摘要
本文主要研究牛顿方程的周期解与拟周期解.全文内容共分四章,第一章是绪论,从第二章到第四章为论文主体部分.
第二章,主要研究在推广的Lazer型非共振性条件下2k阶时滞牛顿方程2π周期解的存在惟一性.我们将二阶时滞牛顿方程周期解方面的结果加以推广,应用Schauder不动点定理和傅氏分析技术,分别针对时滞,多时滞,n维系统情况,在推广的Lazer型非共振性条件下,得到了2k阶时滞牛顿方程2π周期解的存在惟一性.
第三章,主要研究2k+1阶时滞牛顿方程2π周期解的存在性问题.我们应用延拓定理,并利用先验估计与截断函数技术,分别针对时滞与多时滞两种情况,得到了2k+1阶时滞牛顿方程2π周期解存在的充分条件.
第四章,主要研究拟周期摆型方程的拉格朗日稳定性.在频率w满足通常的Diophantus条件下,建立了拟周期摆型方程拉格朗日稳定的充分必要条件,从而解决了拟周期情况下的Moser问题.
The delay differential equation is the powerful tool to examine the periodic-resonance regularity with time-delay phenomena. In the sixties and seventies of the twentieth century, there has been widely concerned on Newton's equation with time-delay in the promotion of the application. It is focusing on the existence of periodic solutions to the equation in the classic non-resonance conditions from the actual background of physics and mechanics. The Work in this area can be found in Layton at 1980 in [42]. In the non-resonance condition of the promotion Lazer-type, Layton in [42] studied the periodic solutions of the equation(?) (1)After that, many scholars had studied the periodic solutions of the Newton's equations with the second order delay and higher order delay, in [25, 33, 34, 75, 23, 101, 88, 72, 73, 21, 84, 35, 63, 58, 60, 83, 87, 22] and so on.
In recent years, the second order delay Newton's equations are researched depthly in the classic non-resonant conditions, but there are few results for the existence of periodic solutions to the high order delay Newton's equation. A natural question is whether there is the similar result for the high order delay Newton's equations? Furtherly, are there some sufficient conditions when the high order delay Newton's equations have the 2π-periodic solutions?
Inspired by literatures in [43, 52, 11, 12, 13, 42], this paper has two parts to research these issues depthly.
In the first part, we will extend the results of the periodic solutions of the second order delay Newton's Equations. The proof relies on Schauder's fixed point theorem and Fourier analysis techniques. And it is proved that there is the existence and uniqueness of periodic solution to the Newton's equation with the (2κ)th-order delay in non-resonant conditions of the Lazer-type, separately for one-delay and multiple delays circumstances. We firstly discuss Newton's equation with the (2κ)th-order delay(?) (2)(?) (3) and take the assumptions as following:(H1) The function f is continuous and differentiate. For (?)(t, u, 0)∈×R×R, we have
where M_1 is a positive constant. (H2) There are an integer m and two positive constants p, q, satisfied the following inequality(H5) The function H is continuous on R~(4k+1), and is 2π-periodic on the first variable. In addition, there is a positive number M_2 satisfied the following inequality
Theorem 2.2.1 Assumed (H1) and (H2) are satisfied, the equation (2) has at least one 2π-periodic solution.
Theorem 2.2.4 Provided (H1), (H2) and (H5) are satisfied, the equation (3) has at least one 2π-periodic solution.
Futhermore, we also discussed the (2κ)th-order Newton's equation with n-dimensional delay(?) (4)and the (2κ)th-order Newton's equation with multiple delays(?) (5)We have the similar results to the previous conclusions in the non-resonance conditions of the promotive Lazer.
In the second part, we research the existence of the perodic solutions for the (2κ+1)th-order delay Newton's equation. We obtain some sufficient conditions for the existence of periodic solutions for the (2k + 1)th-order delay Newton's equations, separately for one-delay and multiple delays cases. The proof relies on Extension Theorem and uses the technology of the priori estimates and the cut-off function.
In this part, we consider the (2k+1)th-order delay Newton's equations(?) (6)and the (2k + 1)th-order Newton's equations with multiple delays(?). (7)In this paper, we always assume that(A1) There is a positive constantσsuch that(A2) There are positive constants b_0 > 2σand c_i,i = 0,1,…,2k, such that
For |x_i|≥a_i, (x_(i+1),…, x_(2k))∈R~(2k-i),we have
For |x_0|≥a_0, (x_1,…, x_(2k))∈R~(2k), we have(A3) There are constants b_0, c_i, i = 0,1,…, 2k, such that (A4) For (x_0,x_1,…,x_(2k))∈R~(2k+1), there are positive constants d_0 > 2σand e_j,j = 0,1,…,2k, such that(A5) There are positive constants d_0, e_j, j = 0,1,…, 2k, such thatOur main result is the following:
Under hypothesis (A1-A3) or (A1,A4,A5), the equation (6) or (7) has at least one 2π-periodic solution.
It is widely concerned on the Lagrange stability to the pendulum-type's equations in today's mathematical physics and applied mathematics. This problem was first raised by the Moser [76]. Moser[77], Levi[46] and You[99] studied the periodic situation separately. In [3], Bibikov developed the KAM theorem of almost integrable Hamiltonian systems with a degree of freedom at quasi-periodic perturbations. Bibikov applied this theorem to discuss the balance stability for a class of second-order nonlinear differential equations. In [55], Lin and Wang studied the double quasi-periodic system, and changed the Diophantine conditions which is stronger than the usual conditions, and had the result that was a sufficient condition to ensure the Lagrange stability. In [47],[100], the authors developed the Quasi-periodic Monotony Reverse theorem.
In the third part, we study the stability of pendulum-type equations with quasi-periodic conditons. We have some necessary and sufficient conditions to the Lagrange stability of the pendulum-type's equations when the frequencies satisfies the usual Diophantine conditions, so as to solve the Moser's problem in quasi-periodic case.
In this part, we research the Pendulum-type equation(?), (8)where p(t,x + 1) = p(t,x), p(t,x) is quasi-periodic function on t, andω= (ω_1,…,ω_n), is the undamental frequency that is,p(t,x) = f(wt,x),where f(θ, x)has the definition on T~n×T~1, and T~n = R~n/Z~n is an n-dimensional torus. Assuming f(θ, x) is a real-analytic function on T~n×T~1, and the frequency ? satisfied the following Diophantine condition:(?), (9)whereν> 0 is a given arbitrary constant, <,> is the usual inner product, and we have the following main result:
Theorem 4.1 Under hypothesis (9), the equation (8) is Lagrange stable if and only if(?). (10)Furthermore, under the conditions (9) and (10) holded, the equation (8) has an infinite number of quasi-periodic solutions with n+1 fundamental frequencies includingω_1,…,ω_n.
引文
[1] AFUWAPE A U, OMARI P, ZANOLIN F. Nonlinear perturbations of differential operators with nontrivial Kernel and applications to third-order periodic boundary value problems[J]. J. Math. Anal. Appl, 1989, 143(1): 35-56.
[2] BECKENBACH E F, BELLMAN R. Inequalities [M]. Springer-Verlag, 1983.
[3] BIBIKOV YU N. On stability of zero solution of essential nonlinear Hamiltonian and reversible systems with one degree of freedom[J]. Differentsial'nye Uravneniia, 2002. 38(5): 579-584.
[4] BROWN K J, LIN S S. Periodically perturbed conservative systems and a global inverse function theorem[J]. Nonlinear Analysis, 1980, 4(1): 193-201.
[5] CAPIETTO A, MAWHIN J, ZANOLIN F. A continuation approach to superlinear periodic boundary value problems[J]. J.Differential Equations, 1990, 88(2): 347-395.
[6] CESARI L. Nonlinear problems across a point of resonance for nonselfadjoint systems[C]// CESARI L, KANNAN R, WEINBERGER H F. Nonlinear Analysis, a Collection of Papers in Honor of Erich H. Rothe. New York San Francisco London: A Subsidiary of Harcourt Brace Jovanovich, c1978: 43-67.
[7] 陈凤德．具无限时滞的非线性积分微分方程的周期解[J]．应用数学学报， 2003，26(1)：141-148．
[8] CHEN H B. LI Y. Rate of decay of stable periodic solutions of Duffing equations[J]. J. Differential Equations, 2007, 236: 493-503.
[9] CHEN H B, LI Y, HOU X J. Exact multiplicity for periodic solutions of Duffing type[J]. Nonlinear Analysis, 2003, 55: 115-124.
[10] CHEN J H, LI W G. Periodic solutions for 2kth boundary value problem with resonance[J]. J. Math. Anal. AppL, 2006, 314: 661-671.
[11] CONG F Z. Periodic solutions for 2kth order ordinary differential equations with nonres- onance[J]. Nonlinear Analysis, 1998, 32: 787-793.
[12] CONG F Z, HUANG Q D, SHI S Y. Existence and uniqueness of periodic solutions for (2n+l)th-order differential equations[J]. J. Math. Anal. AppL, 2000, 241: 1-9.
[13] CONG F Z. Existence of periodic solutions of (2n+1)th-order ordinary differential equations[J]. Appl. Math. Letters, 2004, 17: 727-732.
[14] COSTER C D, FALLY C, MUNYAMARERE F. Nonresonace conditions for fourth order nonlinear boundary value problems[J]. Internat. J. Math and Math. Sci., 1994, 17(4): 725-740.
[15] DING T R. An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance[J]. Proc. Am. Math. Soc, 1982, 86(1): 47-54.
[16] DING T R. Nonlinear oscillations at point of resonance [J]. Scientia Sinica, 1982, 25A: 918-931.
[17] 丁同仁．非线性振动的若干问题[J]．南京大学学报数学半年刊， 1987，常微分方程专辑： 1-7．
[18] DING T R, ZANOLIN F. Time-maps for the solvability of periodically perturbed nonlinear Duffing equations[J]. Nonlinear Analysis, 1991, 17(7): 635-653.
[19] 丁同仁．常微分方程定性方法的应用[M]．北京：高等教育出版社，2004．
[20] 董玉君．关于Duffing方程周期解的一个基本定理的简化证明[J]．应用数学和力学，2001, 22(9)：997-1000．
[21] FAN Y H, QUAN H S. 2π-Periodic solution of the equation u"+bu~r+g(u(t-τ)) = p(t)[J]. Mathematics Applicata(应用数学), 2001, 14(2): 48-51.
[22] FAN Q Y, WANG W T, YI X J. Anti-periodic solutions for a class of nonlinear nth-order differential equations with delays [J]. Journal of Computational and Applied Mathematics, 2009, in press.
[23] GAINES R E, MAWHIN J L. Coincidence degree and nonlinear differential equations[M]. Lecture Notes in Math., vol 568, Berlin-New York: Springer-Verlag, 1977.
[24] 葛渭高．n维Duffing型方程 x''+cx'+g(t，x)=p(t)的2π周期解[J]．数学年刊，1988，9A(4)：499-505．
[25] 葛渭高．非线性常微分方程边值问题[M]．北京：科学出版社，2007．
[26] 郭大钧．非线性泛函分析[M]．济南：山东科学技术出版社， 2004．
[27] GUO Z M, YU J S. Multiplicity results for periodic solutions to delay differential equations via critical point theory[J]. J.Differential Equations, 2005, 218(1): 15-35.
[28] HABETS P, NKASHAMA M N. On periodic solutions of nonlinear second order vector differential equations[J]. Proc. Roy. Soc. Edinb, 1986, 104A(1-2): 107-125.
[29] HABETS P, METZEN G. Existence of periodic solutions of Duffing equations[J]. J.Differential Equations, 1989, 78(1): 1-32.
[30] HE X M, GE W G, SHAN W R. Existence of periodic solutions for generalized Lienard-typed functional differential equations [J]. Journal of Beijing Institute of Technoiogy, 2002, 11(3): 325-328.
[31] HUANG H. Unbounded solutions of almost periodically forced pendulum-type equations[J]. Acta Math. Sinica, 2001, 17(3): 391-396.
[32] HUANG X. On the existence of periodic solutions for the third-order nonlinear ordinary differential equations[J]. Appl. Math. JCU., 1999, 14:125-150.
[33] HUANG X K, XIANG Z G. The existence of 2π-periodic solutions for Duffing equation x" + g(x(t - τ)) = p(t) with delay [J]. Chinese Science Bulletin, 1994, 39(16): 201-203.
[34] IANNACCI R, NKASHAMA M N. On periodic solutions of forced second order differential equations with a deviating argument[M]. Lecture Notes in Math., 1151, Springer-Verlag, 1984, 224-232.
[35] IYASE S A, NIGERIA. Non-resonant oscillations for some fourth-order differential equations with delay[J]. Math. Proc. Roy. Irish Acad, 1999, 99A(1): 113-121.
[36] 贾秀利．四阶Duffing方程的周期解[D].长春：吉林大学数学研究所，2006.
[37] JIANG D Q, WEI J J. Existence of positive periodic solutions for nonautomomous delay differential equations[J]. Chinese Annals. Math., 1999, 20A: 715-720.
[38] 江泽坚，孙善利．泛函分析[M]．北京：高等教育出版社， 1994.
[39] KAPLAN J L, YORKE J A. Ordinary differential equations which yields periodic solutions of differential delay equations[J]. J. Math. Anal. Appl., 1974, 48: 317-324.
[40] KRASNOSEL'SKII M A. Topological methods in the theory of nonlinear integral equations[J]. New York: Macmillan Co., 1964.
[41] KRASNOSEL'SKII A M, MAWHIN J. On some higher order boundary value problems at resonance[J]. Nonlinear Analysis, 1995, 24(8): 1141-1148.
[42] LAYTON W. Periodic solutions of nonlinear delay equations [J]. J. Math. Anal. Appl., 1980, 77: 198-204.
[43] LAZER A C, SANCHEZ D A. On periodically perturbed conservative systems[J]. Michigan Math. J., 1969, 16: 193-200.
[44] LAZER A C, LEACH D. Bounded perturbations of forced harmonic oscillators at resonance[J]. Ann. Mat. Pura Appl., 1969, 82: 49-68.
[45] LAZER A C. Application of lemma on bilinear forms to a problem in nonlinear oscillations[J]. Proceedings of the American Mathematical Society, 1972, 33: 89-94.
[46] LEVI M. KAM theory for particles in periodic potentials[J]. Ergodic Theory and Dynamical Systems, 1990, 10: 777-785.
[47] LEVI M, ZEHNDER E. Boundedness of solutions for quasiperiodic potentials [J]. SIAM J. Math. Anal, 1995, 26(5): 1233-1256.
[48] LI W G. Periodic solutions for 2kth order ordinary differential equations with resonance[J]. J. Math. Anal. Appl., 2001, 259: 157-167.
[49] LI W G, SHEN Z H. Constructive proof on the existence of periodic solution of Duffing equation[J]. Chinese Science Bulletin, 1997, 42(22): 1870-1874.
[50] 李勇．向量方程的边值问题[J]_吉林大学自然科学学报， 1989：4：33-36．
[51] LI Y, LV X R. Continuation theorems for boundary value problems [J]. J. Math. Anal. Appl., 1995, 190: 32-49.
[52] 李勇，王怀中．高阶Duffing方程的周期解[J].高校应用数学学报，1991，6：407-412．
[53] LI Y, WANG H Z. Existence and uniqueness of periodic solutions for Duffing equations across many points of resonance [J]. J. Differential Equations, 1994, 108(1): 152-169.
[54] LI Y K. Periodie solutions of a periodci deyal predator-prey system[J]. Proc. Amer. Math. Soc, 1999, 127: 1331-1335.
[55] LIN S S, WANG Y Q. Lagrangian stability of a nonlinear quasi-periodic system[J]. J. Math. Anal. Appl., 2004, 293(1): 258-268.
[56] LIU B. An existence theorem for harmonic solutions of semi-linear DufBng equations [J]. Acta Math. Sin., 1991, 34: 165-170.
[57] 刘斌，庾建设．一类非线性中立型时滞微分方程周期解的存在性[J].高校应用数学学报，2001，16A(3)：276-282．
[58] LIU B W, HUANG L H. Periodic solutions for nonlinear nth order differential equations with delays[J]. J. Math. Anal. Appl., 2006, 313: 700-716.
[59] 刘炳文，黄立宏，历亚．三阶泛函微分方程的周期解的存在性[J].应用数学学报，2006，29(2)：226-233．
[60] 刘炳文，黄立宏，张正球．一类n阶非线性时滞微分方程周期解的存在性[J].数学物理学报，2007，27A(2)：343-350．
[61] 刘文斌．Duffing方程及高阶周期边值问题解的存在性[D]．长春：吉林大学数学研究所，2002．
[62] 刘文斌，李勇．高阶Duffing方程周期解的存在性[J].数学学报，2003，46(1)：49-56.
[63] LIU Y J, YANG P H, GE W G. Periodic solutions of higher-order delay differential equations[J]. Nonlinear Analysis, 2005, 63(1): 136-152.
[64] LOUD W S. Periodic solutions of nonlinear differential equations of Duffing type In: Poc. US-Japan Seminar on Differential and Functional Equations[M]. New York:. Benjamin, 1967: 199-224.
[65] 鲁世平，葛渭高．中立型泛函微分方程周期解问题[J]．应用数学和力学， 2002，23(12)：1269-1275．
[66] LU S P, GE W G. On the existence of periodic solutions for neutral functional differential equation[J]. Nonlinear Analysis, 2003, 54: 1285-1306.
[67] LU S P, GE W G, ZHENG Z X. Periodic solutions to a kind of neutral functional differential equation in the critical case[J]. J. Math. Anal. Appl., 2004, 293: 462-475.
[68] 鲁世平，葛渭高，郑祖庥．一类中立型泛函微分方程周期解问题[J]．数学年刊，2004，25A(5)：645-652．
[69] LU S P, GE W G. Periodic solutions for a kind of Lienard equations with deviating argument[J]. J. Math. Anal. Appl., 2004, 289: 231-243.
[70] 马如云．半线性Duffing方程周期边值问题的可解性[J].数学年刊， 1993，14A(4)：393-399．
[71] MASSERA J L. The existence of periodic solution of systems differential equations[J]. Duke Math. J., 1950, 17: 457-475.
[72] MA S W, WANG Z C, YU J S. Coincidence degree and periodic solutions of Duffing equations[J]. Nonlinear Analysis, 1998, 34: 443-460.
[73] 马世旺，廖晓昕，庾建设．一类二阶泛函微分方程的周期解[J].应用数学，1999，12(3)：77-80．
[74] MAWHIN J, WARD J R. Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced Lienad and Duffing equations[J]. Rocky Mount. J. Math., 1982, 12: 643-654.
[75] METZEN G. Existence of periodic solutions of second order differential equations with delay [J]. Proceedings of the American Mathematical Society, 1988, 103(3): 765-772.
[76] MOSER J. Stable and random motions in dynamical systems[M]. Princeton Tokyo: Annals of Mathematics Studies, 1973.
[77] MOSER J. A stability theorem for minimal foliations on a torus[J]. Ergodic Theory and Dynamical Systems, 1988, 8: 251-281.
[78] NUSSBAUM R D. Periodic solutions of special differential equations: an example in nonlinear functional analysis[J]. Proc. Roy. Soc. Edinburgh, 1978, 81 A: 131-151.
[79] OMARI P, ZANOLIN F. A note on nonlinear oscillations at resonance[J]. Acta. Math. Sinica. (N.S.), 1987, 3: 351-361.
[80] OMARI P, ZANOLIN F. On forced nonlinear oscillations in nth order differential systems with geometric conditions [J]. Nonlinear Analysis, 1984, 8(7): 723-748.
[81] OPIAL Z. Sur les an solutions periodiques del equation differentielle: x" + g(u) = p(t)[J]. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astr. Phys, 1960, 8: 151-156.
[82] 潘吉勋．同胚定理和非线性时变系统的周期解[J]．吉林大学自然科学学报，1984，3：19-25．
[83] PAN L J. Periodic solutions for higher order differential equations with deviating argument[J]. J. Math. Anal. Appl., 2008, 343: 904-918.
[84] PENG L Q. Existence and uniqueness of periodic solutions for a kind of Duffing equation with two deviating arguments [J]. Mathematical and Computer Modelling, 2007, 45: 378-386.
[85] QIAN D B, YOU J G. Periodic solutions of forced second order equations with the oscillatory time map[J]. Diff. Integral Equ, 1993, 6(4): 793-806.
[86] REISSIG R. Contraction mappings and periodically perturbed non-conservative systems[J]. Atti Accad. Naz. Lincei (Rend. Cl. Sci.) Fis. Mat. Natur., 1975, 58: 696-702.
[87] REN J L, CHENG Z B. On high-order delay differential equation[J]. Computers and Mathematics with Applications, 2009, 57: 324-331
[88] 任景莉，葛渭高．一类二阶泛函微分方程周期解存在性问题[J]．数学学报，2004，47(3)：569-578．
[89] 任景莉，任保献，葛渭高．一类非线性中立型时滞微分方程周期解的存在性[J].应用数学学报，2004，27(1)：89-98．
[90] SEDA V, NIETO J J, GERA M. Periodic boundary value problems for nonlinear higher order ordinary differential equations [J]. Appl. Math. Comput., 1992, 48(1): 71-82.
[91] SHEN Z H. On the periodic solution to the newtonian equation of motion[J]. Nonlinear Analysis, 1989, 13(2): 145-149.
[92] WANG D. On the existence of 2π-periodic solutions of differential equation x" + g(u) =e(t)[J]. Chinese Ann. Math., 1984, 5A(1): 61-72.
[93] WANG H, LI Y. Periodic solutions for Duffing equations[J]. Nonlinear Analysis, 1995, 24(7): 961-979.
[94] 王克．泛函微分方程的全局稳定周期解[J]．数学学报，1994，37(4)：570-573．
[95] WANG Y Y, LIAN H R. GE W G. Periodic solutions for a second order nonlinear functional differential equation [J]. Appl. Math. Letters, 2007, 20: 110-115.
[96] 王在洪．跨共振点Duffing方程周期解的多解性[J]．数学年刊，1997，18A(4)：421-432．
[97] 伍卓群，李勇．常微分方程[M]．北京：高等教育出版社， 2004．
[98] YOSHIZAWA T. Stablity theory by Lyapunov's second methods[M]. Tokyo: Publications of the Mathematical Society of Japan, 1966.
[99] YOU J G. Invariant tori and Lagrange stability of pendulum-type equations[J]. J. Differential Equations, 1990, 85(1): 54-65.
[100] ZHARNITSKY V. Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem[J]. Nonlinearity, 2000, 13: 1123-1136.
[101] 张正球，庾建设．一类时滞Duffing型方程的周期解[J]．高校应用数学学报，1998，13A(4)：389-392．