On Perturbations Of C₀ Semigroups And Application

A C₀ semigroup {T(t):t≥0} on a Banach space X is called eventually normcontinuous, if there exists t₀≥0 such that the function t→T(t) is norm continousfrom (t₀,∞) into L(X), especially, if t₀=0, this semigroup is called immediately normcontinuous. As is shown in [1], an eventually norm continuous semigroup satisfiesthe spectrum determining growth assumption which is a very impartant propertyconcerned with the exponential stability of linear dynamical systems. So it is a veryimportant thing to study eventually norm continous semigroups. In this paper, weintroduce some fundamental conceptions and conclusions about operator semigrouptheory in Chapter 1. In Chapter 2, let A, B be the generators of strongly continuoussemigroups {S(t):t≥0} and {T(t):t≥0} on a Banach space X, respectively.We mainly consider the continuity ofΔ(t):=T(t)-S(t) in this section. In Chapter3, we mainly discuss eventual differentiability of solutions of delay equations by thedefinition of eventual differentiability.In the following, we state the main results of this thesis concretely.In the section 2 of Chapter 2, they are mostly on bounded perturbations, i.e, asB=A+K(K∈B(X)), whetherΔ(t):=T(t)-S(t) is continuous or not.The main results are the following.Theorem 2.2.1. IfΔ(t) is compact for t＞t₀(t₀≥0), thenΔ(t) is right normcontinuous for t＞t₀.Corollary 2.2.2. If (S(t))_t≥0 is norm continuous for t＞t₀(t₀≥0) andΔ(t) iscompact for t＞t₀, thenΔ(t) is norm continuous for t＞t₀.In the section 3 of Chapter 2, we go on discussing the norm continuity ofΔ(t)when the perturbing operator is unbounded.We can state the main results as the following:Theorem 2.3.2. IfΔ(t) is compact for t＞t₀ and norm continuous at t=0,thenΔ(t) is right norm continuous for t＞t₀.Corollary 2.3.3. IfΔ(t) is compact for t＞t₀, norm continuous at t=0, andS(t) is norm continuous for t＞t₀, thenΔ(t) is norm continuous for t＞t₀.Theorem 2.3.7. Suppose that A and B generate strongly continuous semigroups(S(t))_t≥0 and (T(t))_t≥0 on a Hilbert space H, respectively, such that‖T(t)‖,‖S(t)‖≤Me^βt(t≥0) for some constants M,β≥0. LetΔ(t):=T(t)-S(t). ThenΔ(t) isnorm continuous for t＞t₀ if and only if for allτ＞β, (?)‖R(Ï„+iω, B)T(t₀)- R(Ï„+iω, A)S(t₀)‖=0.Theorem 2.3.9. If A and K satisfy the condition (A) and (B),(see page 11)there exists a function q(t): R⁺→R⁺ satisfying q(t)→0, as t→0 such that integral from 0 to t(‖KS(s)xds‖)≤q(t)‖x‖for all x∈D. S(t) is norm continuous for t＞t₀. Let (T(t))_t≥0 be the stronglycontinuous semigroup generated by B=:(?). IfΔ(t)=T(t)-S(t) is compactfor t＞t₀, thenΔ(t) is norm continuous for t＞t₀ and R(λ,B)T(t₀)-R(λ,A)S(t₀) iscompact forλ∈ρ(A)∩ρ(B).Corollary 2.3.10. If a and K satisfy the condition (A) and (B),(see page 11)there exists constants M＞0, p∈[1,∞) and 0≤α＜∞such that integral from 0 toα(‖KS(t)x‖^pdt)≤M‖x‖^p,and S(t) is norm continuous for t＞t₀. Then the result of Theorem 2.3.9. hold.In Chapter 3, we mainly discuss eventual differentiability of solutions of delayequations (ADDE) u′(t)=Au(t)+Φu_t (t≥0), u₀=f, (ADDE)where u_t(θ)=u(t+θ)(t≥0,-1≤θ≤0) and f∈C([-1, 0], X), and we assumethat A generates a C₀ semigroup {T(t):t≥0} on X andΦ:C([-1,0],X)→X is abounded linear operator.We have the following results.Theorem 3.2.3. A generates a immediately differentiable semigroup {T(t):t≥0} on X,Φ: C([-1,0],X)→X is a bounded linear operator, and the domainofΦ,R(Φ)(?)D(A), then the solutions of delay equations (ADDE) are eventuallydifferentiable.