带严格单调算子的微分方程的概周期型解
|
摘要
丹麦数学家H. Bohr于1925-1926年间建立了概周期函数理论。概周期函数是拥有某种结构性质的连续函数,是周期函数的一般化。经过许多数学工作者的努力,概周期函数理论得到了长足的发展。一方面是关于概周期型函数的扩充和性质的研究,另一方面是概周期型函数理论在相应学科中的应用,特别是方程概周期型解的存在性条件。
本文主要包括两部分内容的研究:一部分是关于几种一阶概周期型微分方程解的存在唯一性的研究讨论,另一部分是关于一类带梯度算子的二阶概周期型微分方程解的存在唯一性的讨论。
对于带有严格单调算子的微分方程的概周期解的存在唯一性已经得到了一些数学工作者的研究,并且有了一些重要的结果。Sarason定义了缓慢振荡函数,但至今没有人将它应用到带有严格单调算子的微分方程中。1941年,Fre′chet最先研究引入了渐近概周期函数。渐近概周期函数是概周期函数的一个推广。但到目前为止,还无人对带有严格单调算子的微分方程的渐近概周期解的存在唯一性进行研究。
带有梯度算子的二阶微分方程是一类特殊的非线性方程,这种方程在化工和电子等方面有重要的作用。九十年代初利用变分法人们解决了这种方程解的存在性问题。利用函数的凸性,方程的概周期解的存在性和唯一性问题也已经得到了解决,这以后人们深入地研究更广泛的带梯度算子的二阶方程,在它的概周期解的存在性和唯一性方面得到了一些很好的结果。
近几年中,越来越多的人致力于带有半群无穷小生成元的微分方程的概周期解的研究,但对于其缓慢振荡解的研究是很少的。
本文主要解决了以下几个问题:
1.本文考虑将缓慢振荡函数和带有严格单调算子的微分方程相结合,利用反证法给出了带有严格单调算子的一阶微分方程缓慢振荡解存在和唯一的一个充分条件和一个必要条件,并且举例说明了,这两个条件都不可能成为充分必要条件。特别地对于此类方程的一类特殊情形,我们给出了缓慢振荡解存在和唯一的充分必要条件。
2.本文利用渐近概周期函数的性质给出了带有严格单调算子的一阶微分方程渐近概周期解存在和唯一的一个充分条件和一个必要条件,并且举例说明了,这两个条件都不可能成为充分必要条件。特别地对于此类方程的一类特殊情形,我们给出了渐近概周期解存在和唯一的充分必要条件。并且我们把一些结果推广到了一类二阶微分方程。
3.本文解决了带梯度算子的二阶方程渐近概周期解的存在和唯一性的问题。我们利用渐近概周期函数的性质可以得到这种方程的渐近概周期解在R+上的存在性。另一方面,我们可以把原方程转化,利用迭代法和相关的线性常微分方程的渐近概周期解的存在性和唯一性条件,得到此方程的渐近概周期解的存在和唯一性。
4.本文利用Banach不动点定理考虑了带有半群无穷小生成元的一阶微分方程的缓慢振荡解存在唯一的条件。
The theory of almost periodic functions was mainly created by the Danish math-ematician H. Bohr during 1925-1926. Almost periodic functions are the class of con-tinuous functions possessing certain structural properties and are a generalization ofpure periodicity. The theory was developed further by some mathematicians. Onedirection is the broader study of functions of almost periodic type, another directionis the application in certain subject, in particular, the existence of the almost periodictype solutions of equations.
The paper consists of two parts:one part concerns the existence and uniquenessof the solutions of some one-order almost periodic type differential equations, theother part concerns the existence and uniqueness of the solutions of some second-order almost periodic type differential equations with gradient operators.
Some people discussed the almost periodic solutions of differential equationswith strictly monotone operators, and they have given some important results. Sara-son defined slowly oscillating functions. To our knowledge, nobody has applied suchfunctions to the theory of differential equations with strictly monotone operators. In1941, asymptotically almost periodic functions was originally introduced by Fre′chet.Asymptotically almost periodic functions are a generalization of almost periodic func-tions. To our knowledge, nobody has studied the existence and uniqueness of asymp-totically almost periodic solutions of the differential equations with strictly monotoneoperators.
Second-order equations with gradient operators is a kind of special nonlineardifferential equations. The equations have applications in certain chemical industryand electron. By means of variational principle, at the beginning of 1990s peopleprovided results about the existence of the solutions of such equations. Because ofthe convexity of functions, the existence and uniqueness of almost periodic solutionsof such equations have been solved. In the following years people deeply study widersecond-order equations with gradient operators and get many perfect results of theexistence and uniqueness of almost periodic solutions of such equations.
In recent year, more and more people have given important contributions to the questions of almost periodic solutions of the differential equations with infinitesimalgenerators of semigroups. But slowly oscillating solutions of these differential equa-tions have seldom been discussion.
In this paper, we mainly solve the following questions:
Firstly, we give the relation of slowly oscillating functions and differential equa-tions with strictly monotone operators. By means of reduction to absurdity, the au-thors discuss a necessary and a sufficient conditions for the existence and uniquenessof slowly oscillating solutions of the equations. Then we give examples to show thatthey can’t be necessary and sufficient conditions. Particularly, as a special class, theauthors give necessary and sufficient conditions for the existence and uniqueness ofslowly oscillating solutions of the differential equations with gradient operators.
Secondly, By means of the properties of asymptotically almost periodic func-tions, the authors discuss a necessary and a sufficient conditions for the existence anduniqueness of asymptotically almost periodic solutions of the equations with strictlymonotone operators. Then we give examples to show that they can’t be necessary andsufficient conditions. Particularly, as a special class, the authors give necessary andsufficient conditions for the existence and uniqueness of asymptotically almost peri-odic solutions for the differential equations with gradient operators. Then we extendthe results to some second-order equations.
Thirdly, we solve the problem of existence and uniqueness of asymptoticallyalmost periodic solutions of the second-order equations with gradient operators. Bymeans of the properties of asymptotically almost periodic functions, we get the exis-tence on R+ of asymptotically almost periodic solutions of the equations. On the otherhand, we transform the equations. Thus we present existence and uniqueness theoremfor asymptotically almost periodic solutions of the equations by means of iterationand the conditions for the existence and uniqueness of almost periodic solutions ofcorrelative linear differential equations.
Fourthly, we discuss some conditions for the existence and uniqueness of slowlyoscillating (mild) solutions of the differential equations with infinitesimal generatorsof semigroups by Banach fixed point theorem.
本文主要包括两部分内容的研究:一部分是关于几种一阶概周期型微分方程解的存在唯一性的研究讨论,另一部分是关于一类带梯度算子的二阶概周期型微分方程解的存在唯一性的讨论。
对于带有严格单调算子的微分方程的概周期解的存在唯一性已经得到了一些数学工作者的研究,并且有了一些重要的结果。Sarason定义了缓慢振荡函数,但至今没有人将它应用到带有严格单调算子的微分方程中。1941年,Fre′chet最先研究引入了渐近概周期函数。渐近概周期函数是概周期函数的一个推广。但到目前为止,还无人对带有严格单调算子的微分方程的渐近概周期解的存在唯一性进行研究。
带有梯度算子的二阶微分方程是一类特殊的非线性方程,这种方程在化工和电子等方面有重要的作用。九十年代初利用变分法人们解决了这种方程解的存在性问题。利用函数的凸性,方程的概周期解的存在性和唯一性问题也已经得到了解决,这以后人们深入地研究更广泛的带梯度算子的二阶方程,在它的概周期解的存在性和唯一性方面得到了一些很好的结果。
近几年中,越来越多的人致力于带有半群无穷小生成元的微分方程的概周期解的研究,但对于其缓慢振荡解的研究是很少的。
本文主要解决了以下几个问题:
1.本文考虑将缓慢振荡函数和带有严格单调算子的微分方程相结合,利用反证法给出了带有严格单调算子的一阶微分方程缓慢振荡解存在和唯一的一个充分条件和一个必要条件,并且举例说明了,这两个条件都不可能成为充分必要条件。特别地对于此类方程的一类特殊情形,我们给出了缓慢振荡解存在和唯一的充分必要条件。
2.本文利用渐近概周期函数的性质给出了带有严格单调算子的一阶微分方程渐近概周期解存在和唯一的一个充分条件和一个必要条件,并且举例说明了,这两个条件都不可能成为充分必要条件。特别地对于此类方程的一类特殊情形,我们给出了渐近概周期解存在和唯一的充分必要条件。并且我们把一些结果推广到了一类二阶微分方程。
3.本文解决了带梯度算子的二阶方程渐近概周期解的存在和唯一性的问题。我们利用渐近概周期函数的性质可以得到这种方程的渐近概周期解在R+上的存在性。另一方面,我们可以把原方程转化,利用迭代法和相关的线性常微分方程的渐近概周期解的存在性和唯一性条件,得到此方程的渐近概周期解的存在和唯一性。
4.本文利用Banach不动点定理考虑了带有半群无穷小生成元的一阶微分方程的缓慢振荡解存在唯一的条件。
The theory of almost periodic functions was mainly created by the Danish math-ematician H. Bohr during 1925-1926. Almost periodic functions are the class of con-tinuous functions possessing certain structural properties and are a generalization ofpure periodicity. The theory was developed further by some mathematicians. Onedirection is the broader study of functions of almost periodic type, another directionis the application in certain subject, in particular, the existence of the almost periodictype solutions of equations.
The paper consists of two parts:one part concerns the existence and uniquenessof the solutions of some one-order almost periodic type differential equations, theother part concerns the existence and uniqueness of the solutions of some second-order almost periodic type differential equations with gradient operators.
Some people discussed the almost periodic solutions of differential equationswith strictly monotone operators, and they have given some important results. Sara-son defined slowly oscillating functions. To our knowledge, nobody has applied suchfunctions to the theory of differential equations with strictly monotone operators. In1941, asymptotically almost periodic functions was originally introduced by Fre′chet.Asymptotically almost periodic functions are a generalization of almost periodic func-tions. To our knowledge, nobody has studied the existence and uniqueness of asymp-totically almost periodic solutions of the differential equations with strictly monotoneoperators.
Second-order equations with gradient operators is a kind of special nonlineardifferential equations. The equations have applications in certain chemical industryand electron. By means of variational principle, at the beginning of 1990s peopleprovided results about the existence of the solutions of such equations. Because ofthe convexity of functions, the existence and uniqueness of almost periodic solutionsof such equations have been solved. In the following years people deeply study widersecond-order equations with gradient operators and get many perfect results of theexistence and uniqueness of almost periodic solutions of such equations.
In recent year, more and more people have given important contributions to the questions of almost periodic solutions of the differential equations with infinitesimalgenerators of semigroups. But slowly oscillating solutions of these differential equa-tions have seldom been discussion.
In this paper, we mainly solve the following questions:
Firstly, we give the relation of slowly oscillating functions and differential equa-tions with strictly monotone operators. By means of reduction to absurdity, the au-thors discuss a necessary and a sufficient conditions for the existence and uniquenessof slowly oscillating solutions of the equations. Then we give examples to show thatthey can’t be necessary and sufficient conditions. Particularly, as a special class, theauthors give necessary and sufficient conditions for the existence and uniqueness ofslowly oscillating solutions of the differential equations with gradient operators.
Secondly, By means of the properties of asymptotically almost periodic func-tions, the authors discuss a necessary and a sufficient conditions for the existence anduniqueness of asymptotically almost periodic solutions of the equations with strictlymonotone operators. Then we give examples to show that they can’t be necessary andsufficient conditions. Particularly, as a special class, the authors give necessary andsufficient conditions for the existence and uniqueness of asymptotically almost peri-odic solutions for the differential equations with gradient operators. Then we extendthe results to some second-order equations.
Thirdly, we solve the problem of existence and uniqueness of asymptoticallyalmost periodic solutions of the second-order equations with gradient operators. Bymeans of the properties of asymptotically almost periodic functions, we get the exis-tence on R+ of asymptotically almost periodic solutions of the equations. On the otherhand, we transform the equations. Thus we present existence and uniqueness theoremfor asymptotically almost periodic solutions of the equations by means of iterationand the conditions for the existence and uniqueness of almost periodic solutions ofcorrelative linear differential equations.
Fourthly, we discuss some conditions for the existence and uniqueness of slowlyoscillating (mild) solutions of the differential equations with infinitesimal generatorsof semigroups by Banach fixed point theorem.
引文
1 H. Bohr. Zur Theorie Der Fastperiodischen Funktionen I-III. Acta. Math. 1925-1926, 45,29-127: 46,101-214: 47,237-281
2 H. Bohr. Almost Periodic Functions. Chelsea,New York, 1951
3 S. Bochner. Beitra¨ge Zur Theorie Der Fastperiodischen Funktionen. Math. Ann.1927, 96(1):119–147
4 S. Bochner. A New Approach to Almost Periodicity. Proc. Nat. Acad. Sci. U.S. A. 1962, 48:2039–2043
5 S. Bochner, J. V. Neumann. Almost Periodic Functions in Groups II. Trans.Amer. Math. Soc. 1935, 37(1):21–50
6 A. S. Besicovitch. Almost Periodic Functions. Dover Publications Inc., NewYork, 1955
7 J. Favard. Sur Les e′quations Differentielles Line′aires a` Coefficients Presque-pe′riodiques. Acta. Math. 1928, 51(1):31–81
8 W. Stepanoff. u¨eber Einige Verallgemeinerungen Der Fast Periodischen Func-tionen. Math. Ann. 1926, 95(1):473–498
9 H. Weyl. Integralgleichungen Und Fastperiodische Functionen. Math. Ann.1927, 97(1):338–356
10 A. S. Besicovitch. On Parseval’s Theorem for Dirichlet Series. Proc. LondonMath. Soc. 1927, 26:25–34
11 R. A. Satyanarayan. On the Stepanov Almost Periodic Solutions of a Second-order Infinitesimal Generator of Differential Equations. Internat. J. Math. Math.Sci. 1991, 14(4):757–761
12 Z. Hu. Contributions to the Theory of Almost Periodic Differential Equations.Carleton University, Ph.D. thesis. 2001
13 M. Fre′chet. Les Fonctions Asymptotiquement Presque-periodiques Continues.C. R. Acad. Sci. Paris. 1941, 213:520–522
14 M. Fre′chet. Les Fonctions Asymptotiquement Presque-pe′riodiques. Revue Sci.1941, 79:341–354
15 W. F. Eberlein. Abstract Ergodic Theorems and Weak Almost Periodic Func-tions. Trans. Amer. Math. Soc. 1949, 67:217–240
16 C. Corduneanu. Almost Periodic Functions. Chelsea Publishing Company, NewYork, 1989
17 J. F. Berglund, H. D. Junghenn, P. Milnes. Analysis on Semigroup: FunctionSpaces. Compactifications,Representations. Wiley, New York, 1989
18 A. M. Fink. Almost Periodic Differential Equations. Springer-verlag, Berlin1974
19 B. M. Levitan. Almost Periodic Functions. Higher Education Press, 1956
20 S. Zaidman. Almost Periodic Functions in Abstract Spaces. Pitman,Boston,MA, 1985
21 O. Onicescu, V. Istratescu. Approximation Theorems for Random Functions.Rend. Mat. 1975, 8:65–81
22 O. Onicesu, G. Cenusa, I. Sacuiu. Random Functions Almost Periodic in Prob-ability. Editura Academiei Republicii Socialiste Roumania, Bucharest, 1983
23 D. Sarason. Toeplitz Operators with Semi-almost Periodic Symbols. DukeMath. J. 1977, 44(2):357–364
24 D. Sarason. Remotely Almost Periodic Function. Contemporary Mathematics.1984, 32:237–242
25 D. Sarason. Toeplitz Operators with Piecewise Quasicontinuous Symbols. In-diana Univ. Math. J. 1977, 26(5):817–838
26 C. Zhang. Pseudo Almost Periodic Functions and Their Applications. Univer-sity of Western Ontario, Ph.D. thesis. 1992
27 C.Zhang. Pseudo Almost Periodic Solutions of some Differential Equations.J.Math.Anal.Appl. 1994, 181(1):62–76
28 C.Zhang. Pseudo Almost Periodic Solutions of some Differential Equations(ii).J.Math.Anal.Appl. 1995, 192(2):543–561
29 C. Zhang. Integration of Vector-valued Pseudo Almost Periodic Functions. Pr-oc. Amer. Math. Soc. 1994, 121(1):167–174
30 C. Zhang. Vector-valued Pseudo Almost Periodic Functions. CzechoslovakMath. J. 1997, 47(122)(3):385–394
31 C. Zhang. Almost Periodic Type Functions and Ergodicity. Science Press/Klu-wer, New York, Beijing, 2003
32 C. Zhang, H. Yao. Converse Problems of Fourier Expansion and Their Appli-cations. Nonlinear Analysis TMA. 2004, 56(5):761–779
33 G. M. N’Guerekata. Almost Periodicity in Linear Topological Spaces and Ap-plications to Abstract Differential Equations. Internat. J. Math. Math. Sci. 1984,7(3):529–540
34 G. M. N’Guerekata. Almost Periodic Solutions of Certain Differential Equa-tions in Fre′chet Spaces. Riv. Mat. Univ. Parma (5). 1993, 2:301–303
35 G. M. N’Guerekata. Almost Periodicity of some Solutions to Linear AbstractEquations. Libertas Math. 1996, 16:145–148
36 G. M. N’Guerekata. Almost Automorphic and Almost Periodic Functions inAbstract Spaces. Kluwer Academic, Plenum Publishers, New York, 2001
37 D. Bugajewski, G. M. N’Guerekata. Almost Periodicity in Fre′chet Spaces. J.Math. Anal. Appl. 2004, 299(2):534–549
38 K. Cooke, J. Kapplan. A Periodicity Threshold Theorem for Epidemics andPopulation Growth. Math. Biosci. 1976, 31(1-2):87–104
39 A. Fink, J. Gatiga. Positive Almost Periodic Solutions of some Delay IntegralEquations. J. Differential Equations. 1990, 83(1):166–178
40 E. A. Dads, K. Ezzinbi. Positive Almost Periodic Solution for some NonlinearDelay Integral Equation. Nonlinear Stud. 1996, 3(1):85–101
41 R. Torrejon. Positive Almost Periodic Solutions of a State-dependent DelayNonlinear Integral Equation. Nonlinear Analysis TMA. 1993, 20(12):1383–1416
42 K. Ezzinbi, M. A. Hachimi. Existence of Positive Almost Periodic Solutions ofFunctional Equations Via Hilbert Projective Metric. Nonlinear Analysis TMA.1996, 26(6):1169–1176
43 B. Xu, R. Yuan. On the Positive Almost Periodic Type Solutions for someNonlinear Delay Integral Equations. J. Math. Anal. Appl. 2005, 304(1):249–26844 E. A. Dads, K. Ezzinbi. Almost Periodic Solution for some Neutral NonlinearIntegral Equations. Nonlinear Analysis TMA. 1997, 28(9):1479–1489
45 E. A. Dads, K. Ezzinbi. Existence of Positive Pseudo Almost Periodic Solu-tion for some Nonlinear Infinite Delay Integral Equations Arising in EpidemicProblems. Nonlinear Analysis TMA. 2000, 41:1–13
46 E. A. Dads, K. Ezzibi, O. Arino. Pseudo Almost Periodic Solutions for someDifferential Equations in a Banach Space. Nonlinear Analysis TMA. 1997,28(7):1141–1155
47 D. Piao. Pseudo Almost Periodic Solutions of the Systems of DifferentialEquations with Piecewise Constant Argument [t]. Sci. in China (A). 2001,44(9):1156–1161
48何崇佑.概周期微分方程.高等教育出版社, 1992, 1–246
49 M. Biroli. Sur Les Solutions Borne′es Ou Presque-pe′riodiques Des e′quationsD’e′volution Multivoques Sur Un Espace De Hilbert. Ricerche Mat. 1972,21:17–47
50 C. M. Dafermos. Almost Periodic Process and Almost Periodic Solutionsof Evolutions Equations. Dynamical systems (Proc. Internat. Sympos., Univ.Florida, Gainesville, Fla.,1976), 43-57. Academic Press, New York, 1977
51 A. Haraux. Nonlinear Evolution Equation-global Behavior of Solutions. Lec-ture Notes in Mathematics, 841 Springer-Verlag, Berlin, New York, 1981
52 F. L. Huang. Existence of Almost Periodic Solutions for Dissipative DifferentialEquations. Ann. Differential Equations. 1990, 6:271–279
53 H. Ishii. On the Existence of Almost Periodic Complete Trajectories for Con-tractive Almost Periodic Processes. J. Differential Equations. 1982, 43:66–72
54 P. Cieutat. Necessary and Sufficient Conditions for Existence and Uniquenessof Bounded Or Almost Periodic Solutions for Differential Sytems with ConvexPotential. Differential and Integral Equations. 2004, 18:361–378
55 V. E. Slyusarchuk. Necessary and Sufficient Conditions for Existence andUniquenss of Bounded and Almost Periodic Solutions of Nonlinear DifferentialEquations. Acta Appl. Math. 2001, 65:333–341
56 M.S.Berger, Y.Y.Chen. Forced Quasiperiodic and Almost Periodic Oscillationson Nonlinear Duffing Equations. Nonlinear Analysis TMA. 1992, 19:249–257
57 M.S.Berger, Y.Y.Chen. Forced Quasiperiodic and Almost Periodic Oscillationsfor Nonlinear Systems. Nonlinear Analysis TMA. 1993, 21:949–965
58 C. Carminati. Forced Systems with Almost Periodic and Quasiperiodic ForcingTerm. Nonlinear Analysis TMA. 1998, 32:727–739
59 M. S. Berger, L. Zhang. New Method for Large Quasiperiodic Nonlinear Os-cillations with Fixed Frequencies for the Nondissipative Second Type DuffingEquation. Topological Methods Nonlinear Anal. 1995, 6:283–293
60 M. S. Berger, L. Zhang. A New Method for Large Quasiperiodic Nonlinear Os-cillations with Fixed Frequencies for the Nondissipative Conservative Systems.Commun. Appl. Nonlinear Anal. 1995, 2:79–106
61 P. Cieutat. Almost Periodic Solutions of Second-order Systems with MonotoneField on a Compact Subset. Nonlinear Analysis TMA. 2003, 53:751–763
62 S. F. Zakharin, I. O. Parasyuk. Generalized and Classical Almost Periodic So-lutions of Lagrangian Systems. Funkcial. Ekvac. 1999, 42:325–338
63 P. Cieutat. Almost Periodic Solutins of Forced Vectorial Lie′nard Equations. J.Differential Equations. 2005, 209:302–328
64 P. Cieutat. Bounded and Almost Periodic Solutins of Convex Lagrangian Sys-tems. J. Differential Equations. 2003, 190:108–130
65 J. Bolt. Calculus of Variations on Mean and Convex Lagaragians. J. Math.Anal. Appl. 1988, 134:312–321
66 J. Bolt, P. Cieutat, J. Mawhin. Almost Periodic Oscillations of MonotoneSecond-order Systems. Adv. Differential Equations. 1997, 2:693–714
67 J.Bolt. Oscillations Presque-pe′riodiques Force′es D’e′quations D’euler-lagrange.Bull. Soc. Math. France. 1994, 122:285–304
68 J. P. Chen. Almost Periodic Solutions of Higher-dimensional Lie′nard Systems.Atti Sem. Mat. Fis. Univ. Modena. 1990, 38:123–135
69 F. Lin. Stability and Existence of Periodic Solutions and Almost Periodic Solu-tions on Lie′nard Systems. Ann. Differential Equations. 1997, 13:248–253
70 K. Wang. Almost Periodic Solutions of Forced Line′nard Equations. Chinese J.Contemp. Math. 1995, 16:253–260
71 J. Zhou. The Existence of Periodic Solutions and Almost Periodic Solutions ofFoeced Line′nard-type Equations. Acta. Math. Sinica. 1999, 42:571–576
72 K. Schmitt, J. R. Ward. Almost Periodic Solutions of Nonlinear Second OrderDifferential Equations. Results in Mathematics. 1992, 21:191–199
73 C. Cuevas, M. Pinto. Existence and Uniqueness of Pseudo Almost PeriodicSolutions of Semilinear Cauchy Problems with Non-dense Domain. NonlinearAnalysis TMA. 2001, 45(1):73–83
74 T. Diagana, C. M. Mahop, G. M. N Gue′re′kata. Pseudo Almost Periodic Solutionto some Semilinear Differential Equations. Math. Comput. Modelling. 2006,43(2):89–96
75 T. Diagana, C. M. Mahop, G. M. N Gue′re′kata. Existence and Uniqueness ofPseudo Almost Periodic Solutions to some Classes of Semilinear DifferentialEquations and Applications. Nonlinear Analysis TMA. 2006, 64(11):2442–2453
76 H. X. Li, F. L. Huang, J. Y. Li. Composition of Pseudo Almost PeriodicFunctions and Semilinear Differential Equations. J. Math. Anal. Appl. 2001,255(2):436–446
77 E. Hernandez, M. L. Pelicer, J. P. C. dos Santos. Asymptotically Almost Pe-riodic and Almost Periodic Solutions for a Class of Evolution Equations. J.Differential Equations. 2004, 61:1–15
78 E. Herna′ndez, M. L. Pelicer. Asymptotically Almost Periodic and Almost Peri-odic Solutions for Partial Neutral Differential Equations. Applied MathematicsLetters. 2005, 18:1265–1272
79 T. Diagana. Existence and Uniqueness of Pseudo-almost Periodic Solutions tosome Classes of Partial Evolution Equations. Nonlinear Analysis TMA. 2007,66:384–395
80 T. Diagana, E. Herna′ndez, M. Rabello. Pseudo Almost Periodic Solutionsto some Non-autonomous Neutral Functionsl Differential Equations with Un-bounded Delay. Mathematical and Computer Modelling. 2007, 45:1241–1252
81 T. Diagana, E. M. Herna′dez. Existence and Uniqueness of Pseudo Almost Pe-riodic Solutions to some Abstract Partial Neutral Functional-differential Equa-tions and Applications. J. Math. Anal. Appl. 2007, 327:776–791
82 C. Cuevas, M.Pinto. Existence and Uniqueness of Pseudo Solutions of Semilin-ear Cauchy Problems with Nondense Domain. Nonlinear Analysis TMA. 2001,45(1):73–83
83 T. Diagina. Pseudo Almost Periodic Solutions to some Differential Equations.Nonlinear Analysis TMA. 2005, 60(7):1277–1286
84 H. R. Henr′?quez, C. H. Va′squez. Almost Periodic Solutions of Abstract Re-tarded Functional-differential Equations with Unbounded Delay. Acta Appl.Math. 1999, 57(2):105–132
85 N. M. Man, N. V. Minh. On the Existence of Quasi Periodic and Almost Pe-riodic Solutions of Neutral Functional Differential Equations. Commun. PureAppl. Anal. 2004, 3(2):291–300
86 R. Yuan. Existence of Almost Periodic Solutions of Neutral Functional-differntial Equations Via Liapunov-razumikhin Function. Z. Angew. Math.Phys. 1998, 49(1):113–136
87 E. A. Dads, K. Ezzinbi. Pseudo Almost Periodic Solutions of some Delay Dif-ferential Equations. J. Math. Anal. Appl. 1996, 201(3):840–850
88 D. Piao. Periodic and Almost Periodic Solutions for Differential Equations withRe?ection of the Argument. Nonlinear Analysis TMA. 2004, 57(4):633–637
89 Z. Teng. The Almost Periodic Kolmogorov Competitive Systems. NonlinearAnalysis TMA. 2000, 42(7):1221–1230
90 G. Seifert. Second-order Neutral Delay Differential Equations with PiecewiseConstant Time Dependence. J. Math. Anal. Appl. 2003, 281(1):1–9
91 R. P. Agarwal, J. Hong, Y. Liu. A Class of Ergodic Solutions of NonlinearDifferential Equations and Numerical Treatment. Math. Comput. Modelling.2000, 32(3-4):493–506
92 A. I. Alonso, J. Hong, R. Obaya. Almost Periodic Type Solutions of Differ-ential Equations with Piecewise Constant Argument Via Almost Periodic TypeSequences. Applied Mathematics Letters. 2000, 13(2):131–137
93 R. Yuan. Stability and Almost Periodic Solutions of Neutral Functional Dif-ferential Equations with Infinite Delay. Acta Mathematics Applicatae Sinica.1998, 14(1):68–73
94 J. Hong, C. Nunez. The Almost Periodic Type Difference Equations. Math.Comput. Modelling. 1998, 28(12):21–31
95 B. Guo, R. Yuan. On Existence of Almost Periodic Solution of Ginzburg-landauEquation. Communications in Nonlinear Science and Numerical Simulation.2000, 5(2):74–79
96 O. Rafael, T. Massimo. Almost Periodic Linear Differential Equations withNon-separated Solutions. Journal of Functional Analysis. 2006, 237(2):402–42697 D. Toka, H. Eduardo, R. Marcos. Pseudo Almost Periodic Solutions to someNon-autonomous Neutral Functional Differential Equations with UnboundedDelay. Math. Comput. Modelling. 2007, 45(9-10):1241–1252
98 C. Zhang, F. Yang. Remotely Almost Periodic Solutions of Parabolic InverseProblems. Nonlinear Analysis TMA. 2006, 65(8):1613–1623
99 D. Toka. Pseudo Almost Periodic Solutions to some Differential Equations.Nonlinear Analysis TMA. 2005, 60(7):1277–1286
100 R. Yuan. On Almost Periodic Solutions of Logistic Delay Differential Equa-tions with Almost Periodic Time Dependence. J. Math. Anal. Appl. 2007,330(2):780–798
101 H. M. Eduardo, L. P. Maurício. Asymptotically Almost Periodic and AlmostPeriodic Solutions for Partial Neutral Differential Equations. Applied Mathe-matics Letters. 2005, 18(11):1265–1272
102 C. F. Muckenhoupt. Almost Periodic Functions and Vibrating Systems. J. Math.Phys. 1929, 8:163–198
103 S. Bochner. Fastperiodische Lo¨sungen Der Wellengleichung. Acta. Math. 1933,62(1):227–237
104 T. Naito, N. V. Minh. Evolution Semigroups and Spectral Criteria for AlmostPeriodic Solutions of Periodic Evolution Equations. J. Differential Equations.1999, 152(2):358–376
105 C. Zhang. Ergodicity and Its Applications-part One: Basic Properties. ActaAnalysis Functionalis Applicata. 1999, 1(1):28–39
106 C. Zhang. Ergodicity and Its Applications in Regularity and Solutions of PseudoAlmost Periodic Equations. Nonlinear Analysis TMA. 2001, 46(4):511–523
107 C. Zhang. Ergodicity and Asymptotically Almost Periodic Solutions of someDifferential Equations. Internat. J. Math. Math. Sci. 2001, 25(12):787–800
108 J. Hong, R. Obaya, A. M. Sanz. Ergodic Solutions Via Ergodic Sequence. Non-linear Analysis TMA. 2000, 40:265–277
109 J. Hong, R. Obaya. Ergodic Type Solutions of some Differential Equations.Kluwer Academic Publishers, 2001, 135–152
110 J. Hong, R. Obaya, A. S. Gil. Exponential Trichotomy and a Class of ErgodicSolutions of Differential Equations with Ergodic Perturbations. Applied Math-ematics Letters. 1999, 12(1):7–13
111 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
112 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
113 W. Shen, Y. Yi. Dynamics of Almost Periodic Scalar Parabolic Equations. J.Differential Equations. 1995, 122(1):114–136
114 J. A. Sanders, F. Verhulst. Averaging Methods in Nonlinear Dynamical Sys-tems. Spring-Verlag. New York, 1985
115 C. Zhang. Ergodicity and Its Application II: Averaging Method of some Dy-namical Systems. Acta Analysis Functionalis Applicata. 1999, 1(2):146–159
116 R. Bitmead, C. Johnson. Discrete Averaging Principles in Robust AdaptiveIdentification. Control and Dynamic Systems. 1987, 25:237–271
117 C. Zhang. New Limit Power Function Spaces. IEEE Trans. Auto. Control.2004, 49:763–766
118 C. Zhang. Strong Limit Power Functions. J. Four.Anal. Appl. 2006, 12:291–307119 C. Zhang, C. Meng. C?-algebra of Strong Limit Power Functions. IEEE Trans.Auto. Control. 2006, 51:828–831
120 H. Bre′zis. Ope′rateurs Maximaux Momonotones Et Semi-groupes De Contrac-tions Dans Les Espaces De Hilbert. North-Holland, Amsterdam, 1973
121 S. Zaidman. Abstract Differential Equations. Research Notes in Mathematics36,Pitman, Boston London, 1979
122 S. Zaidman. Topics in Abstract Differential Equations. Pitman Research Notesin Mathematics Series 304,New York, 1994
123 S. Zaidman. Topics in Abstract Differential Equations II. Pitman ResearchNotes in Mathematics Series 321, New York, 1995
124 M. S. Berge. Nonlinearity and Functional Analysis. Academic Press, NewYork, San Francisco, London, 1977
125周鸿兴,王连文.线性算子半群理论及应用.山东科学技术出版社, 1994
126 A. Pazy. Semigroups of Linear Operators and Applications to Partial Differen-tial Equations. Springer-Verlag, New York, 1983
127 J. A. Goldstein. Semigroups of Linear Operators and Applications. OxfordUniv. Press, 1985
2 H. Bohr. Almost Periodic Functions. Chelsea,New York, 1951
3 S. Bochner. Beitra¨ge Zur Theorie Der Fastperiodischen Funktionen. Math. Ann.1927, 96(1):119–147
4 S. Bochner. A New Approach to Almost Periodicity. Proc. Nat. Acad. Sci. U.S. A. 1962, 48:2039–2043
5 S. Bochner, J. V. Neumann. Almost Periodic Functions in Groups II. Trans.Amer. Math. Soc. 1935, 37(1):21–50
6 A. S. Besicovitch. Almost Periodic Functions. Dover Publications Inc., NewYork, 1955
7 J. Favard. Sur Les e′quations Differentielles Line′aires a` Coefficients Presque-pe′riodiques. Acta. Math. 1928, 51(1):31–81
8 W. Stepanoff. u¨eber Einige Verallgemeinerungen Der Fast Periodischen Func-tionen. Math. Ann. 1926, 95(1):473–498
9 H. Weyl. Integralgleichungen Und Fastperiodische Functionen. Math. Ann.1927, 97(1):338–356
10 A. S. Besicovitch. On Parseval’s Theorem for Dirichlet Series. Proc. LondonMath. Soc. 1927, 26:25–34
11 R. A. Satyanarayan. On the Stepanov Almost Periodic Solutions of a Second-order Infinitesimal Generator of Differential Equations. Internat. J. Math. Math.Sci. 1991, 14(4):757–761
12 Z. Hu. Contributions to the Theory of Almost Periodic Differential Equations.Carleton University, Ph.D. thesis. 2001
13 M. Fre′chet. Les Fonctions Asymptotiquement Presque-periodiques Continues.C. R. Acad. Sci. Paris. 1941, 213:520–522
14 M. Fre′chet. Les Fonctions Asymptotiquement Presque-pe′riodiques. Revue Sci.1941, 79:341–354
15 W. F. Eberlein. Abstract Ergodic Theorems and Weak Almost Periodic Func-tions. Trans. Amer. Math. Soc. 1949, 67:217–240
16 C. Corduneanu. Almost Periodic Functions. Chelsea Publishing Company, NewYork, 1989
17 J. F. Berglund, H. D. Junghenn, P. Milnes. Analysis on Semigroup: FunctionSpaces. Compactifications,Representations. Wiley, New York, 1989
18 A. M. Fink. Almost Periodic Differential Equations. Springer-verlag, Berlin1974
19 B. M. Levitan. Almost Periodic Functions. Higher Education Press, 1956
20 S. Zaidman. Almost Periodic Functions in Abstract Spaces. Pitman,Boston,MA, 1985
21 O. Onicescu, V. Istratescu. Approximation Theorems for Random Functions.Rend. Mat. 1975, 8:65–81
22 O. Onicesu, G. Cenusa, I. Sacuiu. Random Functions Almost Periodic in Prob-ability. Editura Academiei Republicii Socialiste Roumania, Bucharest, 1983
23 D. Sarason. Toeplitz Operators with Semi-almost Periodic Symbols. DukeMath. J. 1977, 44(2):357–364
24 D. Sarason. Remotely Almost Periodic Function. Contemporary Mathematics.1984, 32:237–242
25 D. Sarason. Toeplitz Operators with Piecewise Quasicontinuous Symbols. In-diana Univ. Math. J. 1977, 26(5):817–838
26 C. Zhang. Pseudo Almost Periodic Functions and Their Applications. Univer-sity of Western Ontario, Ph.D. thesis. 1992
27 C.Zhang. Pseudo Almost Periodic Solutions of some Differential Equations.J.Math.Anal.Appl. 1994, 181(1):62–76
28 C.Zhang. Pseudo Almost Periodic Solutions of some Differential Equations(ii).J.Math.Anal.Appl. 1995, 192(2):543–561
29 C. Zhang. Integration of Vector-valued Pseudo Almost Periodic Functions. Pr-oc. Amer. Math. Soc. 1994, 121(1):167–174
30 C. Zhang. Vector-valued Pseudo Almost Periodic Functions. CzechoslovakMath. J. 1997, 47(122)(3):385–394
31 C. Zhang. Almost Periodic Type Functions and Ergodicity. Science Press/Klu-wer, New York, Beijing, 2003
32 C. Zhang, H. Yao. Converse Problems of Fourier Expansion and Their Appli-cations. Nonlinear Analysis TMA. 2004, 56(5):761–779
33 G. M. N’Guerekata. Almost Periodicity in Linear Topological Spaces and Ap-plications to Abstract Differential Equations. Internat. J. Math. Math. Sci. 1984,7(3):529–540
34 G. M. N’Guerekata. Almost Periodic Solutions of Certain Differential Equa-tions in Fre′chet Spaces. Riv. Mat. Univ. Parma (5). 1993, 2:301–303
35 G. M. N’Guerekata. Almost Periodicity of some Solutions to Linear AbstractEquations. Libertas Math. 1996, 16:145–148
36 G. M. N’Guerekata. Almost Automorphic and Almost Periodic Functions inAbstract Spaces. Kluwer Academic, Plenum Publishers, New York, 2001
37 D. Bugajewski, G. M. N’Guerekata. Almost Periodicity in Fre′chet Spaces. J.Math. Anal. Appl. 2004, 299(2):534–549
38 K. Cooke, J. Kapplan. A Periodicity Threshold Theorem for Epidemics andPopulation Growth. Math. Biosci. 1976, 31(1-2):87–104
39 A. Fink, J. Gatiga. Positive Almost Periodic Solutions of some Delay IntegralEquations. J. Differential Equations. 1990, 83(1):166–178
40 E. A. Dads, K. Ezzinbi. Positive Almost Periodic Solution for some NonlinearDelay Integral Equation. Nonlinear Stud. 1996, 3(1):85–101
41 R. Torrejon. Positive Almost Periodic Solutions of a State-dependent DelayNonlinear Integral Equation. Nonlinear Analysis TMA. 1993, 20(12):1383–1416
42 K. Ezzinbi, M. A. Hachimi. Existence of Positive Almost Periodic Solutions ofFunctional Equations Via Hilbert Projective Metric. Nonlinear Analysis TMA.1996, 26(6):1169–1176
43 B. Xu, R. Yuan. On the Positive Almost Periodic Type Solutions for someNonlinear Delay Integral Equations. J. Math. Anal. Appl. 2005, 304(1):249–26844 E. A. Dads, K. Ezzinbi. Almost Periodic Solution for some Neutral NonlinearIntegral Equations. Nonlinear Analysis TMA. 1997, 28(9):1479–1489
45 E. A. Dads, K. Ezzinbi. Existence of Positive Pseudo Almost Periodic Solu-tion for some Nonlinear Infinite Delay Integral Equations Arising in EpidemicProblems. Nonlinear Analysis TMA. 2000, 41:1–13
46 E. A. Dads, K. Ezzibi, O. Arino. Pseudo Almost Periodic Solutions for someDifferential Equations in a Banach Space. Nonlinear Analysis TMA. 1997,28(7):1141–1155
47 D. Piao. Pseudo Almost Periodic Solutions of the Systems of DifferentialEquations with Piecewise Constant Argument [t]. Sci. in China (A). 2001,44(9):1156–1161
48何崇佑.概周期微分方程.高等教育出版社, 1992, 1–246
49 M. Biroli. Sur Les Solutions Borne′es Ou Presque-pe′riodiques Des e′quationsD’e′volution Multivoques Sur Un Espace De Hilbert. Ricerche Mat. 1972,21:17–47
50 C. M. Dafermos. Almost Periodic Process and Almost Periodic Solutionsof Evolutions Equations. Dynamical systems (Proc. Internat. Sympos., Univ.Florida, Gainesville, Fla.,1976), 43-57. Academic Press, New York, 1977
51 A. Haraux. Nonlinear Evolution Equation-global Behavior of Solutions. Lec-ture Notes in Mathematics, 841 Springer-Verlag, Berlin, New York, 1981
52 F. L. Huang. Existence of Almost Periodic Solutions for Dissipative DifferentialEquations. Ann. Differential Equations. 1990, 6:271–279
53 H. Ishii. On the Existence of Almost Periodic Complete Trajectories for Con-tractive Almost Periodic Processes. J. Differential Equations. 1982, 43:66–72
54 P. Cieutat. Necessary and Sufficient Conditions for Existence and Uniquenessof Bounded Or Almost Periodic Solutions for Differential Sytems with ConvexPotential. Differential and Integral Equations. 2004, 18:361–378
55 V. E. Slyusarchuk. Necessary and Sufficient Conditions for Existence andUniquenss of Bounded and Almost Periodic Solutions of Nonlinear DifferentialEquations. Acta Appl. Math. 2001, 65:333–341
56 M.S.Berger, Y.Y.Chen. Forced Quasiperiodic and Almost Periodic Oscillationson Nonlinear Duffing Equations. Nonlinear Analysis TMA. 1992, 19:249–257
57 M.S.Berger, Y.Y.Chen. Forced Quasiperiodic and Almost Periodic Oscillationsfor Nonlinear Systems. Nonlinear Analysis TMA. 1993, 21:949–965
58 C. Carminati. Forced Systems with Almost Periodic and Quasiperiodic ForcingTerm. Nonlinear Analysis TMA. 1998, 32:727–739
59 M. S. Berger, L. Zhang. New Method for Large Quasiperiodic Nonlinear Os-cillations with Fixed Frequencies for the Nondissipative Second Type DuffingEquation. Topological Methods Nonlinear Anal. 1995, 6:283–293
60 M. S. Berger, L. Zhang. A New Method for Large Quasiperiodic Nonlinear Os-cillations with Fixed Frequencies for the Nondissipative Conservative Systems.Commun. Appl. Nonlinear Anal. 1995, 2:79–106
61 P. Cieutat. Almost Periodic Solutions of Second-order Systems with MonotoneField on a Compact Subset. Nonlinear Analysis TMA. 2003, 53:751–763
62 S. F. Zakharin, I. O. Parasyuk. Generalized and Classical Almost Periodic So-lutions of Lagrangian Systems. Funkcial. Ekvac. 1999, 42:325–338
63 P. Cieutat. Almost Periodic Solutins of Forced Vectorial Lie′nard Equations. J.Differential Equations. 2005, 209:302–328
64 P. Cieutat. Bounded and Almost Periodic Solutins of Convex Lagrangian Sys-tems. J. Differential Equations. 2003, 190:108–130
65 J. Bolt. Calculus of Variations on Mean and Convex Lagaragians. J. Math.Anal. Appl. 1988, 134:312–321
66 J. Bolt, P. Cieutat, J. Mawhin. Almost Periodic Oscillations of MonotoneSecond-order Systems. Adv. Differential Equations. 1997, 2:693–714
67 J.Bolt. Oscillations Presque-pe′riodiques Force′es D’e′quations D’euler-lagrange.Bull. Soc. Math. France. 1994, 122:285–304
68 J. P. Chen. Almost Periodic Solutions of Higher-dimensional Lie′nard Systems.Atti Sem. Mat. Fis. Univ. Modena. 1990, 38:123–135
69 F. Lin. Stability and Existence of Periodic Solutions and Almost Periodic Solu-tions on Lie′nard Systems. Ann. Differential Equations. 1997, 13:248–253
70 K. Wang. Almost Periodic Solutions of Forced Line′nard Equations. Chinese J.Contemp. Math. 1995, 16:253–260
71 J. Zhou. The Existence of Periodic Solutions and Almost Periodic Solutions ofFoeced Line′nard-type Equations. Acta. Math. Sinica. 1999, 42:571–576
72 K. Schmitt, J. R. Ward. Almost Periodic Solutions of Nonlinear Second OrderDifferential Equations. Results in Mathematics. 1992, 21:191–199
73 C. Cuevas, M. Pinto. Existence and Uniqueness of Pseudo Almost PeriodicSolutions of Semilinear Cauchy Problems with Non-dense Domain. NonlinearAnalysis TMA. 2001, 45(1):73–83
74 T. Diagana, C. M. Mahop, G. M. N Gue′re′kata. Pseudo Almost Periodic Solutionto some Semilinear Differential Equations. Math. Comput. Modelling. 2006,43(2):89–96
75 T. Diagana, C. M. Mahop, G. M. N Gue′re′kata. Existence and Uniqueness ofPseudo Almost Periodic Solutions to some Classes of Semilinear DifferentialEquations and Applications. Nonlinear Analysis TMA. 2006, 64(11):2442–2453
76 H. X. Li, F. L. Huang, J. Y. Li. Composition of Pseudo Almost PeriodicFunctions and Semilinear Differential Equations. J. Math. Anal. Appl. 2001,255(2):436–446
77 E. Hernandez, M. L. Pelicer, J. P. C. dos Santos. Asymptotically Almost Pe-riodic and Almost Periodic Solutions for a Class of Evolution Equations. J.Differential Equations. 2004, 61:1–15
78 E. Herna′ndez, M. L. Pelicer. Asymptotically Almost Periodic and Almost Peri-odic Solutions for Partial Neutral Differential Equations. Applied MathematicsLetters. 2005, 18:1265–1272
79 T. Diagana. Existence and Uniqueness of Pseudo-almost Periodic Solutions tosome Classes of Partial Evolution Equations. Nonlinear Analysis TMA. 2007,66:384–395
80 T. Diagana, E. Herna′ndez, M. Rabello. Pseudo Almost Periodic Solutionsto some Non-autonomous Neutral Functionsl Differential Equations with Un-bounded Delay. Mathematical and Computer Modelling. 2007, 45:1241–1252
81 T. Diagana, E. M. Herna′dez. Existence and Uniqueness of Pseudo Almost Pe-riodic Solutions to some Abstract Partial Neutral Functional-differential Equa-tions and Applications. J. Math. Anal. Appl. 2007, 327:776–791
82 C. Cuevas, M.Pinto. Existence and Uniqueness of Pseudo Solutions of Semilin-ear Cauchy Problems with Nondense Domain. Nonlinear Analysis TMA. 2001,45(1):73–83
83 T. Diagina. Pseudo Almost Periodic Solutions to some Differential Equations.Nonlinear Analysis TMA. 2005, 60(7):1277–1286
84 H. R. Henr′?quez, C. H. Va′squez. Almost Periodic Solutions of Abstract Re-tarded Functional-differential Equations with Unbounded Delay. Acta Appl.Math. 1999, 57(2):105–132
85 N. M. Man, N. V. Minh. On the Existence of Quasi Periodic and Almost Pe-riodic Solutions of Neutral Functional Differential Equations. Commun. PureAppl. Anal. 2004, 3(2):291–300
86 R. Yuan. Existence of Almost Periodic Solutions of Neutral Functional-differntial Equations Via Liapunov-razumikhin Function. Z. Angew. Math.Phys. 1998, 49(1):113–136
87 E. A. Dads, K. Ezzinbi. Pseudo Almost Periodic Solutions of some Delay Dif-ferential Equations. J. Math. Anal. Appl. 1996, 201(3):840–850
88 D. Piao. Periodic and Almost Periodic Solutions for Differential Equations withRe?ection of the Argument. Nonlinear Analysis TMA. 2004, 57(4):633–637
89 Z. Teng. The Almost Periodic Kolmogorov Competitive Systems. NonlinearAnalysis TMA. 2000, 42(7):1221–1230
90 G. Seifert. Second-order Neutral Delay Differential Equations with PiecewiseConstant Time Dependence. J. Math. Anal. Appl. 2003, 281(1):1–9
91 R. P. Agarwal, J. Hong, Y. Liu. A Class of Ergodic Solutions of NonlinearDifferential Equations and Numerical Treatment. Math. Comput. Modelling.2000, 32(3-4):493–506
92 A. I. Alonso, J. Hong, R. Obaya. Almost Periodic Type Solutions of Differ-ential Equations with Piecewise Constant Argument Via Almost Periodic TypeSequences. Applied Mathematics Letters. 2000, 13(2):131–137
93 R. Yuan. Stability and Almost Periodic Solutions of Neutral Functional Dif-ferential Equations with Infinite Delay. Acta Mathematics Applicatae Sinica.1998, 14(1):68–73
94 J. Hong, C. Nunez. The Almost Periodic Type Difference Equations. Math.Comput. Modelling. 1998, 28(12):21–31
95 B. Guo, R. Yuan. On Existence of Almost Periodic Solution of Ginzburg-landauEquation. Communications in Nonlinear Science and Numerical Simulation.2000, 5(2):74–79
96 O. Rafael, T. Massimo. Almost Periodic Linear Differential Equations withNon-separated Solutions. Journal of Functional Analysis. 2006, 237(2):402–42697 D. Toka, H. Eduardo, R. Marcos. Pseudo Almost Periodic Solutions to someNon-autonomous Neutral Functional Differential Equations with UnboundedDelay. Math. Comput. Modelling. 2007, 45(9-10):1241–1252
98 C. Zhang, F. Yang. Remotely Almost Periodic Solutions of Parabolic InverseProblems. Nonlinear Analysis TMA. 2006, 65(8):1613–1623
99 D. Toka. Pseudo Almost Periodic Solutions to some Differential Equations.Nonlinear Analysis TMA. 2005, 60(7):1277–1286
100 R. Yuan. On Almost Periodic Solutions of Logistic Delay Differential Equa-tions with Almost Periodic Time Dependence. J. Math. Anal. Appl. 2007,330(2):780–798
101 H. M. Eduardo, L. P. Maurício. Asymptotically Almost Periodic and AlmostPeriodic Solutions for Partial Neutral Differential Equations. Applied Mathe-matics Letters. 2005, 18(11):1265–1272
102 C. F. Muckenhoupt. Almost Periodic Functions and Vibrating Systems. J. Math.Phys. 1929, 8:163–198
103 S. Bochner. Fastperiodische Lo¨sungen Der Wellengleichung. Acta. Math. 1933,62(1):227–237
104 T. Naito, N. V. Minh. Evolution Semigroups and Spectral Criteria for AlmostPeriodic Solutions of Periodic Evolution Equations. J. Differential Equations.1999, 152(2):358–376
105 C. Zhang. Ergodicity and Its Applications-part One: Basic Properties. ActaAnalysis Functionalis Applicata. 1999, 1(1):28–39
106 C. Zhang. Ergodicity and Its Applications in Regularity and Solutions of PseudoAlmost Periodic Equations. Nonlinear Analysis TMA. 2001, 46(4):511–523
107 C. Zhang. Ergodicity and Asymptotically Almost Periodic Solutions of someDifferential Equations. Internat. J. Math. Math. Sci. 2001, 25(12):787–800
108 J. Hong, R. Obaya, A. M. Sanz. Ergodic Solutions Via Ergodic Sequence. Non-linear Analysis TMA. 2000, 40:265–277
109 J. Hong, R. Obaya. Ergodic Type Solutions of some Differential Equations.Kluwer Academic Publishers, 2001, 135–152
110 J. Hong, R. Obaya, A. S. Gil. Exponential Trichotomy and a Class of ErgodicSolutions of Differential Equations with Ergodic Perturbations. Applied Math-ematics Letters. 1999, 12(1):7–13
111 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
112 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
113 W. Shen, Y. Yi. Dynamics of Almost Periodic Scalar Parabolic Equations. J.Differential Equations. 1995, 122(1):114–136
114 J. A. Sanders, F. Verhulst. Averaging Methods in Nonlinear Dynamical Sys-tems. Spring-Verlag. New York, 1985
115 C. Zhang. Ergodicity and Its Application II: Averaging Method of some Dy-namical Systems. Acta Analysis Functionalis Applicata. 1999, 1(2):146–159
116 R. Bitmead, C. Johnson. Discrete Averaging Principles in Robust AdaptiveIdentification. Control and Dynamic Systems. 1987, 25:237–271
117 C. Zhang. New Limit Power Function Spaces. IEEE Trans. Auto. Control.2004, 49:763–766
118 C. Zhang. Strong Limit Power Functions. J. Four.Anal. Appl. 2006, 12:291–307119 C. Zhang, C. Meng. C?-algebra of Strong Limit Power Functions. IEEE Trans.Auto. Control. 2006, 51:828–831
120 H. Bre′zis. Ope′rateurs Maximaux Momonotones Et Semi-groupes De Contrac-tions Dans Les Espaces De Hilbert. North-Holland, Amsterdam, 1973
121 S. Zaidman. Abstract Differential Equations. Research Notes in Mathematics36,Pitman, Boston London, 1979
122 S. Zaidman. Topics in Abstract Differential Equations. Pitman Research Notesin Mathematics Series 304,New York, 1994
123 S. Zaidman. Topics in Abstract Differential Equations II. Pitman ResearchNotes in Mathematics Series 321, New York, 1995
124 M. S. Berge. Nonlinearity and Functional Analysis. Academic Press, NewYork, San Francisco, London, 1977
125周鸿兴,王连文.线性算子半群理论及应用.山东科学技术出版社, 1994
126 A. Pazy. Semigroups of Linear Operators and Applications to Partial Differen-tial Equations. Springer-Verlag, New York, 1983
127 J. A. Goldstein. Semigroups of Linear Operators and Applications. OxfordUniv. Press, 1985