带逐段常变量微分方程的概周期解及谱分析
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摘要
丹麦数学家H. Bohr于1925-1926年间建立了概周期函数理论。概周期函数是周期函数的一般化,具有优于周期函数的空间结构。在实际生活当中,概周期现象比周期现象更加普遍。从概周期函数的发展状况来看,概周期函数的推广和概周期型函数在方程方面的应用成为主要的研究方向,其中,概周期型函数在方程方面的应用主要体现在讨论方程的概周期型解的存在性、唯一性、稳定性等。
带逐段常变量微分方程是由K. Cooke、J. Wiener、S. Shah等人首先提出并研究的。此类方程结合了微分方程和差分方程的性质,具有连续和离散动力系统的混和形式,在控制理论、生物模型理论及双曲动力系统等方面有重要的应用。基于以上原因,带逐段常变量微分方程的概周期型解的存在性及唯一性问题日益得到人们的关注。
本文研究了两类,即带[·]项和带[·+2 1]项的逐段常变量微分方程的概周期解的存在性、唯一性及谱的性质。主要工作如下:
1.研究了带[·+2 1]项逐段常变量微分方程的概周期弱解及概周期解的存在性。这类方程在特殊条件下的概周期型解的存在性及唯一性已经有了较为详尽的研究。若去掉特殊条件,此类方程的概周期型解的存在情况还没有文献讨论过。本文第一部分研究了不加特殊条件时方程的概周期弱解及解的存在性。作者首先将方程离散化得到差分方程,利用平移算子的性质证明了差分方程的概周期序列解的存在性,然后利用概周期序列解逐段的构造方程的概周期弱解及解,从而证明了不加特殊条件时方程的概周期弱解及解的存在性,除此之外,作者还给出一些例子说明了这些解可能不唯一,同时说明了弱解和解之间的关系。
2.虽然关于带[·+2 1]项逐段常变量微分方程的概周期型解的存在性及唯一性已有很多结论,但是,还没有文献讨论过方程概周期解的谱的性质。本文第二部分工作研究了带[·+2 1]项逐段常变量微分方程概周期解的谱的性质。作者首先给出了方程相应差分方程的概周期序列解的表达式和方程的概周期解的表达式,然后利用近似定理研究了概周期序列解的谱的性质,在此基础上研究了概周期解的谱的性质。
3.证明了带[·]项逐段常变量微分方程概周期解的存在性及唯一性,给出了概周期解的具体的表达式,在此基础上研究了概周期解的谱的性质。关于带[·]项逐段常变量微分方程概周期解的存在性及唯一性,已经有文献研究过,本文的研究方法和该文献的研究方法不同。关于带[·]项逐段常变量微分方程概周期解的谱的性质,虽然也有文献研究过,但是,本文给出了反例说明该文献中得到的结果不正确,并对该文献的结果进行了改正。
The theory of almost periodic functions was mainly created by the Danish mathe-matician H. Bohr during 1925-1926. Almost periodic functions, with a superior spatialstructure, are a generalization of periodic functions. Moreover, in real life, almost periodicphenomenon is more common than periodic phenomenon. From the development processof almost periodic functions, we know that the broader study of almost periodic functionsand the application of almost periodic type functions in equations, which mainly focus onthe discussions of the existence, uniqueness, stability of almost periodic type solutions ofequations, are the main directions.
Differential equations with piecewise constant argument were proposed and studiedfirstly by K. Cooke, J. Wiener, S. Shah, et. al. Such equations combine the propertiesof differential equations and difference equations, and usually describe hybrid dynamicalsystems, have important applications in the control theory, biological model theory andhyperbolic dynamic systems. For these reasons, more and more people are concernedabout the existence and uniqueness of almost periodic type solutions of differential equa-tions with piecewise constant argument.
In this paper, we study the existence, uniqueness and the spectrum property of almostperiodic solutions of two kind of differential equations with piecewise constant argument,i.e. differential equations with piecewise constant argument [·] and differential equationswith piecewise constant argument [·+2 1]. The main work is as follows:
Firstly, the existence of almost periodic weak solutions and almost periodic solu-tions of differential equations with piecewise constant argument [·+2 1] are studied. Undera special condition, the existence and uniqueness of almost periodic type solutions forthese equations already have been studied in detail. If remove this special condition,there is no paper yet available to study the existence of almost periodic type solutions ofsuch equations. The first part of this paper is to study the existence of almost periodicweak solutions and solutions of such equations without the special condition. First ofall, we obtain difference equations from differential equations, using some property ofshift operators, the existence of almost periodic sequence solutions of difference equa-tions is obtained, then, we construct almost periodic weak solutions and solutions of this equations piecewise by almost periodic sequence solutions, therefore, the existence ofalmost periodic weak solutions, almost periodic solutions are proved. Meanwhile, someexamples are given to show that these solutions may not be unique, and also illustrate therelationship between weak solution and solution.
Secondly, although, there are many conclusions about the existence and uniquenessof almost periodic type solutions for differential equations with piecewise constant argu-ment [·+2 1], there is no result about the spectrum property of almost periodic solutions ofsuch equations. The second part of this paper is to study the spectrum property of almostperiodic solutions for such equations. We firstly give the expression of almost periodicsequence solutions of difference equations corresponding to such equations, and the ex-pression of almost periodic solutions of such equations. By means of an approximationtheorem, the spectrum property of almost periodic sequence solutions is obtained, basedon this, the spectrum property of almost periodic solutions is obtained.
Thirdly, the existence and uniqueness of almost periodic solutions for differentialequations with piecewise constant argument [·] are prove, furthermore, the expression ofalmost periodic solutions is obtained. Based on this, the spectrum property of the almostperiodic solutions is studied. As for the existence and uniqueness of almost periodic so-lutions of such equations, there have been literatures published on this, different methodsare used in this paper. As for the spectrum property of almost periodic solutions of suchequations, there already has a literature concerning this, however, some counterexamplesare constructed to show the results obtained in the literature are not correct. We correct it.
带逐段常变量微分方程是由K. Cooke、J. Wiener、S. Shah等人首先提出并研究的。此类方程结合了微分方程和差分方程的性质,具有连续和离散动力系统的混和形式,在控制理论、生物模型理论及双曲动力系统等方面有重要的应用。基于以上原因,带逐段常变量微分方程的概周期型解的存在性及唯一性问题日益得到人们的关注。
本文研究了两类,即带[·]项和带[·+2 1]项的逐段常变量微分方程的概周期解的存在性、唯一性及谱的性质。主要工作如下:
1.研究了带[·+2 1]项逐段常变量微分方程的概周期弱解及概周期解的存在性。这类方程在特殊条件下的概周期型解的存在性及唯一性已经有了较为详尽的研究。若去掉特殊条件,此类方程的概周期型解的存在情况还没有文献讨论过。本文第一部分研究了不加特殊条件时方程的概周期弱解及解的存在性。作者首先将方程离散化得到差分方程,利用平移算子的性质证明了差分方程的概周期序列解的存在性,然后利用概周期序列解逐段的构造方程的概周期弱解及解,从而证明了不加特殊条件时方程的概周期弱解及解的存在性,除此之外,作者还给出一些例子说明了这些解可能不唯一,同时说明了弱解和解之间的关系。
2.虽然关于带[·+2 1]项逐段常变量微分方程的概周期型解的存在性及唯一性已有很多结论,但是,还没有文献讨论过方程概周期解的谱的性质。本文第二部分工作研究了带[·+2 1]项逐段常变量微分方程概周期解的谱的性质。作者首先给出了方程相应差分方程的概周期序列解的表达式和方程的概周期解的表达式,然后利用近似定理研究了概周期序列解的谱的性质,在此基础上研究了概周期解的谱的性质。
3.证明了带[·]项逐段常变量微分方程概周期解的存在性及唯一性,给出了概周期解的具体的表达式,在此基础上研究了概周期解的谱的性质。关于带[·]项逐段常变量微分方程概周期解的存在性及唯一性,已经有文献研究过,本文的研究方法和该文献的研究方法不同。关于带[·]项逐段常变量微分方程概周期解的谱的性质,虽然也有文献研究过,但是,本文给出了反例说明该文献中得到的结果不正确,并对该文献的结果进行了改正。
The theory of almost periodic functions was mainly created by the Danish mathe-matician H. Bohr during 1925-1926. Almost periodic functions, with a superior spatialstructure, are a generalization of periodic functions. Moreover, in real life, almost periodicphenomenon is more common than periodic phenomenon. From the development processof almost periodic functions, we know that the broader study of almost periodic functionsand the application of almost periodic type functions in equations, which mainly focus onthe discussions of the existence, uniqueness, stability of almost periodic type solutions ofequations, are the main directions.
Differential equations with piecewise constant argument were proposed and studiedfirstly by K. Cooke, J. Wiener, S. Shah, et. al. Such equations combine the propertiesof differential equations and difference equations, and usually describe hybrid dynamicalsystems, have important applications in the control theory, biological model theory andhyperbolic dynamic systems. For these reasons, more and more people are concernedabout the existence and uniqueness of almost periodic type solutions of differential equa-tions with piecewise constant argument.
In this paper, we study the existence, uniqueness and the spectrum property of almostperiodic solutions of two kind of differential equations with piecewise constant argument,i.e. differential equations with piecewise constant argument [·] and differential equationswith piecewise constant argument [·+2 1]. The main work is as follows:
Firstly, the existence of almost periodic weak solutions and almost periodic solu-tions of differential equations with piecewise constant argument [·+2 1] are studied. Undera special condition, the existence and uniqueness of almost periodic type solutions forthese equations already have been studied in detail. If remove this special condition,there is no paper yet available to study the existence of almost periodic type solutions ofsuch equations. The first part of this paper is to study the existence of almost periodicweak solutions and solutions of such equations without the special condition. First ofall, we obtain difference equations from differential equations, using some property ofshift operators, the existence of almost periodic sequence solutions of difference equa-tions is obtained, then, we construct almost periodic weak solutions and solutions of this equations piecewise by almost periodic sequence solutions, therefore, the existence ofalmost periodic weak solutions, almost periodic solutions are proved. Meanwhile, someexamples are given to show that these solutions may not be unique, and also illustrate therelationship between weak solution and solution.
Secondly, although, there are many conclusions about the existence and uniquenessof almost periodic type solutions for differential equations with piecewise constant argu-ment [·+2 1], there is no result about the spectrum property of almost periodic solutions ofsuch equations. The second part of this paper is to study the spectrum property of almostperiodic solutions for such equations. We firstly give the expression of almost periodicsequence solutions of difference equations corresponding to such equations, and the ex-pression of almost periodic solutions of such equations. By means of an approximationtheorem, the spectrum property of almost periodic sequence solutions is obtained, basedon this, the spectrum property of almost periodic solutions is obtained.
Thirdly, the existence and uniqueness of almost periodic solutions for differentialequations with piecewise constant argument [·] are prove, furthermore, the expression ofalmost periodic solutions is obtained. Based on this, the spectrum property of the almostperiodic solutions is studied. As for the existence and uniqueness of almost periodic so-lutions of such equations, there have been literatures published on this, different methodsare used in this paper. As for the spectrum property of almost periodic solutions of suchequations, there already has a literature concerning this, however, some counterexamplesare constructed to show the results obtained in the literature are not correct. We correct it.
引文
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94 M. S. Berger, Y. Y. Chen. Forced Quasiperiodic and Almost Periodic Oscillationson Nonlinear Duffing Equations. Nonlinear Analysis: Theory, Methods and Appli-cations. 1992, 19(3): 249-257
95 M. S. Berger, Y. Y. Chen. Forced Quasiperiodic and Almost Periodic Oscillations forNonlinear Systems. Nonlinear Analysis: Theory, Methods and Applications. 1993,21(12): 949-965
96 C. Carminati. Forced Systems with Almost Periodic and Quasiperiodic ForcingTerm. Nonlinear Analysis: Theory, Methods and Applications. 1998, 32(6): 727-739
97 M. S. Berger, L. Zhang. New Method for Large Quasiperiodic Nonlinear Oscilla-tions with Fixed Frequencies for the Nondissipative Second Type Duffing Equation.Topol. Methods Nonlinear Anal. 1995, 6: 283-293
98 M. S. Berger, L. Zhang. A New Method for Large Quasiperiodic Nonlinear Oscil-lations with Fixed Frequencies for the Nondissipative Conservative Systems. Com-mun. Appl. Nonlinear Anal. 1995, 2: 79-106
99 E. Herna′ndez, M. L. Pelicer. Asymptotically Almost Periodic and Almost PeriodicSolutions for Partial Neutral Differential Equations. Appl. Math. Lett. 2005, 18(11):1265-1272
100 T. Diagana. Existence and Uniqueness of Pseudo-almost Periodic Solutions to SomeClasses of Partial Evolution Equations. Nonlinear Analysis: Theory, Methods andApplications. 2007, 66(2): 384-395
101 J. K. Hale. Periodic and Almost Periodic Solutions of Functional Differential Equa-tions. Arch. Rat. Mech. Anal. 1964, 15(4): 289-304
102 W. M. Ruess, V. Q. Phong. Asymptotically Almost Periodic Solutions of EvolutionEquations in Banach Spaces. J. Differential Equations. 1995, 122(2): 282-301
103 T. Diagana, E. Herna′ndez. Existence and Uniqueness of Pseudo Almost PeriodicSolutions to Some Abstract Partial Neutral Functional Differential Equations andApplications. J. Math. Anal. Appl. 2007, 327(2): 776-791
104 Q. Wang, B. Dai. Three Periodic Solutions of Nonlinear Neutral Functional Differ-ential Equations. Nonlinear Analysis: Real World Applications. 2008, 9(3): 977-984
105 X. Chen, F. Lin. Almost Periodic Solutions of Neutral Functional Differential Equa-tions. Nonlinear Analysis: Real World Applications. 2010, 11(2): 1182-1189
106 T. Diagana, E. Herna′ndez, M. Rabello. Pseudo Almost Periodic Solutions to SomeNon-autonomous Neutral Functional Differential Equations with Unbounded Delay.Math. Comput. Model. 2007, 45(9-10): 1241-1252
107 H. R. Henr′?quez. Asymptotically Almost Periodic Solutions of Abstract RetardedFunctional Differential Equations of First Order. Nonlinear Analysis: Real WorldApplications. 2009, 10(4): 2441-2454
108 E. Herna′ndez, H. R. Henr′?quez. Pseudo-almost Periodic Solutions for Nonauto-nomous Neutral Differential Equations with Unbounded Delay. Nonlinear Analysis:Real World Applications. 2008, 9(2): 430-437
109 N. B. Hacene, K. Ezzinbi. Weighted Pseudo Almost Periodic Solutions for SomePartial Functional Differential Equations. Nonlinear Analysis: Theory, Methods andApplications. 2009, 71(9): 3612-3621
110 C. Cuevas, M. Pinto. Existence and Uniqueness of Pseudo Almost Periodic Solu-tions of Semilinear Cauchy Problems with Non-dense Domain. Nonlinear Analysis:Theory, Methods and Applications. 2001, 45(1): 73-83
111 C. Cuevas, E. Herna′ndez. Pseudo-almost Periodic Solutions for Abstract PartialFunctional Differential Equations. Appl. Math. Lett. 2009, 22(4): 534-538
112 S. Abbas, D. Bahuguna. Almost Periodic Solutions of Neutral Functional Differen-tial Equations. Comp. Appl. Math. 2008, 55(11): 2593-2601
113 M. Adimy, K. Ezzinbi, A. Ouhinou. Variation of Constants Formula and AlmostPeriodic Solutions for Some Partial Functional Differential Equations with InfiniteDelay. J. Math. Anal. Appl. 2006, 317(2): 668-689
114 B. Liu, L. Huang. Existence and Uniqueness of Periodic Solutions for a Kind ofSecond Order Neutral Functional Differential Equations. Nonlinear Analysis: RealWorld Applications. 2007, 8(1): 222-229
115 C. Zhang, F. Yang. Remotely Almost Periodic Solutions of Parabolic Inverse Prob-lems. Nonlinear Analysis: Theory, Methods and Applications. 2006, 65(8): 1613-1623
116 O. Rafael, T. Massimo. Almost Periodic Linear Differential Equations with Non-separated Solutions. Journal of Functional Analysis. 2006, 237(2): 402-426
117 T. Diagana, C. M. Mahop, G. M. N G′uerekata. Pseudo Almost Periodic Solution toSome Semilinear Differential Equations. Math. Comput. Model. 2006, 43(2): 89-96
118 D. Toka. Pseudo Almost Periodic Solutions to Some Differential Equations. Non-linear Analysis: Theory, Methods and Applications. 2005, 60(7): 1277-1286
119 N. M. Man, N. V. Minh. On the Existence of Quasiperiodic and Almost PeriodicSolutions of Neutral Functional Differential Equations. Commun. Pure Appl. Anal.2004, 3(2): 291-300
120 X. Tang, Z. Jiang. Periodic Solutions of First-order Nonlinear Functional Differen-tial Equations. Nonlinear Analysis: Theory, Methods and Applications. 2008, 68(4):845-861
121 E. A. Dads, P. Cieutat, K. Ezzinbi. The Existence of Pseudo-almost Periodic So-lutions for Some Nonlinear Differential Equations in Banach Spaces. NonlinearAnalysis: Theory, Methods and Applications. 2008. 69(4): 1325-1342
122 S. Murakami, T. Naito, N. V. Minh. Massera’s Theorem for Almost Periodic So-lutions of Functional Differential Equations. J. Math. Soc. Japan. 2004, 56(1):247-268
123 H. Li, F. Huang, J. Li. Composition of Pseudo Almost Periodic Functions and Semi-linear Differential Equations. J. Math. Anal. Appl. 2001, 255(2): 436-446
124 C. Zhang. Ergodicity and Its Application II: Averaging Method of Some DynamicalSystems. Acta Analysis Functionalis Applicata. 1999, 1(2): 146-159
125 C. Zhang. Ergodicity and Its Applications in Regularity and Solutions of PseudoAlmost Periodic Equations. Nonlinear Analysis: Theory, Methods and Applications.2001, 46(4): 511-523
126 C. Zhang. Ergodicity and Asymptotically Almost Periodic Solutions of Some Dif-ferential Equations. Internat. J. Math. Math. Sci. 2001, 25(12): 787-800
127 W. Shen, Y. Yi. Almost Automorphic and Almost Periodic Dynamics in Skew-product Semi?ows, Part II. Skew-product Semi?ows. Mem. Amer. Math. Soc.1998, 136(647): 171-183
128 W. Shen, Y. Yi. Almost Automorphic and Almost Periodic Dynamics in Skew-product Semi?ows,Part III. Applications to Differential Equations. Mem. Amer.Math. Soc. 1998, 647: 5671-5682
129 W. Shen, Y. Yi. Dynamics of Almost Periodic Scalar Parabolic Equations. J. Differ-ential Equations. 1995, 122(1): 114-136
130 J. A. Sanders, F. Verhulst. Averaging Methods in Nonlinear Dynamical Systems.Spring-Verlag, New York, 1985
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91 J. Hong, R. Obayab, A. Sanzb. Almost Periodic Type Solutions of Some Differ-ential Equations with Piecewise Constant Argument. Nonlinear Analysis: Theory,Methods and Applications. 2001, 45(6): 661-688
92 D. Piao. Pseudo Almost Periodic Solutions for the Systems of Differential Equationswith Piecewise Constant Argument [t+1/2]. Sci. in China (A). 2003, 33(11): 220-22593 E. A. Dads, K. Ezzinbi. Pseudo Almost Periodic Solutions of Some Delay Differen-tial Equations. J. Math. Anal. Appl. 1996, 201(3): 840-850
94 M. S. Berger, Y. Y. Chen. Forced Quasiperiodic and Almost Periodic Oscillationson Nonlinear Duffing Equations. Nonlinear Analysis: Theory, Methods and Appli-cations. 1992, 19(3): 249-257
95 M. S. Berger, Y. Y. Chen. Forced Quasiperiodic and Almost Periodic Oscillations forNonlinear Systems. Nonlinear Analysis: Theory, Methods and Applications. 1993,21(12): 949-965
96 C. Carminati. Forced Systems with Almost Periodic and Quasiperiodic ForcingTerm. Nonlinear Analysis: Theory, Methods and Applications. 1998, 32(6): 727-739
97 M. S. Berger, L. Zhang. New Method for Large Quasiperiodic Nonlinear Oscilla-tions with Fixed Frequencies for the Nondissipative Second Type Duffing Equation.Topol. Methods Nonlinear Anal. 1995, 6: 283-293
98 M. S. Berger, L. Zhang. A New Method for Large Quasiperiodic Nonlinear Oscil-lations with Fixed Frequencies for the Nondissipative Conservative Systems. Com-mun. Appl. Nonlinear Anal. 1995, 2: 79-106
99 E. Herna′ndez, M. L. Pelicer. Asymptotically Almost Periodic and Almost PeriodicSolutions for Partial Neutral Differential Equations. Appl. Math. Lett. 2005, 18(11):1265-1272
100 T. Diagana. Existence and Uniqueness of Pseudo-almost Periodic Solutions to SomeClasses of Partial Evolution Equations. Nonlinear Analysis: Theory, Methods andApplications. 2007, 66(2): 384-395
101 J. K. Hale. Periodic and Almost Periodic Solutions of Functional Differential Equa-tions. Arch. Rat. Mech. Anal. 1964, 15(4): 289-304
102 W. M. Ruess, V. Q. Phong. Asymptotically Almost Periodic Solutions of EvolutionEquations in Banach Spaces. J. Differential Equations. 1995, 122(2): 282-301
103 T. Diagana, E. Herna′ndez. Existence and Uniqueness of Pseudo Almost PeriodicSolutions to Some Abstract Partial Neutral Functional Differential Equations andApplications. J. Math. Anal. Appl. 2007, 327(2): 776-791
104 Q. Wang, B. Dai. Three Periodic Solutions of Nonlinear Neutral Functional Differ-ential Equations. Nonlinear Analysis: Real World Applications. 2008, 9(3): 977-984
105 X. Chen, F. Lin. Almost Periodic Solutions of Neutral Functional Differential Equa-tions. Nonlinear Analysis: Real World Applications. 2010, 11(2): 1182-1189
106 T. Diagana, E. Herna′ndez, M. Rabello. Pseudo Almost Periodic Solutions to SomeNon-autonomous Neutral Functional Differential Equations with Unbounded Delay.Math. Comput. Model. 2007, 45(9-10): 1241-1252
107 H. R. Henr′?quez. Asymptotically Almost Periodic Solutions of Abstract RetardedFunctional Differential Equations of First Order. Nonlinear Analysis: Real WorldApplications. 2009, 10(4): 2441-2454
108 E. Herna′ndez, H. R. Henr′?quez. Pseudo-almost Periodic Solutions for Nonauto-nomous Neutral Differential Equations with Unbounded Delay. Nonlinear Analysis:Real World Applications. 2008, 9(2): 430-437
109 N. B. Hacene, K. Ezzinbi. Weighted Pseudo Almost Periodic Solutions for SomePartial Functional Differential Equations. Nonlinear Analysis: Theory, Methods andApplications. 2009, 71(9): 3612-3621
110 C. Cuevas, M. Pinto. Existence and Uniqueness of Pseudo Almost Periodic Solu-tions of Semilinear Cauchy Problems with Non-dense Domain. Nonlinear Analysis:Theory, Methods and Applications. 2001, 45(1): 73-83
111 C. Cuevas, E. Herna′ndez. Pseudo-almost Periodic Solutions for Abstract PartialFunctional Differential Equations. Appl. Math. Lett. 2009, 22(4): 534-538
112 S. Abbas, D. Bahuguna. Almost Periodic Solutions of Neutral Functional Differen-tial Equations. Comp. Appl. Math. 2008, 55(11): 2593-2601
113 M. Adimy, K. Ezzinbi, A. Ouhinou. Variation of Constants Formula and AlmostPeriodic Solutions for Some Partial Functional Differential Equations with InfiniteDelay. J. Math. Anal. Appl. 2006, 317(2): 668-689
114 B. Liu, L. Huang. Existence and Uniqueness of Periodic Solutions for a Kind ofSecond Order Neutral Functional Differential Equations. Nonlinear Analysis: RealWorld Applications. 2007, 8(1): 222-229
115 C. Zhang, F. Yang. Remotely Almost Periodic Solutions of Parabolic Inverse Prob-lems. Nonlinear Analysis: Theory, Methods and Applications. 2006, 65(8): 1613-1623
116 O. Rafael, T. Massimo. Almost Periodic Linear Differential Equations with Non-separated Solutions. Journal of Functional Analysis. 2006, 237(2): 402-426
117 T. Diagana, C. M. Mahop, G. M. N G′uerekata. Pseudo Almost Periodic Solution toSome Semilinear Differential Equations. Math. Comput. Model. 2006, 43(2): 89-96
118 D. Toka. Pseudo Almost Periodic Solutions to Some Differential Equations. Non-linear Analysis: Theory, Methods and Applications. 2005, 60(7): 1277-1286
119 N. M. Man, N. V. Minh. On the Existence of Quasiperiodic and Almost PeriodicSolutions of Neutral Functional Differential Equations. Commun. Pure Appl. Anal.2004, 3(2): 291-300
120 X. Tang, Z. Jiang. Periodic Solutions of First-order Nonlinear Functional Differen-tial Equations. Nonlinear Analysis: Theory, Methods and Applications. 2008, 68(4):845-861
121 E. A. Dads, P. Cieutat, K. Ezzinbi. The Existence of Pseudo-almost Periodic So-lutions for Some Nonlinear Differential Equations in Banach Spaces. NonlinearAnalysis: Theory, Methods and Applications. 2008. 69(4): 1325-1342
122 S. Murakami, T. Naito, N. V. Minh. Massera’s Theorem for Almost Periodic So-lutions of Functional Differential Equations. J. Math. Soc. Japan. 2004, 56(1):247-268
123 H. Li, F. Huang, J. Li. Composition of Pseudo Almost Periodic Functions and Semi-linear Differential Equations. J. Math. Anal. Appl. 2001, 255(2): 436-446
124 C. Zhang. Ergodicity and Its Application II: Averaging Method of Some DynamicalSystems. Acta Analysis Functionalis Applicata. 1999, 1(2): 146-159
125 C. Zhang. Ergodicity and Its Applications in Regularity and Solutions of PseudoAlmost Periodic Equations. Nonlinear Analysis: Theory, Methods and Applications.2001, 46(4): 511-523
126 C. Zhang. Ergodicity and Asymptotically Almost Periodic Solutions of Some Dif-ferential Equations. Internat. J. Math. Math. Sci. 2001, 25(12): 787-800
127 W. Shen, Y. Yi. Almost Automorphic and Almost Periodic Dynamics in Skew-product Semi?ows, Part II. Skew-product Semi?ows. Mem. Amer. Math. Soc.1998, 136(647): 171-183
128 W. Shen, Y. Yi. Almost Automorphic and Almost Periodic Dynamics in Skew-product Semi?ows,Part III. Applications to Differential Equations. Mem. Amer.Math. Soc. 1998, 647: 5671-5682
129 W. Shen, Y. Yi. Dynamics of Almost Periodic Scalar Parabolic Equations. J. Differ-ential Equations. 1995, 122(1): 114-136
130 J. A. Sanders, F. Verhulst. Averaging Methods in Nonlinear Dynamical Systems.Spring-Verlag, New York, 1985
131 R. Bitmead, C. Johnson. Discrete Averaging Principles in Robust Adaptive Identifi-cation. Control and Dynamic Systems. 1987, 25: 237-271