Asymptotic Properties of Mild Solutions of Nonautonomous Evolution Equations with Applications to Retarded Differential Equations

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation (d/dt)u(t)=Au(t)


Introduction
Consider the inhomogeneous nonautonomous evolution equation where A(t), t ∈ R, are (unbounded, linear) operators on a Banach space X and f ∈ L 1 loc (R, X).Assume that the homogeneous equation is well posed in the sense that the solutions of (1.2) define a uniquely determined evolution family (U (t, s)) t≥s of bounded operators on X.In that case solutions u : R → X of the integral equation can be interpreted as mild solutions of (1.1).It has been shown in [5,22,23] that for each f ∈ C b (R, X) (respectively, C 0 (R, X)) equation ( 1.1) has a unique mild solution u ∈ C b (R, X) (respectively, C 0 (R, X)) if and only if the evolution family (U (t, s)) t≥s has an exponential dichotomy (see also [12,24] when the operators A(t), t ∈ R, are bounded).For a detailed account of the numerous other results in this direction we refer to [7,22].Now assume that (1.2) is p-periodic, that is, A(t + p) = A(t), t ∈ R. It has been shown in [6,19,27,30] that under a certain spectral condition (nonresonance condition) on the monodromy operator U(p, 0) and the inhomogeneity f there is a p-periodic (respectively, almost periodic) mild solution of (1.1) provided that f has the same property.Moreover, u is unique subject to certain spectral assumptions.If (U (t, s)) t≥s has an exponential dichotomy, then the nonresonance condition is always satisfied and we obtain existence and uniqueness of a p-periodic (respectively, almost periodic) mild solution of (1.1) for every p-periodic (respectively, almost periodic) inhomogeneity f .We point out that in [10,31] related results are discussed for Volterra equations (see also [32]).
In the present paper, we study the modified equation where (A, D(A)) is a Hille-Yosida operator on the Banach space X, B(t), t ∈ R, is a family of operators in ᏸ(D(A), X), and f ∈ L 1 loc (R, X).We stress that, in general, X 0 = D(A) is a proper subspace of X from which the main difficulties arise.Our approach is based on the theory of extrapolation spaces associated with the operator A (see Section 2 and [26]).In particular, it is used in our definition of mild solutions of (2.3).Moreover, it allows to show that under a certain boundedness and measurability condition on the family B(t), t ∈ R, there is a (unique) evolution family (U B (t, s)) t≥s on X 0 associated with the homogeneous equation (cf. [9,33]).The evolution family (U B (t, s)) t≥s is used to derive another representation of the mild solutions of (2.3) (see Theorem 2.2).This representation is crucial for the investigations in Section 3.There we extend the above-mentioned results on the existence and uniqueness of mild solutions of (1.1) satisfying a particular asymptotic behavior to mild solutions of (2.3).We point out that in the autonomous case, that is, B(t) = B, similar results are obtained in [2].In Section 4, we discuss asymptotic properties of mild solutions of the semilinear nonautonomous equation where the nonlinearity F : R × X 0 → X satisfies a standard Lipschitz condition.In Section 5, the advantage of our approach becomes visible when we study inhomogeneous nonautonomous retarded differential equations G. Gühring and F. Räbiger 171 on a Banach space Y .A standard procedure (cf.[16,33,39]) allows to transform (5.1) into an equation of the form (2.3) on a different Banach space.Now the results of Section 3 can be applied to investigate asymptotic properties of mild solutions of (5.1).For a special periodic retarded differential equation we derive a characteristic equation which makes it easier to verify the spectral conditions in our results (see Theorem 5.9).Finally, we point out that in finite dimensions asymptotic properties of solutions of inhomogeneous retarded differential equations have been studied in [37] under the assumption that the corresponding homogeneous equation admits an exponential dichotomy (see also [17,Section 6.6.2],[29]).

Mild solutions and extrapolation spaces
We first recall some properties of Hille-Yosida operators and extrapolation spaces.For more details we refer to [26] and the references therein.Throughout the whole paper X denotes a Banach space and (A, D(A)) is a Hille-Yosida operator on X, that is, A is linear and the resolvent set where Typical examples of Hille-Yosida operators appearing in partial differential equations can be found, for example, in [11], see also Section 5. On X 0 we introduce the norm x −1 = R(λ 0 , A 0 )x , where λ 0 ∈ ρ(A) is fixed.A different choice of λ 0 ∈ ρ(A) leads to an equivalent norm.The completion X −1 of X 0 with respect to • −1 is called the extrapolation space of X 0 with respect to A. The extrapolated semigroup (T −1 (t)) t≥0 consists of the unique continuous extensions T −1 (t) of the operators T 0 (t), t ≥ 0, to X −1 .The semigroup (T −1 (t)) t≥0 is strongly continuous and its generator A −1 is the unique continuous extension of A 0 to ᏸ(X 0 , X −1 ).Moreover, X is continuously embedded in X −1 and R(λ, A −1 ) is the unique continuous extension of R(λ, A) to X −1 for λ ∈ ρ(A).Finally, A 0 and A are the parts of A −1 in X 0 and X, respectively.It follows from [26,Proposition 3.3 (2.2) We consider the inhomogeneous nonautonomous evolution equation where (A,D(A)) is a Hille-Yosida operator on the Banach space X and B(t) ∈ ᏸ(X 0 , X), t ∈ R, is a family of operators such that t → B(t)x is strongly measurable for every x ∈ X 0 and B(•) ≤ b(•) for a function b ∈ L 1 loc (R).For our purposes the notion of a mild solution of (2.3) is most useful.We point out that our definition of a mild solution coincides with that given in [8], the F -solutions in [11], the weak solutions in [13] and the integral solutions in [39].
Under our assumptions on A and B(t), t ∈ R, it follows that for f ∈ L 1 loc (R, X) and s ∈ R there is a unique mild solution (cf. [15] or Theorem 2.2).Mild solutions of the homogeneous equation

Proof. Let λ > ω and set
(2.11) Then (2.7) leads to (2.12) uniformly for t ≥ s in compact intervals.Thus if > 0 and I ⊆ R is a compact interval, then by (2.13) there is a constant M depending only on the length of I such that Since A is a Hille-Yosida operator it follows from the definition of w λ that sup{ w λ (t, s) : λ > ω + 1; t ≥ s in I } < ∞.Hence, by (2.12) and Lebesgue's dominated convergence theorem, we have Now consider the function (2.18) By (2.17) and (2.7), we obtain Hence u is a mild solution of (2.3).

3) if and only if
G. Gühring and F. Räbiger 175 In our next result we improve the convergence of the integrals considered in Theorem 2.2.By BUC r (R, X) we denote the space of bounded, uniformly continuous functions f from R into X such that f has relatively compact range.
Then, for fixed s > 0, the limit exists uniformly for t in R.
Proof.We claim that the function (2.24) has relatively compact range.In fact, fix > 0. There exists δ = s/n > 0 for an n ∈ N and a function g : R → X such that g is constant on each interval [kδ, x ∈ K}, and hence, K 0 is compact.On the other hand, by (2.2), there is a constant N independent of t ∈ R such that (2.27) Thus the range of ψ is contained in K 0 + Ns B X 0 , where B X 0 denotes the closed unit ball of X 0 .In particular, the range of ψ is totally bounded, which proves the claim.
Since ψ has relatively compact range we obtain uniformly for t ∈ R.This completes the proof.
The following lemma will be used in Section 3.
Proof.If t ≥ s, then the representation of u obtained in Theorem 2.2 leads to (2.30) Another application of Theorem 2.2 establishes the result.

Asymptotic properties of solutions of inhomogeneous equations
In this section, we discuss conditions on the evolution family (U B (t, s)) t≥s and the inhomogeneity f ∈ L 1 loc (R, X) which ensure that (2.3) has a (unique) mild solution u with a prescribed asymptotic behavior.For the rest of the paper we impose the following condition on the perturbation (B(t)) t∈R . (

Note that (B) implies exponential boundedness of the evolution family (U B (t, s)) t≥s (see (2.9)).
At first we discuss the case where (U B (t, s)) t≥s has an exponential dichotomy.We recall the following notion (see [12,18,21,23,24,25,36]). Definition 3.1.An evolution family (U (t, s)) t≥s on the Banach space Z has an exponential dichotomy with constants α > 0, L ≥ 1 if there exists a bounded, strongly continuous family of projections (P (t)) t∈R ⊆ ᏸ(Z) such that for t ≥ s (i) P (t)U(t,s) = U (t, s)P (s), (ii) the map U | (t, s) : In that case the family ( (t, s)) ( The existence of an exponential dichotomy for the evolution family (U (t, s)) t≥s on the Banach space Z allows to connect asymptotic properties of the solution u(•, f ) ∈ C(R, Z) of the integral equation We will show a corresponding result on asymptotic properties of the mild solutions of the inhomogeneous equation (2.3).We stress that in our case the evolution family (U B (t, s)) t≥s given by equation (2.7) consists of operators on the Banach space X 0 whereas the inhomogeneity f has values in the larger space X.The following lemma plays a central role.By L 1 loc,u (R, X) we denote the space of uniformly locally integrable functions from R into X equipped with the norm f 1,loc,u = sup t∈R t t−1 f (σ ) dσ .Lemma 3.4.Assume that (U B (t, s)) t≥s has an exponential dichotomy with constants α > 0, L ≥ 1, and projections where ( B (t, s)) (t,s)∈R 2 is the Green's operator function corresponding to (U B (t, s)) t≥s .
Since (U B (t, s)) t≥s has an exponential dichotomy and A is a Hille-Yosida operator we obtain for t ∈ R and λ ≥ ω + 1 where C is a constant independent of f .This proves assertion (i) and the continuity of u λ follow.
We come to our first main result.It is an analogue of Theorem 3.3 and connects asymptotic properties of mild solutions of (2.3) with the existence of an exponential dichotomy for the evolution family (U B (t, s)) t≥s .In the special case where B(t) = B is constant a similar result has been shown in [2] by completely different methods.
(i) The evolution family (U B (t, s)) t≥s has an exponential dichotomy.

3). In that case the function u = u(•, f ) is given by
where ( B (t, s)) (t,s)∈R 2 is the Green's operator function corresponding to (U B (t, s)) t≥s .
Proof.(i)⇒(ii).Assume that (U B (t, s)) t≥s has an exponential dichotomy and let f ∈ L 1 loc,u (R, X).Lemma 3.4 implies that the limit function u = u(•, f ) in (3.13) is defined and u ∈ C b (R, X).We claim that u(•, f ) is a mild solution of (2.3).In fact, if t ≥ s, then In particular, Remark 3.6.The arguments in the proof of (iii)⇒(iv) can be used to simplify parts of the proof of [22,Theorem 2.1] considerably.Now we assume that the evolution family (U B (t, s)) t≥s is p-periodic, in the sense that there exists p > 0 such that U B (t + p, s + p) = U B (t, s) for t ≥ s.From formula G. Gühring and F. Räbiger 181 (2.7) we see that (U B (t, s)) t≥s is p-periodic provided that t → B(t) is p-periodic, that is, B(t) = B(t +p).We call U B (p, 0) the monodromy operator of the evolution family (U B (t, s)) t≥s .On C(R, X 0 ) we define the operator T by 3), then the representation formula for u obtained in Theorem 2.2 leads to We need the notion of the spectrum sp(f ) of a Banach space-valued function f : R → Z (cf.[3,20,23,32,35]) where φ denotes the Fourier transform of φ and φ f is the convolution of φ and f .Moreover, we set We obtain the following extension of [6, Theorem .
Theorem 3.7.Assume that the evolution family Then there exists a mild solution u ∈ Ᏺ(R, X 0 ) of (2.3), and u has relatively compact range.
Proof.In order to prove (a) consider In [6, proof of Theorem 3.8] it is shown that the operator T defined in (3.16) maps ᏹ into itself and the restriction T |ᏹ of T to ᏹ is bounded and satisfies 1 ∈ ρ(T |ᏹ ).The invertibility of I d − T |ᏹ and (3.17) show that there is at most one mild solution u of (2.3) contained in ᏹ.
For the proof of (b) let there is a (unique) mild solution u λ ∈ Ᏺ(R, X 0 ) of (1.4) with f λ instead of f such that sp(u λ ) ⊆ p (f λ ) ⊆ p (f ), and u λ has relatively compact range.Let Since f ∈ BUC r (R, X) Proposition 2.5 implies that w(t) = lim λ→∞ w λ (t) exists uniformly for t in R. From (3.17) we obtain (I d −T |ᏺ )u λ = w λ , λ > ω.In particular, w λ ∈ ᏺ for λ > ω, and From Theorem 2.2 and the fact that each u λ is a mild solution of (1.4) with f replaced by f λ it follows that the limit function u is a mild solution of (2.3).Moreover, since each u λ has relatively compact range also u has relatively compact range.This completes the proof.
Recall that a function h ∈ BUC(R, Z) is almost periodic if the set of translates {h(• + t) : t ∈ R} is relatively compact in BUC(R, Z).By AP (R, Z) we denote the space of almost periodic, Z-valued functions.Theorem 3.7 has the following immediate consequence (cf.[6, Corollary 3.9]).
(ii) For every f ∈ AP (R, X) there is a unique mild solution u ∈ AP (R, X 0 ) of (2.3).Proof.Note that (i) is equivalent to the existence of an exponential dichotomy for (U B (t, s)) t≥s (see [19,Theorem 3.2.2],[18,Theorem 7.2.3]).Hence if (i) is satisfied and f ∈ AP (R, X), the existence of a mild solution u ∈ AP (R, X 0 ) of (2.3) follows from Corollary 3.8, whereas the uniqueness is a consequence of Theorem 3.5.The converse implication (ii)⇒(i) follows immediately from [27, Lemma 4].
By P p (R, Z) we denote the space of p-periodic, continuous, Z-valued functions on R.
(ii) For every f ∈ P p (R, X), there exists a unique mild solution u ∈ P p (R, X 0 ) of (2.3).

The semilinear equation
In this section, we apply the results of Section 3 to the semilinear equation where A and B(t), t ∈ R, are as in the previous sections and F : R×X 0 → X is jointly continuous and Lipschitz continuous in the second variable with Lipschitz constant l independent of t and x.Moreover, we assume that t → F (t,0) is a bounded function on R. Our definition of a mild solution of (4.1) is similar to Definition 2.1.
The following conditions will be needed.(H1) The evolution family (U B (t, s)) t≥s has an exponential dichotomy with constants α > 0, L ≥ 1, and projections (P B (t)) t∈R , and l < α/2LC, where By our assumption (2CL/α)l < 1. Hence S is a contraction, and by Banach's fixed point theorem there is a unique function u ∈ C b (R, X 0 ) such that Theorem 3.5 implies that u is the unique mild solution of (4.1) contained in C b (R, X 0 ).
In the same way, the following two results can be derived from Theorem 3.5 and Corollary 3.9, respectively.Proposition 4.3.Assume that condition (H1) holds and that lim t→±∞ F (t,y) = 0 uniformly for y in compact sets in X 0 .Then there exists exactly one mild solution u ∈ C 0 (R, X 0 ) of (4.1).Proposition 4.4.Assume that condition (H1) holds and that the evolution family (U B (t, s)) t≥s is p-periodic.If F (•, x) is almost periodic uniformly for x in compact sets in X 0 , that is, for every compact set K in X 0 and every sequence (t n ) in R there is a subsequence (s n ) of (t n ) such that (F (t + s n , x)) converges uniformly for (t, x) in R × K, then there is exactly one mild solution u ∈ AP (R, X 0 ) of (4.1).
The following result is the semilinear version of Corollary 3.10.Theorem 4.5.Assume that condition (H2) holds and that F (t + p, x) = F (t,x) for every t ∈ R and every x ∈ X 0 .Then there exists exactly one mild solution u ∈ P p (R, X 0 ) of (4.1).

Proof. For f
(4.6)By Proposition 2.5 and Remark 3.11, S is well-defined and maps P p (R, X 0 ) into itself.If f, g ∈ P p (R, X 0 ), then G. Gühring and F. Räbiger 185 Since CpCl < 1, the map S is contractive and there is a unique function v ∈ P p (R, X 0 ) such that (4.8)By Corollary 3.10, there is a unique mild solution u ∈ P p (R, X 0 ) of (1.4) where f is replaced by the function F (•, v(•)).The representation of u obtained in Remark 3.11 shows that u = v, and hence v is a mild solution of (4.1).On the other hand, it follows from (3.17) and Remark 3.11 that each p-periodic mild solution of (4.1) satisfies (4.8).Hence v is the only p-periodic mild solution of (4.1).

Nonautonomous retarded differential equations
In this section, we apply the results obtained for (1.4) to retarded differential equations.Throughout the whole section Y is a fixed Banach space.We consider the inhomogeneous nonautonomous retarded differential equation d dt w(t) = Cw(t) + K(t)w t + h(t), t ∈ R, (5.1) where (C, D(C)) is a Hille-Yosida operator on Y and h ∈ L 1 loc (R, Y ).The part C 0 of C on Y 0 = D(C) generates a C 0 -semigroup (S 0 (t)) t≥0 on Y 0 , and by (S −1 (t)) t≥0 we denote the corresponding extrapolated C 0 -semigroup on the extrapolation space Y −1 .

. 2 )Theorem 3 . 3 .
with asymptotic properties of the function f ∈ C(R, Z).We recall the following result in[22, Theorem 2.1], (see also[23,  Section 10.2, Theorem 1],[5, Theorem 4]).By C b (R, Z) we denote the set of all bounded, continuous, Z-valued functions on R, and C 0 (R, Z) is the space of all functions in C b (R, Z) vanishing at ±∞.Let (U (t, s)) t≥s be an exponentially bounded evolution family on the Banach space Z and let Ᏺ(R, Z) be the space C 0 (R, Z) or C b (R, Z).Then (U (t, s)) t≥s has an exponential dichotomy if and only if for every f ∈ Ᏺ(R, Z) there exists a unique solution u(•, f ) ∈ Ᏺ(R, Z) of(3.2).In that case u(•, f ) is given by

(3. 14 )
By Theorem 2.2, u(•, f ) is a mild solution of (2.3).To show that u(•, f ) is the only mild solution of (2.3) belonging to C b (R, X) we can assume that f ≡ 0 and repeat the arguments in[22,  proof of Proposition 1.2].Since C b (R, X) ⊆ L 1 loc,u (R, X) the implication (ii)⇒(iii) is obvious.(iii)⇒(iv).From the definition of a mild solution it follows immediately that the operator G assigning to each f ∈ C b (R, X) the unique mild solution u

Definition 4 . 1 .
A function u ∈ C(R, X 0 ) is called a mild solution of (4.1) if

Theorem 4 . 2 .
If condition (H1) holds, then there exists exactly one mild solution u

. 3 )
By Lemma 3.4 and the boundedness of F (•, 0), S is well defined and maps C

.
Proposition 5.4.If (U B (t, s)) t≥s is the evolution family on E determined by the variation-of-parameters formula ∈ E, then each mild solution w ∈ C(R, Y 0 ) of (5.1), with h(t) = 0 for all t, satisfies w t = U B (t, s)w s for t ≥ s.