For the proof of the main result, we need the following lemmas.
Lemma 3.1 Let and . If the derivative is uniformly continuous, then , thus .
Remark 3.2 Lemma 3.1 is well know for , . In the scalar case, for , this result is proved in [12], Corollary 5.6, p.59.
Proof Consider the function defined by for . Since is a translation invariant vectorial space, then . The equality shows that the uniform continuity of implies that the sequence with values in the Banach space converges uniformly to on ℝ. Then , and from the definition of , we obtain . □
Lemma 3.3 Let and . If the spectrum of A does not intersect the imaginary axis, then for all , there exists a unique solution in of the differential equation
(3.1)
Moreover, the solution u is in .
Proof Applying Theorem 4.1, p.81 in [20] (or Theorem 4 in [21]), Equation (3.1) admits a unique bounded solution on ℝ which is given by the formula
(3.2)
where G is the principal Green function for Equation (3.1). The Green function is continuous on , and there exist and such that for all .
Now we prove that the bounded solution u defined by (3.2) belongs to .
When (respectively ), this result is a straightforward consequence of Theorem 3.8 in [15] (respectively Theorem 3.9 in [16]) and when , this result is proved in [21], Proposition 3. Then we deduce the result for (respectively ).
Note that the case of the pseudo almost periodic (respectively pseudo almost automorphic) functions is a special case of the weighted pseudo almost periodic (respectively weighted pseudo almost automorphic) functions by taking for all ; remark that the associated measure is exactly the Lebesgue measure.
And so it suffices to prove the cases of weighted pseudo almost periodic functions and of weighted pseudo almost automorphic functions.
When (respectively ), this result is a straightforward consequence of Theorem 3.8 in [15] (respectively Theorem 3.9 in [16]). Consequently, , and from the definition of , we deduce that . Since u satisfies Equation (3.1), then , and from the definition of , we obtain . □
Lemma 3.4 Let X and Y be two finite-dimensional Banach spaces, and let be a continuous mapping. Then the Nemytskii operator , defined by , is continuous.
Remark 3.5 Contrary to the asymptotically almost periodic case and, in particular, for the almost periodic case, when the dimension of the Banach spaces X and Y is infinite, Lemma 3.4 does not hold for the pseudo almost periodic case, and thus for the weighted pseudo almost periodic case, without additional assumptions. This is due to the fact that the range of a pseudo almost periodic function is only bounded, but not relatively compact, contrary to the asymptotically almost periodic case. This last observation still holds when the word almost periodic is replaced by almost automorphic.
Proof When and , replacing by ℝ, this result is a variation of Theorem 8.4 in [22].
When and , replacing by ℝ, the inclusion is a variation of Theorem 2.15 in [9]. Moreover, using Lemma 1 in [23], we know that is continuous, and so its restriction to is also continuous.
When and (respectively and ), this result is a straightforward consequence of Theorem 4.1 (respectively Theorem 4.2) in [24].
When and (respectively and ), using Corollary 4.12 in [15] (respectively Corollary 5.10 in [16]), we know that (respectively ). Moreover, using Lemma 1 in [23], we know that is continuous, and so its restriction to (respectively ) is also continuous. □
Lemma 3.6 Let X and Y be two finite-dimensional Banach spaces, and let be a continuously Fréchet-differentiable mapping. Then the Nemytskii operator is continuously Fréchet-differentiable on , and we have for all .
Proof When and , replacing by ℝ, this result is a variation of Theorem 8.5 in [22].
When and , using Lemma 1 in [23], we know that is of class and that we have when . Now, using Theorem 2.15 in [9], we know that and that when . And so and the announced formula for its Fréchet-differential is proven.
As in the proof of Lemma 3.3, the case of (respectively ) is a corollary of the case (respectively ). And so it suffices to prove the cases of the weighted pseudo almost periodic functions and of the weighted pseudo almost automorphic functions.
To prove the result in the case where and (respectively and ), note that, using Lemma 1 in [23], we know that is of class and that we have when . Now, using Corollary 4.12 in [15] (respectively Corollary 5.10 in [16]), we know that (respectively ) and that (respectively ) when (respectively ).
Consequently, we obtain (respectively ) and the announced formula for its Fréchet-differential is proven. □
Lemma 3.7 Let , and . If x is a solution of () in , that is, and x satisfies (), then .
Proof Lemma 3.2 in [4] asserts that if x is a solution of () in , then , therefore the derivative is uniformly continuous, and by help of Lemma 3.1, we obtain . By using Lemma 3.4, the functions and are in . Applying again Lemma 3.4 to and using the continuous function defined by , we obtain that . Since , then , and from the definition of , we obtain that . □