset
. However, under the convexity of two adjacent subsets
and
, which guarantees the uniqueness of the corresponding best proximity in
point to its adjacent subset
(see [
1], Theorem 3.10 for
), it is guaranteed the convergence of the distances generated from any initial
as well as the convergence of sequences to one of the best-proximity points in the cyclic disposal through the cyclic contractive self-mapping
on
. This fact is addressed in the subsequent result valid for
:
Corollary 1. In Theorem 2, assume that two adjacent subsets and are strictly convex for some . Then, Property (i) of Theorem 2 holds. Furthermore,
- (a)
There are unique best-proximity points to in the strictly convex set and in to , that is, and such that .
- (b)
The best proximity sets
for
are nonempty and there exist ;
(
for
are not non-necessarily singletons) such that ;. The set
is also the unique set of fixed points of
at each of the subsets
for . The above properties also hold if
is an extended -cyclic contraction.
Sketch of Proof. If
then the mapping
is a strict contraction and
is the unique fixed point of
and the set
consists of the fixed point
z and the proof is immediate. Proceed now with the case
. Since
and are strictly convex ([
1], Theorem 3.10) then
for a unique finite best-proximity point
as in the corresponding property of Theorem 2 so that
is a singleton. Similarly,
is also a singleton. Since
are all closed for
, all the sets of best-proximity points
are nonempty for any
and since
is non-expansive, then Lipschitz continuous,
are finite for
. Take any
which is necessary in some unique
for some
since
so that
if
and
if
. Then, from Theorem 2, one has for the given
that
and
is a singleton. Furthermore, from Property (i) of Theorem 2 and Lemma 4, one has the following convergence properties of distances at each subset of the cyclical disposal:
;
as
;
and, since
exists and it is the unique point in
such that
.
Then, since is unique and the above properties of convergence of distances of sequences irrespective of the initial conditions, it follows that for . Thus, are the unique best-proximity points in each which are reachable through the mapping , that is, those fulfilling ; (irrespective of being singletons or not if . □
The next auxiliary result concerned with the uniqueness of the best-proximity point of a strictly convex subset to its adjacent one is later used. The result relies only on the (strict) convexity of the subset of the uniformly convex Banach space and it is not related to the cyclic contractive self-mapping.
Proposition 1. Assume that for some , the nonempty and closed adjacent subsets and of the uniformly convex Banach space satisfy, furthermore, that is strictly convex. Then, the best proximity set to in is a singleton .
Proof. Define
for any given
and
. Since
is a nonempty, strictly convex and closed, the function
for
defines a segment
of points in
of given extreme points
and
, such that
since
is strictly convex. Note that
and that the distance of the segment
, being itself a subset of the uniformly convex Banach space
, to
is
. Since the absolute value of the derivative of a norm of a differentiable function with respect to an argument is less than or equal to the norm of the derivative, one has that:
satisfies the set of inequalities:
which is zero if
which gives the minimum of the distance function from
to
. That, is the minimum distance to
from any segment
of the nonempty, bounded and strictly convex set
occurs when the segment is a point
. Now, assume that there is
such that
and the best proximity set to
in
is a singleton
. Take the segment
, with
, the above previous discussion concludes that
occurs if and only if
. Thus, the best proximity set of
in
is a singleton
. □
Corollary 1 is now re-addressed under Proposition 1, by invoking the strict convexity of just one of the adjacent subsets so that the best proximity set in such a subset to its adjacent one is a singleton . Then, Theorem 1(i) holds, the relevant sequences generated through in from any initial condition in are Cauchy and converge to . Then, their successive images in are (non-necessarily unique) best-proximity points in .
Corollary 2. In Theorem 2, assume that is strictly convex for some . Then, Property (i) of Theorem 2 holds and there is a unique best-proximity point to in to , that is, such that . Furthermore, the best proximity sets for are nonempty and there exist ;(for are not non-necessarily singletons) such that ;. The set is also the unique set of fixed points of at each of the subsets for . These properties also hold if is an extended -cyclic contraction.
Sketch of Proof. It follows directly from Corollary 1 and Proposition 1 by using the fact that the contractive self-mapping is single-valued. Then, (Proposition 1), and for with from Corollary 1. □
Corollary 2 is reformulated below leading to achieved similar properties while, instead of assuming that that the space is a uniformly Banach space, this suffices to assume that is a metric space, that one of the subsets is boundedly compact, instead of convex, and that its best-proximity point is a singleton. The proof is directly supported by Lemmas 3 and 4, Theorem 2 and Corollary 1.
Corollary 3. Assume that are nonempty and closed subsets of a metric space with ; and that is a -cyclic contraction self-mapping. Assume also that the subset is boundedly compact for some and that (so that and ;). Then, all the best proximity sets for are nonempty and there exist ;(for are not non-necessarily singletons) such that ;. The set consists of the unique set of fixed points of at each one of the subsets for . The above properties also hold if is an extended -cyclic contraction.
Proof. For the case , see the sketch of proof of Corollary 1. Consider the case . From (4), note that as ; , that is, for any in any arbitrary subset ; . Also, is bounded from Theorem 1(iii); for any given and any . For each given such that , it always exists such that and such a sequence is bounded. Since is boundedly compact, there exists some subsequence , for a strictly increasing sequence , which is convergent . Now, proceed by contradiction by assuming that all convergence points for any convergent subsequence of any sequence of the form for any given are not , the unique best-proximity point to in . Then, the sequence of distances would not converge to , a contradiction. Therefore, . Note that . If then so that . As a result, is the unique fixed point of the composed self-mapping in . Since for any and any , one gets in a similar way that for . Then, and since is single-valued, the above set , subject to the property for , consists of the unique set of fixed points of at each of the subsets for which are also the best-proximity points reachable through cyclic contractions on . Thus, they fulfil (but the are not necessarily singletons for .
The above properties also hold if is an extended -cyclic contraction. □
It is possible to comment the following observations concerned with the feasibility of the best proximity set to be a singleton (see Lemma 4, Theorem 2, Corollary 1 and Corollary 3). The assumption that the distance from any point of the space to one of its nonempty closed subsets is reached at a unique point holds if the metric space is a uniformly convex Banach space and the considered subset is strictly convex, its set of best-proximity points to its adjacent subset is a singleton if such and adjacent subset is also strictly convex. Both proximity sets are singletons which fix the distance between such adjacent subsets while they are the unique points which set such a distance. However, in some cases, the convexity of the considered set may be overcome. For instance, assume that the focused on subset is imperfect, but still closed, that is, it has at least one isolated point which is obviously an adherent point to the subset which is not either, also obviously, an accumulation point. If is located in the Banach space such that then and then while, obviously, is not convex.
The subsequent result is a slight extension of parallel results in [
1] when the cyclic contraction involves more than two sunsets of a uniformly convex Banach space.
Lemma 5. Let be a uniformly convex Banach space with nonempty sets for with norm-induced metric defined by and then satisfying ; . Then, the following properties hold:
(i)Assume that, for any given , is closed and convex and is closed. Let and be sequences in and be a sequence in satisfying ; as . Then, as .
(ii) Assume that
is an extended -cyclic contraction and that, for any given ,
and
for some arbitrary , if;
as for any thenfor any .
Proof. If
,
, the proof of Property (i) follows directly from ([
1], Lemma 2.5). The proof of Property (ii) follows from Property (i) as follows. Since
then
if
and
if
. Thus,
.
Under close reasoning, one concludes that and .
Then, Property (ii) follows directly from Property (i) since and ; . □
The next result relies on the fact that the basic results of Lemma 5 hold if just one of the subsets of the cyclic disposal is closed and boundedly compact but not necessarily convex.
Lemma 6. Let be a uniformly convex Banach space with nonempty sets for with norm- induced metric defined by and then satisfying ; . Assume that is boundedly compact and that for some . Then, Property (ii) of Lemma 5 holds, and, furthermore, (which are not necessarily singletons for ) for all .
The result remains valid if is an extended -cyclic contraction.
Proof. First, note that the convergence of distances of sequences allocated in adjacent subsets to
is direct as a result of the first part of Lemma 5 (ii), proved in [
1], which does not require the convexity of the subsets.
One proceeds now to prove the convergence to zero of distances of sequences in the same subset whose composition wit their images in adjacent subsets converge to
. First, note that, since
is boundedly compact since
is bounded,
and there are subsequences
and
for some strictly increasing subsequences
and
which are convergent in
to some
and, respectively,
. Then, one has from Property (ii), since
is non-expansive, then Lipschitz-continuous, that
and
and for
, being the unique best proximity to
in
, that is,
, note that
(at this stage is not still proved that
) since
is boundedly compact and
is a convergent subsequence of a bounded sequence, and one has that
since
as
is single-valued then
. If
then
while
with
, a contradiction then
, that is, the limit of the convergent subsequence is the next proximity point. Proceeding in a similar way, one concludes that
. Thus,
And the above subsequences in
fulfil the claimed property. Now, note that
as
with
; then,
so that
. In the same way, one proves that
and combining this property with the above one and using the triangle inequality for distances yields:
Furthermore,
, that is, the best-proximity point of
in
is the fixed point in
of the composed self-mapping
. Otherwise, one would get the following contradiction
using the asymptotic regularity, the non-expansivity, and the consequent continuity of the self-mappings
on
and
on
. Thus, Property (iii) is proved if
and
. Now, since
is single-valued, one has that
for any
with
and, one has, in a similar way, that for any
,
if
and
if
. □
3. Main Results on Combined Iterative Procedures Involving Mixed Compositions of the Self-Mappings T and S
This section considers the combined application of the self-mappings and in any order of composition and with any number of consecutive compositions of any of them before potential switching to the other one. Some of the main obtained results are supported by those given in the former section.
The composed iterations are built by defining recursively the sequence of composed mappings
for any
as follows:
The above notation reflects that and can be applied in any order and under any amount of consecutive applications of any of them along the construction process of the iterations ; .
Consider a
-th iteration
such that:
where all the above amounts are non-negative integers, and
is the number of iterations happened for at the -th iteration, that is, the number of total complete tours of on the whole cyclic disposal;
is the number of iterations occurred for at the -th iteration with so that iterations of have not completed a whole tour around the complete cyclic disposal;
is the number of iterations happened for up until and included the -th iteration , with being the number of iterations of fulfilling either the first condition of (2) or (3) up until the -th iteration while being that fulfilling the second condition of (2) up until the -th iteration .
Remark 3. It is found that denotes that the self-mappings for all are identical for and . If there is, for instance a different mapping for the sequence initialized at and for that initialized at at the -th iteration while the preceding ones are generated with the same previous iterative sequence of self-mappings, this is reflected with the notation .
Remark 4. Note from (20) that . Thus, if , then and so that two of the amounts in this sum are null. Note also that if , then ; then, if , but it can happen that and . In this last case, the number of contractive iterations of is finite as .
Remark 5. Since ;, then if and then and so that, if , then .
In view of (2) and (19), (20), one has:
for , where Note from (22) that as .
The following result is given without explicit proofs since it is a direct consequence of the definitions of and and (19)–(20):
Lemma 7. Assume that for are non-empty closed subsets of a metric space with , and with is a -cyclic self-mapping on such that ;. Assume also that for some .
Then, the following properties hold for all :
- (1)
for some .
- (2)
and for the same of Property 1 for any arbitrary .
- (3)
and for the same of Property 1 for any arbitrary .
- (4)
and for the same of Property 1 for any arbitrary
and any , where if ; if ; and if .
- (5)
where and
and any
.
Remark 6. Note from Lemma 7 that:
- -
Property 1 is a particular version of Property 2 if .
- -
Property 2 is a particular version of Property 3 if .
- -
Property 3 is a particular version of Property 4 if .
- -
Property 5 trivially generalizes Property 4.
The following result establishes that, if either or for any , with either or , then for any , if . This means that the distances which lie outside the interval decrease strictly at any iteration performed under any of the two self-mappings irrespective if both distance arguments are either in the same set or in two adjacent subsets of the cyclic disposal. In that way, the distances which lie outside behave locally as weak contractions. If the mapping is applied then the contraction is, furthermore, strict for distances of points in the same subset.
Lemma 8. Assume that for are non-empty closed subsets of a metric space , such that ; , with , and be self-mappings which satisfy (1)–(3), where is a contraction of contractive constant and is a -cyclic self-mapping on of contractive constant ;.
The following properties hold:
(i) Assume that for any and that for some . Then, if .
(ii) Assume that for any , and that either or for some , and assume also that . Then, if .
(iii) Assume that
for any , that and that . If then . If then .
(iv) If for any and , and for some
then and if then .
Proof. Proof of Property (i): If
and
, then, from (3)
Proof of Property (ii): If
, note from the first condition of (2) with
that the above inequality still holds for
and
since
. Also, if
, it follows from (1) for
if
for any
that
Proof of Property (iii): Note that if
and
, one has under the first condition of (2) that there is some
such that:
On the other hand, if , the first constraint arising from for any with , then from the first condition of (2).
Proof of Property (iv): If
and
for any
with
, one gets from (1) that if
; then,
If , then one has from the above set of inequalities for that . □
The following result is of interest concerning the convergence of distance sequences of iterations constructed with composite mappings of and , (19), (20) and Lemma 8:
Lemma 9. Assume that for are non-empty closed subsets of a metric space , such that ;, and are self-mappings with being a (strict) contraction and being a -cyclic self-mapping on of contractive constant ;. The following properties hold:
(i) If then ; .
(ii) If and ; for some arbitrary then .
(iii) If commute, , ; , ;
for some , and ; . Then, ; . If is replaced with in the above formulas, then the result still holds.
If , then the above limit becomes .
(iv) Assume and , for some arbitrary . If with ; , one has that , ; , and . If , then the above limit becomes .
Also, if, ; , one has
, in particular, if
(v) Assume that ;
and ; , where is the total number of whole iterations of on the whole cyclic disposal up until the -th iteration in , and
and
the number of iterations of
fulfilling, respectively, the first condition of (2) or (3) and the second condition of (2) up until the -th iteration. Thus, if
and for some
then for any , one has ; , and
(v.1) for any as if .
(v.2) for any as if .
(v.3) The limits (v.1)–(v.2) also hold as , for not necessarily diverging, any .
Proof. To prove Property (i), note that if
, then
. Thus,
;
, and, from (1) and the first condition of (2), or from (3), depending on to which set(s)
the points
and
belong to. Thus, one has from (21), (22):
Then, if , (Remark 4). Since ;, that is the number of iterations involving the mapping up until the -th iteration of and, since as from (7), then , as and . Property (i) has been proved.
To prove Property (ii), note that if
, then
and
for some arbitrary
, then
;
(that is, only the self-mapping
is involved in building the whole set of sequences which are in
for some
) so that one has from (3),
If
, then
and, one gets from (24) that
and Property (ii) has been proved.
To prove Property (iii), note that the conditions of Property (ii) are kept from the initial conditions up until the
-th iteration, while there is an application of the self-mapping
on
at the
-th iteration for the sequence generated from the initial point
. From the hypotheses, for
, one determines that composite mappings
and
for all
only differ in
and
and
;
,
for all
:
for some
Since for
,
and
are in disjointed adjacent sets
and
, and since
commute
, one has from (21), which derives from (1)–(3), that
which implies that
as
with
;
,
and
for sufficiently large
. From the first condition of (2), there is some finite non-negative integer
such that for
,
; then, for
, one has
and then
for
so that one gets as a result:
It is obvious that if is replaced with in the (27), the result still holds. Then, Property (iii) has been proved.
To prove Property (iv), note that
for some
. Take
,
and
;
. Then, it follows that
and one gets from (6), (7) that
;
, since, if
for
then either
or
for any
, and note that
Thus, one has
- (b)
if then the above result follows as in Property (iii).
In the same way, it is obtained if, in addition, ; , that .
Property (iv) has been proved.
Property (v) is proved easily under a close reasoning as the one used in the proof of Property (iv). □
Remark 7. Note that it is relevant in Lemma 9 that:
- (a)
Any two iterative processes which only involve compositions of
with itself in the same subset
generate sequences which are contained in this subset.
- (b)
If two iterations only involve compositions of with itself but they are initialized in adjacent subsets, these generate sequences which always remain each in its corresponding subset but the maximum allowed distances at each iteration step is the distance between disjointed adjacent subsets. That means that the self-mapping might be non-single valued from the second equation of (2), forced to be applied to this end if the distances exceed .
- (c)
Any two iterations with initial points in the same subset generate sequences which lie asymptotically in adjacent subsets in the event that they mutually differ in the concourse of
times the self-mapping
for some given . The case when both of them just differ in a single contribution in the iterative composition of maps is sufficient for that property to hold. If the initial points are in adjacent subsets then both generated sequences lie asymptotically in adjacent subsets if they mutually differ in the concourse of
times the self-mapping
for some given . The number of contributions of to the composed mappings in any of the mentioned iterations do not modify the above conclusions since the self-mapping with domain in any subset has its image in the same subset.
The following assumptions will be invoked in the following as linked to results in the previous section and used to extend them to the iterative processes which involve compositions for any of the self-mappings , subject to (19) and (20), where all the subsets of the metric space are nonempty and closed.
Assumption 1. for are non-empty closed subsets of a metric space , where , with ; ; and are self-mappings which satisfy (1)–(3), where is a contraction of contractive constant and is a -cyclic self-mapping on of contractive constant ;.
Assumption 2. The unique fixed points of the contractive self-mapping satisfy ; .
Assumption 3. There is at least one such that is boundedly compact and is a singleton.
Assumption 4. At least one of the subsets of in the cyclic disposal is strictly convex for some .
Assumption 2 states that the fixed points of in each subset are also best-proximity points of this subset to its adjacent subset. Also, from Assumption 3 (which holds directly if is convex, but it can also hold without convexity need in some cases if is not perfect) and Corollary 2, it follows that the best-proximity points to of in for (also fixed points of ; from Assumption 1) satisfy for and for . That is, if contributes with infinitely many tours on the whole cyclic disposal in iterative processes, the sequences of distances of iteration in adjacent subsets converge to (Lemma 9(iv)). From Corollary 3, the sequences in adjacent subsets involved in such distances converge to the unique fixed points of for all which are the, in general, non unique (except for ) best-proximity points of and the unique fixed points of ; under Assumption 2. Assumption 4 guarantees the uniqueness of and then that of the fixed points of for all .
The next result establishes the main result of this section which is supported by Theorem 2, Corollarys 1 and 3, Lemmas 4–6 and 9.
Theorem 3. Let be a uniformly convex Banach space under Assumptions 1–3, where is the norm-induced metric defined by ; , and consider an iterative process which involve compositions for any of the self-mappings , subject to (19)–(20). Then, the following properties hold for any given and :
(i) The convergence of distances of Lemma 9(v) holds for ;
and ; , that is, irrespective of the order of the contributions of the self-mappings
and
to the iterative process , the following holds for any ,
for any : for any as if
or if .
for any as if
or if provided that if
or
, otherwise.
for any as if
or if provided that if
or
, otherwise.
Property (i) also holds if is an extended -cyclic contraction self-mapping.
(ii) The set of respective fixed points of from to is unique and it is also the set of proximity points of to which all the sequences generated by the contractive self-mapping converge at each one of the subset if (that is, if there is at lest a complete tour of in the iterative process as .
Property (ii) also holds if is an extended -cyclic contraction.
Proof. Property (i) follows directly from Lemma 9(v) since a uniformly convex Banach space is a complete metric space and from Lemma 5(ii). Note that if , for any :
for any as if or if , and
for any as if . If then
for any as as from (3) since since is a strict contraction. Finally, if for (at least) a while , then for some from (2), (3) and Assumption 2. The result also holds from Lemma 4(ii) if is an extended -cyclic contraction.
Property (ii) follows from Assumption 3, Lemma 4(i) and Lemma 6, and from Assumption 3, Lemma 4(ii) and Lemma 6 if is an extended -cyclic contraction with at least a contribution of one complete tour of as . □
The next result establishes the second main result of this section. The proof is similar by replacing Assumption 3 by Assumption 4, related to the strict convexity of one of the subsets of the cyclic disposal, and by taking into account Lemma 5 and Proposition 1.
Theorem 4. Let be a uniformly convex Banach space under Assumptions 1, 2 and 4. Assume that is strictly convex for some . Also, let be a -cyclic contraction with contractive constant and take for some arbitrary . Then, the following properties hold:
(i) Property (i) of Theorem 3 holds.
(ii) There is a unique best-proximity point to in the strictly convex set (Assumption 4), that is, , such that ; and there exists ; (such that for are not empty and non-necessarily singletons) such that ; .
(iii) Propositions [(i)–(ii)] also hold if
is an extended -cyclic contraction.
4. Examples
Example 1. Consider the metric space , where is the Euclidean metric, and consider the real intervals and such that . Note that the Euclidean metric in this case coincides with the taxi-cab metric and that the metric space is a uniformly convex metric space and then complete. Define the self-mapping on by which is a non-expansive 2-cyclic self-mapping, since for , with and , and and are the best-proximity points with in to and in to .
Consider a contractive self-mapping
on
satisfying
and
such that for some ,
if
or if
with
and , otherwise, with Note that the unique best-proximity points Zi in Ai to are the respective unique fixed points of in S in
for . Note that:
- (1)
If only composed mappings of contribute in an iteration process , then and as if for
- (2)
If only composed mappings of contribute in an iteration process , then , with , for all any but the iterative process does not converge and as while and as . In particular, if , then and for ; .
- (3)
If a finite even number of compositions of contribute to an iterative process which has infinitely many compositions of , and irrespective of the order of occurrence of the whole set of compositions, if ; . If a finite odd number of compositions of contribute to the iterative process, then if ; . In both cases, as .
- (4)
If infinitely many compositions of contribute to an iterative process which has either a finite number or infinitely many compositions, of , and irrespective of the order of occurrence of the whole set of compositions, and if ; ; and the sequence does not converge while and as .
Example 2. Modify Example 1 with , where , and is continuous, with for .
Thus, , and for and , one has that is -cyclic contractive if Assume that and , then and , then ; equivalently, the necessary constraint , is guaranteed if . Thus, for some real constant , one has that is -cyclic contractive ifwhich holds if Whose second inequality follows since
obtained from the former constraints
and . As a result, is -cyclic contractive if for , for . The properties of Example 1 hold for iterative processes of the above cyclic contractive self-mapping and contractive self-mapping , whose fixed points at each subset is the best-proximity point to its adjacent subset as in Example 1.
Examples 1 and 2 are easily extendable to the real vector case as follows:
Example 3. Consider the complete metric space , where is the supremum metric of all the -vector components obtained from the component-wise taxi-cab metric, and consider the subsets for of , where and such that . The best-proximity points are real -vectors ; with all their respective components being, respectively, and . The contractive self-mapping on are anyone with fixed points being the best-proximity points between adjacent subsets. Then, the derivations and conclusions of Example 2, inherited from Example 1, hold.
There are dynamic systems present in the real word which have several potential configurations which can be activated at different time instants as long as the process under study evolves in different production phases or because of the concourse of certain events which activate switching. Switching actions can cause instability if they occur repeatedly in very short time intervals even if the various alternative configurations are stable. Also, switching actions can translate into stability of the switched configuration if the switching map runs properly through time or is governed by events, for instance, if signals exceed a certain threshold, the switching might be activate to prevent against instability, even if the individual configurations are instable, [
20,
21,
22,
23]. The following two examples link those problems with the combined iterations involving cyclic self-maps and strict contractions. Two sets of the metric space are defined for the state evolution of a time-varying linear dynamic system of
-th order with constant distinct parameterizations in each of both subsets. The two individual parameterizations are assumed to be stable and governed by a strict contraction, while they keep either non-negative solutions or non-positive ones depending on which set has been activated.
Example 4. Let ; ; be real Metzler stability matrices, that is, if and have all their eigenvalues with negative real parts. Since they are Metzler, they are stable iff , that is, they are non-singular with non-negative entries, [24,25,26]. Define the -th order differential systems with initial conditions ;. Since the matrices of dynamics are Metzler, then the fundamental matrices of the corresponding differential systems are non-singular with non-negative entries for all time and the solutions satisfy for if . In the same way, for if . This feature is equivalently denoted by for if .
Consider the Banach space
and consider also consider the first hyperorthant
of and its opposed one
as subsets of. Clearly and both subsets are unbounded, closed and convex if . Rename and so that all the solutions of are in for all time under any given non-positive initial conditions and all those of are in for all time under any given non-negative initial conditions as a result of the properties of Metzler matrices.
Furthermore, as a consequence of the properties of fundamental matrix functions of stability matrices, one has that there exist positive constants and
for
such that
for all
and any , where
for is norm-dependent and
is the maximum real part of the dominant eigenvalue of the stability Metzler matrix , for
, if it has multiplicity equal to one, and any greater negative real number, otherwise.
Thus, in terminology of systems theory, note that both differential systems are globally exponentially stable. Note also that
is a complete metric space if
is the norm-induced metric and that . Fix a sampling period
for picking-up the discrete samples of the solutions for and
.
Assume that the initial condition of the differential system is either in
or in
. Now, define strictly increasing sequences of non-negative integer numbers for such that a self-mapping from to is defined by:if and for ;If , for some , all , and some . That is, is a non-expansive self-mapping on
which is also a -cyclic contraction in at the sampling instants and it governs switching in-between
and .
A self-mapping
on
which defines the solution trajectory in
is contractive in both
and
if the sequences
for
satisfy
for , equivalently, the sampling interval between consecutive values of the sequence
is large enough to satisfy the constraints:for and some . That is, the restriction of
to any of the subsets
and is contractive if there is a sufficiently large number of samples at each of the parameterizations
and before switching (activated by the mapping ), respectively to and . The reason for those constraints is that the constantsmight exceed unity so that small residence times at a parameterization before switching in-between
and can translate into either expansiveness of the solutions or into the achievement of non-expansive (although non-contractive) properties. Note by direct inspection that is the unique fixed point of and the unique best-proximity point of to in and vice versa. Note also that because of the definition of the mapping the solution trajectory never switches in-between in at time instants in an open interval defined by consecutive sampling instants and so an identical sign for all the components of is always kept except at switching time instants which take place at sampling instants accordingly to (28)–(30).
In summary, we have proved the following result:
Proposition 2 (for Example 4). Consider a switched linear time-varying dynamic system described by when the solution trajectory is in and by when the solution trajectory is in , where both matrices of dynamics are stability Metzler matrices and that is either in or in . Assume that the self-mappings and for the discretization of the solution trajectory satisfy (28)–(30) for a given sampling period . Then, the solution trajectory is bounded for all time for any given finite initial conditions and as and as .
Example 5. In Example 4, the stabilization of the cyclic configuration without requiring convergence can be governed just by monitoring the self-mapping without requiring the contractiveness of . That is, even if one (or both) differential system(s) is not asymptotically stable (and even if both of them are unstable), that is, if the Metzler matrices of dynamics fail to fulfil ; , or if they are still stability matrices while the residence time of the solution trajectory within each of the subsets violates the constraint (30).
The following concerned immediate result is pertinent:
Proposition 3 (for Example 5). Assume that at least one of the Metzler matrices (i=1,2) is not a stability matrix and that, for some prefixed arbitrary real constant , the self-mapping on is defined such as to satisfy: if with for any given prescribed minimum sample threshold for any given real constant . Then, there is some real constant such that if , or if , and , then .
Proof. is bounded if
since, if
for some
, then
and
. Otherwise, if
, then
, a contradiction. Since the solution trajectory at sampling instants is uniformly bounded, the solution trajectory in-between consecutive sampling instants is also bounded according to:
for all
and any
. Thus, the solution trajectory is uniformly bounded for all time since
. □
Example 6. Consider the infinite-dimensional Banach space of real -vector functions for any given and any given real with , and consider a set . Note that, since , the distance is zero. Define the two-cyclic self-mapping by , for any , where is a diagonal matrix with diagonal entries in the interval . Consider the sequence generated by; . Then, clearly
, for initial value; and any
, so that converges to the null real -vector function in
as .