Almost Automorphic Solutions in the Sense of Besicovitch to Nonautonomous Semilinear Evolution Equations

: In this paper, we study the existence of almost automorphic solutions in the sense of Besicovitch for a class of semilinear evolution equations. Firstly, we study some basic properties of almost automorphic functions in the sense of Besicovitch, including the composition theorem. Then, by using the Banach ﬁxed point theorem, the existence of almost automorphic solutions in the sense of Besicovitch to the semilinear equation is obtained. Finally, we give an example of partial differential equations to illustrate the applicability of our results.


Introduction
Consider the nonautonomous semilinear evolution equation: where for each t ∈ R, A(t) : D(A(t)) ⊂ X → X is a closed and densely defined linear operator on D = D(A(t)) satisfying the so-called Acquistapace and Terreni conditions: (S 1 ) There are constants λ 0 ≥ 0, θ ∈ ( π 2 , π) and K 1 ≥ 0 such that Σ θ ∪ {0} ⊂ ρ(A(t) − λ 0 ) and for all x ∈ Σ 0 ∪ {0}, t ∈ R, (H 1 ) Function f ∈ AAB p (R × X, X) satisfies the Lipschitz condition with respect to its second argument and uniformly in its first argument, that is, there exists a positive constant L such that for all x, y ∈ X and t ∈ R, and f (t, 0) = 0. (H 2 ) The evolutionary family {U(t, s), t ≥ s} is exponentially stable, that is, there exist numbers M, λ > 0 such that U(t, s) ≤ Me −λ(t−s) , t ≥ s; (H 3 ) U(t, s)x ∈ BAA(R × R, X) uniformly for x in any bounded subset of X; (H 4 ) The constant κ := ML λ < 1, where M is mentioned in (H 3 ). (1) is a continuous function x : R → X satisfying:
If U(t, s) is exponentially stable, then we have: The concept of almost automorphic functions as a generalization of the concept of almost periodic functions was introduced into mathematics by Bochner [2], and once this concept was put forward, its various generalizations were constantly put forward, such as asymptotic almost automorphic functions, pseudo almost automorphic functions, Stepanov almost automorphic functions and so on [3][4][5][6]. At the same time, the study of almost periodic solutions and almost automorphic solutions of differential equations has become an important part of the qualitative theory of differential equations. Almost automorphy in the sense of Besicovitch is a generalization of those concepts mentioned above, but so far, the results of almost automorphic solutions in the sense of Besicovitch of differential equations are still very rare [7]. It is worth mentioning that Reference [7] studies the existence of 1-almost automorphic solutions in the sense of Besicovitch for a class of first-order nonautonomous linear differential equations. However, Reference [7] does not involve the concept of uniform almost automorphic functions in the sense of Besicovitch, nor does it give the composite theorem of almost automorphic functions in the sense of Besicovitch, which are necessary concepts and tools to study the existence of almost automorphic solutions in the sense of Besicovitch for nonlinear differential equations. Therefore, it is very meaningful to study the composition theorem of almost automorphic functions in the sense of Besicovitch and the existence of almost automorphic solutions in the sense of Besicovitch of nonlinear differential equations.
Motivated by the above discussion, in this paper, we first give the concepts of almost automorphic and uniform almost automorphic functions in the sense of Besicovitch defined by Bochner property, and study some of their basic properties, including composition theorems. Then, we study the existence of p-almost automorphic solutions in the sense of Besicovitch of system (1) by using the Banach fixed point theorem. The results of our paper are new.
The rest of this paper is arranged as follows: in Section 2, we study some basic properties of almost automorphic functions in the sense of Besicovitch. In Section 3, we use the results obtained in Section 2 and the Banach fixed point theorem to establish the existence of almost automorphic solutions in the sense of Besicovitch of system (1). In Section 4, we provide an example to illustrate the applicability of our results. Finally, in Section 5, we present a brief conclusion.

Besicovitch Almost Automorphic Functions and Their Some Properties
Let (X, · X ) be a Banach space, C(R, X) be the set of all continuous functions from R to X and BC(R, X) be the space of all bounded continuous functions from R to X. We denote by L ∞ (R, X) the set of all functions f : R → X that are measurable and essentially bounded. The space L ∞ (R, X) is a Banach space with the norm Definition 2 ([8]). Let f ∈ C(R, X), then f is called (Bohr) almost periodic if for each ε > 0, there exists l = l(ε) > 0 such that in every interval of length l of R one can find a number τ ∈ (a, a + l) with the property (Bohr property): The collection of such Bohr almost periodic functions will be denoted by AP(R, X).

Definition 3 ([5])
. A function f ∈ C(R, X) is almost automorphic in Bochner's sense if for every sequence of real numbers (S n ) n∈N , there exists a subsequence (S n ) n∈N such that: is well defined for each t ∈ R, and for each t ∈ R. The set of all such functions will be denoted by AA(R, X). If

Definition 4 ([9]).
A continuous function f : R × R → X is said to be bi-almost automorphic if for every sequence of real numbers, (S n ) n∈N , there exists a subsequence (S n ) n∈N such that: is well defined in t, s ∈ R, and for each t, s ∈ R. The collection of all such functions will be denoted by BAA(R × R, X).
For p ∈ [1, ∞), let f ∈ L p loc (R, X) be the collection of all locally p-integrable functions from R to X. For f ∈ L p loc (R, X), we consider the following seminorm: We denote by BB p (R, X) the set of all such functions.
Similar to the definitions of the corresponding concepts in [5,7,8,[10][11][12], we give the following definition: Definition 6. A function f ∈ BB p (R, X) is said to be p-almost automorphic in the sense of Besicovitch, if for every sequence of real numbers (S n ) n∈N , there exists a subsequence (S n ) n∈N such thatf We denote by AAB p (R, X) the collection of all such functions.
From Definitions 2, 3 and 6, it is easy to see that:

Remark 1.
Obviously, the convergence in Definition 6 is uniformly in t ∈ R, so the almost automorphy defined in Definition 6 is corresponding to the compact almost automorphy of Definition 3.

Remark 2.
Because the functions that are asymptotic to zero and the functions whose integral averages are zero belong to the zero space of semi norm · B p . Therefore, for the almost automorphic functions in the sense of Besicovitch, there are no concepts of asymptotic almost automorphic functions and pseudo almost automorphic functions.

Example 1.
According to Example 4.4 in [8], we see that: Since f B p = 0 and g B p = 0, where f (t) = e −|t| and g(t) = 1 1+t 2 , we have: If a function f ∈ BB p (R, X) is almost automorphic in the sense of Besicovitch, thenf is B p -bounded, wheref is mentioned in Definition 6.
Proof. Since f ∈ BB p (R, X), for every sequence of real numbers (S n ) n∈N , we can extract a subsequence (S n ) n∈N such that for each t ∈ R and ε = 1, we have The proof is complete.
Proof. Since f , g ∈ AAB p (R, X), for every sequence of real numbers (S n ) n∈N , we can extract a subsequence (S n ) n∈N such that for each t ∈ R and any ε > 0, there exists N 1 (t, ε) ∈ N, when n > N 1 , Meanwhile, for (S n ) n∈N , there exist a subsequence (S n ) n∈N of (S n ) n∈N and N 2 (t, ε) ∈ N, when n > N 2 , Consequently, we arrive at, for n > N 0 , Similarly, for n > N 0 , one can get The proof of λ f ∈ AAB p (R, X) is trivial and we will omit it here. The proof of Lemma 3 is complete.
Proof. Since f ∈ AAB p (R, X), for every sequence of real numbers (S n ) n∈N , we can extract a subsequence (S n ) n∈N such that Theorem 1. If f ∈ C(X, X) satisfies the Lipschitz condition and x ∈ AAB p (R, X), then f (x(·)) belongs to AAB p (R, X).
Proof. It is easy to obtain that f (x(·)) ∈ BB p (R, X). Since x ∈ AAB p (R, X), for every sequence of real numbers (S n ) n∈N , there exists a subsequence (S n ) n∈N such that for any ε > 0 and each t ∈ R, there exists a positive number N(ε, t), when n > N, Consequently, Similarly, one can prove that The proof is completed.
Lemma 5. Let X be a Banach algebra. If f ∈ AP(R, X) and g ∈ AAB p (R, X), then f · g ∈ AAB p (R, X).

Proof.
Noting that: we have f · g ∈ BB p (R, X).
On the other hand, since f ∈ AP(R, X), by Bochner property of Bohr almost periodic functions [8], for any sequence (S n ) n∈N of real numbers, there exists a subsequence (S n ) n∈N of (S n ) n∈N such that for every ε > 0, there is N 1 > 0 such that for n > N 1 . Moreover, since g ∈ AAB p (R, X), there exists a subsequence (S n ) n∈N of (S n ) n∈N and N 2 > 0 such that Noting that By (4), (5) and Lemma 2, we derive that: which implies that: Similarly, we can get: Consequently, f · g ∈ AAB p (R, X). The proof is completed.

Lemma 6 ([8]
). The space ((BB p (R, X), · B p ) is a linear space, which is complete with respect to the seminorm · B p . Lemma 7. Let ( f n ) n∈N be a sequence of p-almost automorphic functions in the sense of Besicovitch such that lim n→∞ f n (t) = f (t) uniformly in t ∈ R with respect to the seminorm · B p . Then f is p-almost automorphic in the sense of Besicovitch.
Proof. Let (S k ) n∈N be an arbitrary sequence of real numbers. By the diagonal procedure we can extract a subsequence (S k ) n∈N of (S k ) n∈N such that: f n (t + S k ) −f n (t) B p = 0 as k → ∞ for each n ∈ N and each t ∈ R. Noting that (6) For any ε > 0. By the uniform convergence of ( f n ) n∈N , we can find a positive integer N such that when n, m > N, for all t ∈ R and all k ∈ N, we have This, combined with (6), implies that (f n ) n∈N is a Cauchy sequence. Hence, by Lemma 6, we can deduce the pointwise convergence of the sequence (f n ) n∈N , say to a functionf (t).
Let us prove now that: pointwise on R.
Indeed, for each n ∈ N, we get: Again, by the uniform convergence of ( f n ) n∈N , for an arbitrary ε > 0, we can find some positive integer N 0 such that for every t ∈ R and k ∈ N, Consequently, for every t ∈ R and k ∈ N, we have Now for every t ∈ R, we can find some positive integer N 1 = N(t, N 0 ) such that Finally, we get: for n >N, whereN is some positive integer depending on t and ε. That is, we have proven that By the same method, one can prove that for each t ∈ R. The proof is complete.

Theorem 2.
The space ((AAB p (R, X), · B p ) is a linear space, which is complete with respect to the seminorm · B p .
Proof. According to Lemma 6, BB p (R, X) is complete with respect to the seminorm · B p . Since AAB p (R, X) ⊂ BB p (R, X), by Lemma 7, AAB p (R, X) is a closed subset of BB p (R, X). Consequently, AAB p (R, X) is complete with respect to the seminorm · B p . The proof is completed.
x) with f (·, x) ∈ BB p (R, X) for each x ∈ X, is said to be Besicovitch almost automorphic in t ∈ R uniformly in x ∈ X if for every sequence of real numbers (S n ) n∈N , there exists a subsequence (S n ) n∈N such that is well defined for each t ∈ R, and for each t ∈ R, uniformly in x ∈ X. That is, for each t ∈ R, uniformly in x ∈ X. The collection of these functions will be denoted by AAB p (R × X, X).
Theorem 3. If f ∈ AAB p (R × X, X) satisfies the Lipschitz condition respect to its second argument and uniformly in its first argument, and g ∈ AAB p (R, X), then f (·, g(·)) belongs to AAB p (R, X).
Proof. In view of the definition of the seminorm · B p and by the Lipschitz condition, one can easily get f (·, g(·)) ∈ BB p (R, X). Since g ∈ AAB p (R, X) and f ∈ AAB p (R × X, X), for every sequence of real numbers (S n ) n∈N , one can extract a subsequence (S n ) n∈N ⊂ (S n ) n∈N such that for every ε > 0, t ∈ R and every bounded subset B ⊂ X, there exists a positive number N(ε, t, B) satisfying for n > N, for each t ∈ R and x ∈ B. Therefore, that is, lim n→∞ f (t + S n , g(t + S n )) −f (t,g(t)) B p = 0. Similarly, one can prove that lim n→∞ f (t − S n ,g(t − S n )) − f (t, g(t)) B p = 0.
The proof is completed. Remark 3. Theorems 1 and 3 are called the composition theorems.

Besicovitch Almost Automorphic Solutions
Before stating and proving our existence theorem, we need to give two lemmas.
Proof. Let { f n ; n ≥ 1} be an arbitrary Cauchy sequence in W. Since (L ∞ (R, X), · ∞ ) is a Banach space and { f n } ⊂ L ∞ (R, X) it follows that there exists f ∈ L ∞ (R, X) such that Hence, to show (W, · W ) is a Banach space, it suffices to show f ∈ AAB p (R, X)}. Noting that f n − f B p ≤ f n − f ∞ , consequently, By Lemma 7, we conclude that f ∈ AAB p (R, X). The proof is complete.
belongs to W.
Proof. In view of Lemma 9, we know that f (·, x(·)) ∈ W. Our first task is to show that the integral on the right hand of formula (9) exists. Since (H 2 ) and (H 3 ), we have: which yields that that is to say, (9) is well defined and, as a byproduct, we have obtained that Φx ∈ L ∞ (R, X). Next, we will show that Φx ∈ AAB p (R, X). Because x ∈ W(R, X), then by Lemma 9, we have f (·, x(·)) ∈ W ⊂ AAB p (R, X), which combines with (H 3 ), for given any bounded subset Ω ⊂ X and every sequence {S n }, we can select a subsequence {S n } ⊂ {S n } such that: and Obviously, Ψ is well-defined and maps W into W according to Lemma 10. We just have to show that Ψ : W → W is a contraction mapping. In fact, for any x, y ∈ W, which, combined with (H 4 ), yields Hence, Ψ is a contraction mapping from W to W. Noting the fact that (W, · W ) is a Banach space; therefore, according to the Banach fixed point theorem, Ψ has a unique fixed point z * ∈ W such that Tz * = z * . Consequently, system (10) has a unique p-almost automorphic mild solution z * in the sense of Besicovitch. The proof is complete.

An Example
Consider the following partial differential equation with Dirichlet boundary conditions: u(t, ξ) Take X = L 2 [0, π] with norm · and inner product (·, ·) 2 . Define: A : D(A) ⊂ X → X given by According to [13], we know that A is the infinitesimal generator of an analytic semigroup {T(t)} t≥0 on X satisfying T(t) ≤ e −3t for t > 0.
In addition, where y n (x) = 2 π sin(nx). Define a family of linear operators A(t) by D(A(t)) = D(A), Then, the system has an associated evolution family {U(t, s)} t≥s on X, which can be explicitly express by It is easy to see that for any sequence {S n } ⊂ R, we have: Since, as mentioned before, the function sin is almost automorphic, U(t, s) is bi-almost automorphic. Thus, (H 3 ) is verified. Moreover, U(t, s) ≤ e −2(t−s) for t ≥ s.
Consequently, by Theorem 4, system (14) has a unique almost automorphic mild solution in the sense of Besicovitch.

Conclusions
In this paper, some basic properties of almost automorphic functions in the sense of Besicovitch defined by Bochner properties are studied, and on this basis, the existence of palmost automorphic solutions in the sense of Besicovitch for a class of semilinear evolution equations is established. The results of this paper are new. At the same time, the results and methods of this paper can be used to study the existence of p-almost automorphic solutions in the sense of Besicovitch for other types of semilinear evolution equations. For example, semilinear evolution equations with time delays and semilinear differential integral equations.  Acknowledgments: The authors would like to thank the Editor and the anonymous referees for their helpful comments and valuable suggestions regarding this article.

Conflicts of Interest:
The authors declare no conflict of interest.