Composition of pseudo almost periodic functions and Cauchy problems with operator of non dense domain

In this work, we give a generalization, to Banach spaces, for Zhang’s result concerning the pseudo-almost periodicity of the composition of two pseudo-almost periodic functions. This result is used to investigate the existence of pseudo-almost periodic solutions of semilinear Cauchy problems with operator of non dense domain in original space


Introduction
In this paper, we study the existence, uniqueness and pseudo-almost peri- odicity of the solution to the following semilinear Cauchy problem x'(t) = Ax(t) + j(t,x(t», t E ~, (1) where A is an unbounded linear operator, assumed of Hille-Yosida with negative type and non necessarily dense domain on a Banach space X and f : 1R x X -~--~ X, is a continuous function.First, we begin by studying the inhomogeneous Cauchy problem x'(t) = Ax(t) + f (t), t E ~, (2) which will be used to get our goal.
To study the pseudo-almost periodicity of (1), we need to give a generalization, to Banach spaces, for Zhang's result in which he proved that the composition of two pseudo-almost periodic (p.a.p.) functions in finite dimensional spaces is p.a.p.More precisely, for f : : R x Y --~ X and h : ~ --; Y which are p.a.p. we prove that the function is also p.a.p..One can find this result in Section 3.
The notion of pseudo-almost periodicity has been introduced by Zhang (1992) (see [14]).He has studied in [15] the existence of p.a.p. solutions of (1) in the finite dimensional spaces case.In the case of Banach spaces, in our knowledge, there is only one work [1], concerning the study of the existence of a unique p.a.p. solution of (2), where A is the generator of Co-semigroup.
(resp.AP(R x Y, X )) the set of almost periodic functions from R into X (resp.from 1R x Y into X), where X and Y are two Banach spaces, and defines the sets PAP0(IR, X) and PAP0(IR x Y, X ) by PAP0(IR, X ) : -03C6 E Cb(IR, X ), lim 1 / ~03C6(t) ~ dt = 0 PAP(R, X) (resp.PAP(IR x Y, X ) ) denotes the subset of (resp.C(~ x Y, X )) of all pseudo-almost periodic functions from jZR into X (resp.from IR x Y into X). .We have the following result which will be used in the sequel Proposition 1 Let f E AP(IR x Y, X) and h E AP(IR, Y), then the func- tion f (~, h(~)) E AP(R, X ).
The proof of this proposition is similar to the one given in ( [6], Thm.2.11).
In this subsection, we fix some notations and recall some basic results on extrapolation spaces of Hille-Yosida operators.For more complete account we refer to [10], ~11J, where the proofs are given.
Let X be a Banach space and A be a linear operator with domain D(A).
We say that A is a Hille-Yosida operator on X if there exist w ~ IR such that (c~, +00) C p (A) ( p (A) is the resolvent set of A) and A > cv, r~ > o} oo. ( The infinimum of such w is called the type of A.
For the rest of this section we assume without loss of generality that (A, D(A)) is a Hille-Yosida operator of negative type on X.This implies that 0 E p(A), i.e., A-1 E L(X).
On the space Xo we introduce a new norm by ~ E X~.
The completion of (Xo, ~.~-1) will be called the extrapolation space of X o associated to Ao and will be denoted by X-1.
One can show easily that, for each t > 0, the operator To(t) can be extended to a unique bounded operator on X-i denoted by T-l(t).The family is a C0-semigroup on X-I, which will be called the extrapolated semigroup of The domain of its generator A_1 is equal to Xo.

Main results
We state the fundamental lemma, which will be crucial for our aim.
Lem m a 3 Let A be a Hille-Yosida operator of negative type, 03C9 E 03C1(A), 03C9 0 and f E X }.The following properties hold t (z} / --8) E X~, for all t E R.
(ii) There exist C independent from f such that for every t E IR a t 3' 1 {t ----a) Ce03C9t t-~e-03C9s ~f {g) ~ds.
It is easy to see that t t / -T 1(t - _'_'' / -co -03C3)f(03C3)d03C3 in X-1 and consequently, X0 X_ 1 implies Then, we obviously have (i).For (it) ; it follows immediately from the esti- mation satisfied by fn.Finally, (iii ) can be obtained easily.from (it) 1   Our main results consists of the study of the existence of a unique bounded and pseudo-almost periodic solution to the inhomogeneous Cauchy problem, the generalization of Zhang's result and to use these results to in- vestigate the semilinear Cauchy problem case.

(~)
where A is a Hille-Yosida operator on .x of negative type and f E Cb(IR, X).By using the Lemma3, we show easily that the unique bounded mild solution x(~) of this problem is given by t x(t) -(T f )(t) := s) f(s)ds.' for all t E 1R (5) = ~0-~T -1(-s)ft(s)ds. (6) If we assume that f E PAP(IR, X), then there are g E AP(IR, X ) and cp E PAP0(IR, X ), such that f = g + cp.It is easy to show that cp E Cb(R, X), thus x = Tg+T03C6.The operator T is bounded and commutes with translation group, then it's easy to see that Tg E AP(IR, X ).Furthermore, Lemma 3 implies, for r > 0, that 2r r -r T0 3 C 6 ( t ) ~dt ~ 2r -r ewt e-03C9s~03C6(s) ~ds dt C 2 r r _ r t-~ e -0 3 C 9 s 0 3 C 6 ( s + t ~ds dt ~ C -oo e-03C9s 2r -r ~03C6s(t)~d t] ds, (*)   where w E p(A) such that ;~ o.
We show, by simple computation.that the set PAP0(IR, X ) is invariant under the translation group.Hence, using Lebesgue's theorem, (*) goes to zero, as r ~ +~.This proves the following theorem.Theorem 4 Let A be a Hille-Yosida operator on X of negative type and f E X ) pseudo almost periodic.Then (4) admits a unique bounded pseudo almost periodic mild solution given by (5).

Composition of two pseudo almost periodic functions
Let us consider two Banach spaces X and Y, and a continuous function The generalization of Zhang's result is announced in the following theorem.Theorem 5 Let f E PAP(IR x Y, X) satisf y the Lipschitz condition ~f(t,x)-f(t,y)~ ~ L~x-y~. ,for all x, y ~ Y and t ~ IR.

Semilinear Cauchy problem
Let A be a Hille-Yosida operator of negative type 03C9 on a Banach space X.
We can now state the following main result.
Theorem 6 Under the above assumptions, if f E PAP(IR x X0, X ) then Equation (10) admits one and only one bounded mild solution on IR, which is pseudo-almost periodic.
Proof.Let f E PAP(IR x Xo, X ) and y be a function in PAP(R, X0).
Hence, since (-Ck 03C9) I, there is a unique bounded and pseudo-almost periodic solution of which is a bounded pseudo-almost periodic mild solution of (10). .
To finish this work, we give the following example as an application of our previous abstract results.Example.
Consider the following partial differential equation ~ ~tu(t, x) = ~ ~xu(t, x) !u(t, x) + x)), t, x E IR, (11)  where p, is a positive constant and f : 1R x I~t ---~ IR is continuous and lipschitzian function with respect to x uniformly in t.
Let X := with the supremum norm and the operator A defined on X by A f := , for f E D(A) := ~ f EX: f is absolutely continuous and f' E X ~ .
We can easily show that A is a Hille-Yosida operator of type 03C9=- 0, with non dense domain (see [5]).

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From the above abstract results, if f(.,.) E PAP(IR x D(A), X ), then the semilinear Cauchy problem (12) has one and only one bounded p.a.p. mild solution.Consequently the partial differential equation ( 11) admits a unique bounded p.a.p. solution with respect to the ~°°(~)-norm.

k=1
For k (1 ~ k ~ m), the set is open and IR = U Bk. Let Ek = BkB Bi and E1 = B1.Then Et n Ej = km ==x