脉冲方程和Rayleigh方程的周期解
|
摘要
脉冲方程和Rayleigh方程是两类重要的微分方程模型。关于它们的周期解以及相关问题的研究，一直受到关注。
本文的特点是综合泛函和几何的方法。在抽象的泛函框架中，把方程的解投影到相平面上，分析它们所表现出来的凡何性质。利用这样的分析，得到应用拓扑度所需要的先验估计。在符号条件和一些单侧增长条件下证明了Rayleigh方程的周期解的存在性。
关于脉冲方程的周期解的研究，已有的结果大多是在次线性脉冲的条件下得到的，并且也没有无穷多个调和解的结果。本文把脉冲方程看成流和映射的复合，在相平面上研究其Poincaré映射的几何性质。对同胚脉冲给出了由脉冲引起的旋转角的解析定义。利用Poincaré-Birkhoff扭转定理证明了具有多项式同胚脉冲的超线性方程的无穷多个周期解。对于一般的非同胚脉冲情形，我们将Poincaré-Birkhoff扭转定理进行改造，得到部分扭转定理，利用这个定理讨论脉冲项有一个线性退化脉冲时方程的周期解。最后，我们给出一个脉冲方程的扰动引理，在这个引理的基础上，通过迭合度来讨论一般的带脉冲的非保守的二阶微分方程的周期解。
Impulsive differential equations and Rayleigh equations are two important models of ordinary differential equations. Attetions are paled to study the periodic solutions of these two equations and correlated problems.
The characteristic of this article is the integration of functional method and geometry. In the abstract functional frame, we project the solutions onto phase-plane, analysis their geometry character. Based on this geometry point, we get the priori estimate necessary for coincidence degree, and then get periodic solutions of Rayleigh equations which increase on one side, and the periodic solutions of Rayleigh equations under the signed conditions.
With regard to the study of periodic solutions of impulsive equations, most results were gotten under the conditions that the impulse is sublinear, and there aren't any results about infinite periodic solutions. In this article, impulsive equations are regarded as the composition of the flow and the maps, we study the the geometry character of Poincare map on phaseplane, when the impulse is homeomorphism, we put out the analytic definition of twist angle arosed by impulse ingeniously, when the impulse is polynomial homeomorphism, we get infinite periodic solutions by Poincare-Birkhoff twist theorem. As for the more general case, that is to say, the impulse isn't homeomorphism, we improve the Poincare-Birkhoff twist theorem, and get partial twist theorem, by this, we study the linear impulsive differential equations that one of the impulse terms is linear and degenerative. In the end, we get a disturbance lemma of impulsive differential equations, based on this lemma, we study the periodic solutions of nonconserative impulsive differential equations.
本文的特点是综合泛函和几何的方法。在抽象的泛函框架中，把方程的解投影到相平面上，分析它们所表现出来的凡何性质。利用这样的分析，得到应用拓扑度所需要的先验估计。在符号条件和一些单侧增长条件下证明了Rayleigh方程的周期解的存在性。
关于脉冲方程的周期解的研究，已有的结果大多是在次线性脉冲的条件下得到的，并且也没有无穷多个调和解的结果。本文把脉冲方程看成流和映射的复合，在相平面上研究其Poincaré映射的几何性质。对同胚脉冲给出了由脉冲引起的旋转角的解析定义。利用Poincaré-Birkhoff扭转定理证明了具有多项式同胚脉冲的超线性方程的无穷多个周期解。对于一般的非同胚脉冲情形，我们将Poincaré-Birkhoff扭转定理进行改造，得到部分扭转定理，利用这个定理讨论脉冲项有一个线性退化脉冲时方程的周期解。最后，我们给出一个脉冲方程的扰动引理，在这个引理的基础上，通过迭合度来讨论一般的带脉冲的非保守的二阶微分方程的周期解。
Impulsive differential equations and Rayleigh equations are two important models of ordinary differential equations. Attetions are paled to study the periodic solutions of these two equations and correlated problems.
The characteristic of this article is the integration of functional method and geometry. In the abstract functional frame, we project the solutions onto phase-plane, analysis their geometry character. Based on this geometry point, we get the priori estimate necessary for coincidence degree, and then get periodic solutions of Rayleigh equations which increase on one side, and the periodic solutions of Rayleigh equations under the signed conditions.
With regard to the study of periodic solutions of impulsive equations, most results were gotten under the conditions that the impulse is sublinear, and there aren't any results about infinite periodic solutions. In this article, impulsive equations are regarded as the composition of the flow and the maps, we study the the geometry character of Poincare map on phaseplane, when the impulse is homeomorphism, we put out the analytic definition of twist angle arosed by impulse ingeniously, when the impulse is polynomial homeomorphism, we get infinite periodic solutions by Poincare-Birkhoff twist theorem. As for the more general case, that is to say, the impulse isn't homeomorphism, we improve the Poincare-Birkhoff twist theorem, and get partial twist theorem, by this, we study the linear impulsive differential equations that one of the impulse terms is linear and degenerative. In the end, we get a disturbance lemma of impulsive differential equations, based on this lemma, we study the periodic solutions of nonconserative impulsive differential equations.
引文
[1]D.Bainov, P.Simenov, Impulsive Differential Equations:Periodic Solutions and Applications, Longman Scientific and Technical, United States, 1993.
[2]C. Cooke, J. Kroll, The existence of periodic solutions to certain impulsive differential equations, Computers and Mathematics with Applications, 44(2002),667-676.
[3]T. Ding, R.Iannacci, F.Zanolin, On periodic solutions of sublinear duffing equations, J. Math. Anal. Appl., 58(1991),316-332.
[4]T. Ding, F. Zanolin, Periodic solutions of Duffing’s equation with superquadratic potential, J.Differential Equations, (1992), 328-378.
[5]丁同仁，常微分方程定性方法的应用，高等教育出版社，北京，2004．
[6]丁伟岳，扭转映射的不动点与常微分方程的周期解，数学学报，25(1982)，227-235．
[7]R. Iannacci, M. kashama, P. Omari, F. Zanolin, Periodic solutions of the forced Lienard equations with jumping nonlinearities under nonuniform Conditions, Proc. Roy. soc. Edinburgh, sect. A, 110(1988),183-198.
[8]J. Li, J. Nieto, J. Shen, Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl., 2006.
[9]J.Mawhin,Topological degree methods in nonlinear boundary value problems, CBMS.Conf. in Math, Providence,RI, Amer. Math. Soc, 40(1979).
[10]J. Nieto, Basic theory for nonresonace impulsive periodic problems of first order, J.Math. Anal. Appl., 205(1997), 423-433.
[11]J. Nieto, Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Analysis TMA, 51(2002),1223-1232.
[12]P. Omari, G. Villari, F.Zanolin, Periodic Solutions of the Lienard equation with one-sided growth restrictions, J. Differential Equations, 67(1987),278-293.
[13]D. Qian, X. Li, Periodic solutions for ordinary differential equations with sulinear impulsive effects, J. Math. Anal. Appl., 303(2005), 288-303.
[14]D. Qian, Of forced nonlinear oscillations for the second order equations with semi-quadratic potential, Nonlinear Analysis TMA, 26(1996), 1715-1731.
[15]唐磊清，具有次线性脉冲的超线性Dufffing方程的调和解，苏州，苏州大学，硕士学位论文，2006.
[16]Z. Wang, On the existence of periodic solutions of rayleigh equations, Z. Angew. Math.Phys, 56(2005), 592-608.
[17]张芷芬，丁同仁等，微分方程定性理论，科学出版社，北京，1997．
[18]钟承奎，范先令，陈文原，非线性泛函分析引论，兰州大学出版社，兰州，2004．
[19]周凯，关于隐函数全局存在性的若干结果，苏州大学，本科学位论文，苏州，2003．
[2]C. Cooke, J. Kroll, The existence of periodic solutions to certain impulsive differential equations, Computers and Mathematics with Applications, 44(2002),667-676.
[3]T. Ding, R.Iannacci, F.Zanolin, On periodic solutions of sublinear duffing equations, J. Math. Anal. Appl., 58(1991),316-332.
[4]T. Ding, F. Zanolin, Periodic solutions of Duffing’s equation with superquadratic potential, J.Differential Equations, (1992), 328-378.
[5]丁同仁，常微分方程定性方法的应用，高等教育出版社，北京，2004．
[6]丁伟岳，扭转映射的不动点与常微分方程的周期解，数学学报，25(1982)，227-235．
[7]R. Iannacci, M. kashama, P. Omari, F. Zanolin, Periodic solutions of the forced Lienard equations with jumping nonlinearities under nonuniform Conditions, Proc. Roy. soc. Edinburgh, sect. A, 110(1988),183-198.
[8]J. Li, J. Nieto, J. Shen, Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl., 2006.
[9]J.Mawhin,Topological degree methods in nonlinear boundary value problems, CBMS.Conf. in Math, Providence,RI, Amer. Math. Soc, 40(1979).
[10]J. Nieto, Basic theory for nonresonace impulsive periodic problems of first order, J.Math. Anal. Appl., 205(1997), 423-433.
[11]J. Nieto, Periodic boundary value problems for first-order impulsive ordinary differential equations, Nonlinear Analysis TMA, 51(2002),1223-1232.
[12]P. Omari, G. Villari, F.Zanolin, Periodic Solutions of the Lienard equation with one-sided growth restrictions, J. Differential Equations, 67(1987),278-293.
[13]D. Qian, X. Li, Periodic solutions for ordinary differential equations with sulinear impulsive effects, J. Math. Anal. Appl., 303(2005), 288-303.
[14]D. Qian, Of forced nonlinear oscillations for the second order equations with semi-quadratic potential, Nonlinear Analysis TMA, 26(1996), 1715-1731.
[15]唐磊清，具有次线性脉冲的超线性Dufffing方程的调和解，苏州，苏州大学，硕士学位论文，2006.
[16]Z. Wang, On the existence of periodic solutions of rayleigh equations, Z. Angew. Math.Phys, 56(2005), 592-608.
[17]张芷芬，丁同仁等，微分方程定性理论，科学出版社，北京，1997．
[18]钟承奎，范先令，陈文原，非线性泛函分析引论，兰州大学出版社，兰州，2004．
[19]周凯，关于隐函数全局存在性的若干结果，苏州大学，本科学位论文，苏州，2003．