Almost automorphic delayed differential equations and Lasota-Wazewska model

Existence of almost automorphic solutions for abstract delayed differential equations is established. Using ergodicity, exponential dichotomy and Bi-almost automorphicity on the homogeneous part, sufficient conditions for the existence and uniqueness of almost automorphic solutions are given.


Introduction
The development of the theory of almost periodic type functions has been strongly stimulated by problems arising in differential equations, stability theory, dynamical systems and many other areas of science. Nowadays, also there exists a wide range of applications starting from the basic mathematical models based on linear ordinary differential equations, including nonlinear linear ordinary differential equations, differential equations in Banach space and also partial differential equations. Moreover, there exists several related concepts which arise as generalizations of the almost periodic concept. For instance the notions of almost automorphic, asymptotically almost periodic, asymptotically almost automorphic and pseudo almost periodic. Since there are plenty of results in literature, let us just quote, for their applications in engineering and life science, for example asymptotically almost periodic functions [17,18,19,20,27,29,30,31,32,33,40], and pseudo almost periodic functions [9,11,28]. Moreover, we recall that N'Guérékata has given a huge impulse to the study of almost automorphic solutions of differential equations [1,7,12,14,15,21,23,25,26]. For a recent results of almost automorphic differential equations see also [5,6].
In this paper, we are initially motivated by a biological-mathematical model [13,16,22,37,38,41] which is a delayed differential equation of the following type y ′ (t) = −δ(t)y(t) + p(t)g(y(t − τ )), with τ > 0, δ and p almost automorphic functions and g a Lipschitz function. Then, we focus our attention in the existence and uniqueness of solutions of the following delayed differential equation y ′ = A(t)y + f (t) + g(t, y(t − τ )) with τ ≥ 0, (1.1) under several assumptions on A, f and g. Naturally, the assumptions on A and f are related with an almost automorphic behavior and the assumptions on g are mainly related with a Lipschitz requirement. We note that (1.1) naturally, includes as particular cases the following equations 3) y ′ = A(t)y + f (t) + g(t, y(t)).
(1.4) Thus, following a natural sequence of the classical systemic study of ordinary differential equations we start by analizing the homogeneous linear equation (1.2). Then, we develop the theory for the non-homogeneous linear equation ( The main contributions and the organization of the paper are given as follows. In section 2 we introduce the general assumptions, recall the concepts of almost automorphicity and ergodic functions, define a convolution operator and get the results for (1.2). To be a little more precise in this section, we obtain some conditions for the exponential dichotomy using ergodic functions and we prove the Bi-almost periodicity and Bi-almost automorphicity of the Green function when the evolution operator commute with the projection. We note that, the integral Bi-almost automorphicity property of the Green function is fundamental to obtain the main results. In section 3, almost automorphicity of solutions of nonautonomous systems (1.3), (1.4) and (1.1) are obtained. Here, the results of almost automorphicity of the differential equation solutions are obtained by assuming that A and f are almost authomorphic and g satisfies (2.1). Finally, in section 4 we study a biological model establishing explicit condition under which there exists a unique almost automorphic solution of the Lasota-Wazewska equation.

Preliminaries
In this section we present some general assumptions, precise the concepts related with the almost automorphic notion, we recall the notion of ergodic functions, we define a convolution operator and the α-exponential dichotomy and introduce several results for the homogeneous equation (1.2) in the scalar, system and abstract case.

General assumptions.
Here we present two general assumptions. Firstly, throughout of the paper (V, · V ) will be a Banach space and let (BC(R, V ), · ∞ ) will be used to denote the Banach space of bounded continuous functions from R into V endowed with the sup norm ϕ ∞ = sup t∈R ϕ(t) V . Second, concerning to the assumptions on the coefficients A, f and g for equations (1.1)-(1.4) we comment that it will be specifically done on the hypothesis of each result. However, in order to give a unified presentation, we introduce some notation related with the assumption of the local Lipschitz behavior of g. Indeed, given a function g, it is assumed that: the open ball centred in a given (fix) function ϕ 0 : R → V and with radius ρ ∈ R + , i.e., ∆(ϕ 0 , ρ) = ϕ : R → V ϕ − ϕ 0 ∞ ≤ ρ . In particular, in subsections 3.2-3.3 will be assumed that ϕ 0 is of the form ϕ 0 (t) = R G(t, s)f (s)ds with G the Green function defined on (3.17); (g 2 ) There exists a positive constant L such that the inequality g(t, y 1 ) − g(t, y 2 ) ≤ L y 1 − y 2 holds for all (t, y 1 , This set of conditions (g 0 )-(g 2 ) appear in several parts of the paper and essentially when we study the nonlinear equations in subsections 3.2-3.3.

2.2.
Almost automorphic notion and related concepts. We recall that the almost automorphic functions have been developed by Bochner [2,3] as a generalization of almost periodic functions. We recall that a function f ∈ BC(R, V ) is called Bohr almost periodic [8] if for each ǫ > 0, there exists l ǫ > 0 such that every interval of length l ǫ contains a number ξ with the property: The set of Bohr almost periodic will be denoted by BC(R, V ). Then, we precise the concept of almost automorphic functions and matrices.
Definition 2.1. Consider V a Banach space. Then, (i) A continuous function ψ : R → V is called an almost automorphic function if for any sequence of real numbers {τ n } ∞ n=1 , there exists a subsequence {τ n } ∞ n=1 of {τ n } ∞ n=1 such that the limit of the sequence {ψ(t +τ n )} ∞ n=1 , denoted byψ(t), is well defined for all t ∈ R and the sequence {ψ(t −τ n )} ∞ n=1 converges pointwise on R to ψ(t), or equivalentlỹ are well defined for all t ∈ R. The collection of all almost automorphic functions from R to V is denoted by AA(R, V ). (ii) A matrix valued function A : R → C d1×d2 is called an almost automorphic matrix valued function or equivalently (most of the time by briefness) A(t) ∈ C d1×d2 is called an almost automorphic matrix if for any sequence {ξ ′ n } ∞ n=1 ⊂ R, there exists a subsequence {ξ n } ∞ n=1 of {ξ ′ n } ∞ n=1 and a matrix B(t) ∈ C d1×d2 such that the sequences {A(t + ξ n )} ∞ n=1 and {B(t − ξ n )} ∞ n=1 converges pointwise to B(t) and A(t), respectively. We note that the convergence in (2.2) is pointwise. Then, the functionψ in (2.2) is measurable, but not necessarily continuous. Moreover, we note if we consider that convergence in definition 2.1 is uniform on R instead of pointwise convergence, we get that the function ψ is Bochner almost periodic. It is well known that both definitions of almost periodicity (Bohr and Bochner) are equivalents, see for instance [8]. Now, we note that AP (R, V ) and AA(R, V ) are vectorial space and AP (R, V ) is a proper subspace of AA(R, V ), since for instance ψ(t) = cos [2 + sin(t) + sin(t √ 2)] −1 is an almost periodic function but not almost authomorphic. Similarly, it is proven that the inclusion AP (R, V ) ⊂ BC(R, V ), for an extensive discussion consult [4,3,15,14,21,23,24,25,26,34,35,42,44,10,1,39,43,12].
To close this subsection we introduce two additional facts. Firstly, we note that the simpler equation (1.3) with A ≡ 0, i.e., y ′ (t) = f (t), with f ∈ AA(R, V ) has not necessarily a solution y ∈ AA(R, V ). However, this fact is true in a uniformly convex Banach space V and hence in every Hilbert space, see Theorem 2.1. In the second place we need a composition result [7], which will be fundamental for the analysis of (1.4) and (1.1), see Proposition 2.2.
Theorem 2.1. Denote by C 0 the vectorial space formed by the functions which vanishes at infinity. Consider that V is a Banach space which does not contain C 0 as an isomorphic subspace and let f ∈ AA(R, V ). Then, the function F (t) = t 0 f (s)ds is in AA(R, V ) if and only if it is bounded. Such Banach space with this property for F will be called a Banach space with the Bohl-Bohr property.

Ergodic functions.
Here we introduce the concept of ergodic functions and deduce that these types of functions implies naturally an exponential behavior (or α-exponential dichotomy to be more precisely).
exists uniformly with respect to ξ ∈ R and its value is independent of ξ. The complex number M (f ) is called the mean of the function f .
The mean of an ergodic function has several properties. Here, among of these useful basic properties, we only recall two since both will be used frequently in the proofs given below in this paper, a complete list of properties may be consulted in [45]. Firstly, by the definition of M (f ) independent of ξ we have that M (f ) can be rewritten equivalently as follows The second property is well known as the translation invariance property of M (f ), which set that M (f ) satisfies the following identity where f ξ denotes a ξ-translation of f , i.e. f ξ (t) = f (t + ξ) for all t ∈ R and any arbitrarily given ξ ∈ R (but fixed).
Lemma 2.3. Consider µ ∈ BC(R, C) an ergodic function with Re(M (µ)) = 0 and also consider α ∈ R + . Then, there exists two positive constants T 0 (big enough) and c such that the two assertions given below are valid: then the following inequalities hold true: then the following inequalities hold true: Proof. Let us assume that µ ∈ BC(R, C) is an ergodic function. Then, by the Definition 2.2 and the translation invariance property of M (f ) (see (2.3)) we have that Here and throughout of the paper o(1) corresponds to the well known Bachmann-Landau notation, and the proof of (2.4) follows immediately. The proof of (i) is a consequence of the exponential function increasing behavior. Thus, the proof of item (i) is completed. Now, the proof of item (ii) is similar and we omit it.
2.4. The convolution operator. Let us denote by L 1 (R) and L ∞ (R) the spaces of Lebesgue integrable functions on R and essentially bounded functions on R, respectively. Then, the convolution operator on L : L ∞ (R) → L ∞ (R) is defined as the operator such that for all ϕ ∈ L ∞ (R). Some properties of L , which will be needed in the proof the main results, are summarized in the following lemma.
Proof. Let us select ϕ ∈ AA(R). Then, by Definition 2.1, given an arbitrary sequence of real 2) is equivalently rewritten as follows (2.13) Indeed, this fact can be proved by application of Lebesgue's dominated convergence theorem, sincẽ Let us consider ϕ ∈ BC(R), then we deduce (2.11) by application of the Hölder inequality. Now, from (2.11) we follow the invariance of BC(R).
We note that, if we define for some α ∈ R + and denote by L 1 and L 2 the corresponding convolution operators associated with h 1 and h 2 , respectively. Then, we get an interesting result by application of Lemma 2.4. More precisely we have the following Corollary.
Corollary 2.5. Consider α ∈ R + and the operators L i , i = 1, 2 defined by respectively. Then, the spaces BC(R), AP (R) and AA(R) are invariants under the operator L i , i = 1, 2. Moreover, the inequalities (2.11) and (2.12) are satisfied with L i , i = 1, 2, instead of L .
2.5. Some concepts and properties related to equation (1.2). In this subsection we study the equation (1.2). In order to introduce the concepts and results, we recall the standard notation of fundamental matrix and flow associated with (1.2), which are denoted by Φ A and Ψ A , respectively. More precisely Given a matrix A(t), then the notation Φ A = Φ A (t) and Ψ A are used for a fundamental matrix of the system (1.2) and for the application defined as follows Lemma 2.6. Consider the notation (2.15). Then, the identities are satisfied for all (t, s, ξ) ∈ R 3 .
Proof. Let us denote by H the function defined by the following correspondence rule H(t, s) = Ψ A (t, s) − Ψ B (t, s). Then, by partial differentiation of H with respect to the first variable and making some rearrangements, we get Now, by multiplying to the left by Ψ B (t, r) and simplifying, we deduce that Thus, by integration over the interval [s, t], we have that which implies (2.16) by noticing that Ψ B (t, t) = I and H(s, s) = 0. Now, the proof (2.17) follows by similar arguments or by direct application of (2.16).
Lemma 2.7. Consider the notation (2.15) and the sets − → respectively. Assume that the following three statements are true: A(t) is an almost periodic matrix (see definition 2.1), P is a constant projection matrix that commutes with Φ A and for some given positive constants c and α the inequality In particular, if ξ is an ǫ-almost period of A the inequality Proof. Since the proofs with (t, s, ξ) ∈ − → R 2 × R or with (t, s, ξ) ∈ ← − R 2 × R are analogous, we consider only one of the cases. Then, in order to fix ideas, let us consider (t, s, ξ) ∈ − → R 2 × R and recall the notation ∆ ξ defined by ∆ ξ F (x) = F (x + ξ) − F (x) for any function F . (2.21) In particular, for instance, we have that ∆ ξ Φ(t, s) = Φ(t + ξ, s + ξ) − Φ(t, s) and ∆ ξ A(t) = A(t + ξ) − A(t). From (2.16) and the hypothesis that P is a constant projection matrix, i.e. P 2 = P , which commutes with Φ(t) for every t ∈ R, we have that    Now, from the identity (2.16), we deduce that the assumption (2.18) implies the following bounds for all (t, s, n) ∈ R 2 × N and some real constants c 1 > 0 and α ′ ∈]0, α[. Then, by applying four times the Lebesgue's dominated convergence theorem we deduce the integrally Bi-almost automorphic property of Ψ A . Indeed, firstly by the first limit given in (2.25) we deduce that for each (s, t) ∈ R 2 the integral t s (A ξn − B)(r) dr converges to 0 when n → ∞. Then, with this convergence in mind, in a second application, from (2.26) we get that for each t ∈ R the integral t −∞ (Ψ A ξn − Ψ B )(t, s)P ds converges to 0 when n → ∞. Similarly, applying twice more the Lebesgue's theorem (in the second integral (2.25) and in the inequality (2.27)) we deduce that for each (s, t) ∈ R 2 the integral t s (B −ξn − A)(r) dr converges to 0 when n → ∞ and for each t ∈ R the integral t −∞ (Ψ B −ξn − Ψ A )(t, s)P ds converges to 0 when n → ∞. Thus, (2.23) holds and Ψ A is integrally Bi-almost automorphic. The proof of (2.24) can be obtained by similar arguments.
Definition 2.4. Consider the notation (2.15). The linear system (1.2) has an α-exponential dichotomy if there exists a projection P and two positive constants c and α such that for all (t, s) ∈ R 2 the estimate

28)
is satisfied. The matrix G A is called the Green matrix associated with the dichotomy.
Lemma 2.9. Consider the notation (2.15) and define the Green operator Γ as follows Assume that A(t) is an almost automorphic matrix and (1.2) has an exponentially dichotomy such the its projection commutes with the fundamental matrix Φ A . Then, the following assertions are satisfied: (i) The Green matrix G A is integrally Bi-almost automorphic.
(ii) The spaces BC(R, V ), AP (R, V ) and AA(R, V ) are invariants under the operator Γ. Moreover, there exists two positive constants c 1 and c 2 such that the following inequalities are satisfied. Here L i and ∆ ξ denotes the operators defined on (2.14) and (2.21), respectively.
Proof. The proofs of (i) and (ii) are straightforward. Indeed, for (i), let us consider A(t), B(t) and {ξ n } ∞ n=1 as given in the proof of Lemma 2.8. Now, by the assumptions we can deduce that where G A is the Green matrix defined on (2.28). Thus, we can follow that G A is integrally Bialmost automorphic. Meanwhile, we follow the proof of (ii) by application of Lemmas 2.3, 2.4 and 2.9-(i).

Main Results
In this section we present several results of Massera type for (1.3) and (1.1) and related with the almost authomorphic behavior of the A and f . Results for (1.3). Here we present a result for the scalar abstract case, see Theorem 3.1. Then, we extend this result can to linear triangular systems and general linear constant systems, see Theorem 3.2. We also, present simple and useful relation between finite and infinite dimension is deduced from Theorem 3.3. Finally, we present to results on the general case , see Theorems 3.4 and 3.5.

3.1.
Proof. (i) Before of prove the item we deduce two estimates (see (3.5) and (3.6)) and introduce some notation (see (3.7) to (3.8)). Firstly, by Lemma 2.3 we can deduce that the scalar equation x has an α-exponential dichotomy. Indeed, we note that by the hypothesis M (Re(µ)) = 0 we can always select α satisfying |M (Re(µ))| ≥ α > 0. Then, by application of Lemma 2.3, we have that there exists a positive constant c such that |g(t, s)| ≤ ce −α|t−s| with g defined in (3.1). (3.4) Moreover, by application of Lemma 2.7 and integration on s, we deduce that there exists c 1 ∈ R + and α ′ ∈]0, α[ such the following inequalities Similarly, given the sequence {ξ ′ n } ∞ n=1 by the hypothesis f ∈ AA(R, V ), there exists a subsequence {ξ ′′ n } ∞ n=1 of {ξ ′ n } ∞ n=1 and the functionf such that lim Now we develop the proof of the item. Indeed, we consider that y is defined by (3.2) and we prove that y is belongs to AA(R, V ). Let us start by considering the notation y ± andỹ ± for the functions defined as follows (3.10) respectively. Then, by algebraic rearrangements we deduce that (3.12) Now, by using Lebesgue dominated convergence theorem we get that the four integrals on (3.11)-(3.12) converges to 0 when n → ∞. Indeed, by (3.5) and (3.7) we follow that the first integral in (3.11) converges to 0 when n → ∞. We see that the second integral in (3.11) vanishes when n → ∞ by consequence of (3.8). Meanwhile, we note that both integrals in (3.12) converge to 0 when n → ∞ by application of (3.6), (3.7) and (3.8). Consequently, we have that lim n→∞ (y ± ) ξn (t) =ỹ ± (t) for all t ∈ R. Similarly, we can prove that (ỹ ± ) −ξn (t) → y(t) for all t ∈ R and when n → ∞. Hence y ∈ AA(R, V ).
(ii) Noticing that h(s) = exp (i s 0 a)f (s) ∈ AA(R, V ) and the Banach space V has the Bohl-Bohr property we follow the proof by application of Theorem 2.1. (3.14) Assume that the Banach space V has the Bohl-Bohr property. Then any solution y of system (1.3) is bounded if and only if y is belongs AA(R, V p ).
(3. 15) We note that the p-th equation in (3.15) can be analyzed by application of Theorem 3.1. Indeed, by Theorem 3.1-(i), we have that there exists y p ∈ AA(R, V ) given for Similarly, by substituting y p ∈ AA(R, V ) in (p−1)−th equation of (3.15) and by a new application of Theorem 3.1-(i) we can find an explicit expression for y p−1 (t). This argument can be repeated to construct y p−2 (t), y p−3 (t), . . . , y 2 (t) and y 1 (t) by backwards substitution and application of Theorem 3.1-(i) in the system (3.15). Hence, we can construct y(t) an also get that the conclusion of the theorem 3.2-(i) is valid.
(ii) The proof of this item is similar to the proof of the precedent item (i) of the Theorem 3.2. In a broad sense, in this case we apply Theorem 3.1-(ii) instead of Theorem 3.1-(i) and similarly we use backwards substitution.
In the general case, a formula for the bounded solutions can be also obtained.
Proof. If {µ i } p i=1 are distinct, the constant system (1.2) is similar to a diagonal system. Then, without loss of generality, we can suppose that A is an upper triangular matrix. Hence, the result (3.16) follows by application of the Theorem 3.2-(i).
Assume that A has an α-exponential dichotomy, i.e., there exists two constants c and α such that 3) has a unique solution y ∈ AA(R, V ) given by and satisfying the following estimate Indeed, for t ≥ s we have that G(t, s)y(s) ≤ ce −α(t−s) y ∞ , which implies that G(t, s)y(s) → 0 when s → −∞. Similarly, we get that G(t, s)y(s) vanishes when s → ∞. We note that, for t ∈ R (fix), applying T (t − s) on the identity y ′ (s) = Ay(s) + f (s) and using the fact that A commutes with T (t) on the domain of A, we get Now, the proof consists of three main parts: (a) we prove that the solution y of (1.3) is given by (3.19); (b) we prove that y ∈ AA(R, V ) and (c) we prove the uniqueness.
(a). Proof of that the solution y of (1.3) is given by (3.19). Firstly, we note that a formal integration of (3.22) on s ∈ (−∞, t) gives the following identity Then an integration on s ∈ [r, t] implies the following relation Now, by (3.21) letting r → −∞, we deduce that Here, we note that a integration of (3 and a integration of (3.24) on s ∈ [t, r] yields Thus, we conclude the proof of (3.19).
(b). Proof y ∈ AA(R, V ). The proof of this property follows by (3.30) the hypothesis f ∈ AA(R, V ) and application of Lemma 2.9. Indeed, if x ∈ BC(R, V ) is a solution of the linear system we have that x(t) = P x(t)+(I −P )x(t) = x 1 (t) + x 2 (t). Note that x 1 → ∞ as t → −∞ and x 2 → −∞ as t → ∞, by the exponential dichotomy.
Finally, we note that (3.20) is a consequence of (3.30).
Theorem 3.5. Assume that A ∈ AA(R, C p×p ) and (1.2) has an exponential dichotomy with a projection P that commutes with the fundamental matrix Φ A (t). Assume that f ∈ AA(R, V p ). Then, the linear non-homogeneous equation (1.3) has a unique AA(R, V p ) solution given by satisfying (3.20).
Proof. By application of Lemma 2.9.
Theorem 3.6. Consider V be a Hilbert space and A a linear compact operator on V . Suppose that V = ⊕ ∞ k=1 V k is a Hilbert sum such that V k is a finite dimensional subspace of V for each k ∈ N. Suppose that each orthogonal projection P k on V k commutes with A. If f ∈ AA(R, V ), then every bounded solution y of (1.3) is belongs AA(R, V ).
Proof. Noticing that for any y ∈ V , we have that y = ∞ k=1 P k y = ∞ k=1 y k . Then, by the fact that A is bounded on V , we deduce that (3.31) Now, from the hypothesis that f ∈ AA(R, V ), we have that for any subsequence {τ n } ∞ n=1 ⊂ R, there exists a subsequence {τ n } ∞ n=1 ⊂ {τ n } ∞ n=1 and a functionf such that f τn (t) →f (t) and f −τn (t) → f (t) pointwise on R when n → ∞. Then, by compactness of A we deduce that Af τn (t) → Af (t) and Af −τn (t) → Af (t) pointwise on R when n → ∞. Now, choosing y k (t) = P k y(t) and assuming that y is solution of equation (1.3) we can deduce that or equivalently y k satisfies the equation (1.3) in the finite dimensional space V k with P k f (t) ∈ AA(R, V k ) since P k is a bounded linear operator. Thus, y k is bounded if and only if y k ∈ AA(R, V k ). Now, if y(t) is bounded the set Ay(t)|t ∈ R is relatively compact in V . Hence ∞ k=1 P k Ay(t) = Ay(t) uniformly on R.
On the other hand P k Ay(t) ∈ AA(R, V k ) since P k Ay(t) = AP k y(t) = Ay k (t). Then, Ay(t) ∈ AA(R, V ) and y ′ (t) ∈ AA(R, V ) since y(t) satisfies the equation (1.3). Therefore, using the Theorem 2.1, y ∈ AA(R, V ) since y ∈ BC(R, V ) and V is a Hilbert space. (1.4). Before start we recall the notation ∆(ϕ 0 , ρ) and ϕ 0 given on (2.1). Here, in this subsection, we present two results for (1.4) assuming fundamentally that g satisfies the assumptions given on (2.1) and f if a function such that the inequality
Theorem 3.7. Consider A(t) ∈ AA(R, C p×p ) such that (1.2) has an α-exponential dichotomy with a projection P that commutes with Φ A . Assume that g satisfies the assumptions given on (2.1) and f is selected such that the (3.32) holds. Then, if 4cL < α the equation (1.4) has a unique solution belongs AA(R, V p ).

Application to the Delayed Lasota-Wazewska Model
The Lasota-Wazewska model is an autonomous differential equation of the form y ′ (t) = −δy(t) + pe −γy(t−τ ) , t ≥ 0. (4.1) It was occupied by Wazewska-Czyzewska and Lasota [36] to describe the survival of red blood cells in the blood of an animal. In this equation, y(t) describes the number of red cells bloods in the time t, δ > 0 is the probability of death of a red blood cell; p, γ are positive constants related with the production of red blood cells by unity of time and τ is the time required to produce a red blood cell.
In this section, we study the following delayed model: where τ > 0, δ(·), p(·) are positive almost automorphic functions and g(·) is a positive Lipschitz function with Lipschitz constant γ. Equation (4.2) models several situations in the real life, see [22]. We will assume the following condition (D) The mean of δ satisfies M (δ) > δ − > 0. In this section, the principal goal is the following Theorem: Theorem 4.1. In the above conditions, for γ sufficiently small, the equation (4.1) has a unique almost automorphic solution.
By Lemma 2.3, the linear part of equation (4.1) has an exponential dichotomy. Let ψ(t) be a real almost automorphic function and consider the equation y ′ (t) = −δ(t)y(t) + p(t)g(ψ(t − τ )). The homogeneous part of equation of (4.3) has an exponential dichotomy and since δ is almost automorphic function, by Lemma 2.8, it is integrally Bi-almost automorphic. Therefore, Theorem 4.1 follows from Theorem 3.7.