As the phase space for equations with finite delay one usually takes the space of continuous functions. This is justifiable because the section xt of the solution becomes a continuous function fort≥σ+r, whereσis the initial time andris the delay time of the equation, even if the initial functionxσis not continuous. However, the situation is different for equations with infinite delay. The section xt contains the initial functionxσas its part for everyt≥σ. There are many candidates for the phase space of equations with infinite delay. However, we can discuss many problems
independently of the choice of the phase space. This can be done by extracting the common properties of phase spaces as the axioms of an abstract phase spaceB. We will use the following fundamental axioms, due to Hale and Kato [79].
The space B is a Banach space consisting of E-valued functions φ, ψ,ã ã ã, on (−∞,0] satisfying the following axioms.
(B) If a function x: (−∞, σ+a)→E is continuous on [σ, σ+a) and xσ ∈ B, then, fort∈[σ, σ+a),
i) xt∈ Bandxtis continuous in B,
ii) H−1|x(t)| ≤ |xt| ≤K(t−σ) sup{|x(s)|:σ≤s≤t}+M(t−σ)|xσ|, whereH >
0 is constant, K, M : [0,∞)→[0,∞) are independent of x, K is continuous, M is measurable, and locally bounded.
Now we consider several examples for the spaceBof functionsφ: (−∞,0]→E.
Letg(θ), θ≤0,be a positive, continuous function such thatg(θ)→ ∞asθ→ −∞.
The space U Cg is a set of continuous functionsφ such thatφ(θ)/g(θ) is bounded and uniformly continuous forθ≤0. Set
|φ|= sup{|φ(θ)|/g(θ) :θ∈(−∞,0]}.
Then this space satsifies the above axioms. The space Cg is the set of continuous functionsφsuch that φ(θ)/g(θ) has a limit inE as θ→ −∞. Thus,Cg is a closed subspace of U Cg and satisfies the above axioms with respect to the same norm.
The spaceLg is the set of strongly measurable functionsφsuch that|φ(θ)|/g(θ) is integrable over (−∞,0]. Set
|φ|=|φ(0)|+ Z 0
−∞
|φ(θ)|/g(θ)dθ.
Then this space satisfies the above axioms.
Next, we present several fundamental properties of B. Let BC be the set of bounded, continuous functions on (−∞,0] toE, andC00be its subset consisting of functions with compact support. Forφ∈BC put
|φ|∞= sup{|φ(θ)|:θ∈(−∞,0]}.
Every functionφ∈C00is obtained asxr=φfor somer≥0 and for some continuous functionx:R→E such that x(θ) = 0 forθ≤0. Since x0 = 0∈ B, Axiom (B) i) implies thatxt∈ Bfort≥0. As a resultC00is a subspace ofB, and
kφkB≤K(r)kφk∞, φ∈C00,
provided suppφ⊂[−r,0].For everyφ∈BC, there is a sequence{φn} in C00 such that φn(θ)→ φ(θ) uniformly for θ on every compact interval, and that kφnk∞ ≤ kφk∞. From this observation, the space BCis contained inBunder the additional axiom (C).
(C) If a uniformly bounded sequence{φn(θ)}inC00converges to a functionφ(θ) uniformly on every compact set of (−∞,0], thenφ∈ Band limn→∞|φn−φ|= 0.
In fact, BC is continuously imbedded into B. The following result is found in [107].
Lemma 1.2 If the phase spaceB satisfies the axiom (C), then BC⊂ B and there is a constantJ >0 such that|φ|B≤Jkφk∞ for allφ∈BC.
For each b∈E, define a constant function ¯b by ¯b(θ) =b forθ ∈(−∞,0]; then
|¯b|B ≤J|b|from Lemma 1.2. Define operatorsS(t) :B → B, t≥0, as follows : [S(t)φ](θ) =
φ(0) −t≤θ≤0, φ(t+θ) θ≤ −t.
Let S0(t) be the restriction of S(t) to B0 := {φ ∈ B : φ(0) = 0}. If x : R → E is continuous on [σ,∞) and xσ ∈ B, we take a function y : R → E defined by y(t) =x(t), t≥σ;y(t) =x(σ), t≤σ. From Lemma 1.2yt∈ B fort≥σ, andxt is decomposed as
xt=yt+S0(t−σ)[xσ−x(σ)] for t∈[σ,∞).
Using Lemma 1.2 and this equation, we have an inequality
|xt| ≤Jsup{|x(s)|:σ≤s≤t}+|S0(t−σ)[xσ−x(σ)]|.
The phase spaceBis called a fading memory space [107] if the axiom (C) holds and S0(t)φ→0 ast→ ∞for eachφ∈ B0. IfBis such a space, thenkS0(t)kis bounded fort≥0 by the Banach Steinhaus theorem, and
|xt| ≤Jsup{|x(s)|:σ≤s≤t}+M|xσ|,
whereM = (1 +HJ) supt≥0kS0(t)k.As a result, we have the following property.
Proposition 1.1 Assume that B is a fading memory space. If x : R → E is bounded, and continuous on[σ,∞)andxσ∈ B, thenxtis bounded in Bfort≥σ.
In addition, ifkS0(t)k →0 ast→ ∞, thenBis called a uniform fading memory space. It is shown in [107, p.190], that the phase spaceBis a uniform fading memory space if and only if the axiom (C) holds andK(t) is bounded and limt→∞M(t) = 0 in the axiom (B).
For the spaceU Cg, we have that
kS0(t)k= sup{g(s)/g(s−t) :s≤0},
and this space is a uniform fading memory space if and only if it is a fading memory space, cf. [107, p.191].
LetAbe the infinitesimal generator of aC0-semigroup onEsuch thatkT(t)k ≤ M ewt, t ≥0. Suppose that F(t, φ) is an E-valued continuous function defined for t≥σ, φ∈ B, and that there exists a locally integrable functionN(t) such that
|F(t, φ)−F(t, ψ)| ≤N(t)|φ−ψ|, t≥σ, φ, ψ∈ B.
Every continuous solutionu: [σ−r, σ+a)→E of the equation u(t) =T(t−σ)u(σ) +
Z t σ
T(t−s)F(s, us)ds σ≤t < σ+a, (1.22) will be called a mild solution of the functional differential equation
u0(t) =Au(t) +F(t, ut) (1.23) on the interval [σ, σ+a).
As in the equations with finite delay, the mild solution exists uniquely forφ∈ B, and the norm|ut(φ)|is estimated in the similar manner in terms of the functions K(r), M(r) appearing in the axiom (B). We refer the reader to [206], [108] for more details on the results of this section. The compact property of the orbit in B of a bounded solution follows from the following lemmas (for the proofs see [108]).
Lemma 1.3 Let S be a compact subset of a fading memory spaceB. Let W(S)be a set of functions x:R→E having the following properties :
i) x0∈S.
ii) The family of the restrictions ofxto[0,∞) is equicontinuous.
iii) The set{x(t) :t≤0, x∈W(S)}is relatively compact in E.
Then the setV(S) :={xt:t≥0, x∈W(S)} is relatively compact in B.
Lemma 1.4 In Eq.(1.22) letB be a fading memory space, (T(t))t≥0 be a compact C0-semigroup, and F(t, φ) be such that for everyB >0
sup
t≥0,|φ|≤B
{|F(t, φ)|}<+∞.
Then, for every solutionu(t)of Eq.(1.22) bounded on[0,+∞), the orbit{ut:t≥0}
is relatively compact inB.