几类非线性微分方程的周期、概周期解的存在性
|
摘要
随着科学技术的发展和社会的进步,常微分方程的应用不断扩大与深入,在自然科学和社会科学的诸多领域都有广泛的应用,自动控制理论、无线电技术、火箭的飞行、导弹的发射、机器的运转、电子管振荡器的震荡、化学反应稳定性的研究、神经网络、生物技术、图像处理、军备竞赛、人口问题、传染病问题和金融问题等数学模型往往可化为常微分方程。因而常微分方程的研究具有实际意义。自然界和社会生活中的各种各样的现象,有的现象可通过数学模型描述出来,其中有一类是通过微分方程的形式描述的,而微分方程往往又是非线性的,在形形式式的诸多现象中,有一类特殊的现象,周期现象,在非线性微分方程中,表示为方程的周期解,从法国著名数学家Poincare[1]认识到周期解在常微分方程定性理论研究中的重要性之后,很多数学家和物理学家也开始关注非线性方程的周期解,希望以这种特殊而重要的解的研究为突破口来搞清楚未知的微分方程的解的一些性态,从而能够进一步加深人们对自然界中广泛存在的各种各样的自然现象的认识和理解,并为人们利用自然和改造自然提供强有力的理论依据.因此对非线性微分方程的周期解的研究具有重要的科学意义和应用价值.
本文研究了几类非线性微分方程的周期解的存在性,也涉及到一些周期解的稳定性,研究的系统主要有：高维非线性微分系统,高维里卡提微分系统,非线性多项式微分系统,阿贝尔方程,里卡提方程,非线性Logistic系统以及非线性Lotka-Volterra生态竞争系统。
第一章介绍了研究周期解的常用的数学工具,不动点定理,指数型二分性理论,周期解的存在性的一个定理,稳定性理论,李雅普诺夫第二方法等概念。
第二章讨论了高维非线性微分方程,在高维系统周期解的研究中,主要用的方法的矩阵的特征值理论,利用压缩映射原理得到周期解的存在唯一性,利用李雅普诺夫函数法得到周期解的稳定性,推广了前人的一些相关研究成果；利用高维系统周期解存在性的一些理论,研究了高维里卡提方程,得到了其周期解的存在性和唯一性的一些充分性条件。
第三章研究了非线性多项式微分系统,讨论了方程可积的一些列充分性条件,并讨论了非线性多项式微分系统的三个周期解的存在性,其中两个周期解的稳定性；接着,讨论了阿贝尔方程和里卡提方程的周期解的存在性和稳定性,得到了一些新的结论.
第四章讨论了一类非线性系统,利用不动点定理得到了系统概周期解的存在性,并讨论了概周期解的稳定性.
第五、六、七和第八章讨论了一些较为流行的生态系统的周期、概周期解的存在性和稳定性,主要有：时滞单种群生态模型,利用重合度理论得到了该系统周期正解的存在性；两种群的非线性的Votarra生态模型,得到了其周期解的存在唯一性的一些充分性条件；非线性的Votarra生态竞争模型,得到了其概周期解的存在唯一性的一些充分性条件；具反馈控制非线性的Votarra生态竞争模型,得到了其概周期解的存在唯一性的一些充分性条件.
第九章总结和展望
With the development of science and technology and the progress of society, the application of ordinary differential equations expanding depth, have a wide range of applications in many fields of natural science and social science, automatic control theory, radio technology, the flight of the rocket, missile launch, the mathematical model of the operation of the machine, the shock of the vacuum tube oscillator, the stability of the chemical reaction, neural networks, bio-technology, image processing, the arms race, the population problem, the problem of infectious diseases and financial problems often approach into ordinary differential equations. Thus, it is much practical significance to study ordinary differential equations. In the nature and social life, there are various phenomena, some phenomena can be descriped through mathematics model, which has category through differential equation, and differential equation is often nonlinear. In shaped forms of many phenomena, there ia a special phenomenon, that is cycle phenomenon, in the nonlinear differential equation, it is said cycle solutions of differential equation in mathematics. cycle solutions corresponds to differential equation, since France famous mathematician Poincare[1], he awared that cycle solutions are very important in differential equation qualitative theory research, Many mathematician and physicist also began concern cycle solutions of nonlinear equation, hope to this special and important of solutions of research for breakthrough to got clearly unknown of solutions of differential equation of some sexual State, furthermore, it deepened people in nature in the awareness of various of natural phenomenon and understanding the nature widely, and for people, using natural and natural provides strong of rationale transformation. Therefore, it has important of science meaning and application value.
In this paper, the existence of periodic solutions of some class of nonlinear differential equations is studied, also related to the stability of periodic solutions, we study the following system:high-dimensional nonlinear differential systems, the high Riccati system nonlinear entry systems, Abel equations, nonlinear logistic system and ecosystems.
The first chapter introduces the mathematical tools to study periodic solution, fixed point theorem, exponential dichotomy theory, a theorem of the existence of periodic solutions, stability theory, Lyapunov second method.
The second chapter discusses the high-dimensional nonlinear differential equations, in the study of periodic solutions of high-dimensional systems, the main method is matrix eigenvalue theory, and the contraction mapping principle, by these methods, the existence and uniqueness of periodic solution of the equation are obtained and using Lyapunov function, we obtained the stability of the periodic solution, these results promoted some related achievements of their predecessors; also, we study the high-dimensional system, and obtained the existence and stability of periodic solutions, Some sufficient conditions are derived.
In third chapter, we studied the non-linear multinomial system, and discussed some sufficient conditions which guranteed the general solutions of the equation, and discussed the non-linear multinomial system, it has three periodic solutions, and two periodic solutions are stable; then, we discussed the Abbel equation and Riccati equation, the existence and the stability of the systems are obtained, some new conclusions are drawn.
The fourth chapter discusses a class of nonlinear systems, by using fixed point theorem, we obtained the existence and uniqueness of the almost periodic solution are obtained.
In fifth, sixth, seventh and eighth chapters, we discuss some more popular ecosystem, and these systems exist the periodic solutions under certain conditions and the stability of the periodic solutions, In a delay population ecology model, by using the coincidence degree theorem theorem, we obtained the existence of the positive periodic solution; In nonlinear Lotka-volterra ecological model, we obtained some sufficient conditions which guaranteed the positive almost periodic solution; In nonlinear Lotka-volterra ecological model with feedback control, the existence and uniqueness of positive almost periodic solutions are obtained.
At last,that is to say, ninth chapter is summary and outlook.
本文研究了几类非线性微分方程的周期解的存在性,也涉及到一些周期解的稳定性,研究的系统主要有：高维非线性微分系统,高维里卡提微分系统,非线性多项式微分系统,阿贝尔方程,里卡提方程,非线性Logistic系统以及非线性Lotka-Volterra生态竞争系统。
第一章介绍了研究周期解的常用的数学工具,不动点定理,指数型二分性理论,周期解的存在性的一个定理,稳定性理论,李雅普诺夫第二方法等概念。
第二章讨论了高维非线性微分方程,在高维系统周期解的研究中,主要用的方法的矩阵的特征值理论,利用压缩映射原理得到周期解的存在唯一性,利用李雅普诺夫函数法得到周期解的稳定性,推广了前人的一些相关研究成果；利用高维系统周期解存在性的一些理论,研究了高维里卡提方程,得到了其周期解的存在性和唯一性的一些充分性条件。
第三章研究了非线性多项式微分系统,讨论了方程可积的一些列充分性条件,并讨论了非线性多项式微分系统的三个周期解的存在性,其中两个周期解的稳定性；接着,讨论了阿贝尔方程和里卡提方程的周期解的存在性和稳定性,得到了一些新的结论.
第四章讨论了一类非线性系统,利用不动点定理得到了系统概周期解的存在性,并讨论了概周期解的稳定性.
第五、六、七和第八章讨论了一些较为流行的生态系统的周期、概周期解的存在性和稳定性,主要有：时滞单种群生态模型,利用重合度理论得到了该系统周期正解的存在性；两种群的非线性的Votarra生态模型,得到了其周期解的存在唯一性的一些充分性条件；非线性的Votarra生态竞争模型,得到了其概周期解的存在唯一性的一些充分性条件；具反馈控制非线性的Votarra生态竞争模型,得到了其概周期解的存在唯一性的一些充分性条件.
第九章总结和展望
With the development of science and technology and the progress of society, the application of ordinary differential equations expanding depth, have a wide range of applications in many fields of natural science and social science, automatic control theory, radio technology, the flight of the rocket, missile launch, the mathematical model of the operation of the machine, the shock of the vacuum tube oscillator, the stability of the chemical reaction, neural networks, bio-technology, image processing, the arms race, the population problem, the problem of infectious diseases and financial problems often approach into ordinary differential equations. Thus, it is much practical significance to study ordinary differential equations. In the nature and social life, there are various phenomena, some phenomena can be descriped through mathematics model, which has category through differential equation, and differential equation is often nonlinear. In shaped forms of many phenomena, there ia a special phenomenon, that is cycle phenomenon, in the nonlinear differential equation, it is said cycle solutions of differential equation in mathematics. cycle solutions corresponds to differential equation, since France famous mathematician Poincare[1], he awared that cycle solutions are very important in differential equation qualitative theory research, Many mathematician and physicist also began concern cycle solutions of nonlinear equation, hope to this special and important of solutions of research for breakthrough to got clearly unknown of solutions of differential equation of some sexual State, furthermore, it deepened people in nature in the awareness of various of natural phenomenon and understanding the nature widely, and for people, using natural and natural provides strong of rationale transformation. Therefore, it has important of science meaning and application value.
In this paper, the existence of periodic solutions of some class of nonlinear differential equations is studied, also related to the stability of periodic solutions, we study the following system:high-dimensional nonlinear differential systems, the high Riccati system nonlinear entry systems, Abel equations, nonlinear logistic system and ecosystems.
The first chapter introduces the mathematical tools to study periodic solution, fixed point theorem, exponential dichotomy theory, a theorem of the existence of periodic solutions, stability theory, Lyapunov second method.
The second chapter discusses the high-dimensional nonlinear differential equations, in the study of periodic solutions of high-dimensional systems, the main method is matrix eigenvalue theory, and the contraction mapping principle, by these methods, the existence and uniqueness of periodic solution of the equation are obtained and using Lyapunov function, we obtained the stability of the periodic solution, these results promoted some related achievements of their predecessors; also, we study the high-dimensional system, and obtained the existence and stability of periodic solutions, Some sufficient conditions are derived.
In third chapter, we studied the non-linear multinomial system, and discussed some sufficient conditions which guranteed the general solutions of the equation, and discussed the non-linear multinomial system, it has three periodic solutions, and two periodic solutions are stable; then, we discussed the Abbel equation and Riccati equation, the existence and the stability of the systems are obtained, some new conclusions are drawn.
The fourth chapter discusses a class of nonlinear systems, by using fixed point theorem, we obtained the existence and uniqueness of the almost periodic solution are obtained.
In fifth, sixth, seventh and eighth chapters, we discuss some more popular ecosystem, and these systems exist the periodic solutions under certain conditions and the stability of the periodic solutions, In a delay population ecology model, by using the coincidence degree theorem theorem, we obtained the existence of the positive periodic solution; In nonlinear Lotka-volterra ecological model, we obtained some sufficient conditions which guaranteed the positive almost periodic solution; In nonlinear Lotka-volterra ecological model with feedback control, the existence and uniqueness of positive almost periodic solutions are obtained.
At last,that is to say, ninth chapter is summary and outlook.
引文
[1]H.poincare, Les, methods nouvelles dela mecaniqueceleste[J], Gauthiers-Villare,Parli,1892.
[2]范进军,常微分方程续论[M].济南：山东大学出版社,2009,7.
[3]W. A. Coppel, Periodic propeties of ordinary dierentialequations[J]. Ann. Math. Pura. Appl.76,1976,27-50.
[4]Wang Ke, Periodic solutions to a class of differential equations with deviating arguments[J]. ACTA MATHEMATICASINICA,1994,3(37):409-413.
[5]林发兴,线性系统的指数型二分性[M].安徽大学出版社,1999：25。
[6]Hale J K, Ordinary Differential Equations[M]. Wiley-Interscience,1969.
[7]Cai Sui Lin, Ordinary Differential Equations[M]. Wuhan:Wuhan University Press,2003:301.
[8]翁爱芝,中国工程数学学报[J],2005；22(5)：893-897。
[9]秦元勋,周期系数Riccati微分方程的周期解[J],科学通报(英文),1979；24(23)：1062-1066。
[10]赵怀忠,周期系数Riccati微分方程,科学通报(英文)[J],1990；35(4)：314-315。
[11]李鸿翔,关于几类Riccati方程和二阶微分方程的周期解(英文)[J],应用数学和力学,1982；12(2)203-208。
[12]Andersenk M,Sandquist A, On the cross-ratio of four solutions of a first order ordinary differential equation.J. Differential Equations,1994,108:89-100.
[13]武津刚,关于周期系数Riccati方程周期解的存在性准则[J],系统科学与数学.1990；10(1)：24-30.
[14]何崇佑,概周期微分方程[M],北京：高等教育出版社,1992.
[15]李雄,周期系数Riccati方程的周期解[J],数学进展,1999；28(4)313-322.
[16]黄建吾,一类高维Riccati方程的解(英文)[J],数学研究,2001；34(4)：374-378.
[17]Hale J K,常微分方程[M].。威利跨学科,1969.
[18]蔡燧林,常微分方程[M].。武汉：武汉大学出版社,2003：301.
[19]J.P.M. Lebrun, On two coupled Abel-type differential equations arising in a magnetostatic problem[J]. Ⅱ Nuovo Cimento,1990,103A:1369-1379.
[20]Borghero and A. Melis, On Szebehely's problem for holonomic systems involving generalized potential functions[J]. Celestial Mechanics and Dynamical Astronomy.1990,49:273-284.
[21]S.S. Bayn, Solutions of Einstein's field equations for static fluid spheres[J]. Phys. Rev. 1978,D18:2745-2751.
[22]A. Garcia, A. Macias and E.W. Mielke,. Stewart-Lyth second order approach as an Abel equation for recon-structing inflationary dynamics[J]. Phys. Lett.1997, A229:32-36.
[23]V.R. Gavrilov, V.D. Ivashchuk and V.N. Melnikov, Multidimensional integrable vacuum cosmology with two curvatures[J]. Class. Quantum Gray.1996,13:3039-3056.
[24]G. Haager and M. Mars, A self-similar inhomogeneous dust cosmology [J]. Glass. Quantum Gray. 1998,15:1567-1580.
[25]M.K.Mak and T.Harko, Full causal bulk-viscous cosmological models[J]. J. Math. Phys. 1998,39:5458-5476.
[26]M.K. Mak and T. Harko, Exact causal viscous cosmologies[J]. Gen. Rel. Gray.1998,30:1171-1186.
[27]M.K. Mak and T. Harko, Addendum to "Exact causal viscous cosmologies"[J]. Gen. Rel. Gray. 1999,31:273-274.
[28]Y.Matsuno,Two-dimensional dynamical system associated with Abel's nonlinear differential equation[J], J. Math. Phys.1992,33:412-421.
[29]G.L.Strobel and J.L.Reid, Nonlinear superposition rule for Abel's equation[J], Phys. Left.1982, 91A:209-210.
[30]J.L.Reid and G.L, Strobel. The nonlinear superposition theorem of Lie and Abel's differential equations[J]. Lettere Al Nuovo Cimento.1983,38:448-452.
[31]M.K. Mak, H.W., Chan and T. Harko. Solutions generating technique for Abel-type nonlinear ordinary differential equations[J]. Computers Math. Applic.2001,41:1395-1401.
[32]M.K.Mak and T.Harko, New Method for Generating General Solution of Abel Differential Equation[J]. Computers and Mathematics with Applications.2002,43:91-94.
[33]C.Y. He, Almost Periodic Functions and Differential Equations[M], Higher education press, Beijing, China,1992.
[34]C. R. Li and S. J. Lu, "The qualitative analysis of n-species periodic coefficient, nonlinear.
[35]张剑锋,系数变号时一类阿贝尔方程的极限环.数学年刊,18 A：3(1997),271-278.
[36]张剑锋,二次系统对应的阿贝尔方程的周期解.数学年刊,17A：6(1996),693-702
[37]周大勇,孙建梅,复域上阿贝尔方程可积条件初步.大连交通大学学报,2009,30(1)：94-97.
[38]刘靖,管克英,第一类阿贝尔方程可积性的初步研究.北京交通大学学报,2006,80(3).
[39]NI Hua,TAN Li-xin and ZHANG Hong. The Existence and Stability of the Periodic Solutions on Abel Differential Equation[J]. MATHEMATICA APPLJCATA,25(4):854-862.
[40]魏凤英,王克,渐近周期的Logistic方程.生物数学学报,2005,20(4)：399-405.
[41]Hirsch, M. W. and Smale, S.Differential Equations,Dynamical Systems and Linear Algebra[J], Aeademic Press, New YOrk,1974.
[42]Hale, J. K. and Kocak, H., Dynamics and bifurcations[M],Springer-Verlag, NewYork,1991.
[43]Chen, Y..Periodic solutions of adelayed periodic Logistic equation[J], APPI. Math. Comput., 139(2003),311-321.
[44]Chen, Y., Existence of positive periodic solutions for a periodic Logistic equation[J], Applied Math. Lett.,16(2003),1047-1051.
[45]Liu, X. and Chen, L., Global dynamics of the periodic logistic system with periodic impulsive perturbations[J],J. Math. Anal. Appl.,289(2004),279-291.
[46]De Mottoni, P. and Sehiaffino, A., Completion systems with periodic coeffieients:A geometrieal approach[J],J. Math. Biol.,11(1981),11-20.
[47]Teng, Z., On the positive almost periodic solution of a class of Lotka-Volterra type systems with delays[J],J. Math. Aoal. Appl.,249(2000),433-444
[48]Xia, Y.,Lin, M. and Cao, J., The existence of almost periodic solutions of certain perturbation systems[J],J. Math. Aoal. Anal.,310(2005),81-96.
[49]Nkashama, M. N., Bounded and almost periodic solutions of nonlocal perturbations of nonau-tonomousLogistic-type equations[J], Acta Appl. Math.,65(2001),25-293.
[50]Zhu, W. Z., Ji, S. G. and Liu, B. F., The existence of periodic and almost periodic solutions of Logistic type equation,Northeast[J]. Math. J.,23(4)(2007),298-310.
[51]Chen Fengde, Lin Faxin, Chen Xiaoxing. Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control [J]. Appiled Mathematics and Computation,2004,158(1):45-68.
[52]Chen Fengde, Shi Jinlin. Periodicity in a Logistic type system with several delays [J]. Computer and Mathematics with Applications,2004,48(1-2):35-44.
[53]Huang Zhenkun, Chen Fengde. Almost periodic solution of two species model with feedback regulation and infinite delay[J]. Chinese Journal of Engineering Mathematics,2004,21(1):33-40.
[54]Li Yonhkun. Attractivity of a positive periodic solution for all other positive solution in a delay population model [J]. Applied Mathematics-JCU,1997,12(3):279-282.
[55]陈凤德,陈晓星,林发兴,等.状态依赖时滞单种群对数模型的正周期解[J].福州大学学报(自然科学版),2003,31(3)：261-264.
[56]Gaines R E, Mawhin J L. Coincidence degree and nonlinear differential equations [M]. Berlin, Springer-Verlag,1997
[57]Gopalsmay K. Global asymptotic stability in an almost periodic Lotka-Volterra system[J]. J. Austral. Math.Sac. Ser.,1986,28(13):346-360.
[58]文贤章.多种群生态竞争-捕食时滞系统正周期解的全局吸引性[J].数学学报,2002,45(1)：83-92.
[59]Gyori I, Ladas G. Oscillation Theory of Delay Di(?)erential Equation[M]. Oxford:Oxford Science Publication1991.
[60]Fan Meng, Wang Ke. Global existence of positive periodic solutions of periodic predator-prey system with infinite delay[J]. J. Math. Appl.,2001,262(1):1-11.
[61]杨喜陶,朱焕然.具有无穷时滞的生态竞争系统概周期解存在性[J].生物数学学报,2005,20(4)：424-428.
[62]王全义.具无限时滞的积分微分方程的周期解的存在性、唯一性及稳定性[J].应用数学学报,1998,21(2)：312-318.
[63]黄启吕.具无限时滞的泛函微分方程的周期解的存在性[J].中国科学：A辑,1984(10)：882-889.
[64]Fink A M. Almost Periodic Differential Equation[M].Lecture Notes in Mathemat-ics,377,Berlin:Springerverlag,1974,1-15.
[65]Yoshizawa T. Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Berlin:Springer-Verlag,1975.
[66]Teng Z D. On the positive almost periodic solutions of a class of Lotka-Volterra type system with delays[J] Journal of Mathematical Analysis and Applications,2000,249(2):433-444.
[67]C. Wang and J.Shi,"Positive almostperiodicsolutions ofa class of Lotka-Volterratypecompetitive system with delaysand feedback controls," Applied Mathematicsand Computation,vol.193,no.1,pp. 240-252,2007.
[68]谢毅,李宪高.一类Lotka-Volterra系统的止概周期解[J].数学年刊,2009,30A(2)：161-168.
[69]鲁世平,葛渭高,一类多偏差变元的n种群Lotka-Volterra模型的周期正解[J].数学学报,2005,48(3)：427-438.
[70]陈凤德,n种群Lotka-Volterra时滞竞争反馈控制生态系统的全局吸引性[J].数学学报,2006,49(2)：335-346.
[71]Smart D.Fixed Point Theorems[M].London:Cambridge University Press,1974.
[72]Socio A.Ambrosetti. Periodic solutions of nonlinear wave equations with non-monotone forcierng terms. Rend. Mat.Acc.Lincei. s.9,v.16:117-124(2005)
[73]Massimiliano Berti,Luca Biasco, Forced vibrations of wave equations with non-monotone nonlinearities[J], Ann.I.H.Poincare-AN 23(2006)439-474
[74]PIETRO BALDI AND MASSIMILIANO BERTI.FORCED VIBRATIONS OF A OGENEOUS STRING[J].SIAM J. M ATH. A NAL.ol.40, No.1, pp.382-412
[75]Alberto Capozzi,Addolorata Salvatore. On forced vibrations of a semilinear wave equation without assumptions of monotonicity I Nonlinear Analysis[J].71(2009)e1932_e1941
[76]Shu guan Ji, Yong Li。Time Periodic Solutions to the One-Dimensional Nonlinear Wave Equation[J]。 Arch.Rational Mech.Anal.199(2011)435-451.
[77]廖晓昕.稳定性的理论、方法和应用(第二版)[M].武汉：华中科技大学出版社,2010,1.
[78]Gaines R E, Mawhin J L. Coincidence degree and nonlinear differential equations [M]. Berlin, Springer-Verlag,1997.
[2]范进军,常微分方程续论[M].济南：山东大学出版社,2009,7.
[3]W. A. Coppel, Periodic propeties of ordinary dierentialequations[J]. Ann. Math. Pura. Appl.76,1976,27-50.
[4]Wang Ke, Periodic solutions to a class of differential equations with deviating arguments[J]. ACTA MATHEMATICASINICA,1994,3(37):409-413.
[5]林发兴,线性系统的指数型二分性[M].安徽大学出版社,1999：25。
[6]Hale J K, Ordinary Differential Equations[M]. Wiley-Interscience,1969.
[7]Cai Sui Lin, Ordinary Differential Equations[M]. Wuhan:Wuhan University Press,2003:301.
[8]翁爱芝,中国工程数学学报[J],2005；22(5)：893-897。
[9]秦元勋,周期系数Riccati微分方程的周期解[J],科学通报(英文),1979；24(23)：1062-1066。
[10]赵怀忠,周期系数Riccati微分方程,科学通报(英文)[J],1990；35(4)：314-315。
[11]李鸿翔,关于几类Riccati方程和二阶微分方程的周期解(英文)[J],应用数学和力学,1982；12(2)203-208。
[12]Andersenk M,Sandquist A, On the cross-ratio of four solutions of a first order ordinary differential equation.J. Differential Equations,1994,108:89-100.
[13]武津刚,关于周期系数Riccati方程周期解的存在性准则[J],系统科学与数学.1990；10(1)：24-30.
[14]何崇佑,概周期微分方程[M],北京：高等教育出版社,1992.
[15]李雄,周期系数Riccati方程的周期解[J],数学进展,1999；28(4)313-322.
[16]黄建吾,一类高维Riccati方程的解(英文)[J],数学研究,2001；34(4)：374-378.
[17]Hale J K,常微分方程[M].。威利跨学科,1969.
[18]蔡燧林,常微分方程[M].。武汉：武汉大学出版社,2003：301.
[19]J.P.M. Lebrun, On two coupled Abel-type differential equations arising in a magnetostatic problem[J]. Ⅱ Nuovo Cimento,1990,103A:1369-1379.
[20]Borghero and A. Melis, On Szebehely's problem for holonomic systems involving generalized potential functions[J]. Celestial Mechanics and Dynamical Astronomy.1990,49:273-284.
[21]S.S. Bayn, Solutions of Einstein's field equations for static fluid spheres[J]. Phys. Rev. 1978,D18:2745-2751.
[22]A. Garcia, A. Macias and E.W. Mielke,. Stewart-Lyth second order approach as an Abel equation for recon-structing inflationary dynamics[J]. Phys. Lett.1997, A229:32-36.
[23]V.R. Gavrilov, V.D. Ivashchuk and V.N. Melnikov, Multidimensional integrable vacuum cosmology with two curvatures[J]. Class. Quantum Gray.1996,13:3039-3056.
[24]G. Haager and M. Mars, A self-similar inhomogeneous dust cosmology [J]. Glass. Quantum Gray. 1998,15:1567-1580.
[25]M.K.Mak and T.Harko, Full causal bulk-viscous cosmological models[J]. J. Math. Phys. 1998,39:5458-5476.
[26]M.K. Mak and T. Harko, Exact causal viscous cosmologies[J]. Gen. Rel. Gray.1998,30:1171-1186.
[27]M.K. Mak and T. Harko, Addendum to "Exact causal viscous cosmologies"[J]. Gen. Rel. Gray. 1999,31:273-274.
[28]Y.Matsuno,Two-dimensional dynamical system associated with Abel's nonlinear differential equation[J], J. Math. Phys.1992,33:412-421.
[29]G.L.Strobel and J.L.Reid, Nonlinear superposition rule for Abel's equation[J], Phys. Left.1982, 91A:209-210.
[30]J.L.Reid and G.L, Strobel. The nonlinear superposition theorem of Lie and Abel's differential equations[J]. Lettere Al Nuovo Cimento.1983,38:448-452.
[31]M.K. Mak, H.W., Chan and T. Harko. Solutions generating technique for Abel-type nonlinear ordinary differential equations[J]. Computers Math. Applic.2001,41:1395-1401.
[32]M.K.Mak and T.Harko, New Method for Generating General Solution of Abel Differential Equation[J]. Computers and Mathematics with Applications.2002,43:91-94.
[33]C.Y. He, Almost Periodic Functions and Differential Equations[M], Higher education press, Beijing, China,1992.
[34]C. R. Li and S. J. Lu, "The qualitative analysis of n-species periodic coefficient, nonlinear.
[35]张剑锋,系数变号时一类阿贝尔方程的极限环.数学年刊,18 A：3(1997),271-278.
[36]张剑锋,二次系统对应的阿贝尔方程的周期解.数学年刊,17A：6(1996),693-702
[37]周大勇,孙建梅,复域上阿贝尔方程可积条件初步.大连交通大学学报,2009,30(1)：94-97.
[38]刘靖,管克英,第一类阿贝尔方程可积性的初步研究.北京交通大学学报,2006,80(3).
[39]NI Hua,TAN Li-xin and ZHANG Hong. The Existence and Stability of the Periodic Solutions on Abel Differential Equation[J]. MATHEMATICA APPLJCATA,25(4):854-862.
[40]魏凤英,王克,渐近周期的Logistic方程.生物数学学报,2005,20(4)：399-405.
[41]Hirsch, M. W. and Smale, S.Differential Equations,Dynamical Systems and Linear Algebra[J], Aeademic Press, New YOrk,1974.
[42]Hale, J. K. and Kocak, H., Dynamics and bifurcations[M],Springer-Verlag, NewYork,1991.
[43]Chen, Y..Periodic solutions of adelayed periodic Logistic equation[J], APPI. Math. Comput., 139(2003),311-321.
[44]Chen, Y., Existence of positive periodic solutions for a periodic Logistic equation[J], Applied Math. Lett.,16(2003),1047-1051.
[45]Liu, X. and Chen, L., Global dynamics of the periodic logistic system with periodic impulsive perturbations[J],J. Math. Anal. Appl.,289(2004),279-291.
[46]De Mottoni, P. and Sehiaffino, A., Completion systems with periodic coeffieients:A geometrieal approach[J],J. Math. Biol.,11(1981),11-20.
[47]Teng, Z., On the positive almost periodic solution of a class of Lotka-Volterra type systems with delays[J],J. Math. Aoal. Appl.,249(2000),433-444
[48]Xia, Y.,Lin, M. and Cao, J., The existence of almost periodic solutions of certain perturbation systems[J],J. Math. Aoal. Anal.,310(2005),81-96.
[49]Nkashama, M. N., Bounded and almost periodic solutions of nonlocal perturbations of nonau-tonomousLogistic-type equations[J], Acta Appl. Math.,65(2001),25-293.
[50]Zhu, W. Z., Ji, S. G. and Liu, B. F., The existence of periodic and almost periodic solutions of Logistic type equation,Northeast[J]. Math. J.,23(4)(2007),298-310.
[51]Chen Fengde, Lin Faxin, Chen Xiaoxing. Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control [J]. Appiled Mathematics and Computation,2004,158(1):45-68.
[52]Chen Fengde, Shi Jinlin. Periodicity in a Logistic type system with several delays [J]. Computer and Mathematics with Applications,2004,48(1-2):35-44.
[53]Huang Zhenkun, Chen Fengde. Almost periodic solution of two species model with feedback regulation and infinite delay[J]. Chinese Journal of Engineering Mathematics,2004,21(1):33-40.
[54]Li Yonhkun. Attractivity of a positive periodic solution for all other positive solution in a delay population model [J]. Applied Mathematics-JCU,1997,12(3):279-282.
[55]陈凤德,陈晓星,林发兴,等.状态依赖时滞单种群对数模型的正周期解[J].福州大学学报(自然科学版),2003,31(3)：261-264.
[56]Gaines R E, Mawhin J L. Coincidence degree and nonlinear differential equations [M]. Berlin, Springer-Verlag,1997
[57]Gopalsmay K. Global asymptotic stability in an almost periodic Lotka-Volterra system[J]. J. Austral. Math.Sac. Ser.,1986,28(13):346-360.
[58]文贤章.多种群生态竞争-捕食时滞系统正周期解的全局吸引性[J].数学学报,2002,45(1)：83-92.
[59]Gyori I, Ladas G. Oscillation Theory of Delay Di(?)erential Equation[M]. Oxford:Oxford Science Publication1991.
[60]Fan Meng, Wang Ke. Global existence of positive periodic solutions of periodic predator-prey system with infinite delay[J]. J. Math. Appl.,2001,262(1):1-11.
[61]杨喜陶,朱焕然.具有无穷时滞的生态竞争系统概周期解存在性[J].生物数学学报,2005,20(4)：424-428.
[62]王全义.具无限时滞的积分微分方程的周期解的存在性、唯一性及稳定性[J].应用数学学报,1998,21(2)：312-318.
[63]黄启吕.具无限时滞的泛函微分方程的周期解的存在性[J].中国科学：A辑,1984(10)：882-889.
[64]Fink A M. Almost Periodic Differential Equation[M].Lecture Notes in Mathemat-ics,377,Berlin:Springerverlag,1974,1-15.
[65]Yoshizawa T. Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Berlin:Springer-Verlag,1975.
[66]Teng Z D. On the positive almost periodic solutions of a class of Lotka-Volterra type system with delays[J] Journal of Mathematical Analysis and Applications,2000,249(2):433-444.
[67]C. Wang and J.Shi,"Positive almostperiodicsolutions ofa class of Lotka-Volterratypecompetitive system with delaysand feedback controls," Applied Mathematicsand Computation,vol.193,no.1,pp. 240-252,2007.
[68]谢毅,李宪高.一类Lotka-Volterra系统的止概周期解[J].数学年刊,2009,30A(2)：161-168.
[69]鲁世平,葛渭高,一类多偏差变元的n种群Lotka-Volterra模型的周期正解[J].数学学报,2005,48(3)：427-438.
[70]陈凤德,n种群Lotka-Volterra时滞竞争反馈控制生态系统的全局吸引性[J].数学学报,2006,49(2)：335-346.
[71]Smart D.Fixed Point Theorems[M].London:Cambridge University Press,1974.
[72]Socio A.Ambrosetti. Periodic solutions of nonlinear wave equations with non-monotone forcierng terms. Rend. Mat.Acc.Lincei. s.9,v.16:117-124(2005)
[73]Massimiliano Berti,Luca Biasco, Forced vibrations of wave equations with non-monotone nonlinearities[J], Ann.I.H.Poincare-AN 23(2006)439-474
[74]PIETRO BALDI AND MASSIMILIANO BERTI.FORCED VIBRATIONS OF A OGENEOUS STRING[J].SIAM J. M ATH. A NAL.ol.40, No.1, pp.382-412
[75]Alberto Capozzi,Addolorata Salvatore. On forced vibrations of a semilinear wave equation without assumptions of monotonicity I Nonlinear Analysis[J].71(2009)e1932_e1941
[76]Shu guan Ji, Yong Li。Time Periodic Solutions to the One-Dimensional Nonlinear Wave Equation[J]。 Arch.Rational Mech.Anal.199(2011)435-451.
[77]廖晓昕.稳定性的理论、方法和应用(第二版)[M].武汉：华中科技大学出版社,2010,1.
[78]Gaines R E, Mawhin J L. Coincidence degree and nonlinear differential equations [M]. Berlin, Springer-Verlag,1997.