# Stepanov-Like Asymptotical Almost Periodic Functions and an Application.

1. Introduction

Almost periodic functions, which are an important generalization of periodic functions, were introduced into the field of mathematics by Bohr [1, 2]. From the very beginning, the concept of almost periodic function has attracted extensive attention of mathematicians and has led to various extensions and variations of this concept. For example, Stepanov proposed a weaker concept of almost periodic functions in the sense of Bohr. For more details about Stepanov's almost periodic functions, see [3-11].

On the one hand, due to the fact that almost periodic phenomena exist in the real world, more and more scholars are interested in the almost periodicity and its various generalizations. For example, Diagana  introduced Stepanovlike pseudo almost periodicity in 2007. The Stepanov-like pseudo almost periodicity is a generalization of the classical pseudo almost periodicity . The concept of Stepanovlike weighted pseudo almost periodicity was introduced by Diagana et al. . This notion is more extensive than Stepanov-like pseudo almost periodicity. Moreover, Diagana also introduced Stepanov-like almost automorphic functions which are a generalization of the classical almost automorphic functions; for more details, see . In 2009, Diagana introduced the notion of Stepanov-like pseudo almost automorphy which generalizes the concept of pseudo almost automorphy .

On the other hand, the concept of the asymptotically almost periodicity was introduced into the research field by French mathematician Frechet [17,18]. Such a notion is a natural generalization of the concept of the almost periodicity in the sense of Bohr. Since then, asymptotically almost periodic functions have become a very important function class and to find asymptotically almost periodic solutions for differential equations has been a hot topic for researchers. For the basic properties of asymptotical almost periodic functions, we refer the reader to  and for some recent papers about the existence of asymptotically almost periodic solutions for differential equations arising in theory and application, we refer the reader to [20-25]. However, up to now, few studies have been done on Stepanov-like asymptotical almost periodic functions , but these studies are necessary.

Motivated by the above discussions, in this paper, we first study some basic properties of Stepanov-like asymptotical almost periodic functions. Then, based on these properties and by using the contraction mapping principle, we investigate the existence and uniqueness of Stepanovlike asymptotical almost periodic solutions for a class of semilinear delay differential equations.

2. Preliminaries

In this section, we recall some basis definitions and lemmas of Bohr almost periodic functions and Stepanov's almost periodic functions which are used throughout this paper.

Let (X, [parallel] * [[parallel].sub.x]) be a Banach space and BC(R,X) be the collection of bounded continuous functions from R to X with the norm [mathematical expression not reproducible].

Definition 1 (see ). A function f [member of] BC(R, X) is said to be almost periodic in Bohr sense if for each [member of] > 0 there exists l = l ([member of]) > 0 such that in every interval of length l of R one can find a number [tau] [member of] (a, a + l) with the property

[parallel]f (t + [tau]) - f (t)[ [parallel].sub.X] < [epsilon], t [member of] R. (1)

We denote the space of all such functions by AP(RX); the norm of the space is

[mathematical expression not reproducible]. (2)

Definition 2 (see ). Let A be a set of some almost periodic functions in Bohr sense. Then A is uniformly almost periodic family if it is uniformly bounded, equicontinuous and for every [epsilon] > 0 there exists a number l > 0 such that every interval of length l contains a number [tau] such that

[mathematical expression not reproducible] (3)

Lemma 3 (see ). Function f [member of] AP(R, X) is equivalent to the property of relative compactness for the family F = {f(t + h);h [member of] R}.

Definition 4 (see ). The space M is defined as follows:

[mathematical expression not reproducible] (4)

with the norm defined by

[mathematical expression not reproducible] (5)

Lemma 5 (see ). The space M(R, R) is a Banach space.

Definition 6 (see ). A function f [member of] M(R,R) is said to be Stepanov's almost periodic if for every [epsilon] >0 there exists l = 1([epsilon]) > 0 such that each interval (a, a + l) [subset] R contains a point [tau] with the property

[mathematical expression not reproducible] (6)

We denote the space of all such functions by S(R, R) and the norm of S(R, R) is

[mathematical expression not reproducible] (7)

Lemma 7 (see ). The space S(R, R) is a Banach space.

Lemma 8 (see ). A function f [member of] S(R,R) if and only if [phi]\$(*, *) [member of] AP(R, L([0,1], R)), where

[phi](t,x) = f(t + x), t [member of] R, x [member of] [0,1] [subset] R (8)

Lemma 9 (see ). A function f e S(R,R) if and only if F = [f(t+h);h [member of] R} [subset] S(R,R) is relatively compact in S(R,R).

Lemma 10 (see ). Let [f.sub.1], [f.sub.2], ..., fn be almost periodic functions in Bohr sense from R into Banach space [X.sub.1], [X.sub.2], ..., [X.sub.n], respectively. Then for each [epsilon] > 0, all the functions [f.sub.1], [f.sub.2], ..., [f.sub.n] have a common set of [epsilon]-almost periods.

3. Stepanov-Like Asymptotic Almost Periodic Functions and Their Basic Properties

Let

[mathematical expression not reproducible] (9)

then we give the following definition.

Definition 11 (see ). A function f [member of] M(R,R) is said to be a Stepanov-like asymptotical almost periodic function if it can be expressed as

f(t) = g(t) + h(t), (10)

where g [member of] S(R,R), h [member of] [M.sub.0](R,R). The collection of all such functions will be denoted by AS(R, R).

Remark 12. Several equivalent statements of Definition 11 are given by Theorem 1.6.2 in .

Remark 13. Obviously, if [f.sub.1], [f.sub.2] [member of] AS(R,R) and [lambda] [member of] R, then [f.sub.1] + [f.sub.2], [lambda][f.sub.1] [member of] AS(R,R).

Lemma 14. The space of [M.sub.0](R, R) is a Banach space endowed with the norm

[mathematical expression not reproducible] (11)

Proof. Let [[h.sub.n]} [subset] [M.sub.0] (R,R) [subset] M(R,R) be a Cauchy sequence, then we can find a function h [member of] M(R, R) such that

[mathematical expression not reproducible] (12)

Hence

[mathematical expression not reproducible] (13)

Since [mathematical expression not reproducible], we deduce that

[mathematical expression not reproducible], (14)

that is, h [member of] [M.sub.0](R, R). Thus, [M.sub.0](R, R) is a Banach space. The proof is complete.

Lemma 15. Let f = g + h [member of] AS(R,R),g [member of] S(R,R),h [member of] [M.sub.0](R, R) and g(t + x) [member of] AP(R, L([0,1], R)), h(t + x) : R [right arrow] L([0,1],R),f(t + x) :R [right arrow] L([0,1], R), x [member of] [0,1], then

[mathematical expression not reproducible] (15)

Proof. Assume that (15) does not hold, then there exist [t.sub.0] [member of] R and [epsilon] > 0 such that

[mathematical expression not reproducible] (16)

Since g(t + x) [member of] AP(R, L([0,1], R)), there exists l > 0 and for every n [member of] Z, there exists [[tau].sub.n] [member of] [nl - [t.sub.0], nl - [t.sub.0] + l] such that

[mathematical expression not reproducible] (17)

By using the uniform continuity on R of the almost periodic function g(t + x), there exists [K.sub.0] [member of] N such that [K.sub.0] [greater than or equal to] 2 and for all t [member of] [[t.sub.0] + [[tau].sub.n] - l/[K.sub.0], [t.sub.0] + [[tau].sub.n] + l/[K.sub.0]],

[mathematical expression not reproducible] (18)

From (16)-(18), it follows that

[mathematical expression not reproducible] (19)

Since f = g + h, by (19), we have

[mathematical expression not reproducible] (20)

Thus,

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

Consequently, (15) holds. The proof is complete.

Definition 16. A function g [member of] S(R x X, R) if the following three conditions are true:

(i) for every x [member of] X, g(*, x) [member of] S(R, R),

(ii) the set [g(*, x) | x [member of] X] is uniformly bounded in the S-norm and equicontinuous in the S-norm,

(iii) for every [epsilon] > 0, there exists a number l > 0 such that every interval of length l contains a number [tau] with the property

[mathematical expression not reproducible]. (23)

Lemma 17. For a bounded continuous function f : R [right arrow] X, denote

[mathematical expression not reproducible]. (24)

Then [V.sub.f] satisfies the following properties:

(a) [V.sub.f]([tau]) [greater than or equal to] 0, [V.sub.f] (-[tau]) = [V.sub.f]([tau]), [tau] [member of] R.

(b) [V.sub.f] (0) = 0.

(c) [V.sub.f](t + s) [less than or equal to] [V.sub.f](t) + [V.sub.f](s),s, t [member of] R

(d) f [member of] AP(R,X) if and only if [V.sub.f] [member of] AP(R,R).

Proof. Properties (a) through (c) are easy to show. We only prove the property (d). If f [member of] AP(R, X), then for each [epsilon] >0 there exists l > 0 such that in any interval of length l of R one can find a number t [member of](a, a+ T) with the property

[mathematical expression not reproducible]. (25)

From (c), we have [mathematical expression not reproducible]. That is,

[mathematical expression not reproducible]. (26)

Hence,

[mathematical expression not reproducible] (27)

It follows from (25) and (27) that

[mathematical expression not reproducible] (28)

Since f [member of] AP(R, X), it is uniformly continuous. So, we know that, for every [member of] >0, there exists [delta] > 0 such that if [absolute value of h] < [delta] then [mathematical expression not reproducible]. Therefore, [V.sub.f](h) < [epsilon] and by (27) we get [mathematical expression not reproducible], which implies that [V.sub.f] (t) is continuous. Thus, [V.sub.f] (t) [member of] AP(R, R).

Conversely, if [V.sub.f] [member of] AP(R, R), then for each [epsilon] > 0 there exists l > 0 such that in any interval of length l of R one can find a number [tau] [member of](a,a + l) with the property

[mathematical expression not reproducible]. (29)

Hence, [mathematical expression not reproducible]. Since [V.sub.f] is continuous and [V.sub.f](0) = 0, then for every [epsilon] >0, there exists [delta] > 0 such that if [absolute value of h] < [delta] we have [V.sub.f](h) < e[epsilon]. Therefore, [mathematical expression not reproducible], which implies f is continuous. Thus, f [member of] AP(R, X).

Lemma 18. Let f [member of] [LAMBDA], where A consists of some almost periodic functions in Bohr sense from R to X, [mathematical expression not reproducible] is finite. Then the family [LAMBDA] is uniformly almost periodic if and only if V [member of] AP(R, R).

Proof. It is easy to find that V satisfies the following properties:

(i) V(t) [greater than or equal to] 0,V (-t) = V(t),

(ii) V(0) = 0.

Moreover, by (c) in Lemma 17 we have

[mathematical expression not reproducible] (30)

Similar to the proof of (27), we have

[mathematical expression not reproducible] (31)

If V [member of] AP(R, R), then for each [epsilon] > 0, there exists l > 0 such that in any interval of length l of R one can find a number [tau] [member of] (a,a + l) with the property

[mathematical expression not reproducible]. (32)

Noticing that V(0) = 0,hence, V([tau]) < [member of]. Therefore, [V.sub.f]([tau]) < [epsilon] for all f [member of] [lambda]. So, the family A is a uniformly almost periodic family.

Conversely, suppose that A is a uniformly almost periodic family. Then for each [epsilon] >0 there exists l > 0 such that in any interval of length l of R one can find a number [tau] [epsilon](a, a+ 1) with the property

[mathematical expression not reproducible]. (33)

From (33), we obtain that [V.sub.f]([tau]) < [epsilon], [for all]f [member of] [LAMBDA] and V([tau]) [less than or equal to] e. By (31) we obtain that

[mathematical expression not reproducible]. (34)

Besides, since f [member of] [LAMBDA] is continuous,

[mathematical expression not reproducible] (35)

By (31) we obtain that [mathematical expression not reproducible] as h [right arrow] 0, which implies that V is continuous. Therefore, V [member of] AP(R, R). The proof is completed.

Lemma 19. Let [LAMBDA] be a uniformly almost periodic family in Bohr sense. Then given a sequence {[[alpha]'.sub.n]}, there exists a subsequence [[[alpha]'.sub.n]] [subset] {[[alpha]'.sub.n]} satisfying the following property: for every [epsilon] > 0, there exists a constant N such that

[mathematical expression not reproducible] (36)

n,m > N, for all t [member of] R and all f [member of] A.

Proof. According to Lemma 18, we know that V(t) = [mathematical expression not reproducible] (t) is almost periodic. By Lemma 3 we obtain that, for every sequence {[[alpha].sub.'n]}, there exists a subsequence {[[alpha].sub.n]} satisfying the following property: for each [epsilon] > 0, there exists a constant N > 0 such that

[mathematical expression not reproducible]. (37)

When [mathematical expression not reproducible]. According to the definition of [V.sub.f], we have

[mathematical expression not reproducible] (38)

Hence, it is easy to see that

[mathematical expression not reproducible] (39)

The proof is completed.

Theorem 20. Let g e S(R x X,R). Then for every sequence {[[alpha].sub.'n]}, there exists a subsequence {[[alpha].sub.n]} c {[[alpha].sub.'n]} such that {g(t + [[alpha].sub.'n],x),n> 1} is convergent uniformly with respect to x [member of] X.

Proof. Since g [member of] S(R x X,R), we know that, for every [epsilon] >0, there exists a number l > 0 such that every interval of length l contains a number [tau] satisfying

[mathematical expression not reproducible]. (40)

According to (40), we have

[mathematical expression not reproducible] (41)

Thus,

[mathematical expression not reproducible] (42)

From Lemma 8, we know that, for every fixed x, g(t + u, x) e AP(R, L([0,1], R)). Therefore, (42) implies that {g(t + u,x),x e X} is a uniformly almost periodic family. From Lemma 19, we have that, for every sequence [[[alpha].sub.'n]}, there exists a subsequence {[[alpha].sub.n]} c {[[alpha].sub.'n]} such that, for every [epsilon] > 0,

[mathematical expression not reproducible] (43)

Hence, {g(t + [[alpha].sub.'n], x),n [greater than or equal to] 1} is convergent uniformly with respect to x [member of] X.

Definition 21. A function h [member of] [M.sub.0](R x X,R) if the following two conditions are true:

(i) for every x [member of] X, h(*, x) e [M.sub.0](R, R),

(ii) for every [epsilon] >0, there exists a constant T > 0 such that

[mathematical expression not reproducible]. (44)

Definition 22. A function f [member of] AS(R x X,R) if it can be expressed as

f = g + h, (45)

where g [member of] S(R x X, R), h [member of] [M.sub.0] (R x X, R).

Theorem 23. The space of AS(R, R) is a Banach space endowed with the norm

[mathematical expression not reproducible]. (46)

Proof. Let {[f.sub.n]} [subset] AS(R,R) be a Cauchy sequence; i.e., for each [epsilon] >0, there exists a natural number N >0 such that

[mathematical expression not reproducible]. (47)

Let [mathematical expression not reproducible]. By using Lemma 15, we have

[mathematical expression not reproducible] (48)

Then, for every [bar.[epsilon]] > 0,

[mathematical expression not reproducible] (49)

According to the arbitrariness of [bar.e], we obtain

[mathematical expression not reproducible] (50)

Thus,

[mathematical expression not reproducible] (51)

which means that {[g.sub.n]} [subset] S(R, R) is a Cauchy sequence. So [mathematical expression not reproducible]. Similarly, we can obtain

[mathematical expression not reproducible] (52)

According to Lemma 14, we obtain [mathematical expression not reproducible]. The proof is completed.

4. Stepanov-Like Asymptotical Almost Periodic Solutions of Semilinear Delay Differential Equations

In this section, we investigate the existence and uniqueness of Stepanov-like asymptotical almost periodic solutions for the following semilinear differential equation:

x (t) + A(t)x(t) = f (t,x(t),x (t-[tau])), t [member of] R, (53)

where A [member of] AP(R, R), f [member of] AS(R x R x R, R), and [tau] > 0 is a constant.

We make some assumption:

([H.sub.1]) A [member of] AP(R, R) and A* = [inf.sub.t[member of]R] A(t) > 0.

([H.sub.2]) [mathematical expression not reproducible]

(H3) There exist constants [L.sub.1], [L.sub.2] > 0 such that, for all t [member of] R and for all [u.sub.1], [u.sub.2], [v.sub.1], [v.sub.2] [member of] R,

[mathematical expression not reproducible] (54)

Lemma 24. Let x [member of] AS(R,R) and ([H.sub.2]), ([H.sub.3]) hold; then f(*,x(*),x(*-[tau])) [member of] AS(R,R).

Proof. Since x [member of] AS(R, R), we have x = [x.sub.1] + [x.sub.2], where [x.sub.1] [member of] S(R, R), [x.sub.2] [member of] [M.sub.0](R, R). Then function f(t, x(t), x(t - [tau])) can be written in the form:

[mathematical expression not reproducible] (55)

Step 1. We prove [f.sub.1](*, [x.sub.1](*), [x.sub.1](* - tau])) [member of] S(R, R). Let [epsilon] > 0. By Lemma 9, for every sequence {[h.sub.k]; k [greater than or equal to] 1} [subset] R, there exists a subsequence {[h.sub.1k]; k [greater than or equal to] 1} [subset] {[h.sub.k]; k [greater than or equal to] 1} such that the sequence {[x.sub.1](t+hik);k > 1} is convergent. From Theorem 20, it follows that {[f.sub.1](t + [h.sub.1k], u,v); k [greater than or equal to] 1} is also convergent uniformly with respect to u, v [member of] R. Therefore, for any [epsilon] > 0, there exist positive integers [N.sub.1] and [N.sub.2] such that

[mathematical expression not reproducible] (56)

and

[mathematical expression not reproducible] (57)

Hence

[mathematical expression not reproducible] (58)

for k, p [greater than or equal to] = max {N.sub.l][N.sub.2]

Therefore, [f.sub.1](*, [x.sub.1] (*), [x.sub.1](* - [tau])) [member of] S(R, R).

Step 2. We prove that

[mathematical expression not reproducible] (59)

and

[mathematical expression not reproducible] (60)

From (H3), we have

[mathematical expression not reproducible] (61)

According to the definition of f and Definition 21, we have

[mathematical expression not reproducible] (62)

and

[mathematical expression not reproducible] (63)

Hence, f(*, x(*), x(* - t)) e AS(R, R). The proof is complete.

Theorem 25. Assume ([H.sub.1])-([H.sub.3]) hold. If [DELTA] <1, where [DELTA] = 4(L/][DELTA]*), L = max{[L.sub.1] [,.sub.L]2}, then system (53) has a unique Stepanov-like asymptotical almost periodic solution.

Proof. For any [phi] [member of] AS(R, R), consider the linear differential equation

x (t) + A(t)x (t) = f(t,[phi] (t), [phi](t-[tau])). (64)

Since ([H.sub.1]) holds, by the exponential dichotomy of linear differential equation, (64) has a unique bounded solution

[mathematical expression not reproducible] (65)

Define an operator F : AS(R, R) [right arrow] M(R, R) by setting F[phi] = [x.sup.[phi]] for every [[phi] [member of] AS(R,R).

Step 1 (F is self-mapping). From Lemma 24, we have [mathematical expression not reproducible], and define

[mathematical expression not reproducible] (66)

By Lemmas 8 and 10, there exists l > 0 such that every interval of length l contains a number [theta] such that

[mathematical expression not reproducible] (67)

Then

[mathematical expression not reproducible] (68)

where [sigma] = s-w. Hence, [F.sub.1] [member of] S(R, R).

Since [mathematical expression not reproducible], we have

[mathematical expression not reproducible] (69)

where [xi] = s - w. According to Lebesgue's dominated convergence theorem, we obtain F2 [member of] [M.sub.0](R,R). Therefore, F is self-mapping.

Step 2 (F is a contraction mapping). [for all] [phi][psi] [member of]AS(R,R), we have

[mathematical expression not reproducible] (70)

Hence, T has a unique fixed point in AS(R,R). Therefore, system (53) has a unique Stepanov-like asymptotical almost periodic solution. The proof is complete.

An example: consider the following equation:

[mathematical expression not reproducible] (71)

where [tau] is a positive constant. In this case, A(t) = sin t + 5,

[mathematical expression not reproducible] (72)

Obviously, A* = 4,

[mathematical expression not reproducible] (73)

Hence, 4L/A* = 1/2 < 1. Thus, all the conditions of Theorem 25 are satisfied. By Theorem 25, system (71) has a unique Stepanov-like asymptotical almost periodic solution.

https://doi.org/10.1155/2018/8208323

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of China under Grants No. 11861072 and No. 11361072.

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Yongkun Li (iD), Yaolu Wang, Jianglian Xiang

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Correspondence should be addressed to Yongkun Li; yklie@ynu.edu.cn

Received 20 September 2018; Revised 12 November 2018; Accepted 21 November 2018; Published 4 December 2018