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International Journal of Differential Equations
Volume 2013 (2013), Article ID 268309, 4 pages
On Uniform Exponential Stability and Exact Admissibility of Discrete Semigroups
1Shaheed Benazir Bhutto University Sheringal, Dir Upper 18000, Pakistan
2Government College University, Abdus Salam School of Mathematical Sciences (ASSMS), Lahore 54600, Pakistan
3Department of Mathematics, University of Peshawar, Peshawar 25000, Pakistan
Received 22 April 2013; Revised 19 June 2013; Accepted 20 June 2013
Academic Editor: Sotiris Ntouyas
Copyright © 2013 Aftab Khan et al. 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.
We prove that a discrete semigroup of bounded linear operators acting on a complex Banach space is uniformly exponentially stable if and only if, for each , the sequence belongs to . Similar results for periodic discrete evolution families are also stated.
The solutions of the autonomous discrete systems or lead to the idea of discrete semigroups. There are a lot of spectral criteria which characterize different types of stability (or other types of asymptotic behavior) of the solutions of above systems. For further results on asymptotic behavior of semigroups, we refer to .
New difficulties appear in the study of the 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 . Having in mind the well-known results stated in the continuous case, see for example [2, 3], we can say that this method is a very efficient one. See also [4, 5] for recent developments concerning the semigroups of evolution acting on almost periodic function spaces.
In this note, we study the asymptotic behavior of the discrete semigroups in terms of exact admissibility of the space of almost periodic sequences.
In this regard, we develop the theory of discrete evolution semigroups on a special space of bounded sequences. Results of this type in the continuous case may be found in  and the references therein. However, by contrast with the continuous case, we did not find in the existent literature papers written in the spirit of the present one referring to the discrete evolution semigroups. These results could be new and useful for people whose area of research is restricted to difference equations.
2. Definitions and Preliminary Results
Let be a complex Banach space and the Banach algebra of all linear and bounded operators acting on . The norms in and in will be denoted by . Let be the set of all nonnegative integer numbers. A sequence is said to be almost periodic if for any there exists an integer such that any discrete interval of length contains an integer , such that The integer number is called -translation number of . The set of all almost periodic sequences will be denoted by . For further details about almost periodic functions, we refer to the books [9, 10]. The set of all bounded sequences becomes a Banach space when it is endowed with the “sup” norm denoted by . Clearly, is a subset of . Let be the space of all -periodic ( is an integer number) sequences with . Denote by the set of all sequences for which there exists with and such that Let . Here the closeness is considered in the space .
For a bounded linear operator , acting on , we denote by the spectrum of and by its resolvent set. Recall that a subset of is called discrete semigroup if it satisfies the following conditions: (i), where is the identity operator on . (ii), for all . A discrete semigroup is said to be uniformly exponentially stable if there exist such that The spectral radius of denoted by is defined as It is well known that, see for example [11, page 42], As a consequence of (5), a discrete semigroup is uniformly exponentially stable if and only if .
Having in mind the continuous case, the “infinitesimal generator” of the discrete semigroup denoted by is defined by . For discrete semigroups, the Taylor formula of order one is
A discrete semigroup is said to be exact admissible, if for every the sequence belongs with .
The evolution semigroup associated with on is defined by
The following lemma shows that the associated evolution semigroup acts on .
Lemma 1. Let and be a discrete semigroup of bounded linear operators on . The sequence , given by belongs to .
Proof. First we show that for any . Since there exist with , and , such that Let and set . Clearly is -periodic sequence. It remains to show that If , then and , so If , then and ; hence Thus . Now, from linearity it follows that belongs to whenever . Let now , , and let , such that . Clearly belongs to , and that is, is in . This completes the proof.
Lemma 2. Let be a discrete semigroup of bounded linear operators on , and let be the evolution semigroup associated with on , having as generator. Let . The following two statements are equivalent:(i),(ii), for all .
Proof. : Using the Taylor formula (6), one has Then, for every , one has : For each , one has This completes the proof.
See also , for a variant of this lemma in other space.
The next result is the main ingredient in the proof of Theorem 5 that follows.
Theorem 3 (see ). Let be a discrete semigroup on , and let be a real number. If for every , then is power bounded (i.e., ) and .
As a corollary of this theorem, we state the following.
The result of this paper reads as follows.
Theorem 5. Let be a discrete semigroup on . The following four statements are equivalent:(i) is uniformly exponentially stable.(ii) The evolution semigroup associated with on is uniformly exponentially stable.(iii) The semigroup is exact admissible.(iv), for all .
Proof. : Let be uniformly exponentially stable, and let and be positive constants such that
Then for every in , one has
: Since is uniformly exponentially stable, , that is, is invertible. Then for each in , there exists such that .
On the other hand, by Lemma 2, , for every ; hence is exact admissible.
It is obvious.
Obviously, if and is a real number, then belongs to . Now, we can apply Corollary 4 to finish the proof.
The following example is a concrete application of Theorem 5.
Example 6. Let be a complex Banach space, and let be a bounded linear operator acting on . Consider the following two discrete Cauchy problems:
The solutions of (20) and (21) are (resp.) given by and . Here .
From Theorem 5, the following two statements are equivalent. For each the solution of (20) decays exponentially, or, equivalently, there exist two positive constants and such that For each the solution of (21) belongs to .
In fact, we can state a more general result concerning -periodic discrete evolution families. To establish this result, we recall that a family is said to be -periodic discrete evolution family if it satisfies the following properties.(i) and , for all with , where is the identity operator on .(ii), for all .
It is said to be uniformly exponentially stable if there exist the positive constants and such that Also, the family is said to be exact admissible, if for every the sequence belongs to .
The discrete evolution semigroup associated with the evolution family on is defined by As in Lemma 1 it can be proved that it acts on .
Theorem 7. Let be a -periodic evolution family of bounded linear operators on . The following statements are equivalent: is uniformly exponentially stable. The evolution semigroup associated with is uniformly exponentially stable. is exact admissible., for all .
The proofs of are similar to those in the semigroup case. For the proof of we use the following result from .
If for every and every , one has then the family is uniformly exponentially stable.
The authors would like to thank the anonymous referees for their comments and suggestions on preliminary versions of this paper, which have led to a substantial improvement in its readability. In particular, we have completed the last section of this note at the suggestions of referees.
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