Applied Mathematics and Computation 219 (2013) 10655–10667

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Asymptotically non-expansive self-maps and global stabilitywith ultimate boundedness of dynamic systems

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.04.009

⇑ Corresponding author.E-mail addresses: manuel.delasen@ehu.es (M. De la Sen), Asier.ibeas@uab.cat (A. Ibeas).

M. De la Sen a,⇑, A. Ibeas b

a Institute of Research and Development of Processes, Campus of Leioa, Barrio Sarriena, 48090 Leioa, Bizkaia, Spainb Department of Telecommunications and Systems Engineering, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

a r t i c l e i n f o a b s t r a c t

Keywords:Contractive mapsNon-expansive mapsMetric spaceFixed points

This paper investigates self-maps T : X ? X which satisfy a distance constraint in a metricspace with mixed point-dependent non-expansive properties or, in particular, contractiveones, and potentially expansive properties related to some distance threshold. The abovementioned constraint is feasible in certain real-world problems of usefulness, for instance,when discussing ultimate boundedness in dynamic systems which guarantees Lyapunovstability. This fact makes the proposed analysis to be potentially useful to investigate globalstability properties in dynamic systems in the potential presence of some locally unstableequilibrium points. The results can be applied to stability problems of dynamics systemsand circuit theory as the given examples suggest.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Fixed point theory and related techniques are of increasing interest for solving a wide class of mathematical problemswhere convergence of a trajectory or sequence to some equilibrium set is essential. Recently, the subsequent set of moresophisticated related problems are under strong research activity:

(1) In the, so-called, p-cyclic non-expansive or contractive self-maps map each element of a subset Ai of an either metricor Banach space B to an element of the next subset Ai+1 in a strictly ordered chain of p subsets of B such that Ap+1 = A1. Ifthe subsets do not intersect then fixed points do not exist and their relevance in Analysis is played by best proximitypoints [1,2]. Best proximity points are also of interest in hyperconvex metric spaces [3,4].

(2) The so-called Kannan maps are also being intensively investigated in the last years as well as their relationships withcontractive maps. See for instance [5,6,11].

(3) Although there is an increasing number of theorems about fixed points in Banach or metric spaces, new related recentresults have been proven. Some of those novel results are, for instance, the generalization in [7] of Edelstein fixed pointtheorem for metric spaces by proving a new theorem. Also, an iterative algorithm for searching a fixed point in non-expansive mappings in Hilbert spaces has been proposed in [8]. On the other hand, an estimation of the size of anattraction ball to a fixed point has been provided in [9] for nonlinear differentiable maps.

(4) Fixed point theory can be also used successfully to find oscillations of solutions of differential or difference equationswhich can be themselves characterized as fixed points. See, for instance [9,10,12,13]. The fixed point tools are alsouseful to investigate certain applied problems as, for instance, some ones related to image restoration [14].

10656 M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667

(5) Recent research has been performed in [15] concerning the existence of common fixed points in non-self and non-asymptotically non-expansive mappings. On the other hand, Hyers–Ulam-type stability has been proven in [16] fora general mixed additive-quadratic-cubic-quartic functional equation and fixed point theory has been applied tofuzzy-type characterizations in [17]. On the other hand, relevant recent investigation is being addressed towardsthe study of mixed equilibrium problems linked to the set of non-unique fixed points in nonexpansive mappings[18,19].

(6) The robust stability of uncertain dynamic systems, potentially involving switching in-between different parameteriza-tions, is a very important property to be achieved in most of applications which is often addressed through Lyapunovstability theory often with parallel use of related matrix inequalities. See, for instance, [13,20–24] and references therein. Stability is a basic property or requirement in many real life problems. See, for instance [23–31] concerning thestability of time-delayed, discrete, fractional and hybrid continuous-time/discrete-time systems and adaptive controlas well as references there in. It is also a basic property to be guaranteed by the controller designer in common adap-tive control problems and also in the analysis of the properties of equilibrium points and study of boundedness of thesolutions in epidemic models and a number of real uncertain dynamic systems (see, for instance, [32–37] and refer-ences therein). In this context, one of the weakest, but at the same time useful, global stability concepts for dynamicsystems with a unique equilibrium point is that of global Lyapunov stability with ultimate boundedness. Intuition dic-tates that the map defining the solution trajectory from any set of bounded initial conditions might be in some casesexpansive close to the equilibrium point and contractive far away from the equilibrium point, thus preserving globalstability (in the sense of boundedness of any solution from any bounded set of initial conditions). The dynamic systemexhibits also local instability around the equilibrium point which is not an attractor since the mapping defining thetrajectory is not contractive and it is not even non-expansive close to the equilibrium point.

(7) There are also interesting results available concerning the use of Fixed Point Theory in variational analytic and numer-ical methods and, in particular, in iterative methods based on variational inequalities. Variational inequalities havewell-known relevant application fields in Physics and Control Theory based on Hamiltonian formalisms as it is thecase, for instance, in what is concerning with variational min/max principles in Optics and Mechanics and in controloptimization problems. It is obvious that the Physical variational principles are related to stability since the optimaltrajectories are such that they make extremal a Hamiltonian-type functional involved. In control theory, the optimalcontrol stabilizes the closed-loop system in parallel with making extremal the particular Hamiltonian defining theproblem. A set of results concerning those topics can be found in [38–45] and references therein. Thus, there is someclose research objectives between stability/stabilization studies and associated methods and variational-type princi-ples in many disciplines.

Therefore, it turns out that there are many real-life problems which are characterized by self-maps which can be expan-sive, non-expansive or contractive depending of the size of the distance values within given sets as, for instance, the gener-ation of state trajectory solutions from initial conditions in dynamic systems subject to disturbances of local large sizes (interms of norms or distances) while such sizes are upper-bounded by asymptotically non-strictly increasing functions of thestate trajectory solution norms. In this case, the system can exhibit simultaneous local instability around equilibrium pointswith global norms. In this case the system can exhibit simultaneous local instability around equilibrium points with globalstability under the ultimate boundedness property. In other words, and roughly speaking, a tendency of the state-norm todiverge is neutralized by a contractive property of the state-trajectory solution for associated large state norms or, in Lyapu-nov stability theory terms, by the property of the Lyapunov functional candidate to become strictly decreasing for such largenorms. This manuscript is devoted to investigate self-maps T : X ? X in a metric space (X,d), or in a normed space (X,k k),which satisfy the constraint d(Tx,Ty) � d(x,y) 6 �Kd(x,y) + M, for some real constants K P 0, M P 0. A motivating practicalusefulness of this property is concerned, for instance, with the description of ultimate boundedness in stability dynamic sys-tems. In particular, assume that x,y = Tx, z = Ty = T2x are three selected distinct points of a state-space trajectory solution ofthe dynamic system, ordered according to increasing time, such that the distance in-between the first and second pointsd(x,y) exceeds a sufficiently large threshold. Then, under the above constraint, there is an ultimate boundedness propertyguaranteeing global stability, since the distance in-between the second and third points d(y,z) decreases avoiding divergencewith time of the state-trajectory solution. More formally, it is direct to see that d(Tx,Ty) 6 d(x,y); i.e., T : X ? X is non-expan-sive, if d(x,y) P M/K; "x,y 2 X. Also, if X is bounded then

dðx; yÞ < M=K ) dðTx; TyÞ 6 ð1� KÞdðx; yÞ þM < M=K; 8x; y 2 X ð1:1Þ

Then, the self-map T : X ? X exhibits the following constraint under (1.1) provided that it is continuous: T : Ax y ? Axz whereAx y � X is the open circle of center cx y 2 Xof radius R :¼M/K for each given x,y 2 Ax yand Axz � Xis an open circle of center atsome cxz 2 X also of radius R. Note that Ax y can be distinct from Axz. However, if T : X ? X is not continuous then the exis-tence of the above circles is not ensured; it is only known that (1.1) holds. Note that (1.1) does not guarantee that the self-map T : X ? X is globally non-expansive since it can be expansive for small distances fulfilling d(x,y) < M/K and non-expan-sive if d(x,y) P M/K. This elementary idea is addressed in the following simple technical proposition:

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667 10657

Proposition 1.1. Assume that d(Tx,Ty) � d(x,y) 6 �Kd(x,y) + M; "x,y 2 X. Then

(i) d(Tx,Ty) 6 d(x,y) for any given x,y 2 X such that d(x,y) P M/K.(ii) Assume, in addition, that {(x,y) 2 X � X : d(x,y) < M/K} – ;. Then, a non-empty collection {Aa} of bounded nonempty subsets

of X � X exists such that

dðx; yÞ 6 dðTx; TyÞ 6 ð1� KÞdðx; yÞ þM < M=K; 8ðx; yÞ 2 A; 8A 2 fAag ð1:2Þ

Proof. Assume that Property (i) is not true. Then, one gets:

dðx; yÞ < dðTx; TyÞ 6 ð1� KÞdðx; yÞ þM ð1:3Þ

which implies 0 < �Kd(x,y) + M, equivalently, d(x,y) < M/K what contradicts d(x,y) P M/K. Then, d(x,y) P M/K) d(Tx,Ty) 6 d(x,y) and Property (i) is proved. Property (ii) follows since the last inequality of (1.2) is equivalent tod(Tx,Ty) � d(x,y) 6 �Kd(x,y) + M while the first one holds in any nonempty collection of bounded subsets {Aa} of X fulfillingthe set inclusion condition A # {x,y 2 X : d(x,y) < M/K}; "A 2 Aa. Since {(x,y) 2 X � X : d(x,y) < M/K} – ; by hypothesis, atleast one such a nonempty bounded set always exists being the whole set {(x,y) 2 X � X : d(x,y) < M/K}. Property (ii) has beenproved. h

A simple example to visualize Proposition 1.1 follows:

Example 1.2. Assume X � R and Tx = K0x + mx with mx being point-dependent and bounded fulfilling jmxj 6 g <1, in general.Assume that d(x,y) = jx � yj is the Euclidean distance in R. Assume that K0 2 [0,1) and define M :¼ 2 max (mx : x 2 X) 6 2g.Then, if K = 1 � K0 then K 2 (0,1], and

dðTx; TyÞ 6 ð1� KÞdðx; yÞ þM

so that(1) If x,y 2 X such that jx � yjP M/K then d(Tx,Ty) 6 d(x,y) and the self-map is non-expansive for those values x,y 2 X

(Proposition 1.1 (i)).Constraints (1.2) hold, then d(Tx,Ty) P d(x,y), "(x,y) 2 {x,y 2 X : d(x,y) < M/K} (Proposition 1.1 (ii)). In particular, if

B :¼ {x,y 2 X: (x – y ^ jx � yj < M/K)} then d(Tx,Ty) > d(x,y) and the self-map is expansive for those values x,y 2 X. h

The objective of this paper is the investigation of self-maps T : X ? X which such mixed properties related to some dis-tance threshold. Examples are given to illustrate such properties in the field of nonlinear dynamic systems.

2. Basic distance property and related motivating example

Let (X,d) be a metric space and T a self-map from X to X. Such a self-map is uncertain in the sense that the distance issubject to the following constraint:

dðTx; TyÞ � dðx; yÞ 6 �Kdðx; yÞ þM; 8x; y 2 X ð2:1Þ

for some real constants K P 0, M P 0. In order to discuss the feasibility of (2.1), note the following features:

(1) If M = 0 and K 2 (0,1] then (2.1) is the usual contractive constraint of Banach contraction principle and T: X ? X isstrictly contractive. Note that there is no contractiveness if K = 0. If K = M = 0 then T : X ? X is non-expansive. IfK = M = 0, and the inequality in (2.1) is strict for x,y(–x) 2 X then T : X ? X is weakly contractive in the sense thatd(Tx,Ty) < d(x,y) for any given x,y(–x) 2 X.

(2) If K = 1 then d(Tx,Ty) 6M; "x,y 2 X. The constraint (2.1) is applicable, in particular, to the familyAT :¼ fAi � X : ðdiamðAiÞ 6 M ^ TðAiÞ � Aj; some Aj 2 XÞg of bounded subsets of X. In this case, d(Tjx,Tjy) 6M; "j 2 Z+

provided that x; y 2 Aa 2 bAT and T maps X to some member Ai of bAT for each given x,y 2 X. In other words, the imageof T is restricted as T : X ? XjAi (for some Ai 2 bAT which depends, in general, on x and y) so that d(Tx,Ty) 6M in order to(2.1) to be feasible, i.e. Tx,Ty are in some set of the family bAT if the pair (x,y) in X � X is such that d(x,y) > M. Note thatT : X ? X is not necessarily a retraction from X to some element of bAT since T(Ai) # Aj for Ai;Ajð–AiÞ 2 bAT . Note thatT : X ? XjAi can possess a fixed point if K = 1 and ((2.1) holds. As pointed out by one of the referees, the constraint(2.1) is not limited to families of bounded subsets of X. For instance, Tx = sinx satisfies the condition also in unboundedsubsets of X.

(3) If K > 1 then d(Tx,Ty) 6M if x = y for x,y 2 X or if dðx; yÞP MK�1 for x,y 2 X, and

dðx; yÞP M=ðK � 1Þ ) 0 6 dðTx; TyÞ 6 dðx; yÞ � MK � 1

< dðx; yÞ if x; yð – xÞ 2 X

Thus, if x,y 2 X exist such that dðx; yÞ 2 ð0; MK�1Þ then (2.1) is unfeasible for any self-map T on X since this would imply

d(Tx,Ty) < 0. Fixed points can only exist in trivial cases as, for instance, Xa :¼ fXx : dðx; yÞP MK�1 ;8y 2 Xg � X is a set of iso-

lated points with a minimum pair-wise distance threshold so that T : X ? X is such that Ty = x 2 X; "y 2 X. An example is

10658 M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667

X = R2 endowed with the Euclidean metrics, Xa # bXa :¼ X n C0 0; MK�1

� �� �[ f0g and C0 0; M

K�1

� �is the open circle centred at z = 0

of radius MK�1 excluding the circle center z = 0 (so that such a center z = 0 is in bXa and also in Xa).

(4) The case of interest discussed through this paper concerning the constraint (2.1) is when M > 0 and K 2 [0,1). It isproved that the self-map T : X ? X exhibits contractive properties for sufficiently large distances which exceed a min-imum real threshold while it might possibly be expansive for distances under such a threshold. In other words, such aself-map can be contractive or expansive in different subsets of X. A related motivating example follows.

Example 2.1. Note that (2.1) is equivalent to:

dðTx; TyÞ 6 ð1� KÞdðx; yÞ þM; 8x; y 2 X; for some M P 0 ð2:2Þ

Eq. (2.1) is relevant, for instance, in the following important physical problem. Consider the linear time-invariant nth orderdynamic system:

_xðtÞ ¼ AxðtÞ þ gxðtÞ ð2:3Þ

with A 2 Rn�n being a stability matrix whose fundamental matrix satisfies keAtk 6 K0e�a0t ; 8t P 0 for some positive real con-stants K0 (being norm-dependent) and a0 and g : [0,1) � X ? Rn being an unknown uniformly bounded perturbation (whichcan be of parametrical nature or even eventually include unmodeled dynamics) of bounded essential supremum satisfying:

ess sup1>tP0

kgxðtÞk 6 M0 <1; 8x 2 X:

The unique solutions of (2.3) for x(0) = x0 and y(0) = y0 are:

xðtÞ ¼ eAtx0 þZ t

0eAðt�sÞgxðsÞds ð2:4aÞ

yðtÞ ¼ eAty0 þZ t

0eAðt�sÞgyðsÞds ð2:4bÞ

Direct calculation with (2.4) for the norm-induced distance d(x,y) :¼ kx � yk; "x,y 2 X yields:

dðxðtÞ; yðtÞÞ ¼ kxðtÞ � yðtÞk 6 K0e�a0tkx0 � y0k þK0

a0sup

06s<1kgxðsÞ � gyðsÞk 6 ð1� KÞdðx0; y0Þ þM; ð2:5Þ

"t 2 [h0,1) for any given h0 2 ½0; �h0� and �h0 2 R0þ with K :¼ 1� K0e�a0h0 so that K 2 ½1� K0e�a0�h0 ;1� and1 > M P 2K0M0

a0. Now,

let X � Rn the state space of (2.1), generated by (2.4) for x0 2 X provided that (X,d) is a complete metric space. In accordancewith the previous discussion related to the inequality (2.1), we have:

(a) If K 2 (0,1] and M = 0 then (2.5) is a contraction for t P h0

Define the state transformation Thx(k h) = x[(k + 1)h] on X which generates the sequence of states fxðkhÞg10 being in X ifx0 2 X with h being any real constant which satisfies h P h0. Then, the self-map Th : X ? X satisfies (2.1). Note that thesystem (2.3) is always globally Lyapunov stable for any bounded initial conditions in view of (2.5). If the perturbationis identically zero then the origin is globally asymptotically Lyapunov stable since A is a stability matrix. This propertyalso follows from (2.5) since the self-map Th on X is a contraction which has zero as its unique fixed and equilibriumpoint so that x(k h + s) = eAsx(k h) ? 0 as k ?1; "s 2 [0,h); "h P h0. Thus, x(t) ? 0 as t ?1.

(b) However, in the presence of the perturbation, the origin is not globally asymptotically stable (although the system isglobally stable) and it exhibits ultimate boundedness since for sufficiently large distances dðxðkhÞ; yðkhÞÞP M

K (respec-tively, dðxðkhÞ; yðkhÞÞ > M

K Þ, the self-map is non-expansive (respectively, contractive). Then,0 6 d(Thx(k h),Thy(kh)) 6 d(x(k h),y(k h)) respectively, d(Thx(k h),Thy(k h)) < d(x(k h),y(k h))). But such properties are not guaranteed ifdðxðkhÞ; yðkhÞÞ < M

K which can lead to Th : X ? X being expansive. h

Remark 2.2. Example 2.1 points out the fact that there are many real-world problems whose solutions can be modelledthrough self-maps which do not possess a unique contractive, expansive or non-expansive character but a mixed typedepending on distances between pairs of points. In Example 2.1, this last situation is due to the presence of unknown per-turbations of known prescribed upper-bound. Note that in Example 2.1, the self-map from X to X defining the solution isguaranteed to be point-wise contractive or potentially expansive for each given pair of elements in X accordingly to thedistance between them. Note that the global asymptotic Lyapunov stability relies on a contractive mapping and a zero

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667 10659

equilibrium point which is a global attractor and a fixed point of a certain mapping from initial conditions to subsequentpoints of the state-trajectory solution. However, this is not a requirement for global Lyapunov stability where the abovemapping may be relaxed to be non-contractive or to possess the so-called ultimate boundedness property, which alsoimplies global stability with eventual local instability around the equilibrium, related to boundedness of the state trajectorysolution for arbitrary bounded initial conditions. In this case, the above mapping can be locally expansive and then contrac-tive for trajectory-solution points being sufficiently far away from the equilibrium point.

3. Main results

This section is devoted to formalize the general context of the described problem in the light of Fixed Point Theory. A firstmain result follows:

Theorem 3.1. Assume that K 2 (0,1) and consider any set X0 � X with diam(X0) 6 R. Then, the following properties hold:

(i) Assume any real constant R P M/K. Then, the restricted map TjX0 of T from X to X0 is locally non-expansive (i.e.d(Tx,Ty) 6 d(x,y)) for any pair of elements x,y 2 X0 such that d(x,y) P M/K and locally weakly contractive (i.e.d(Tx,Ty) < d(x,y) ) for any pair of elements x,y(–x) 2 X0 such that d(x,y) > M/K.

(ii) The distance between the iterates Tjx and Tjy is uniformly bounded; "x,y 2 X0,"j 2 Z+ and there exists a bounded subset X1 ofX fulfilling X1 � X0 – X1 such that Tjx,Tjy 2 X1; "j 2 Z+.

(iii) All iterate Tjx enters a compact subset Xa of X of diameter R; "x 2 X0; "j P j0 and some finite integer j0; i.e. the sequence Tj xis permanent; "x 2 X0. Also, there is a finite positive integer j0 = j0(e,R) such that dðTjx; TjyÞ 6 M

K þ e; 8x; y 2 X0.

Proof.

(i) It follows by direct inspection from (2.2) that for any x,y 2 X0:

dðx; yÞP M=Kð () � Kdðx; yÞ 6 �MÞ ) dðTx; TyÞ 6 dðx; yÞ � Kdðx; yÞ þM 6 dðx; yÞ �M þM 6 dðx; yÞ

(ii) Since (1 � K) 2 (0,1), direct recursive calculation with (2.2) for j 2 Z+ and any bounded X0 � X with diam(X0) 6 Ryields:

dðTx; TyÞ 6 ð1� KÞdðx; yÞ þM

dðT2x; T2yÞ 6 ð1� KÞdðTx; TyÞ þM 6 ð1� KÞ2dðx; yÞ þM½ð1� KÞ þ 1�dðT3x; T3yÞ 6 ð1� KÞdðT2x; T2yÞ þM 6 ð1� KÞ3dðx; yÞ þM½ð1� KÞ2 þ ð1� KÞ þ 1�. . .

dðTjx; TjyÞ 6 ð1� KÞjdðx; yÞ þMXj�1

i¼0

ð1� KÞj�1�i

6 ð1� KÞjdðx; yÞ þMX1i¼0

ð1� KÞi �X1i¼j

ð1� KÞi !

¼ ð1� KÞjdðx; yÞ þM1K� ð1� KÞj

K

!

¼ ð1� KÞjdðx; yÞ þMKð1� ð1� KÞjÞ ð3:1Þ

6 dðx; yÞ þMK6 RþM

K<1; 8j 2 Zþ; ð3:2Þ

"x,y 2 X since K 2 (0,1). Furthermore, lim supj!1

dðTjx; TjyÞ 6 M=K for any (x,y) 2 X. Also, for any given z 2 X;

lim supj!1

dðTjx; TjyÞ 6 lim supj!1

dðTjx; zÞ þ lim supj!1

dðz; TjyÞ 6 2M=K; 8ðx; yÞ 2 X. Therefore, the sequences {Tjx} and {Tjy} are

bounded for any (x,y) 2 X and lie in some bounded set X1 � X.(iii) From the first inequality of (3.2),

lim supj!1

dðTjx; TjyÞ 6 MKð1� ð1� KÞjÞ 6 M

K<1; 8x; y 2 X ð3:3Þ

since K 2 (0,1) so that for any given real constant e > 0, it exists a positive integer j01 = j01 (e) such that from (3.2)

dðTjx; TjyÞ 6 MKþ ð1� KÞj01 R�M

K

� �6

MKþ e; 8j P j01; 8x; y 2 X ð3:4Þ

provided that e P ð1� KÞj01 R� MK

� �, or equivalently, j01 P ln e

R�M=K � ln j1� Kj provided that MK < R 6 e

j1�Kj þ MK . Now, assume

that R > ej1�Kj þ M

K . Then, from (3.3) and the definition of limit superior, there exists a finite positive integer j02 = j02(d,R) for

10660 M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667

any arbitrary given positive real constant d such that dðTjx; TjyÞ 6 MK þ d; 8j P j002. Then, choose d ¼ e

j1�Kj for the given e. Thus,

(3.4) holds; 8j P j02 :¼ j01 þ j002. Finally, assume that 0 6 R < MK . Then, from (3.3), it exists j03 = j03(e) such that

dðTjx; TjyÞ 6 MK þ e; 8j P j03. As a result, one has that for any bounded set X0 � X with diam(X0) 6 R, it exists a finite positive

integer j0 = j0(e,R) such that dðTjx; TjyÞ 6 MK þ e; 8x; y 2 X0;8j P j0, for some finite integer j0. Thus, for each given real e > 0,

there is a compact convex subset Xa � X0of X fulfilling diamðXaÞ 6 lim supj!1

diamðTjðX0ÞÞ 6 MK where all the iterates Tjx enter;

"j P j0, for some finite integer j0. Thus, the sequence of sets Tj(X0) converge to a bounded closed set as j ?1. Furthermore, any twoiterates Tjx,Tj y are within a compact subset of Xa of a prescribed diameter M

K þ e; 8x; y 2 X0;8j P j0 for some finite integer j0. h

Remark 3.2. Note that Theorem 3.1(i) does not conclude on the contrary that the self-map T : X ? X is necessarily expansivefor some pair x,y 2 X0 � X (i.e. d(Tx,Ty) > d(x,y)) if d(x,y) < M/K but only that the upper-bound (1 � K)d(x,y) + M of d(Tx,Ty) isupper-bounded by d(x,y) if d(x,y) P M/K. Thus, d(x,y) < M/K for some pair x,y 2 X0 is a necessary (but not sufficient) condi-tion for T : X ? X to be locally expansive for that pair. If d(x,y) P M/K, then the self-map T : X ? Xis non-expansive for such apair and if d(x,y) > M/K then the self-map T : X ? X is weakly contractive for such a pair. However, if d(x,y) is finite forx,y 2 X0, then any iterate d(Tjx,Tjy) is finite for any nonnegative integer j. h

A summarized important formal conclusion from (2.1), equivalently from (2.2), which relies on Theorem 3.1(i), and whichis of usefulness in the context of ultimate boundedness of dynamic systems under parametrical and/or unmodeled dynamics(see Example 2.1), is given below:

Corollary 3.3. Consider any two points x,y 2 X0 and assume that K 2 [0,1]. Then:

dðTx; TyÞ 6 2M if K ¼ 0;dðTx; TyÞ 6 M if K ¼ 1 and dðTx; TyÞ 6 M=K if K 2 ð0;1Þ:

Proof. d(Tx,Ty) 6 2M if K = 0 and d(Tx,Ty) 6M if K = 1 follow from (1.1). Assume that K 2 (0,1). Then d(Tx,Ty) 6 (1 � K)M/K + M 6M/K from (1.1) if d(x,y) 6M/K. If d(x,y) P M/K then d(Tx,Ty) 6M/K from Theorem 3.1(i) (note that not necessarilyd(Tx,Ty) 6 d(x,y)). h

Remark 3.4. Note that X0 in Theorem 3.1 is not required to be convex. Theorem 3.1(ii) guarantees that Tjx,Tjy 2 X1 � X0

although eventually it may not belong to X0. h

Now, define the sets Xe as the subsets of X such that the restricted map TjXe : Xe ? Xe satisfies the constraints:

K1dðx; yÞ 6 dðTx; TyÞ 6 minðð1� KÞdðx; yÞ þM;K2dðx; yÞÞ; 8x; y 2 Xe # X ð3:5Þ

for some real constants K 2 [0,1) M > 0,K1 > max (1 � K,0), K2 P K1. It will be proved later on in Corollary 3.6 the existence ofsubsets of X where the self-map T is contractive, expansive or non-expansive. It would be proven that (3.5) is impossibleeverywhere in X if K1 > 1.

Theorem 3.5. Assume that K 2 [0,1). Then, there is a family of nonempty bounded subsets of X for which (3.5) holds and, thentrivially, a subfamily of nonempty bounded convex subsets of X with the same property.

Proof. The constraints (3.5) are guaranteed under two possibilities for each x,y 2 Xe(x) # X where Xe(x) :¼ {y 2 X:(3.5) holds} is a point-dependent subset of X, namely:

K1dðx; yÞ 6 dðTx; TyÞ 6 ð1� KÞdðx; yÞ þM 6 K2dðx; yÞ; 8x; y 2 XeðxÞ# X; some M > 0 ð3:6Þ

which implies:

K1dðx; yÞ 6 dðTx; TyÞ 6 K2dðx; yÞ 6 ð1� KÞdðx; yÞ þM; 8x; y 2 XeðxÞ# X ð3:7Þ

The constraint (3.6) is subject to the necessary conditions:

dðx; yÞ 2 MK þ K2 � 1

;M

K þ K1 � 1

� ; 8x; y 2 XeðxÞ ð3:8Þ

MK1

K þ K2 � 16 K1dðx; yÞ 6 dðTx; TyÞ

6 ð1� KÞdðx; yÞ þM 6 K2dðx; yÞ 6 MK1

K þ K1 � 1; 8x; y 2 XeðxÞ

) dðTx; TyÞ 2 MK1

K þ K2 � 1;

MK1

K þ K1 � 1

� ; 8x; y 2 XeðxÞ ð3:9Þ

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667 10661

Since T is a self-map on X, any pair x,y 2 Xe(x) has to satisfy simultaneously (3.8) and (3.9) so that

dðx; yÞ 2 MK þ K2 � 1

minð1;K1Þ;M

K þ K1 � 1minð1;K1Þ

� ; 8x; y 2 XeðxÞ ð3:10Þ

under the constraint (3.6). Since K2 P K1, the constraint (3.7) requires

dðx; yÞ 6 MK þ K2 � 1

; 8x; y 2 Xe

dðTx; TyÞ 6 K2dðx; yÞ 6 MK2

K þ K2 � 16

MK1

K þ K1 � 1; 8x; y 2 Xe ð3:11Þ

The last inequality of (3.11) follows directly if K1 > 1 � K since K2 P K1 > 1 � K implies that

K þ K1 � 1K1

¼ 1� 1� KK1

P 1� 1� KK2

¼ K þ K2 � 1K2

ð3:12Þ

Combining (3.11) and (3.12), one gets that (3.7) holds if

dðx; yÞ 2 0;M

K þ K2 � 1minð1;K2Þ

� ; 8x; y 2 XeðxÞ ð3:13Þ

Thus, it is clear the existence of a countable family of nonempty bounded subsets {Xe i(x)} of Xe(x); "x 2 X defined by

XeiðxÞ :¼ y 2 X : dðx; yÞ 6 MK þ K1 � 1

minð1;K1Þ �

� XeðxÞ � X;8x 2 X ð3:14Þ

since Xe(x) :¼ {y 2 X: (3.5) holds} � Xe i(x); "x 2 X. From the above developments, it turns out that there exists a convex sub-set in the family {Xe i(x)} by construction and then a subfamily of the set {Xe i} which also possess such a property. h

Theorems 3.1 and 3.5 lead to the following important conclusion:

Corollary 3.6. Assume that K 2 [0,1]. Then, the following properties hold if (3.5) holds:

(i) If max (1 � K,0) < K1 6 K2 6 1 then T : X ? X is non-expansive.(ii) If max (1 � K,0) < K1 6 K2 < 1, so that K 2 [0,1), then T : X ? X is (strictly) contractive and then it has a fixed point which is

in X if (X,d) is a complete metric space.(iii) If K1 2 [0,1) and K2 > 1 then the restriction of T to bXðxÞ; TjbXðxÞ :¼ ðT : XjbXðxÞ ! XÞ;8x 2 X, is non-expansive wherebXðxÞ :¼ fy 2 X : dðx; yÞP M

Kg � X;8x 2 X but TjbXeiðxÞ is weakly contractive for all sets Xe i(x) defined in (3.14) resulting

to be XeiðxÞ :¼ y 2 X : dðx; yÞ 6 MKþK1�1

n o; 8x 2 X. As a result, T : X ? X is neither contractive nor expansive on X. If K1 > 1

then neither (3.6) nor (3.7) are feasible for any x,y 2 X and (3.5) is not feasible either; "x,y 2 X.(iv) If K2 P K1 = 1 then T : X ? X is not contractive, and

K1dðx; yÞ 6 dðTx; TyÞ 6 ð1� KÞdðx; yÞ þM 6 max K2dðx; yÞ; K1MK þ K1 � 1

� �; 8x; y 2 X ð3:15Þ

Proof. Properties (i)–(ii) follow from Theorem 3.1. Property (iii) follows from Theorem 3.5 since:

K1 > 1) M=K P M=ðK þ K1 � 1Þ ) bXðxÞ \ XeiðxÞ ¼ ;; 8x 2 X

Then, for any x,y 2 X, if y 2 bXðxÞ then y R Xe i(x) and conversely. The constraints (3.15) follow directly from (3.5) and its nec-essary condition dðx; yÞ 6 M

KþK1�1 ; 8x; y 2 X. It is now proven by contradiction that neither (3.6) nor (3.7) are feasible for all

given pair x,y in X if K1 > 1. A necessary condition for (3.6) to hold for each x,y 2 X is that dðx; yÞ 2 MKþK2�1 ;

MKþK1�1

h i. Thus,

T : X ? X is not expansive which contradicts the constraint d(x,y) < K1d(x,y) 6 d(Tx,Ty); "x,y(–x) 2 X if K1 > 1. Also, if (3.7)holds; "x,y 2 X then dðx; yÞ 6 M

KþK2�1 which leads to the same above contradiction if K1 > 1. On the other hand, a necessary

condition for (3.5) to hold is that

K1dðx; yÞ 6 ð1� KÞdðx; yÞ þM ) dðx; yÞ 6 MK þ K1 � 1

<1; 8x; y 2 X

which contradicts K1 > 1. Thus, Property (iii) has been proven. Property (iv) follows from (3.6) and Property (iii) forK2 P K1 = 1. h

The following example discusses the contractive properties for perturbations of large sizes in a dynamic system with de-lay even if the property is not guaranteed for nonzero perturbations of small size implying that some equilibrium point is notlocally stable:

10662 M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667

Example 3.7. (Application to non-linear delayed dynamic systems): Consider the scalar time-delay dynamic system:

_xðtÞ ¼ aðtÞxðtÞ þ a1ðtÞxðt � rÞ þ gðt; xtÞxðtÞ ð3:16Þ

subject to any piecewise continuous function of initial conditions u:[ � r,0] ? R, where x(0) = u(0),a,a1: R0+ ? R(R0+ :¼ R+ [ {0}) are uniformly bounded piecewise continuous parameterizing functions and g : R0+ � R ? R is, in gen-eral, a sate-dependent disturbance function which can describe parameterization errors and unmodeled dynamics withthe strip of solution of (3.16) in the interval [t � r, t] being denoted by xt where r 2 R0+ is a constant delay. From Pi-card–Lindeloff theorem, the solution exists and it is unique on R+. Now, consider a Lyapunov functional candidateV : R0+ � xt ? R+ defined by Vðt; xtÞ :¼ pðtÞx2ðtÞ þ

R tt�r qðsÞx2ðsÞds for some continuous functions q : R0+ ? R0+ and

p : R0+ ? R+, the last one being also everywhere time-differentiable on R+. The time-derivative of the Lyapunov func-tional candidate becomes:

_Vðt; xtÞ ¼ xTðtÞQðtÞxðtÞ þ 2pðtÞgðt; xtÞxðtÞ; 8t 2 R0þ ð3:17Þ

where xðtÞ :¼ ðxðtÞ; xðt � rÞÞT , the superscript ‘‘ T’’ denotes transposition, and

QðtÞ :¼2pðtÞaðtÞ þ _pðtÞ þ qðtÞ pðtÞa1ðtÞ

pðtÞa1ðtÞ �qðt � rÞ

� ; 8t 2 R0þ ð3:18Þ

If _pðtÞ ¼ c� 2pðtÞaðtÞ � qðtÞ and ja1ðtÞj 6ffifficp

qðt�rÞpðtÞ , if a1(t) – 0, and q(t � r) arbitrary positive if a1(t) = 0 then P(t) is negative

definite for all t 2 R0+ if c 2 R+ and p : R0+ ? R+ is strictly positive and uniformly bounded. This is achieved by choosingc P ec þmint2R0þqðsÞ, for some real constant ec 2 R+, so that for any t 2 R0+:

0 < p1 6 ec

Z t

0e2R t

saðs0 Þds0ds 6 pðtÞ ¼ pð0Þe2

R t

0aðsÞds þ

Z t

0e2R t

saðs0Þds0 ðc� qðsÞÞds 6 p2 <1; ð3:19Þ

"t 2 R0+ provided that (1)R t

0 aðsÞds < 0 and (2)R t

0 eR t

saðs0 Þds0ds <1; 8t 2 R0þ. If, in addition, ja1ðtÞj 6

ffifficp

qðt�rÞpðtÞ ; 8t 2 R0þ then it

turns out that if g � 0 then the system (3.16) is globally exponentially Lyapunov stable since Q(t) is negative definite, "t 2 R0+

so that the candidate is in fact a Lyapunov functional with negative time-derivative for all time. However, if the perturbationis non zero then one gets from (3.17):

�kmaxð�QðtÞÞkxðtÞk2 � 2pðtÞjgðt; xtÞjkxðtÞk 6 _Vðt; xtÞ 6 �ðkminð�QðtÞÞkxðtÞk � 2pðtÞjgðt; xtÞjÞkxðtÞk; 8t 2 R0þ ð3:20Þ

where k.k stands for the Euclidean norm and

kmaxðQðtÞÞ ¼ �kminð�QðtÞÞ ¼�ðcþ qðt � rÞÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðcþ qðt � rÞÞ2 þ 4ðp2ðtÞa2

1ðtÞ � cqðt � rÞÞq

2ð3:21Þ

kminðQðtÞÞ ¼ �kmaxð�QðtÞÞ ¼�ðcþ qðt � rÞÞ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðcþ qðt � rÞÞ2 þ 4ðp2ðtÞa2

1ðtÞ � cqðt � rÞÞq

2ð3:22Þ

are the maximum and minimum eigenvalues of Q(t), respectively. Then, if the above three conditions hold and, furthermore,the perturbation fulfils the following worst-case type condition:

lim supt!1

jgðt; xtÞj �12

lim inft!1

kminðQðtÞÞkxðtÞkpðtÞ < 0 ð3:23Þ

which holds if

lim supt!1

jgðt; xtÞj �1p2

lim inft!1

ðkminðQðtÞÞkxðtÞkÞ < 0 ð3:24Þ

then the perturbed system is globally stable with ultimate boundedness (otherwise, a contradiction would follow from(3.23), or (3.24), and (3.20)). However, this feature does not guarantee local stability for sufficiently smallkxðtÞk < 2pðtÞjgðt;xt Þj

kminð�QðtÞÞ since the Lyapunov functional candidate can possess positive time-derivative on some time interval

[t, t + et) of nonzero measure so that it is not guaranteed to be a Lyapunov functional. In spite of this fact, global stabilityis guaranteed. The interpretation of those properties under the metric constraint (2.1) is as follows. First, assume that theperturbation g � 0. Take any real constant h 2 R+ as a sampling period used for the discretization of the problem for analysispurposes. Now, denote for k 2 Z0+ :¼ Z+ [ {0}:

xk :¼ xðkhÞ; Vxk :¼ VðxðkhÞÞ; �xk is a pair ðxk; xkrÞ

xkr strip of solution x:[kh � r,kh] ? R,Vxkr is the corresponding strip of Lyapunov functional

V : ½kh� r; kh� ! R

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667 10663

T : R ? R and TV : R ? R are self-maps defining the real sequence solution of the state-space trajectory solution and that ofthe Lyapunov functional, respectively.

It is easy to see that, since the nominal system (i.e., that defined in the absence of parametrical disturbances) is globallyexponentially stable, then

_Vðxk; xkrÞVðkh; xkÞ

6 Kk :¼ jkminðQ kÞj

kminðPkÞ6 K :¼ min

k2Z0þ

jkminðQkÞjkminðPkÞ

< 1 ð3:25Þ

If the distance is chosen as the Euclidean norm, by taking advantage that the Banach space of the state-trajectory solutionscan be considered as a metric space for the Euclidean metric, then the point-distance between two state trajectories satisfiesthe contraction constraint d(T Vx(k+1),TVy(k+1)) 6 Kd(T Vxk,TVyk); "k 2 Z0+ so that the iterates TkVx0 ? 0 which is the unique

fixed point in R of TV: R ? R. Also kx(t)k? 0 as t ?1 and kxkk? 0 as k(2Z0+) ?1 since kxðtÞk26

1kmaxðPðtÞÞVðt; xtÞ so that

V = 0 is also the unique stable equilibrium point of the dynamic system. On the other hand, note that the nominal state-tra-jectory solution at time instants t = kh is given by:

xkþ1 ¼ Tðxk; xkrÞ ¼ eR ðkþ1Þh

khaðkhþsÞdsxk þ

Z ðjþ1Þh

jheR ðjþ1Þh

jhþsaðs0 Þds0

a1ðsÞxðs� rÞds; ð3:26Þ

"k 2 Z0+. Then T : R ? R is a contractive self-mapping with a unique fixed point, since contractive maps are Lipschitz con-tinuous, which is also the unique stable equilibrium point, at x = 0 provided that:

K :¼ maxk2Z0þ

eR ðkþ1Þh

khaðkhþsÞds þ

Z ðjþ1Þh

jheR ðjþ1Þh

jhþsaðs0 Þds0 ja1ðsÞjds

!< 1 ð3:27Þ

If the perturbation is not identically zero then (3.27) becomes modified as follows:

xkþ1 ¼ Tðxk; xkrÞ ¼ eR ðkþ1Þh

khaðkhþsÞdsxk þ

Z ðjþ1Þh

jheR ðjþ1Þh

jhþsaðs0 Þds0 ða1ðsÞxðs� rÞ þ gðs; xsÞxðsÞÞds ð3:28Þ

so that

jxkþ1j 6 eR ðkþ1Þh

khaðkhþsÞds þ

Z ðjþ1Þh

jheR ðjþ1Þh

jhþsaðs0 Þds0 ðja1ðsÞj þ jgðs; xsÞjÞds

!max06s<h

jxðkhþ sÞj ð3:29Þ

Note that the real discrete sequence

Kk :¼ eR ðkþ1Þh

khaðkhþsÞds þ

Z ðjþ1Þh

jheR ðjþ1Þh

jhþsaðs0Þds0 ðja1ðsÞj þ jgðs; xsÞjÞds ð3:30Þ

may have values exceeding unity so that the state-trajectory solution may be locally non-contractive or even nonex-pansive. However, under conditions (1) to (3) in (3.19), (3.24) is fulfilled and the system is globally Lyapunov stablewith ultimate boundedness so that T : R ? R is necessarily contractive if kxðtÞk grows exceeding a certain thresholdsince TV : R ? R is contractive (otherwise (3.23) and (3.20) would lead to a contradiction implying the lost of globalstability). This is in fact the mechanism guaranteeing global non-asymptotic stability under the given class of pertur-bations, i.e. contractiveness guaranteed for perturbations of large size without guaranteeing contractiveness for nonzerosmall perturbations with some equilibrium point being potentially locally unstable. The same conclusion arises for theself-map TV : R ? R from the given stability conditions under ultimate boundedness. Note that the robust stability ofuncertain dynamic systems, potentially involving switching in-between different parameterizations, is a very importantproperty to be achieved in most of applications which is often addressed through Lyapunov stability theory often com-bined with related matrix inequalities for stability discussion, [18–22]. The approach given in this manuscript focusesthe global stability property as being associated with an ‘‘ad hoc’’ built asymptotically non-expansive mapping associ-ated with the solution which can be locally expansive during the transient. This would include the case of ultimateboundedness which implies global Lyapunov stability while some of the equilibrium points can be locally unstable.The proposed method is easily testable in practice with slight ‘‘a priori’’ knowledge on the parameters and theuncertainties.

4. Numerical examples

This section studies two numerical examples illustrating how mixed-type operators (exhibiting simultaneously contrac-tive and expansive properties) are commonly encountered in many problems related to systems theory. The first example isconcerned with a switched discrete-time system while the second one is a numerical example corresponding to the theo-retical Example 3.7.

Fig. 1. Evolution of kxkk2 through time.

10664 M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667

4.1. Switched discrete-time system

Consider the switched discrete-time system given by:

xkþ1 ¼ ArðkÞxk ð4:1Þ

where xk 2 R3 denotes the state vector at discrete-time k, r : N [ {0} ? {1,2} is a switching signal defining the active subsys-tem at each time k and

A1 ¼�0:05 0:25 0:5�2:05 1:85 2:91:15 �0:95 �1:5

0B@1CA; A2 ¼

�0:55 0:75 1:5�8:55 6:35 9:94:65 �3:45 �5:5

0B@1CA ð4:2Þ

Notice that A1 is a stable matrix (i.e. all its eigenvalues absolute values less than unity) while A2 is unstable. For this example,the switching signal is given by:

rðkÞ ¼1 if kxkk2 P R

2 if kxkk2 < R

ð4:3Þ

where kxkk2 denotes the Euclidean vector norm of xk. With this switching signal, the state trajectory tends to the origin whenkxkk2 P R while goes away from it when kxkk2 < R. This behaviour is captured by the evolution operator U (k,0), representing

the solution of system 4.1, 4.2, 4.3, which is given by Uðk;0Þ ¼Qk�1

‘¼0 Arð‘Þ and xk = U(k,0)x0. The operator U(k,0) exhibits acontractive behavior outside the sphere of radius R centered at the origin and an expansive behavior within that sphere.Thus, it is a mixed-type operator. Fig. 1 shows the evolution through time of the norm of the state vector with R = 1 andxT

0 ¼ 1 1 1½ �. It illustrates both types of behaviors of the operator describing the solution, namely, the expansive andthe contractive one.

Fig. 2 shows the difference between the solutions of the switched system 4.1, 4.2, 4.3 with initial conditionsxT

01 ¼ 1 1 1½ � and xT02 ¼ 1 2 4½ �.

Figs. 1 and 2 illustrate the results introduced in Theorem 3.1(ii) and (iii). On the one hand, Fig. 2 shows how the distancebetween iterates with different initial conditions is bounded as Theorem 3.1(ii) states. Fig. 1 shows that the evolution of thestate-vector is confined to a certain region of the state-space, since its norm never exceeds a maximum attained at the initialmoments, as Theorem 3.1(iii) claims. In this way, mixed-type operators may appear naturally in the study of dynamicsystems.

Fig. 2. Evolution of kxk2 � xk1k2 through time.

Fig. 3. Evolution of the system when g = 0.

M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667 10665

4.2. Simulations regarding Example 3.7

This second example is concerned with a numerical simulation of the system introduced in Example 3.7 above. In thisway, the following equation is considered:

_xðtÞ ¼ �2:5xðtÞ � xðt � 1Þ þ gðt; xtÞxðtÞ

Fig. 4. Evolution of the system when g is given by Eq. (4.4).

10666 M. De la Sen, A. Ibeas / Applied Mathematics and Computation 219 (2013) 10655–10667

with a nonlinear dynamics

gðt; xtÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

jxðt � 1Þj þ 0:1p ð4:4Þ

The equation without the disturbance function g is asymptotically stable and its solution converges to zero as Fig. 3 showswith initial condition x(t) = 1, �1 6 t 6 0:

Therefore, the origin is the unique fixed point of the state-trajectory solution operator TV. However, if the perturbation isnon-zero, then the solution does not converge to zero but possesses expansion and contraction phases, as Fig. 4 depicts:

Then, the system is globally stable with ultimate boundedness as Example 3.7 concludes. Thus, the results from mixed-type operators are of relevant interest for the study of the stability of dynamic and control systems, in particular, especiallyrelated to the case where the state-trajectory solution has not contractive-type properties but mixed expansive and contrac-tive ones with global stability with simultaneous ultimate boundedness and local instability of the equilibrium point.

Acknowledgments

The authors thank the Spanish Ministry of Economy and Competitiveness its support of this work through Grant DPI2012-30651. They are also grateful to the Basque Government and University of the Basque Country by their supports throughGrants GIC07143-IT-269-07 and UFI 11/07. The authors also thank the reviewers for their interesting comments whichhelped them to improve the first version of the manuscript.

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