Research Article | Open Access
Uniform Exponential Stability of Discrete Evolution Families on Space of -Periodic Sequences
We prove that the discrete system is uniformly exponentially stable if and only if the unique solution of the Cauchy problem , , is bounded for any real number and any -periodic sequence with . Here, is a sequence of bounded linear operators on Banach space .
The investigation of difference equations or leads to the idea of discrete evolution family. The main interest is the asymptotic behavior of the solutions and different types of stabilities in the study of such systems. There are a number of spectral criteria for the characterizations of stability of such systems.
New difficulties appear in the study of nonautonomous systems, especially because the part of the solution generated by the forced term , that is, , is not a convolution in the classical sense. These difficulties may be passed by using the so-called evolution semigroups. The evolution semigroups were exhaustively studied in . Clark et al.  developed this efficient method to the study of continuous case. So far, there are few related results regarding the investigation of discrete systems. Recently, the discrete versions of [3, 4] were obtained in [5, 6], respectively.
Buşe et al.  considered the uniform exponential stability of discrete nonautonomous systems on the space of sequences denoted by . The objective of this paper is to extend results obtained in  to the space of -periodic sequences denoted by . The results reported in this paper develop the theory of discrete evolution families on a space of bounded sequences. Similar results of this type for the continuous case may be found in the paper by Buşe and Jitianu  and the references cited therein.
2. Notation and Preliminaries
Let be a real or complex Banach space and let be the Banach algebra of all linear and bounded operators acting on .
We denote by the norms of operators and vectors. Denote by the set of real numbers and by the set of all nonnegative integers.
Let be the space of -valued bounded sequences with the supremum norm, and let be the space of -periodic (with ) sequences with . Then, is a closed subspace of .
Throughout this paper, , denotes the spectrum of , and denotes the spectral radius of . It is well known that . The resolvent set of is defined as , that is, the set of all for which is an invertible operator in .
We give some results in the framework of general Banach space and spaces of sequences as defined above.
Recall that is power bounded if there exists a positive constant such that for all .
We need the following auxiliary lemma.
Lemma 1 (Buşe et al. ). Assume and let Then, .
3. Uniform Exponential Stability of Discrete Evolution Family on Space
The family of bounded linear operators is called -periodic discrete evolution family, for a fixed integer , if it satisfies the following properties:(i), for all ;(ii), for all , ;(iii), for all , ;(iv)the map : is continuous, for all .
It is well known that any -periodic discrete evolution family is exponentially bounded; that is, there exist and an such that When a family is exponentially bounded, its growth bound is the infimum of all for which there exists an such that inequality (2) is fulfilled. It is known that As a matter of fact, A family is termed uniformly exponentially stable if is negative or, equivalently, there exist an and such that , for all . Therefore, we have the following lemma.
Lemma 2. The discrete evolution family is uniformly exponentially stable if and only if .
The map is also called the Poincare map of the evolution family .
Consider the following discrete Cauchy problem: where the sequence is -periodic; that is, for all and a fixed .
Let Then, the family is a discrete -periodic evolution family. Over finite dimensional spaces, the uniform exponential stability of the Cauchy problem in discrete and continuous autonomous cases has been studied in [9, 10].
Let us divide by , that is, for some , where . We consider the following sets which will be useful along this work: If , then define It is clear that With the help of partition (8), we construct the space which consists of all the sequences of the form That is, Obviously, is the subspace of .
Our result is stated as follows.
Theorem 3. Let be a discrete evolution family on . If the sequence is bounded for each real number and each -periodic sequence , then is uniformly exponentially stable.
Proof. Let . By virtue of (9), By virtue of the definition of , we have where Write in the form where . Furthermore, Therefore, Since is bounded, we conclude that is bounded; that is, Using Lemma 1, we deduce that . Hence, by Lemma 2, is uniformly exponentially stable. This completes the proof.
Conflict of Interests
The authors declare that they have no competing interests.
The authors express their sincere gratitude to the editor and referees for the careful reading of the original paper and useful comments that helped to improve the presentation of the results and accentuate important details. This research is supported by the AMEP of Linyi University, China.
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