Pseudo almost automorphic solutions of class r in α-norm under the light of measure theory

where −A : D(A) → X is the in nitesimal generator of compact analytic semigroup of uniformly bounded linear operators on a Banach space X, Cα = C([−r, 0], D(Aα)), 0 < α < 1, denotes the space of continuous functions from [−r, 0] into D(Aα), Aα is the fractional α-power of A. This operator (Aα , D(Aα))will be describe later and ‖φ‖Cα = ‖A φ‖C([−r,0],X). For t ≥ 0, and u ∈ C([−r, a], D(Aα)), r > 0 and ut denotes the history function of Cα de ned by


Introduction
In this work, present a new approach to study weighted pseudo almost automorphic functions in α-norm for the following partial functional di erential equation For t ≥ , and u ∈ C([−r, a], D(A α )), r > and u t denotes the history function of Cα de ned by u t (θ) = u(t + θ) for − r ≤ θ ≤ .
L is a bounded linear operator from Cα into X and f : R → X is a continuous function. Some recent contributions concerning pseudo almost periodic solutions for abstract di erential equations similar to equation (1) have been made.
In [3], the authors present new approach to study weighted pseudo almost automorphic functions using the measure theory. They present a new concept of weighted ergodic functions which is more general than the classical one. Then they establish many interesting results on the functional space of such functions like completeness and composition theorems. Our aim of this work is to generalize the results obtain in [5].
This work is organised as follow, in section 2 we recall some preliminary results about analytic semigroups and fractional power associated to its generator will be used throughout this work. In section 3, we recall some prelimary results on spectral decomposition. In section 4, we recall some prelimary results on α − (µ, ν)-pseudo almost automorphic functions and neutral partial functional di erential equations that will be used in this work. In section 5, we give some properties of α − (µ, ν)-pseudo almost automorphic functions of class r. In section 6, we discuss the main result of this paper. Using the strict contraction principle we show the existence and uniqueness of α − (µ, ν)-pseudo almost automorphic solution of class r for equation (1). Last section is devoted to some applications arising in population dynamics.

Analytic semigroup
Let (X, . ) be a Banach space and α be a constant such that < α < and −A be the in nitesimal generator of a bounded analytic semigroup of linear operator (T(t)) t≥ on X. We assume without loss of generality that ∈ ρ(A). Note that if the assumption ∈ ρ(A) is not satis ed, one can substitute the operator A by the operator (A − σI) with σ large enough such that ∈ ρ(A − σI). This allows us to de ne the fractional power A α for < α < , as a closed linear invertible operator with domain D(A α ) dense in X. The closeness of A α implies that D(A α ), endowed with the graph norm of A α , |x| = x + A α x , is a Banach space. Since A α is invertible, its graph norm |.| is equivalent to the norm |x|α = A α x . Thus, D(A α ) equipped with the norm |.|α, is a Banach space, which we denote by Xα. For < β ≤ α < , the imbedding Xα → X β is compact if the resolvent operator of A is compact. Also, the following properties are well known.
Proposition 2.1. [8] Let < α < . Assume that the operator −A is the in nitesimal generator of an analytic semigroup (T(t)) t≥ on the Banach space X satisfying ∈ ρ(A). Then we have iii) for every t > , A α T(t) is bounded on X and there exist Mα > and ω > such that Recall that A −α is given by the following formula where the integral converges in the uniform operator topology for every α > . Consequently, if T(t) is compact for each t > , then A −α is compact.

Spectral decomposition
To equation (1), we associate the following initial value problem where f : R + → X is a continuous function. For each t ≥ , we de ne the linear operator U(t) on Cα by where v(., φ) is the solution of the following homogeneous equation Then A U is the in nitesimal generator of the semigroup (U(t)) t≥ on Cα.
Let X be the space de ned by where the function X c is de ned by Consider the extension A U de ned on Cα ⊕ X by We make the following assertion: (H 0 ) The operator −A is the in nitesimal generator of an analytic semigroup (T(t)) t≥ on the Banach space X and satis es ∈ ρ(A).
Now, we can state the variation of constants formula associated to equation (2). Theorem 3.3. [1] Assume that (H 0 ) holds. Then for all φ ∈ Cα , the solution u of equation (2) is given by the following variation of constants formula De nition 3.4. We say that a semigroup (U(t)) t≥ is hyperbolic if For the sequel, we make the following assumption: We get the following result on the spectral decomposition of the phase space Cα. Proposition 3.5. [1] Assume that (H 0 ) and (H 1 ) hold. If the semigroup (U(t)) t≥ is hyperbolic, then the space Cα is decomposed as a direct sum Cα = S ⊕ U of two U(t) invariant closed subspaces S and U such that the restriction of (U(t)) t≥ on U is a group and there exist positive constants M and ω such that where S and U are called respectively the stable and unstable space, Π s and Π u denote respectively the projection operator on S and U.

(µ, ν)-Pseudo almost automorphic functions
In this section, we recall some properties about pseudo almost automorphic functions. Let BC(R, X) be the space of all bounded and continuous function from R to X equipped with the uniform topology norm. We denote by AAc(R; X), the space of all such functions.
In view of the above, the proof of the next lemma is straightforward. De nition 4.6. Let X and X be two Banach spaces. A continuous function ϕ : R × X → X is called compact almost automorphic in t ∈ R if every real sequence (sm), there exists a subsequence (sn) such that where the limits are uniform on compact subsets of R for each x ∈ X .
Denote by AAc(R × X ; X ) the space of all such functions.
In the sequel, we recall some preliminary results concerning the α − (µ, ν)-pseudo almost automorphic functions. E(R; Xα , µ, ν) stands for the space of functions To study delayed di erential equations for which the history belongs to C([−r, ]; Xα), we need to introduce the space |u(θ)|α dµ(t) = .
In addition to above-mentioned spaces, we consider the following spaces where in both cases the limit (as τ → +∞) is uniform in compact subset of Xα.
In view of previous de nitions, it is clear that the space E(R; Xα , µ, ν, r) is continuously embedded in E(R; Xα , µ, ν).
On the other hand, one can observe that a ρ-weighted pseudo almost automorphic functions is µ-pseudo almost automorphic, where the measure µ is absolutely continuous with respect to the Lebesgue measure and its Radon-Nikodym derivative is ρ: and ν is the usual Lebesgue measure on R, i.e ν([−τ, τ]) = τ for all τ ≥ .

Example 4.7. [3] Let ρ be a nonnegative B-measurable function. Denote by µ the positive measure de ned by
where dt denotes the Lebesgue measure on R. The function ρ which occurs in equation (3) is called the Radon-Nikodym derivative of µ with respect to the Lebesgue measure on R.

De nition 4.8. A bounded continuous function ϕ
We denote by PAA(R; Xα , µ, ν), the space of all such functions.
We now introduce some new spaces used in the sequel.

De nition 4.10. A bounded continuous function ϕ
We denote by PAAc(R; Xα , µ, ν), the space of all such functions.
We have the following result.
Suppose that f ∈ BC(R; Xα) such that for any ε > , We can assume |f (t)|α ≤ N for all t ∈ R. Using (H 2 ), we have Which implies that Therefore f ∈ E(R; Xα , µ, ν). |z(t)|α and n ∈ N such that for all n ≥ n , zn − z ∞,α < ε. Let n ≥ n , then we have We deduce that From the de nition of PAA(R; Xα , µ, ν, r), we deduce the following result. . Assume that f ∈ BC(R, Xα). Then the following assertions are equivalent: iii) For any ε > , lim .
From above equalities and the fact that ν(R) = +∞, we deduce that ii) is equivalent to iii) ⇒ ii) Denote by A ε τ and B ε τ the following sets Assume that iii) holds, that is From the equality we deduce that for τ su ciently large By using (H 2 ), it follows that |f (θ)|α dµ(t) ≤ δε, for any ε > , consequently (ii) holds.
ii) ⇒ iii) Assume that ii) holds. From the following inequality for τ su ciently large, we obtain equation (4), that is iii).
From µ ∈ M, we formulate the following hypotheses.
(H 3 ) For all a, b and c ∈ R, such that ≤ a < b ≤ c, there exist δ and α > such that (H 4 ) For all τ ∈ R, there exist β > and a bounded interval I such that µ({a + τ : a ∈ A} ≤ βµ(A) when A ∈ B satis es A ∩ I = .
De ne the function g on R by g(t) = ϕ (t, h (t)), then g ∈ AA(R; Xα). In fact since h ∈ AA(R; Xα) then for a given sequence (sm) m∈N of real numbers, we can extract a subsequence (sn) n∈N and v ∈ BC(R; Xα) such that v(t) = lim n→+∞ h (t + sn) is well de ned for each t ∈ R, and lim for each t ∈ R. Since ϕ ∈ AA(R × Xα; Xα), then is well de ned for each t ∈ R, and for each t ∈ R. We conclude that g ∈ AA(R; Xα).
For µ ∈ M and δ ∈ R, we denote µ δ the positive measure on (R, B) de ned by We have the following result.
Proof. Assume that u = g + h where g ∈ AA(R; Xα) and h ∈ E(R; Xα , µ, ν, r). We can see that u t = g t + h t . We want to show that g t ∈ AA(R; Xα) and h t ∈ E(R; Xα , µ, ν, r). Firstly for a given sequence (sm) m∈N of real numbers, x a subsequence (sn) n∈N and v ∈ BC(R; Xα) such that g(s + sn) → v(s) uniformly on compact subsets of R. Let K ⊂ R be an arbitrary compact and L > such that K ⊂ [−L, L]. For ε > , x N ε,L ∈ N such that |g(s + sn) − v(s)|α ≤ ε for s ∈ [−L, L] whenever n ≥ N ε,L . For t ∈ K and n ≥ N ε,L , we have In view of above, g t+sn converges to v t uniformly on K. Similary, one can prove that v t−sn converges to u t uniformly on K. Thus, the function s → gs belongs to AAc(R; Xα). Let us denote by where µ δ and ν δ are the positive measures de ned by equation (8). By using Lemma 5.12, it follows that µ δ and µ are equivalent and ν δ and ν are also equivalent. Then by using Theorem 5.10 we have E(R; Xα , µ δ , ν δ , r) = E(R; Xα , µ, ν, r), therefore h ∈ E(R; X, µ δ , ν δ , r), that is On the other hand, for r > we have which shows using Lemma 5.12 and Lemma 5.13 that ϕ t belongs to PAAc (Cα , µ, ν, r). Thus, we obtain the desired result.

(µ, ν)-pseudo almost automorphic solutions of class r
In what follows, we will be looking at the existence of bounded integral solutions of class r of equation (1).
Proposition 6.1. [1] Assume that (H 0 ) and (H 1 ) hold and the semigroup (U(t)) t≥ is hyperbolic. If f is bounded on R, then there exists a unique bounded solution u of equation (1) on R, given by where B λ = λ(λI − A U ) − for λ > ω, Π s and Π u are the projections of Cα onto the stable and unstable subspaces, respectively.

Proposition 6.2. Let h ∈ AAc(R; X) and Γ be the mapping de ned for t ∈ R by
Then Γh ∈ AAc(R, Xα).
For a given sequence (sm) m∈N of real numbers, x a subsequence (sn) n∈N and v ∈ BC(R; Xα) such that h(t + sn) converges to v(t) and v(t − sn) converges to h(t) uniformly on compact subsets of R. Then, if then by equation (9) and the Lebesgue Dominated Convergence Theorem, it follows that w(t + sn) converges to It remains to prove that the convergence is uniform on all compact subset of R. Let K ⊂ R be an arbitrary compact and let ε > . Fix L > and Then, for each t ∈ K, ones has which proves that the convergence is uniform on K, by the fact that the last estimate is independent of t ∈ K.
Proceeding as previously, one can similarly prove that z(t − sn) converges to w uniformly on compact subsets in R. This completes the proof.
Thus, we obtain the desired result.
For the existence of (µ, ν)-pseudo almost automorphic solution of class r, we make the following assumption.
Our next objective is to show the existence of (µ, ν)-pseudo almost automorphic solutions of class r for the following problem where f : R × Cα → X is continuous.
For the sequel, we make the following assumptions.
(H 7 ) Let µ, ν ∈ M and f : R × Cα → X cl(µ, ν)-pseudo almost automorphic of class r such that there exists a positive constant L f such that This means that H is a strict contraction. Thus by Banach's xed point theorem, H has a unique xed point u in PAAc(R; Xα , µ, ν, r). We conclude that equation (10), has one and only one cl(µ, ν)-pseudo almost automorphic solution of class r.

Application
For illustration, we propose to study the existence of solutions for the following model where G : [−r, ] → R is a continuous function and h : R × R → R is Lipschitz continuous with respect to the second argument. To rewrite equation (11) in the abstract form, we introduce the space X = L ([ , π]; R) vanishing at 0 and π, equipped with the L norm that is to say for all x ∈ X, Lemma 7.1. [9] If y ∈ D(A ), then y is absolutely continuous, y ′ ∈ X and |y ′ | = |A y|.
It is well known that −A is the generator of a compact analytic semigroup semigroup (T(t)) t≥ on X which is given by +∞ n= e −n t (x, en)en , x ∈ X.
We de ne f : R × C → X and L : C → X as follows   It follows that t → arctan t is (µ, ν)-ergodic of class r, consequently, f is uniformly compact (µ, ν)-pseudo almost automorphic of class r. Moreover, L is a bounded linear operator from C to X. Let k be the lipschiz constant of h, then for every φ , φ ∈ C and t ≥ , we have Consequently, we conclude that f is Lipschitz continuous and cl(µ, ν)-pseudo almost automorphic of class r.