Abstract
We study quasi-hyperbolicity of the delay semigroup associated with the equation , where is the history function and is the generator of a quasi-hyperbolic semigroup. We give conditions under which the associated solution semigroup of this equation generates a quasi-hyperbolic semigroup.
1. Introduction
We consider the abstract delay differential equation: where is a Banach space, , , , lies in an appropriate space, and we assume that generates a -semigroup on and that , the delay operator, is bounded and linear. One approach to the study of the abstract theory of such equations is via semigroups of operators, pioneered by Hale [1], Webb [2], and Kraskovi, among others. This involves associating a -semigroup (defined on an appropriate state space) with the above delay equation, and whose orbits correspond to the mild solutions of (DE). Analysis of qualitative and asymptotic behaviour of this semigroup then yields detailed information about the solutions of the delay equation (see [3–6] for more details).
In this paper we study the property of quasi-hyperbolicity of the semigroup associated with the delay equation (DE). Recently, Batty and Tomilov [5] have introduced and studied quasi-hyperbolicity of -semigroups, motivated by questions arising in the context of quasi-Anosov diffeomorphisms. Precisely, a -semigroup is quasi-hyperbolic if there exists (independent of ) such that for all . They show in particular that quasi-hyperbolicity of a -semigroup is closely connected to the generator satisfying certain lower bounds. Quasi-hyperbolicity of a -semigroup generalises the property of hyperbolicity of -semigroup. Hyperbolicity of the delay semigroup has been explained in detail in [3, 4], and we follow their approach here. In particular we prove that if generates a quasi-hyperbolic semigroup, and the delay operator is small in some sense, then the associated delay semigroup remains quasi-hyperbolic. This is the main result, proved in Sections 3 and 4, for Hilbert spaces and Banach spaces, respectively. Section 2 is of a preliminary nature containing known fact about delay equation and delay semigroups. Throughout this paper, we follow the notations of [3].
2. Preliminaries
We collect here some basic results about delay differential equations that will be required in the sequel and refer to [3], for details. Throughout this paper, we shall consider the equation (DE), where is fixed and(1) is a Banach space,(2) is the generator of a -semigroup,(3),(4) is a linear, bounded operator,(5) and , and is defined by , .
Let and define the operator on as follows:where is the generator of the left shift semigroup on . Thus, , and The following relations between the solutions of the abstract Cauchy problemassociated with the operator matrix on the Banach space with initial value and (DE) are well known [3, Lemma 2.2]:(i)If is a solution of (DE), then is a solution of the equation (ACP).(ii)If is a solution of (ACP), then for all and is a solution of (DE).
Further (DE) is well posed if and only if (ACP) is well posed if and only if generates a strongly continuous semigroup on . In the case that (DE) is well posed, the semigroup is called the delay semigroup corresponding to (DE). Then the solutions (both mild and classical) of (DE) are determined by the semigroup . We shall be exploring the properties of the delay semigroup , associated with (DE).
We note here that since in this paper we shall be working with , fixed, we write (DE) instead of , (ACP) instead of and so on, the dependence on being implicit.
3. Quasi Hyperbolic Semigroups and Delay Equations
The delay operator is said to be admissible (see [3]) if the operator is the generator of a strongly continuous semigroup on for each generator on and the function is a bounded analytic function on the half plane for all .
A -semigroup is said to be quasi-hyperbolic [5, Definition 3.1] if there exists (independent of ) such that For -semigroups defined on Hilbert spaces, the following complete characterisation of such a semigroup is available in terms of the generator.
Theorem 1 (see [5, Corollary 3.10]). Let be the generator of a -semigroup on a Hilbert space H. Then is quasi-hyperbolic if and only if for some .
The following result gives conditions under which the delay semigroup is quasi-hyperbolic.
Theorem 2. Let be a Hilbert space and consider the delay equation (DE) with . Assume that is admissible, and the semigroup generates a quasi-hyperbolic -semigroup. If the strict inequalityholds, then generates a quasi-hyperbolic -semigroup.
Proof. Let generate a quasi-hyperbolic semigroup. Therefore, there exists such thatAs is admissible, generates a -semigroup on . From (8), for every , we havewhere is a constant strictly less than . Moreover, the semigroup generated by on the Hilbert space is hyperbolic, hence quasi hyperbolic. Therefore, it follows that for all . Now for , , and above , we have, on using (9) and (10), Thus, for all , and , It follows from Theorem 1 that generates a quasi-hyperbolic -semigroup.
Example 3. Let be a function of bounded variation on such that , for , and be given by the Riemann-Stieltjes integralfor all . For satisfies the conditions of Theorem 2. It has been shown in [3] that extends to a bounded admissible delay operator from to .
Since generates a nilpotent semigroup, is empty. Moreover, for So if we choose , then such a would satisfy Hence, by Theorem 2, generates quasi-hyperbolic -semigroup.
The following example adapted from [5, Example 3.14] illustrates Theorem 2.
Example 4. Define , asThen is a strictly positive, continuous function, satisfying for some . Then Let weighted right shift -semigroup on be given by Then is a -semigroup on with generator say, By [5, Example 3.14], generates a quasi-hyperbolic -semigroup, which is not hyperbolic. Let be as in Example 3. Now, by our Theorem 2 it follows that the delay semigroup , associated with (DE) is quasi-hyperbolic.
Remark 5. It follows from proof of Theorem 2 that lower bounds for imply lower bounds for the delay semigroup generator , even on general Banach spaces, provided condition (8) holds. Precisely if , and (8) holds, then
4. Banach Space Case
In this section we obtain conditions for quasi-hyperbolicity of the delay semigroup associated with (DE) when the underlying space is a general Banach space. Consider (DE) with fixed. Let be a densely defined, closed linear operator on a Banach space , and denote the space of -valued Schwartz function. For and , we write , . Let and be the Fourier transform on . Note that is dense in for every . Further, since generates a hyperbolic semigroup [7, Theorem 2.7], it follows from [4] that is an Fourier multiplier from into itself; that is, for the operator , given byextends to a bounded linear operator on . We recall from [5] the definition of a lower Fourier multiplier.
Definition 6. The linear operator is said to be a lower -Fourier multiplier if there exists such that for every .
Note that Batty and Tomilov [5] have given the following characterisation for quasi-hyperbolic in terms of lower Fourier multipliers.
Theorem 7 (see [5, Theorem 3.9]). Let be a -semigroup on a Banach space X with generator . Then is quasi-hyperbolic if and only if is a lower -Fourier multiplier for all/some p .
We are now able to deduce quasi-hyperbolicity of the delay semigroup on under the assumption of quasi-hyperbolicity of the semigroup generated by , under certain conditions.
Theorem 8. Let X be a Banach space and consider the delay equation (DE), with fixed. Assume that the delay and the operator satisfy the following conditions: (1) is a generator of a -semigroup on whenever generates a -semigroup on . (2) is a Fourier multiplier for some/all from to ; that is, the map , given by extends to a bounded linear operator from to , (3). (4)the semigroup generated by is quasi-hyperbolic. Then generates a quasi-hyperbolic -semigroup.
Proof. Let generate a quasi-hyperbolic semigroup. Then, by Theorem 7, for , there exists such that, for ,We show that is a lower -Fourier multiplier. Observe that since maps , maps . Recall that is the generator of the nilpotent left shift semigroup in .
Since is an Fourier multiplier, therefore, there exists such that, for all ,In particular, for and , letting , it follows from (28) that for all , Now, for , we have, on using (27) and (29), that where we have used the fact that is the generator of a hyperbolic semigroup and therefore also generator of quasi hyperbolic semigroup so that is a lower Fourier multiplier. Thus, It follows similarly thatfor every . It now follows from Theorem 7 that generates a quasi-hyperbolic -semigroup.
We next turn to multipliers. Following Batty and Tomilov [5], we set and say that the linear operator , defined by is a lower -Fourier multiplier if there exists such that for every . Recall from [5] that is an Fourier multiplier for . The following theorem relates the approximate spectrum of an operator from the delay semigroup to that of the semigroup generated by .
Theorem 9. Let be a Banach space and consider the delay equation (DE). Assume that satisfies the following conditions: (1) is a generator of a -semigroup on whenever generates a -semigroup on . (2) is a Fourier multiplier for some/all from to , that is, the map given by extends to a bounded linear operator from to , (3). Let be -semigroup generated by the operator and be the delay semigroup generated by on , fixed. Then implies .
Proof. Suppose . Then by [5, Theorem 3.12], is a lower Fourier multiplier for , and there exists such thatfor every . We show that is lower -Fourier multiplier. Note that . Since is an Fourier multiplier, therefore, there exists and for all , Choosing , , in the above inequality, we getNow, for , we have, for , , on using (37) and (39), that where we have used the fact that is a lower Fourier multiplier. Thus,for some and . It follows from [5, Theorem 3.12] that .
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The first author would like to thank CSIR for JRF Fellowship while the second author acknowledges with gratitude the support from the R & D grant of University of Delhi.