二阶微分方程周期解的存在性与多解性
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摘要
本文考虑Duffing方程x" + g(x) = e(t),周期解的存在性与多解性,其中g,e: R→R是连续函数,e(t)还是以2π为周期的周期函数.设g(x)满足下列条件:
((τ_0))存在常数μ>0和互素的正整数m,n,以及数列{a_k}, {b_k},a_k→+∞,b_k→+∞, (k→+∞),使时间映射τ~+(h)满足本文利用Poincar(?)-Birkhoff扭转定理和Poincar(?)-Bohl不动点定理证明了上述方程周期解的存在性与多解性.
另外,本文还研究了二阶奇异微分方程的周期解的存在性,其中a,e∈L~1[0,2π],.f∈Car([0,2π]×R~+,R),并且假设下列条件:
(H_0)格林函数G(t,s)是非负的,对所有的(t,s)∈[0,2π]×[0,2π]都成立;
(H_1) f(t,x)+e(t)≥0,对于任意的(t,x)∈[0,2π]×(0,+∞);
(H_2)存在非负的函数g(x),h(x),k(t)使得其中g(x)+h(x)是单调不增的;
(H_3)存在常数R满足,R >Φ_* +γ_* > 0 , R≥K~*(h(r) + g(r)) +γ~*.
本文证明了上述奇异方程至少存在一个正周期解.
In this paper, we study the existence of periodic solutions of the second order Duffing equationx" + g(x) = e(t),where g,e : R→R are continuous and e(t) is 2π-periodic. Assume that the following conditions hold,
(τ_0) There exist positive constantμ, positive intergers m, n and (m,n)=1, {a_k}, {b_k},a_k→+∞,b_k→+∞, (k→+∞) such thatWe prove the existence and multiplicity of periodic solutions by using Poincar(?)-Birkhoff theorem and Poincar(?) - Bohl fixed point theorem.
On the other hand, we also study the existence of periodic solutions of second order differential equations with singularity,where a, e∈L~1[0,2π], f∈Car([0,2π]×R~+, R). Moreover, assume the following conditions hold,
(H_0) Green funtion G(t,s) is non-negative for all(t, s)∈[0,2π×[0,2π];
(H_1) f(t, x) + e(t)≥0, for all (t, x)∈[0, 2π]×(0, +∞);
(H_2) there exist continuous non-negative functions g(x), h(x), k(t) such thatf(t,x)≤k(t)[g(x) + h(x)). for all (t,x)∈[0.2π]×(0, +∞)and g(x) + h(x) is non-increasing;
(H_3)there exists a constant R, such that R >Φ_* +γ_* > 0 and R≥K~*(h(r) +g(r)) +γ~*.We prove the exsistence of positive periodic solution of the given equation.
((τ_0))存在常数μ>0和互素的正整数m,n,以及数列{a_k}, {b_k},a_k→+∞,b_k→+∞, (k→+∞),使时间映射τ~+(h)满足本文利用Poincar(?)-Birkhoff扭转定理和Poincar(?)-Bohl不动点定理证明了上述方程周期解的存在性与多解性.
另外,本文还研究了二阶奇异微分方程的周期解的存在性,其中a,e∈L~1[0,2π],.f∈Car([0,2π]×R~+,R),并且假设下列条件:
(H_0)格林函数G(t,s)是非负的,对所有的(t,s)∈[0,2π]×[0,2π]都成立;
(H_1) f(t,x)+e(t)≥0,对于任意的(t,x)∈[0,2π]×(0,+∞);
(H_2)存在非负的函数g(x),h(x),k(t)使得其中g(x)+h(x)是单调不增的;
(H_3)存在常数R满足,R >Φ_* +γ_* > 0 , R≥K~*(h(r) + g(r)) +γ~*.
本文证明了上述奇异方程至少存在一个正周期解.
In this paper, we study the existence of periodic solutions of the second order Duffing equationx" + g(x) = e(t),where g,e : R→R are continuous and e(t) is 2π-periodic. Assume that the following conditions hold,
(τ_0) There exist positive constantμ, positive intergers m, n and (m,n)=1, {a_k}, {b_k},a_k→+∞,b_k→+∞, (k→+∞) such thatWe prove the existence and multiplicity of periodic solutions by using Poincar(?)-Birkhoff theorem and Poincar(?) - Bohl fixed point theorem.
On the other hand, we also study the existence of periodic solutions of second order differential equations with singularity,where a, e∈L~1[0,2π], f∈Car([0,2π]×R~+, R). Moreover, assume the following conditions hold,
(H_0) Green funtion G(t,s) is non-negative for all(t, s)∈[0,2π×[0,2π];
(H_1) f(t, x) + e(t)≥0, for all (t, x)∈[0, 2π]×(0, +∞);
(H_2) there exist continuous non-negative functions g(x), h(x), k(t) such thatf(t,x)≤k(t)[g(x) + h(x)). for all (t,x)∈[0.2π]×(0, +∞)and g(x) + h(x) is non-increasing;
(H_3)there exists a constant R, such that R >Φ_* +γ_* > 0 and R≥K~*(h(r) +g(r)) +γ~*.We prove the exsistence of positive periodic solution of the given equation.
引文
[1] Loud, w. s., Periodic solutions of differential equations of Duffing type,In.Poc.United states-Janpan Seminar on Differential and Functional Equations, New York, 1967, 199-224.
[2] Reissig, R., Contraction mappings and periodically perturbed nonconverservative syetems.Atti Accad. Naz Lincei, 58 (1975), 696-702.
[3] Mawhin, J., Recent trends in nonlinear boundary valueproblems, In: Konferenz uber nichtlineare Schwingungen, Band, Abhandlungen der ADW, Akademie Verlag , Berlin, 1977, 51-70.
[4] Leach, D. E., On Poincare perturbation theorem and a theorem of W. S. Lond, J. Differential Equations 7 (1970), 34-53.
[5] Oman, P. and Zanolin, F., A note on nonlinear oscillations at reaonance, Acat Mathemata Sinica, 3 (1987), 351-361.
[6] Ding, T., nonlinear oscillations at a point of reaonance, Scientia Sinica, 9A (1982), 918-931.
[7] Ding, T., An infinite class of periodic solutions of periodically pertrbed Duffing equation at resonance, Proc. Amer. Math. Soc, 86 (1982), 47-54.
[8] Ding, T. and Zanolin, F., Existence and multiplicity results for Periodic solutions of semilinear Duffing equation, J. Differential Equations 105 (1993), 364-409.
[9] Lloyd, N. G., Degree theory, University press, Carbridge, 1978.
[10] Ding, W. Y., A generalzation of the Poincar(?)-Brikhoff theorem, Proc. Amer. Math. Soc, 88 (1983), 341-346.
[11] Qian, D., Time-map and Duffing equations with resonance-acrossing, Science in china, 5 (1993), 471-479.
[12] Capietto, A, Mawhin, and Zanolin, F., A continuation theorem for periodic boundary value problem with oscillatory nonlinearitties, Nonlinear Differential Equations and Applications, 2 (1995), 133-163.
[13] 王在洪,跨共振点Duffing方程周期解的多解性.数学年刊(A),Vol.4(1997),421-432.
[14] A. C. Lazer and S. Solimini., On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc, 99 (1987), 109-144.
[15] A. Fonda, and F. Zanolin., Subharmonic solutions for some second order differential equations with singularities, SIAM J. Math. Anal, 24(1993), 1294-1311.
[16] A. Fonda., Periodic solutions of scalar second order differential equations with singularities, Mem. Cl. Sciences Acad. R. Belguique(4), 8 (1993), 7-39.
[17] P. J. Torres., Weak singularities may help Periodic solutions to exist , J. Differential Equations, 232 (2007), 277-284.
[18] T. Ding., Applications of Qualitative Methods of Ordinary Differential Equations, Higher Education Press, 2004.
[19] D. Jing, J. Chu and M. Zhang., Multiplicity of positive periodic solutions to superlinear repilsive singular equqtions, J. Differential Equations, 211 (2005), 282-302.
[20] J. Chu and P. J. Torres., Applications of Schauder fixed point theorem to singular differential equations. Bull London Math. Soc. 39 (2007), 653-660.
[21] 丁同仁,常微分方程稳定性方法的应用,高等教育出版社, 2004.
[22] P. J. Torres., Existence of one-signed periodic solutions of some second order differential equations via a Krasnoseskii fixed point theorem, J. Differential Equations, 190 (2003), 643-662.
[23] T. Ding, R. Iannacci and Zanolin, F., Existence and mutiplicity results for periodic solutions of semilinear Duffing equations, J. Differential Equations, 105(1993), 364-409.
[24] W. Ding., Fixed points of twist mappings and periodic solutions of ordinary differential equations. Acta Math. Sinica 25(1982), 227-235.
[25] C. Fabry, J. Mawhin., Oscillations of s forced asymmetric oscillator at resonance, Nonlinearity, 13(2000), 493-505.
[26] P. J. Torres, M. Zhang., A monotone iterative svheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nachr, 251 (2003), 101-107.
[27] P. J. Torres, and M. Zhang., Twist periodic solutions of repulsive singular equations, Nonlinear Anal, 56 (2004), 591-599.
[28] P. J. Torres., Bounded solutions in singular equations of repulsive type, Nonlinear Anal, 32 (1998), 117-284.
[29] P. J. Torres., Twist solutions of a Hill's Equations with singular term, Advanced Nonlinear Studies, 2 (2002), 279-287.
[30] Z. Wang., Periodic solutions of the second order differential equations with singularity, Nonline. Anal, 58 (2004), 319-331.
[2] Reissig, R., Contraction mappings and periodically perturbed nonconverservative syetems.Atti Accad. Naz Lincei, 58 (1975), 696-702.
[3] Mawhin, J., Recent trends in nonlinear boundary valueproblems, In: Konferenz uber nichtlineare Schwingungen, Band, Abhandlungen der ADW, Akademie Verlag , Berlin, 1977, 51-70.
[4] Leach, D. E., On Poincare perturbation theorem and a theorem of W. S. Lond, J. Differential Equations 7 (1970), 34-53.
[5] Oman, P. and Zanolin, F., A note on nonlinear oscillations at reaonance, Acat Mathemata Sinica, 3 (1987), 351-361.
[6] Ding, T., nonlinear oscillations at a point of reaonance, Scientia Sinica, 9A (1982), 918-931.
[7] Ding, T., An infinite class of periodic solutions of periodically pertrbed Duffing equation at resonance, Proc. Amer. Math. Soc, 86 (1982), 47-54.
[8] Ding, T. and Zanolin, F., Existence and multiplicity results for Periodic solutions of semilinear Duffing equation, J. Differential Equations 105 (1993), 364-409.
[9] Lloyd, N. G., Degree theory, University press, Carbridge, 1978.
[10] Ding, W. Y., A generalzation of the Poincar(?)-Brikhoff theorem, Proc. Amer. Math. Soc, 88 (1983), 341-346.
[11] Qian, D., Time-map and Duffing equations with resonance-acrossing, Science in china, 5 (1993), 471-479.
[12] Capietto, A, Mawhin, and Zanolin, F., A continuation theorem for periodic boundary value problem with oscillatory nonlinearitties, Nonlinear Differential Equations and Applications, 2 (1995), 133-163.
[13] 王在洪,跨共振点Duffing方程周期解的多解性.数学年刊(A),Vol.4(1997),421-432.
[14] A. C. Lazer and S. Solimini., On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc, 99 (1987), 109-144.
[15] A. Fonda, and F. Zanolin., Subharmonic solutions for some second order differential equations with singularities, SIAM J. Math. Anal, 24(1993), 1294-1311.
[16] A. Fonda., Periodic solutions of scalar second order differential equations with singularities, Mem. Cl. Sciences Acad. R. Belguique(4), 8 (1993), 7-39.
[17] P. J. Torres., Weak singularities may help Periodic solutions to exist , J. Differential Equations, 232 (2007), 277-284.
[18] T. Ding., Applications of Qualitative Methods of Ordinary Differential Equations, Higher Education Press, 2004.
[19] D. Jing, J. Chu and M. Zhang., Multiplicity of positive periodic solutions to superlinear repilsive singular equqtions, J. Differential Equations, 211 (2005), 282-302.
[20] J. Chu and P. J. Torres., Applications of Schauder fixed point theorem to singular differential equations. Bull London Math. Soc. 39 (2007), 653-660.
[21] 丁同仁,常微分方程稳定性方法的应用,高等教育出版社, 2004.
[22] P. J. Torres., Existence of one-signed periodic solutions of some second order differential equations via a Krasnoseskii fixed point theorem, J. Differential Equations, 190 (2003), 643-662.
[23] T. Ding, R. Iannacci and Zanolin, F., Existence and mutiplicity results for periodic solutions of semilinear Duffing equations, J. Differential Equations, 105(1993), 364-409.
[24] W. Ding., Fixed points of twist mappings and periodic solutions of ordinary differential equations. Acta Math. Sinica 25(1982), 227-235.
[25] C. Fabry, J. Mawhin., Oscillations of s forced asymmetric oscillator at resonance, Nonlinearity, 13(2000), 493-505.
[26] P. J. Torres, M. Zhang., A monotone iterative svheme for a nonlinear second order equation based on a generalized anti-maximum principle, Math. Nachr, 251 (2003), 101-107.
[27] P. J. Torres, and M. Zhang., Twist periodic solutions of repulsive singular equations, Nonlinear Anal, 56 (2004), 591-599.
[28] P. J. Torres., Bounded solutions in singular equations of repulsive type, Nonlinear Anal, 32 (1998), 117-284.
[29] P. J. Torres., Twist solutions of a Hill's Equations with singular term, Advanced Nonlinear Studies, 2 (2002), 279-287.
[30] Z. Wang., Periodic solutions of the second order differential equations with singularity, Nonline. Anal, 58 (2004), 319-331.