This subsection will be devoted to some generalization of the method discussed in the previous ones for functional differential equations of the form
dx(t)
dt =Ax(t) + [Bx](t) +f(t),∀t∈R, (2.38) where the operator A is a linear operator on X and B is assumed to be an au- tonomous functional operator.
We first make precise the notion ofautonomousnessfor functional operatorsB:
Definition 2.11 Let B be an operator, everywhere defined and bounded on the function space BU C(R,X) into itself. B is said to be an autonomous functional operatorif for everyφ∈BU C(R,X)
S(τ)Bφ=BS(τ)φ,∀τ ∈R,
where (S(τ))τ∈R is the translation groupS(τ)x(ã) :=x(τ+ã) inBU C(R,X).
In connection with autonomous functional operators we will consider closed transla- tion invariant subspacesM ⊂BU C(R,X) which satisfy condition H3. Recall that ifB is an autonomous functional operator and M satisfies condition H3, then by definition,Mis left invariant under B.
Definition 2.12 Let A be the generator of a C0-semigroup and B be an au- tonomous functional operator. A function u on R is said to be a mild solution of Eq.(2.38) onRif
u(t) =e(t−s)Au(s) + Z t
s
e(t−ξ)A[(Bu)(ξ) +f(ξ)]dξ, ∀t≥s.
As we have defined the notion of mild solutions it is natural to extend the notion of mild admissibility for Eq.(2.38) in the case where the operator A generates a strongly continuous semigroup. It is interesting to note that in this case because of the arbitrary nature of an autonomous functional operatorBnothing can be said on the “well posedness” of Eq.(2.38). We refer the reader to Chapter 1 for particular cases of “finite delay” and “infinite delay” in which Eq.(2.38) is well posed. However, as shown below we can extend our approach to this case. Now we formulate the main result for this subsection.
Theorem 2.8 Let A be the infinitesimal generator of an analytic strongly contin- uous semigroup, B be an autonomous functional operator on the function space BU C(R,X) and M be a closed translation invariant subspace of AAP(X) which satisfies condition H3. Moreover, assume that
σ(DM)∩σ(A+B) =.
ThenMis mildly admissible for Eq. (2.38), i.e., for everyf ∈ Mthere is a unique mild solutionuf ∈ M of Eq.(2.38).
Proof. SinceMsatisfies condition H3, for everyf ∈ Mwe haveBf ∈ M. Thus, D((A+B)M) = {f ∈ M:Af(ã) +Bf ∈ M}
= {f ∈ M:Af(ã)∈ M}
= D(AM).
Hence
(A+B)M=AM+BM.
AsMsatisfies condition H3 it satisfies condition H1 as well. Thus, by Lemma 2.8, σ(AM)⊂σ(A)⊂σ(A)
and
kR(λ,AM)k ≤ kR(λ, A)k,∀λ∈ρ(A).
Since B is bounded DM and (A+B)M = AM+BM satisfy condition P. From [167, Lemma 2 and the remarks follows] it may be seen thatAMis the infinitesimal generator of the strongly continuous semigroup (T(t))t≥0
T(t)f(ξ) :=etAf(ξ),∀f ∈ M, ξ∈R.
Hence D((A+B)M) =D(AM) is dense everywhere in M. It may be noted that R(λ,A+B) commutes with the translation group. Since Msatisfies condition H3 we can easily show that
σ((A+B)M)⊂σ(A+B).
Applying Theorem 4.10 we get
σ(DM−(A+B)M)⊂σ(DM)−σ((A+B)M).
Hence
0∈ρ(DM−(A+B)M).
On the other hand, sinceBM is bounded onM
DM−(A+B)M = DM− AM− BM
= LM− BM we have
0∈ρ(LM− BM). (2.39)
Ifu, f∈ Msuch that (LM− BM)u=f , then LMu=BMu+f.
By definition of the operatorLM , this is equivalent to the following u(t) =e(t−s)Au(s) +
Z t s
e(t−ξ)A[(BMu)(ξ) +f(ξ)]dξ,∀t≥s,
i.e.,uis a mild solution to Eq.(2.38). Thus (2.39) shows thatMis mildly admissible for Eq.(2.38).
Remark 2.8 Sometime it is convenient to re-state Theorem 2.8 in other forms than that made above. In fact, in practice we may encounter difficulty in computing the spectrumσ(A+B). Hence, alternatively, we may considerD − A − B as a sum of two commuting operatorsD − BandAifBcommutes withA. In subsection 3.4 we again consider this situation.
We formulate here the analogs of Theorems 2.5, 2.7 for higher order functional differential equations
dnx(t)
dtn =Ax(t) + [Bx](t) +f(t). (2.40) Theorem 2.9 Let M be a closed translation invariant subspace of the function space BU C(R,X) which satisfies condition H3, A be a closed linear operator on X with nonempty resolvent set, B be an autonomous functional operator on BU C(R,X)andΛ be a closed subset of the real line. Moreover, let
(iΛ)n∩σ(A+B) =.
Then for every f ∈ M ∩Λ(X) there exists a unique (classical) solution uf to Eq.(2.39) provided one of the following conditions is satisfied:
i) EitherΛ is compact or ii) A is bounded onX.
In particular, in both casesM ∩Λ(X)is admissible for Eq.(2.39).
Proof. The proof can be done in the same way as that of Theorem 2.5. So the details are omitted.
In applications we frequently meet the operator B in the integral form. This implies the commutativeness ofBwith the convolution, i.e.,
B(u∗v) =u∗(Bv),∀u∈L1(R), v∈BU C(R,X).
Hence, as a consequence of Theorem 2.9 we have
Corollary 2.7 LetAbe the generator of aC0-semigroup andB be an autonomous functional operator onBU C(R,X)which commutes with the convolution. Moreover, let u be a bounded uniformly continuous mild solution of Eq.(2.38) with almost periodicf. Then,
spAP(u)⊂iR∩σ(A+B).
Proof. The proof can be done identically as that of Theorem 2.6. So the details are omitted.
Theorem 2.10 Let A be a linear operator on X such that there are positive con- stantsR, θ and
Σ(θ, R)⊂ρ(A)and sup
λ∈Σ(θ,R)
|λ|kR(λ, A)k<∞,
andB be an autonomous functional operator. Furthermore, letMbe a translation invariant closed subspace of BU C(R,X)which satisfies condition H3 such that
σ(D2M)∩σ(A+B) =. Then,M is admissible for the following equation:
d2x(t)
dt2 =Ax(t) + [Bx](t) +f(t).
Proof. The proof can be done as in that of Theorem 2.7. So the details are omitted.
We now study the mild admissibility of a function spaceMfor the nonlinearly perturbed equation
dx(t)
dt =Ax(t) + [Bx](t) + [F x](t),∀t∈R, (2.41) whereF is not necessarily an autonomous functional operator. Note that the notion of mild solutions to Eq.(2.40) in the case whereAis the generator of aC0-semigroup can be extended to Eq.(2.41).
Theorem 2.11 LetAbe the generator of aC0-semigroup andBbe any autonomous functional operator on BU C(R,X), andMbe a closed translation invariant sub- space of BU C(R,X) which satisfies condition H3 and is mildly admissible for Eq.(2.40). Moreover, let F be a (possibly nonlinear) operator defined onM which satisfies the Lipschitz condition
kF(u)−F(v)k ≤δku−vk,∀u, v∈ M.
Then, for sufficiently smallδ, Eq.(2.41) has a unique mild solutionuF ∈ M.
Proof. Under the assumptions of the theorem the closed linear operatorLM−BM is invertible. Thus if we define the normed spaceBto be the setD(LM− BM) with graph normkukB:=k(LM− BM)uk+kuk , for everyu∈D(LM− BM) , thenB becomes a Banach space. Moreover,LM− BM is an isomorphism fromBontoM.
Thus, by the Inverse Lipschitz Continuous Mapping Theorem, for sufficiently small δthere exists the inverse function toLM− BM−F which is Lipschitz continuous.
This proves the theorem.
Remark 2.9 In the case whereB = 0 we can weaken considerably conditions on the function space M (see the previous section). Here the translation invariance and condition H3 are needed to use the differential operatorLM− BM.