Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 4375069, 10 pages
http://dx.doi.org/10.1155/2016/4375069

Research Article

Approaching the Discrete Dynamical Systems by means of Skew-Evolution Semiflows

Codruţa Stoica

Department of Mathematics and Computer Science, Aurel Vlaicu University of Arad, 2 Elena Drăgoi Street, 310330 Arad, Romania

Received 10 April 2016; Accepted 9 June 2016

Academic Editor: Allan C. Peterson

Copyright © 2016 Codruţa Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The aim of this paper is to highlight current developments and new trends in the stability theory. Due to the outstanding role played in the study of stable, instable, and, respectively, central manifolds, the properties of exponential dichotomy and trichotomy for evolution equations represent two domains of the stability theory with an impressive development. Hence, we intend to construct a framework for an asymptotic approach of these properties for discrete dynamical systems using the associated skew-evolution semiflows. To this aim, we give definitions and characterizations for the properties of exponential stability and instability, and we extend these techniques to obtain a unified study of the properties of exponential dichotomy and trichotomy. The results are underlined by several examples.

1. Introduction

The phenomena of the real world, in domains as economics, biology, or environmental sciences, do not take place continuously, but at certain moments in time. Therefore, a discrete-time approach is required. By means of skew-evolution semiflows, we intend to construct a framework that deepens the analysis of discrete dynamical systems.

Playing an outstanding role in the study of stable and instable manifolds and in approaching several types of differential equations and difference equations, the exponential dichotomy for evolution equations is one of the domains of the stability theory with an impressive development. The dichotomy is a conditional stability, due to the fact that the asymptotic properties of the solutions of a given evolution equation depend on the location of the initial condition in a certain subspace of the phase space. Over the last decades, the classic techniques used to characterize asymptotic properties as stability and instability were generalized towards a natural generalization of the classic concept of dichotomy, the notion of trichotomy. The main idea in the study of trichotomy is to obtain, at any moment, a decomposition of the state space in three subspaces: a stable subspace, an instable one, and a third one called the central manifold. We intend to give several conditions in order to describe the behavior related to the third subspace.

A relevant step in the study of evolution equations is due to Henry, who, in [1], studied the property of dichotomy in the discrete setting, in the spirit of the classic theory initiated by Perron in [2].

A special interest is dedicated to the study of dynamic linear systems by means of associated difference equations, as emphasized by Chow and Leiva in [3] and Latushkin and Schnaubelt in [4].

In [5], the uniform exponential dichotomy of discrete-time linear systems given by difference equations is presented, and the results are applied at the study of dichotomy of evolution families generated by evolution equations. In [6], a characterization of exponential dichotomy for evolution families associated with linear difference systems in terms of admissibility is given.

In [7], characterizations for the uniform exponential stability of variational difference equations are obtained and, in [8], the uniform exponential dichotomy of semigroups of linear operators in terms of the solvability of discrete-time equations over is characterized. In [9], new characterizations for the exponential dichotomy of evolution families in terms of solvability of associate difference and integral equations are deduced. In [10, 11], some dichotomous behaviors for variational difference equations are emphasized, such as a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories or characterizations in terms of the admissibility of pairs of sequence spaces over with respect to an associated control system.

The notion of trichotomy was introduced in 1976 by Sacker and Sell and studied for the case of linear differential equations in the finite dimensional setting in [12]. For the first time, a sufficient condition for the existence of the trichotomy, in fact a continuous invariant decomposition of the state space into three subspaces, was given. In the same study, the case of skew-product semiflows was as well approached.

A stronger notion, but still in the finite dimensional case, was introduced by Elaydi and Hajek in [13], the exponential trichotomy for linear and nonlinear differential systems, by means of Lyapunov functions. They prove the fact that the property of exponential trichotomy of a differential system implies the Sacker-Sell type trichotomy. Meanwhile, the notion of invariance to perturbations of the property of trichotomy is given. Thus, in the case of a nonlinear perturbation of a linear exponential trichotomic system, the obtained system preserves the same qualitative behavior as the nonperturbed one.

In [14], a relation between Lyapunov function and exponential trichotomy for the linear equation on time scales is given and, as application, the roughness of exponential trichotomy on time scales is proved. Several asymptotic properties for difference equations were studied in [15–17] and, recently, in [18, 19]. Other asymptotic properties for discrete-time dynamical systems were considered in [20, 21]. A new concept of  -trichotomy for linear difference systems is given in [22], as an extension of the -dichotomy and of the exponential trichotomy in   spaces.

The notion of skew-evolution semiflow considered in this paper and introduced by us in [23] generalizes the concepts of semigroups, evolution operators, and skew-product semiflows and seems to be more appropriate for the study of the asymptotic behavior of the solutions of evolution equations in the nonuniform case, as they depend on three variables. The applicability of the notion has been studied in [24–28].

The case of stability for skew-evolution semiflows is emphasized in [29], and various concepts for trichotomy are studied in [30]. Some asymptotic properties, as stability, instability, and trichotomy for difference equations in a uniform as well in a nonuniform setting, were studied by us in [31–33].

The following sections outline the structure of this paper. In Section 2, the definitions for evolution semiflows, evolution cocycles, and skew-evolution semiflows are given, featured by examples. In Section 3, we present definitions and characterizations for the properties of exponential growth and decay, respectively, for the exponential stability and instability. The main results are stated in Sections 4 and 5, where we give definitions and characterizations for these asymptotic properties in discrete time for skew-evolution semiflows. Finally, some conclusions are emphasized in Section 6. The list of references allows us to build the overall context in which the discussed problem is placed.

2. Preparatory Notions

Let us consider a metric space, a real or complex Banach space, and the family of linear -valued bounded operators defined on . The norm of vectors and operators is . In what follows, we will denote , , and . By we denoted the identity operator on .

Definition 1. The mapping defined by the relation

where has the properties

,  ,,  ,  

and satisfies

,  ,,  , ,

is called skew-evolution semiflow on .

Remark 2. is called evolution semiflow and is an evolution cocycle.

The approach of asymptotic properties in discrete time is of an obvious importance because the results obtained in this setting can easily be extended in continuous time.

Example 3. Let be a Banach space and let be the set of -valued sequences . The mapping

is an evolution semiflow on . We consider the linear system in discrete time:

where . If we denote

where , then every solution of system (3) satisfies the relation

The pair is a skew-evolution semiflow, associated with system (3), where is an evolution cocycle over the evolution semiflow , given by

where denotes the integer part of the term of rank .

Example 4. To every skew-evolution semiflow , one can associate the mapping given by

such that .

3. Preliminary Results

This section aims to emphasize some asymptotic behaviors, as exponential growth and decay and exponential stability and instability, as a foundation for the main results. We give the definitions of these properties in continuous time and we underline the characterizations in discrete time, as results that play the role of equivalent definitions (see [33]).

Definition 5. A skew-evolution semiflow has exponential growth if there exist mappings , , such that

for all and all .

Definition 6. A skew-evolution semiflow is said to be exponentially stable if there exist a constant and a mapping such that

for all and all .

Proposition 7. A skew-evolution semiflow with exponential growth is exponentially stable if and only if there exist a constant and a sequence of real numbers with the property , , such that

for all and all .

Proof. We have the following.
Necessity. It is obtained immediately if we consider in relation (9) and , and if we define

where the existence of and of is given by Definition 6.
Sufficiency. As a first step, if , we denote and . The following relations hold:

We obtain

for all , where functions and are given by Definition 5.
As a second step, for , we have

for all .
Hence, is exponentially stable.

Definition 8. A skew-evolution semiflow has exponential decay if there exist mappings , , such that

for all and all .

Definition 9. A skew-evolution semiflow is said to be exponentially instable if there exist a mapping and a constant such that

for all and all .

Proposition 10. A skew-evolution semiflow with exponential decay is exponentially instable if and only if there exist a constant and a sequence of real numbers with the property , , such that

for all and all .

Proof. We have the following.
Necessity. We take in relation (16) , , and and we define

where the existence of function and of constant is given by Definition 9.
Sufficiency. First step, let us take and we denote and, respectively, . We obtain

It follows that

for all , where the existence of function and of constant is assured by Definition 8.
As a second step, if we consider , we obtain

for all .
Hence,

for all , where we have denoted

which proves the exponential instability of .

4. Nonuniform Discrete Dichotomic Behaviors

Definition 11. A projector on is called invariant relative to a skew-evolution semiflow if the following relation

holds for all and all .

Definition 12. Two projectors and are said to be compatible with a skew-evolution semiflow if

projectors and are invariant on ;for all , the projections and verify the relations

Definition 13. A skew-evolution semiflow is called exponentially dichotomic if there exist functions , , constants , , and two projectors and compatible with such that

for all and all .

Example 14. We denote by the set of all continuous functions , endowed with the topology of uniform convergence on compact subsets of , metrizable relative to the metric

If , then for all we denote , . Let be the closure in of the set , where is a nondecreasing function with the property . Then, is a metric space and the mapping

is an evolution semiflow on .
Let be endowed with the norm

The mapping , given by

is an evolution cocycle over the evolution semiflow . We consider the projectors

As

we obtain that

Similarly, as

it follows that

The skew-evolution semiflow is exponentially dichotomic with characteristics

In what follows, let us denote , where and

In discrete time, we will describe the property of exponential dichotomy as given in the next proposition.

Proposition 15. A skew-evolution semiflow is exponentially dichotomic if and only if there exist two projectors , compatible with , constants , and a sequence of real positive numbers such that

 for all and all .

Proof. We have the following.
Necessity. If we consider for in relation (26) of Definition 13   and and if we define

relation is obtained.
Statement results from Definition 13 for if we consider in relation (27) and and

Sufficiency. It is obtained by means of Proposition 7 for and, respectively, of Proposition 10 for .
Hence, is exponentially dichotomic.

Theorem 16. A skew-evolution semiflow is exponentially dichotomic if and only if there exist two projectors , compatible with such that

there exist a constant and a sequence of real positive numbers such that

there exist a constant and a sequence of real positive numbers such that

 for all and all .

Proof. We have the following.
Necessity. According to Proposition 7, there exist a constant and a sequence of real numbers with the property , . We obtain for and according to Proposition 7

where we have denoted

By Proposition 10, there exist a constant and a sequence of real numbers with the property , . We obtain for

for all and all , where we have denoted

Sufficiency. Let , . We define and . We consider , where we define . We have that

for all , where and are given by Definition 5. We obtain further for

for all , where . Then, there exist and such that

On the other hand, for , we have

We obtain that is stable, where

Hence, there exists a sequence with the property , , such that

which implies the exponential stability of and ends the proof.
According to the hypothesis, if we consider we obtain

for all , which implies the exponential instability of and ends the proof.

5. Nonuniform Discrete Trichotomic Behaviors

Definition 17. Three projectors are said to be compatible with a skew-evolution semiflow if

each projector , is invariant on ;for all , the projections , , and verify the relations

Definition 18. A skew-evolution semiflow is called exponentially trichotomic if there exist the functions , , , , constants , , , and with the properties

and three projectors , , and compatible with such that

for all and all .

Example 19. We consider the evolution semiflow defined in Example 14. Let be endowed with the norm

The mapping , given by

is an evolution cocycle. Then, is a skew-evolution semiflow.
We consider the projections

The skew-evolution semiflow is exponentially trichotomic with characteristics

As in the case of dichotomy, let , where and

In discrete time, the trichotomy of a skew-evolution semiflow can be described as in the next proposition.

Proposition 20. A skew-evolution semiflow is exponentially trichotomic if and only if there exist three projectors compatible with , constants , , , and with the property , and a sequence of positive real numbers such that

 for all and all .

Proof. We have the following.
Necessity. is obtained if we consider for in relation (9) of Definition 6   and and if we define

follows according to Definition 9 for if we consider in relation (16) , , and and

is obtained for out of relation (8) of Definition 5 for , , and and if we define

follows for from relation (15) of Definition 8 for , , and and if we consider

Sufficiency. Let . We denote and and we obtained the relations

According to , we have

for all , where functions and are given as in Definition 5.
For , we have

for all . Hence, relation (57) is obtained.
Let and and, respectively, . It follows that

From , it is obtained that

for all , where and are given by Definition 8.
For , we have

for all . It follows that

for all , where we have denoted

and which implies relation (58).
By a similar reasoning, from relation (59) is obtained, and from relation (60) follows.
Hence, the skew-evolution semiflow is exponentially trichotomic.

Some characterizations in discrete time for the exponential trichotomy for skew-evolution semiflows are given in what follows.

Theorem 21. A skew-evolution semiflow is exponentially trichotomic if and only if there exist three projectors compatible with such that has exponential growth, and has exponential decay and such that the following relations hold:

there exist a constant and a sequence of positive real numbers such that

there exist a constant and a sequence of positive real numbers such that

there exist a constant and a sequence of positive real numbers such that

there exist a constant and a sequence of positive real numbers such that

 for all and all .

Proof. We have the following.
Necessity. As is exponentially trichotomic, Proposition 20 assures the existence of a projector , of a constant , and of a sequence of positive real numbers such holds. Let . If we consider

we obtain

where we have denoted

Sufficiency. As has exponential growth, there exist constants and such that relation

holds for all and all . If we denote , the inequality can be written as follows:

We consider successively the relations. By denoting , we obtain

for all and all . If we define the constant

and the sequence of nonnegative real numbers

relation is obtained.
Similarly, the other equivalences can also be proved.

In order to characterize the exponential trichotomy by means of four projectors, we give the next definition.

Definition 22. Four invariant projectors that satisfy for all the following relations

, and  ,,,,

are called compatible with the skew-evolution semiflow .

Theorem 23. A skew-evolution semiflow is exponentially trichotomic if and only if there exist four projectors compatible with , constants , and a sequence of positive real numbers such that

 for all and all .

Proof. We have the following.
Necessity. As is exponentially trichotomic, according to Proposition 20 there exist three projectors compatible with , constants , and a sequence of positive real numbers such that relations (65)–(68) hold.
We will define the projectors

such that

Projectors , , , and are compatible with . Let us define

Hence, relations (93)–(96) hold.
Sufficiency. We consider the projectors

These are compatible with .
The statements of Proposition 20 follow if we consider

Hence, is exponentially trichotomic, which ends the proof.

6. Conclusions

The paper emphasizes a way to unify the analysis of continuous and discrete asymptotic properties for skew-evolution semiflows, such as the exponential dichotomy and trichotomy. Thus, we give characterizations for the asymptotic behaviors in discrete time, in order to gain necessary instruments in punctuating the properties of the solutions of difference equations.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author gratefully acknowledges helpful suggestions and support from Professor Emeritus Mihail Megan, the Head of the Research Seminar on Mathematical Analysis and Applications in Control Theory at the West University of Timişoara, Romania.

References

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1981.

O. Perron, “Die Stabilitätsfrage bei Differentialgleichungen,” Mathematische Zeitschrift, vol. 32, no. 1, pp. 703–728, 1930.

S.-N. Chow and H. Leiva, “Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces,” Journal of Differential Equations, vol. 120, no. 2, pp. 429–477, 1995.

Y. Latushkin and R. Schnaubelt, “Evolution semigroups, translation algebras, and exponential dichotomy of cocycles,” Journal of Differential Equations, vol. 159, no. 2, pp. 321–369, 1999.

M. Megan, A. L. Sasu, and B. Sasu, “Discrete admissibility and exponential dichotomy for evolution families,” Discrete and Continuous Dynamical Systems, vol. 9, no. 2, pp. 383–397, 2003.

P. H. Anh Ngoc and T. Naito, “New characterizations of exponential dichotomy and exponential stability of linear difference equations,” Journal of Difference Equations and Applications, vol. 11, no. 10, pp. 909–918, 2005.

A. L. Sasu, “New criteria for exponential stability of variational difference equations,” Applied Mathematics Letters, vol. 19, no. 10, pp. 1090–1094, 2006.

A. L. Sasu, “Discrete methods and exponential dichotomy of semigroups,” Acta Mathematica Universitatis Comenianae, vol. 73, no. 2, pp. 197–205, 2004.

B. Sasu, “Uniform dichotomy and exponential dichotomy of evolution families on the half-line,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1465–1478, 2006.

B. Sasu, “On exponential dichotomy of variational difference equations,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 324273, 18 pages, 2009.

B. Sasu, “On dichotomous behavior of variational difference equations and applications,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 140369, 16 pages, 2009.

R. J. Sacker and G. R. Sell, “Existence of dichotomies and invariant splittings for linear differential systems III,” Journal of Differential Equations, vol. 22, no. 2, pp. 497–522, 1976.

S. Elaydi and O. Hajek, “Exponential trichotomy of differential systems,” Journal of Mathematical Analysis and Applications, vol. 129, no. 2, pp. 362–374, 1988.

J. Zhang, “Lyapunov function and exponential trichotomy on time scales,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 958381, 22 pages, 2011.

L. Barreira and C. Valls, “Stability in delay difference equations with nonuniform exponential behavior,” Journal of Differential Equations, vol. 238, no. 2, pp. 470–490, 2007.

S. Elaydi and K. Janglajew, “Dichotomy and trichotomy of difference equations,” Journal of Difference Equations and Applications, vol. 3, no. 5-6, pp. 417–448, 1998.

G. Papaschinopoulos and G. Stefanidou, “Trichotomy of a system of two difference equations,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 216–230, 2004.

C.-L. Mihiţ, M. Megan, and T. Ceauşu, “The equivalence of Datko and Lyapunov properties for ()-trichotomic linear discrete-time systems,” Discrete Dynamics in Nature and Society, vol. 2016, Article ID 3760262, 8 pages, 2016.

X.-q. Song, T. Yue, and D.-q. Li, “Nonuniform exponential trichotomy for linear discrete-time systems in Banach spaces,” Journal of Function Spaces and Applications, vol. 2013, Article ID 645250, 6 pages, 2013.

A. L. Sasu and B. Sasu, “Discrete admissibility and exponential trichotomy of dynamical systems,” Discrete and Continuous Dynamical Systems, vol. 34, no. 7, pp. 2929–2962, 2014.

B. Sasu and A. L. Sasu, “On the dichotomic behavior of discrete dynamical systems on the half-line,” Discrete and Continuous Dynamical Systems—Series A, vol. 33, no. 7, pp. 3057–3084, 2013.

S. Matucci, “The lp trichotomy for difference systems and applications,” Archivum Mathematicum, vol. 36, pp. 519–529, 2000.

M. Megan and C. Stoica, “Exponential instability of skew-evolution semiflows in Banach spaces,” Studia Universitatis Babeş-Bolyai Mathematica, vol. 53, no. 1, pp. 17–24, 2008.

A. J. G. Bento and C. M. Silva, “Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs,” Bulletin des Sciences Mathmatiques, vol. 138, no. 1, pp. 89–109, 2014.

P. Viet Hai, “Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 12, pp. 4390–4396, 2010.

P. Viet Hai, “Discrete and continuous versions of Barbashin-type theorem of linear skew-evolution semiflows,” Applicable Analysis, vol. 90, no. 12, pp. 1897–1907, 2011.

P. Viet Hai, “A generalization for theorems of Datko and Barbashin type,” Journal of Function Spaces, vol. 2015, Article ID 517348, 5 pages, 2015.

T. Yue, X.-Q. Song, and D.-Q. Li, “On weak exponential expansiveness of skew-evolution semiflows in Banach spaces,” Journal of Inequalities and Applications, vol. 2014, no. 1, article 165, pp. 1–11, 2014.

C. Stoica and M. Megan, “On uniform exponential stability for skew-evolution semiflows on Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1305–1313, 2010.

C. Stoica, “Trichotomy for dynamical systems in Banach spaces,” The Scientific World Journal, vol. 2013, Article ID 793813, 8 pages, 2013.

M. Megan and C. Stoica, “On exponential trichotomy for discrete time skew-evolution semiows in Banach spaces,” Scientific Bulletin of the Politehnica University of Timisoara, Mathematics-Physics Series, vol. 51, no. 2, pp. 25–32, 2006.

M. Megan and C. Stoica, “Trichotomy for discrete skew-evolution semiows in Banach spaces,” Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity, vol. 5, pp. 3–9, 2007.

C. Stoica and M. Megan, “Discrete asymptotic behaviors for skew-evolution semiflows on Banach spaces,” Carpathian Journal of Mathematics, vol. 24, no. 3, pp. 348–355, 2008.