# Quasi-Hyperbolicity and Delay Semigroups

Authors: Shard Rastogi; Sachi Srivastava
Journal: Abstract and Applied Analysis
 (2016)

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## Abstract

We study quasi-hyperbolicity of the delay semigroup associated with the equationu′(t)=Bu(t)+Φut, where ut is the history function and (B,D(B)) 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.


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## Body

## 1. Introduction

We consider the abstract delay differential equation:(1)DEu′t=But+Φut,t⩾0,u0=x0,u0=f0,where X is a Banach space, ut(θ)=ut+θ,  -1≤θ≤0,  x0∈X,  f lies in an appropriate space, and we assume that (B,D(B)) generates a C0-semigroup on X and that Φ:W1,p-1,0,X→X, 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 C0-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 C0-semigroups, motivated by questions arising in the context of quasi-Anosov diffeomorphisms. Precisely, a C0-semigroup (T(t))t⩾0 isquasi-hyperbolic if there exists t>0 (independent of x) such that max⁡T2tx,x≥2Ttx for all x∈X. They show in particular that quasi-hyperbolicity of a C0-semigroup is closely connected to the generator satisfying certain lower bounds. Quasi-hyperbolicity of a C0-semigroup generalises the property of hyperbolicity of C0-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 B,DB 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 p∈[1,∞) is fixed and(1)
x0∈X is a Banach space,(2)
B:D(B)⊆X→X is the generator of a C0-semigroup,(3)
f0∈Lp([-1,0],X),(4)
Φ:W1,p([-1,0],X)→X is a linear, bounded operator,(5)
u:[-1,∞)→X and t⩾0, and ut:[-1,0]→X is defined by utσ≔ut+σ,  σ∈[-1,0].LetX=X×Lp([-1,0],X) and define the operator A on X as follows:(2)A≔BΦ0A0,where (3)DA≔xf∈DB×W1,p-1.0,X:f0=x,A0 is the generator of the left shift semigroup (T0(t))t⩾0 on Lp(-1,0,X). Thus, DA0=f∈W1,p-1,0,X:f0=0,  A0f=f′ and (4)T0tfs≔ft+sifs∈-1,0,t+s≤0,0ifs∈-1,0,t+s>0.The following relations between the solutions of the abstract Cauchy problem(5)ACPv′t=Avtt⩾0,v0=v0,associated with the operator matrix A,DA on the Banach space X with initial value v0=x0f0 and (DE) are well known [3, Lemma  2.2]:(i)
Ifu is a solution of (DE), then t↦u(t)ut is a solution of the equation (ACP).(ii)
Ift↦utvt is a solution of (ACP), then vt=ut for all t⩾0 and u is a solution of (DE).Further (DE) is well posed if and only if (ACP) is well posed if and only ifA,DA generates a strongly continuous semigroup Ttt⩾0 on X=X×Lp-1,0,X. In the case that (DE) is well posed, the semigroup Ttt⩾0 is called thedelay semigroup corresponding to (DE). Then the solutions (both mild and classical) of (DE) are determined by the semigroup (T(t))t⩾0. We shall be exploring the properties of the delay semigroup Ttt⩾0, associated with (DE).We note here that since in this paper we shall be working withp∈[1,∞), fixed, we write (DE) instead of (DE)p, (ACP) instead of ACPp and so on, the dependence on p being implicit.


## 3. Quasi Hyperbolic Semigroups and Delay Equations

The delay operatorΦ∈L(W1,p-1,0,X,X) is said to beadmissible (see [3]) if the operator A,DA is the generator of a strongly continuous semigroup on X=X×Lp-1,0,X for each generator B,DB on X and the function λ↦ΦRλ,A0 is a bounded analytic function on the half plane λ∈C:Reλ>ω for all ω∈R.AC0-semigroup Ttt⩾0 is said to bequasi-hyperbolic [5, Definition  3.1] if there exists t>0 (independent of x) such that (6)max⁡T2tx,x≥2Ttx∀x∈X.For C0-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]).
LetB be the generator of a C0-semigroup (T(t))t⩾0 on a Hilbert space H. Then (T(t))t⩾0 is quasi-hyperbolic if and only if (7)B-isx≥cx,∀x∈DBfor some c>0.The following result gives conditions under which the delay semigroup is quasi-hyperbolic.Theorem 2.
LetH be a Hilbert space and consider the delay equation (DE) with p=2. Assume that Φ is admissible, and the semigroup B,DB generates a quasi-hyperbolic C0-semigroup. If the strict inequality(8)sups∈R⁡ΦRis,A0<1holds, then A,DA generates a quasi-hyperbolic C0-semigroup.Proof.
LetB,DB generate a quasi-hyperbolic semigroup. Therefore, there exists c>0 such that(9)B-isx≥cx,x∈DB.As Φ is admissible, A,DA generates a C0-semigroup on H×L2-1,0,H. From (8), for every f∈DA0, we have(10)ΦfH≤mA0-isfL2-1,0,H,where m is a constant strictly less than 1. Moreover, the semigroup generated by (A0,D(A0)) on the Hilbert space L2([-1,0],H) is hyperbolic, hence quasi hyperbolic. Therefore, it follows that (A0-is)f≥m2f for all f∈D(A0). Now for xf∈DA,  s∈R, and above m, we have, on using (9) and (10), (11)A-isxfH×L2-1,0,H=B-isΦ0A0-isxfH×L2-1,0,H=B-isx+ΦfH+A0-isfL2-1,0,H≥B-isxH-ΦfH+A0-isfL2-1,0,H≥cxH-ΦfH+A0-isfL2-1,0,H≥cxH-mA0-isfL2-1,0,H+A0-isfL2-1,0,H≥c′xH+fL2-1,0,H.
Thus, for alls∈R, and xf∈DA, (12)A-isxfH×L2-1,0,H≥c′xfH×L2-1,0,H.It follows from Theorem 1 that A,DA generates a quasi-hyperbolic C0-semigroup.Example 3.
Letη be a function of bounded variation on -1,0 such that η-1,0≤k, for k<1, and Φ:C-1,0,X→X be given by the Riemann-Stieltjes integral(13)Φf=∫-10fdηfor all f∈C-1,0,X. For p=2,Φ satisfies the conditions of Theorem 2. It has been shown in [3] that Φ extends to a bounded admissible delay operator from W1,p-1,0,X to X.
SinceA0 generates a nilpotent semigroup, σ(A0)∩iR is empty. Moreover, for f∈L2-1,0,X(14)ΦRis,A0f=∫-10dησRis,A0fσ=∫-10dησ∫0∞e-istT0tfσdt=∫-10dησ∫σ0eisσ-rfrdr≤∫-10∫σ0frdrdησ≤∫-10∫-10frdrdησ≤f2∫-10dησ=η-1,0f2. So if we choose η-1,0≤k, then such a Φ would satisfy (15)sups∈R⁡ΦRis,A0<1.Hence, by Theorem 2, A,DA generates quasi-hyperbolic C0-semigroup.The following example adapted from [5, Example  3.14] illustrates Theorem 2.Example 4.
Definew:R→R, as(16)ws=e-s,s≤0,es,otherwise.Then w is a strictly positive, continuous function, satisfying (17)iwsws-t≤ceωt,iilimn→+∞wnwn-1=e,iiilimn→-∞wnwn-1=1e for some c,ω∈R. Then (18)limn→±∞wnwn-1=w±,w-<1<w+.Let weighted right shift C0-semigroup Ttt⩾0 on L2(R) be given by (19)Ttfs=wsws-tfs-t.
ThenTtt⩾0 is a C0-semigroup on L2(R) with generator say, B. By [5, Example  3.14], B,DB generates a quasi-hyperbolic C0-semigroup, which is not hyperbolic. Let Φ be as in Example 3. Now, by our Theorem 2 it follows that the delay semigroup Ttt⩾0, associated with (DE) (20)u′t=But+Φut,t⩾0,u0=x0,u0=f0is quasi-hyperbolic.Remark 5.
It follows from proof of Theorem2 that lower bounds for B imply lower bounds for the delay semigroup generator A, even on general Banach spaces, provided condition (8) holds. Precisely if B-isx≥cx,  c>0∀x∈DB and (8) holds, then(21)A-isxf≥cxf,c>0,xf∈DA.


## 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) withp∈1,∞ fixed. Let B be a densely defined, closed linear operator on a Banach space X, and S(R,X) denote the space of X-valued Schwartz function. For φ∈S(R) and x∈D(B), we write φ⊗xt=φ(t)x, (t∈R). Let SBR,X=lin⁡φ⊗x:x∈DB,φ∈SR and F be the Fourier transform on S(R,X). Note that SB(R,X) is dense in Lp(R,X) for every p∈[1,+∞). Further, since A0 generates a hyperbolic semigroup [7, Theorem  2.7], it follows from [4] that Ri·,A0 is an L1 Fourier multiplier from L1R,L1-1,0,X into itself; that is, for Y=Lp-1,0,X the operator M0:SR,Y→SR,Y, given by(22)M0f=F-1Ri·,A0Ff,extends to a bounded linear operator on Lp(R,Y). We recall from [5] the definition of a lower Lp Fourier multiplier.Definition 6.
The linear operator(23)MB-i·:SBR,X⟼SR,X,MB-i·fs=B-isfs,is said to be a lower Lp(R,X)-Fourier multiplier if there exists c>0 such that (24)F-1MB-i·FfLp≥cfLp,for every f∈SB(R,X).Note that(25)F-1MB-i·Ffs=Bfs-f′s.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(T(t))t⩾0 be a C0-semigroup on a Banach space X with generator B. Then (T(t))t⩾0 is quasi-hyperbolic if and only if MB-i· is a lower Lp(R,X)-Fourier multiplier for all/some p ∈1,+∞.We are now able to deduce quasi-hyperbolicity of the delay semigroup onX×Lp([-1,0],X) under the assumption of quasi-hyperbolicity of the semigroup generated by B, under certain conditions.Theorem 8.
Let X be a Banach space and consider the delay equation (DE), withp∈[1,∞) fixed. Assume that the delay Φ and the operator B satisfy the following conditions: (1)
(A,D(A)) is a generator of a C0-semigroup on Z=X×Lp-1,0,X whenever (B,D(B) generates a C0-semigroup on X.(2)
ΦRi·,A0 is a Lp Fourier multiplier for some/all p∈1,∞ from LpR,Y to LpR,X; that is, the map M~p:SR,Y→SR,X, given by (26)M~pf≔F-1ΦRi·,A0Ff,extends to a bounded linear operator fromLp(R,Y) to Lp(R,X),(3)
M~1LL1R,Y,L1R,X<1.(4)
the semigroup generated by(B,D(B)) is quasi-hyperbolic.
Then(A,D(A)) generates a quasi-hyperbolic C0-semigroup.Proof.
LetB,DB generate a quasi-hyperbolic semigroup. Then, by Theorem 7, for p=1, there exists c>0 such that, for Φ∈LW1,p-1,0,X,X,(27)F-1MB-i·Fφ⊗xL1R,X=Bφ⊗x-φ′⊗xL1R,X≥cφ⊗xL1R,X.We show that MA-i· is a lower L1R,Z-Fourier multiplier. Observe that since MB-i· maps SBR,X↦SR,X, MA-i· maps SAR,Z↦SR,Z. Recall that A0,DA0 is the generator of the nilpotent left shift semigroup T0tt⩾0 in LpR,Lp-1,0,X.
SinceΦR(i·,A0) is an L1(R,Y) Fourier multiplier, therefore, there exists 0<m1<1 such that, for all g∈L1(R,Y),(28)M~1gL1R,X≤m1gL1R,Y.In particular, for f∈D(A0) and φ∈S(R), letting g=F-1MA0-i·Fφ⊗f=φ⊗A0f-φ′⊗f∈L1R,Y, it follows from (28) that for all φ∈S(R),  f∈DA0(29)Φφ⊗fL1R,X≤m1F-1MA0-i·Fφ⊗fL1R,Y≤m1A0φ⊗f-φ′⊗fL1R,Y.Now, for Z=X×L1-1,0,X, we have, on using (27) and (29), that (30)F-1MA-i·Fφ⊗xfL1R,Z=BΦ0A0φ⊗xφ⊗f-φ′⊗xφ′⊗fL1R,Z=Bφ⊗x+Φφ⊗f-φ′⊗xA0φ⊗f-φ′⊗fL1R,X×L1R,L1-1,0,X=Bφ⊗x+Φφ⊗f-φ′⊗xL1R,X+A0φ⊗f-φ′⊗fL1R,L1-1,0,X≥Bφ⊗x-φ′⊗xL1R,X-Φφ⊗fL1R,X+A0φ⊗f-φ′⊗fL1R,L1-1,0,X≥cφ⊗xL1R,X+1-m1A0φ⊗f-φ′⊗fL1R,L1-1,0,X=cφ⊗xL1R,X+c′φ⊗fL1R,L1-1,0,X,where we have used the fact that A0 is the generator of a hyperbolic semigroup and therefore also generator of quasi hyperbolic semigroup so that A0-i· is a lower L1 Fourier multiplier. Thus,(31)F-1MA-i·Fφ⊗xfL1R,Z≥c1φ⊗xL1R,X+φ⊗fL1R,L1-1,0,X,F-1MA-i·Fφ⊗xfL1R,Z≥c1φ⊗xfL1R,Z. It follows similarly that(32)F-1MA-i·FgL1R,Z≥c1gL1R,Zfor every g∈SAR,Z=lin⁡φ⊗xf:xf∈DA,φ∈SR. It now follows from Theorem 7 that (A,D(A)) generates a quasi-hyperbolic C0-semigroup.We next turn toLp(T,X) multipliers. Following Batty and Tomilov [5], we set (33)PT,X≔∑k=-nnξkxk:ξ∈T,xk∈X,n∈N,PBT,X≔∑k=-nnξkxk:ξ∈T,xk∈DB,n∈N,and say that the linear operator MB-i·:PB(T,X)↦P(T,X), defined by (34)MB-i·∑k=-nnξkxk≔∑k=-nnξkB-ikxk,is alowerLpT,X-Fourier multiplier if there exists c>0 such that (35)∑k=-nnξkB-ikxkLpT,X≥c∑k=-nnξkxkLpT,Xfor every n∈N,xk∈D(B). Recall from [5] that R(i·,A0) is an Lp(T,Lp([-1,0],X)) Fourier multiplier for p∈[1,∞). The following theorem relates the approximate spectrum of an operator from the delay semigroup to that of the semigroup generated by (B,D(B)).Theorem 9.
LetX be a Banach space and consider the delay equation (DE). Assume that Φ satisfies the following conditions: (1)
(A,D(A)) is a generator of a C0-semigroup on Z=X×Lp([-1,0],X) whenever (B,D(B)) generates a C0-semigroup on X.(2)
ΦR(i·,A0) is a Lp Fourier multiplier for some/all p∈[1,∞) from Lp(T,Y) to Lp(T,X), that is, the map Mp:P(T,Y)→PT,X given by (36)Mp∑k=-nnξkxk≔∑k=-nnξkΦRik,A0xkextends to a bounded linear operator fromLp(T,Y) to Lp(T,X),(3)
M1LL1T,Y,L1T,X<1.
LetTtt⩾0 be C0-semigroup generated by the operator B and (T(t))t⩾0 be the delay semigroup generated by (A,DA) on X×Lp-1,0,X, p∈1,∞ fixed. Then 1∉σapT2π implies 1∉σapT2π.Proof.
Suppose1∉σap(T(2π)). Then by [5, Theorem  3.12], MB-i· is a lower Lp(T,X) Fourier multiplier for p=1, and there exists c>0 such that(37)∑k=-nnξkB-ikxkL1T,X≥c∑k=-nnξkxkL1T,Xfor every n∈N,xk∈D(B). We show that MA-i· is lower Lp(T,Z)-Fourier multiplier. Note that MA-i·:PA(T,Z)↦P(T,Z). Since ΦR(i·,A0) is an L1T,L1-1,0,X Fourier multiplier, therefore, there exists 0<m1<1 and for all n∈N, gk∈L1T,L1-1,0,X(38)M1∑k=-nnξkgkL1T,X≤m1∑k=-nnξkgkL1T,L1-1,0,X.Choosing gk=(ik-A0)fk, fk∈D(A0), in the above inequality, we get(39)Φ∑k=-nnξkfkL1T,X≤m1∑k=-nnξkA0-ikfkL1T,L1-1,0,X.Now, for Z=X×L1-1,0,X, we have, for xk∈DB, fk∈DA0, on using (37) and (39), that (40)∑k=-nnξkA-ikxkfkL1T,Z=∑k=-nnξkB-ikxk+ξkΦfkξkA0-ikfkL1T,Z=∑k=-nnξkB-ikxk+∑k=-nnξkΦfkL1T,X+∑k=-nnξkA0-ikfkL1T,L1-1,0,X≥∑k=-nnξkB-ikxkL1T,X-∑k=-nnξkΦfkL1T,X+∑k=-nnξkA0-ikfkL1T,L1-1,0,X≥c∑k=-nnξkxkL1T,X+1-m1∑k=-nnξkA0-ikfkL1T,L1-1,0,X=c∑k=-nnξkxkL1T,X+c′∑k=-nnξkfkL1T,L1-1,0,X,where we have used the fact that A0-i· is a lower L1 Fourier multiplier. Thus,(41)∑k=-nnξkA-ikxkfkL1T,Z≥c∑k=-nnξkxkfkL1T,Zfor some c>0 and xkfk∈DA. It follows from [5, Theorem  3.12] that 1∉σapT2π.



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