Stepanov-Like Asymptotical Almost Periodic Functions and an Application

Li, Yongkun; Wang, Yaolu; Xiang, Jianglian

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

Journal of Function Spaces

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 8208323 | https://doi.org/10.1155/2018/8208323

Stepanov-Like Asymptotical Almost Periodic Functions and an Application

Yongkun Li,¹Yaolu Wang,¹and Jianglian Xiang¹

Academic Editor: Richard I. Avery

Received20 Sept 2018

Revised12 Nov 2018

Accepted21 Nov 2018

Published04 Dec 2018

Abstract

In this paper, we first study some basic properties of Stepanov-like asymptotical almost periodic functions including the completeness of the space of Stepanov-like asymptotical almost periodic functions. Then, as an application, based on these and the contraction mapping principle, we obtain sufficient conditions for the existence and uniqueness of Stepanov-like asymptotical almost periodic solutions for a class of semilinear delay differential equations.

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 [12] introduced Stepanov-like pseudo almost periodicity in 2007. The Stepanov-like pseudo almost periodicity is a generalization of the classical pseudo almost periodicity [13]. The concept of Stepanov-like weighted pseudo almost periodicity was introduced by Diagana et al. [14]. 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 [15]. In 2009, Diagana introduced the notion of Stepanov-like pseudo almost automorphy which generalizes the concept of pseudo almost automorphy [16].

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 [19] 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 [26], 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 Stepanov-like 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 be a Banach space and be the collection of bounded continuous functions from to with the norm .

Definition 1 (see [27]). A function is said to be almost periodic in Bohr sense if for each there exists such that in every interval of length of one can find a number with the property We denote the space of all such functions by ; the norm of the space is

Definition 2 (see [28]). Let be a set of some almost periodic functions in Bohr sense. Then is uniformly almost periodic family if it is uniformly bounded, equicontinuous and for every there exists a number such that every interval of length contains a number such that

Lemma 3 (see [27]). Function is equivalent to the property of relative compactness for the family .

Definition 4 (see [27]). The space is defined as follows: with the norm defined by

Lemma 5 (see [27]). The space is a Banach space.

Definition 6 (see [27]). A function is said to be Stepanov’s almost periodic if for every there exists such that each interval contains a point with the property We denote the space of all such functions by and the norm of is

Lemma 7 (see [27]). The space is a Banach space.

Lemma 8 (see [27]). A function if and only if , where

Lemma 9 (see [27]). A function if and only if is relatively compact in .

Lemma 10 (see [29]). Let be almost periodic functions in Bohr sense from into Banach space , respectively. Then for each , all the functions , have a common set of -almost periods.

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

Let then we give the following definition.

Definition 11 (see [26]). A function is said to be a Stepanov-like asymptotical almost periodic function if it can be expressed as where , . The collection of all such functions will be denoted by .

Remark 12. Several equivalent statements of Definition 11 are given by Theorem 1.6.2 in [26].

Remark 13. Obviously, if and , then .

Lemma 14. The space of is a Banach space endowed with the norm

Proof. Let be a Cauchy sequence, then we can find a function such that Hence Since , we deduce that that is, . Thus, is a Banach space. The proof is complete.

Lemma 15. Let and , then

Proof. Assume that (15) does not hold, then there exist and such thatSince , there exists and for every , there exists such thatBy using the uniform continuity on of the almost periodic function , there exists such that and for all ,From (16)-(18), it follows thatSince , by (19), we have Thus, which contradicts Consequently, (15) holds. The proof is complete.

Definition 16. A function if the following three conditions are true: (i)for every , ,(ii)the set is uniformly bounded in the -norm and equicontinuous in the -norm,(iii)for every , there exists a number such that every interval of length contains a number with the property

Lemma 17. For a bounded continuous function , denote Then satisfies the following properties: (a).(b)(c)(d) if and only if .

Proof. Properties through are easy to show. We only prove the property . If , then for each there exists such that in any interval of length of one can find a number with the propertyFrom , we have and , then . That is, Hence,It follows from (25) and (27) that Since , it is uniformly continuous. So, we know that, for every , there exists such that if then . Therefore, and by (27) we get , which implies that is continuous. Thus, .
Conversely, if , then for each there exists such that in any interval of length of one can find a number with the propertyHence, . Since is continuous and , then for every , there exists such that if we have Therefore, , which implies is continuous. Thus, .

Lemma 18. Let , where consists of some almost periodic functions in Bohr sense from to , is finite. Then the family is uniformly almost periodic if and only if .

Proof. It is easy to find that satisfies the following properties: (i),(ii). Moreover, by in Lemma 17 we haveSimilar to the proof of (27), we haveIf , then for each , there exists such that in any interval of length of one can find a number with the property Noticing that , hence, . Therefore, for all . So, the family is a uniformly almost periodic family.
Conversely, suppose that is a uniformly almost periodic family. Then for each there exists such that in any interval of length of one can find a number with the propertyFrom (33), we obtain that and . By (31) we obtain that Besides, since is continuous, By (31) we obtain that as , which implies that is continuous. Therefore, . The proof is completed.

Lemma 19. Let be a uniformly almost periodic family in Bohr sense. Then given a sequence , there exists a subsequence satisfying the following property: for every , there exists a constant such that

Proof. According to Lemma 18, we know that is almost periodic. By Lemma 3 we obtain that, for every sequence , there exists a subsequence satisfying the following property: for each , there exists a constant such that When , we obtain . Thus, for all and . According to the definition of , we have Hence, it is easy to see that The proof is completed.

Theorem 20. Let . Then for every sequence , there exists a subsequence such that is convergent uniformly with respect to .

Proof. Since , we know that, for every , there exists a number such that every interval of length contains a number satisfyingAccording to (40), we have Thus,From Lemma 8, we know that, for every fixed , . Therefore, (42) implies that is a uniformly almost periodic family. From Lemma 19, we have that, for every sequence , there exists a subsequence such that, for every , Hence, is convergent uniformly with respect to .

Definition 21. A function if the following two conditions are true: (i)for every , ,(ii)for every , there exists a constant such that

Definition 22. A function if it can be expressed as where , .

Theorem 23. The space of is a Banach space endowed with the norm

Proof. Let be a Cauchy sequence; i.e., for each , there exists a natural number such that Let , where . By using Lemma 15, we have Then, for every , According to the arbitrariness of , we obtain Thus, which means that is a Cauchy sequence. So . Similarly, we can obtain According to Lemma 14, we obtain and . 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:where , , and is a constant.

We make some assumption: and , where , .There exist constants such that, for all and for all ,

Lemma 24. Let and hold; then .

Proof. Since , we have , where . Then function can be written in the form: Step 1. We prove . Let . By Lemma 9, for every sequence , there exists a subsequence such that the sequence is convergent. From Theorem 20, it follows that is also convergent uniformly with respect to . Therefore, for any , there exist positive integers and such that and Hence Therefore, .
Step 2. We prove that and From , we have According to the definition of and Definition 21, we have and Hence, . The proof is complete.

Theorem 25. Assume - hold. If , where , then system (53) has a unique Stepanov-like asymptotical almost periodic solution.

Proof. For any , consider the linear differential equationSince holds, by the exponential dichotomy of linear differential equation, (64) has a unique bounded solution Define an operator by setting for every .
Step 1 ( is self-mapping). From Lemma 24, we have . Let , where , and define By Lemmas 8 and 10, there exists such that every interval of length contains a number such that Then where . Hence, .
Since , we have where . According to Lebesgue’s dominated convergence theorem, we obtain . Therefore, is self-mapping.
Step 2 ( is a contraction mapping). , we have Hence, has a unique fixed point in . Therefore, system (53) has a unique Stepanov-like asymptotical almost periodic solution. The proof is complete.

An example: consider the following equation:where is a positive constant. In this case, , Obviously, , Hence, . Thus, all the conditions of Theorem 25 are satisfied. By Theorem 25, system (71) has a unique Stepanov-like asymptotical almost periodic solution.

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.

References

H. Bohr, “Zur theorie der fast periodischen funktionen,” Acta Mathematica, vol. 45, no. 1, pp. 29–127, 1925.
View at: Publisher Site | Google Scholar | MathSciNet
H. Bohr, “Zur Theorie der Fastperiodischen FUNktionen,” Acta Mathematica, vol. 46, no. 1-2, pp. 101–214, 1925.
View at: Publisher Site | Google Scholar | MathSciNet
A. S. Besicovitch, Almost Periodic Functions, Dover, New York, NY, USA, 1954.
B. Amir and L. Maniar, “Composition of pseudo-almost periodic functions and Cauchy problems with operator of nondense domain,” Annales Math\'Ematiques Blaise Pascal, vol. 6, no. 1, pp. 1–11, 1999.
View at: Google Scholar | MathSciNet
L. Amerio and G. Prouse, Almost-periodic functions and functional equations, Van Nostrand Reinhold Co., New York-Toronto, Ont.-Melbourne, 1971.
View at: MathSciNet
J. Andres and D. Pennequin, “On Stepanov almost-periodic oscillations and their discretizations,” Journal of Difference Equations and Applications, vol. 18, no. 10, pp. 1665–1682, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
J. Andres and D. Pennequin, “On the nonexistence of purely Stepanov almost-periodic solutions of ordinary differential equations,” Proceedings of the American Mathematical Society, vol. 140, no. 8, pp. 2825–2834, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Hu, “Boundedness and Stepanov's almost periodicity of solutions,” Electronic Journal of Differential Equations, No. 35, 7 pages, 2005.
View at: Google Scholar | MathSciNet
Z. Hu and A. B. Mingarelli, “Bochner's theorem and Stepanov almost periodic functions,” Annali di Matematica Pura ed Applicata. Series IV, vol. 187, no. 4, pp. 719–736, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
M. Maqbul and D. Bahuguna, “Almost periodic solutions for Stepanov-almost periodic differential equations,” Differential Equations and Dynamical Systems, vol. 22, no. 3, pp. 251–264, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Li and P. Wang, “Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales,” Discrete and Continuous Dynamical Systems - Series S, vol. 10, no. 3, pp. 463–473, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
T. Diagana, “Stepanov-like pseudo almost periodic functions and their applications to differential equations,” Communications in Mathematical Analysis, vol. 3, no. 1, pp. 9–18, 2007.
View at: Google Scholar | MathSciNet
C. Y. Zhang, “Pseudo-almost-periodic solutions of some differential equations,” Journal of Mathematical Analysis and Applications, vol. 181, no. 1, pp. 62–76, 1994.
View at: Publisher Site | Google Scholar | MathSciNet
T. Diagana, G. M. Mophou, and G. M. N’Guerekata, “Existence of weighted pseudo almost periodic solutions to some classes of differential equations with Sp-weighted pseudo almost periodic coefficients,” Nonlinear Analysis, vol. 72, no. 1, pp. 430–438, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
T. Diagana and G. M. N’Guerekata, “Stepanov-like almost automorphic functions and applications to some semilinear equations,” Applicable Analysis: An International Journal, vol. 86, no. 6, pp. 723–733, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
T. Diagana, “Existence of pseudo almost automorphic solutions to some abstract differential equations with Sp-pseudo almost automorphic coefficients,” Nonlinear Analysis, vol. 70, no. 11, pp. 3781–3790, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
M. Frechet, Les fonctions asymptotiquement presque-periodiques continues, vol. 213, pp. 520-522, C. R. National Academy of Sciences, Paris, 1941.
M. Frechet, “Les fonctions asymptotiquement presque-periodiques,” Revue Science, vol. 79, pp. 341–354, 1941.
View at: Google Scholar
T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, Berlin , Germany, 2013.
View at: MathSciNet
B. de Andrade and C. Lizama, “Existence of asymptotically almost periodic solutions for damped wave equations,” Journal of Mathematical Analysis and Applications, vol. 382, no. 2, pp. 761–771, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
J. Cao, A. Debbouche, and Y. Zhou, “Asymptotically almost periodicity for a class of Weyl-Liouville fractional evolution equations,” Mediterranean Journal of Mathematics, vol. 15, no. 4, Art. 155, 22 pages, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
J. Cao, Q. Yang, Z. Huang, and Q. Liu, “Asymptotically almost periodic solutions of stochastic functional differential equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1499–1511, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Tang and S. Xie, “Global attractivity of asymptotically almost periodic Nicholson's blowflies models with a nonlinear density-dependent mortality term,” International Journal of Biomathematics, vol. 11, no. 6, 1850079, 15 pages, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
C. Huang, B. Liu, X. Tian, L. Yang, and X. Zhang, “Global Convergence on Asymptotically Almost Periodic SICNNs with Nonlinear Decay Functions,” Neural Processing Letters, 2018.
View at: Publisher Site | Google Scholar
L. Yao, “Dynamics of Nicholson's blowflies models with a nonlinear density-dependent mortality,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 64, pp. 185–195, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
D. N. Cheban, Asymptotically Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York, NY, USA, 2009.
View at: MathSciNet
C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, New York, NY, USA, 2009.
View at: Publisher Site | MathSciNet
A. M. Fink, Almost Periodic Differential Equations, Springer-Verlag, Berlin, Germany, 1974.
B. M. Levitan and V. V. Zhikov, Almost-periodic functions and functional differential equations, Cambridge University Press, Cambridge, 1982.

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Copyright © 2018 Yongkun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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