一类具有强迫项的有限时滞Lienard方程周期解的存在唯一性
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摘要
利用重合度的Mawhin延拓定理,构造新算子,使用新技巧,证明一类具有强迫项的有限时滞Lienard方程x″(t)+f_1(x)x′(t)+f_2(x)(x′(t))~2+g(x(t-τ))=e(t)存在唯一周期解的条件,得到了周期解存在唯一的新的结果.
In this paper, we use the Mawhin's continuity theorem to establish and prove new results on the existence and uniqueness of the periodic solution for a class forced and finite delayed Lienard equations of the form x″(t) + f_1(x) x′(t) + f_2(x)(x′(t))~2+g(x)(t-τ))=e(t).
In this paper, we use the Mawhin's continuity theorem to establish and prove new results on the existence and uniqueness of the periodic solution for a class forced and finite delayed Lienard equations of the form x″(t) + f_1(x) x′(t) + f_2(x)(x′(t))~2+g(x)(t-τ))=e(t).
引文
[1] Herden U A. Periodic solutions of a nonlinear second order differential equations with delay[J]. J Math Anal Appl. 1979,70:599-609.
[2] Hale J K. Theory of Functional Differential Equations[M]. New York:Springer-Verlag. 1977.
[3] Hale J K. Introduction to Functional Differential Equations[M]. Berlin:Springer-Verlag, 1977.
[4] Kuang Y. Delay Differential Equations with Applications in Population Dynamics[M]. New York:Academic Press, 1993.
[5] Liao Xiao-xin. Theory and Application of Stability for Dynamical Systems[M]. Beijing:Defence Industrial Press. 2000.(in Chinese)
[6]魏俊杰,黄启昌.关于具有限时滞Lienard方程周期解的存在性[J].科学通报,42, 1997, 42(9):906-909.
[7]陈红斌,李开泰,李东升.Lienard方程周期解的存在唯一与唯二性问题[J].数学学报,2004, 47(3):417-424.
[8]陈世哲,陈仕洲.具有两个偏差变元的Lienard型方程的周期解的存在唯一性[J].科技通报,2012,28(11):11-15.
[9]陈仕洲.一类Lienard型p-Laplacian方程周期解的存在唯一性[J].数学的实践与认识,2013, 43(8):244-253.
[10]陈月红·具有两个偏差变元的Lienard型方程周期解的存在性[J].数学的实践与认识,2014, 44(18):315-320.
[11]黄勇,黄燕革.同时带有强迫项的有限时滞Lienard方程周期解的存在性[J].数学的实践与认识,2016,46(16):213-220.
[12]刘炳文.一类具偏差变元的Lienard型方程周期解的存在与唯一性[J].系统科学与数学,2009, 29(3):360-366.
[13] liu B, Huang L, Existence and uniqueness of periodic solutions for a kind of Lienard equation with a deviating argument[J], Applied Mathematics Letters, 2008, 21(1):56-62.
[14] Lu S, Ge W, Sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument[J]. J Math Anal Appl, 2005, 308:393-419.
[15]黄勇,黄燕革.具有强迫项的有限时滞Lienard方程周期解的存在性[J].四川师范大学学报(自然科学版),2016, 39(1):111-116.
[16] Cheung W S, Ren Jingli, On the existence of periodic solutions for p-Laplacian generalized Lienard equation[J], Nonlinear Anal, 2005, 60(1):65-75.
[17] Omari P, Villari G, and Zanolin F. Periodic solutions of the Lienard equation with one-side growth restrictions[J]. J Diff Equs, 1987, 67:278-293.
[18] Gaines R E. and Mawhin J L. Coincidence Degree and Nonlinear Differential Equations[M].LMN586 Berlin Springer-Verlag, 1977.
[19] Deimling K. Nonlinear functional analysis[M]. Berlin Springer-Verlag, 1985.
[20] JIN Shan, LU Shi-ping. Periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument[J]. Nonlinear Anal TMA, 2008, 69(5/6):1710-1718.
[21] Mawhin. J. Periodic solutions of some vector retarded functional differential equations[J]. Math Anal Appl, 1974, 45:588-603.
[2] Hale J K. Theory of Functional Differential Equations[M]. New York:Springer-Verlag. 1977.
[3] Hale J K. Introduction to Functional Differential Equations[M]. Berlin:Springer-Verlag, 1977.
[4] Kuang Y. Delay Differential Equations with Applications in Population Dynamics[M]. New York:Academic Press, 1993.
[5] Liao Xiao-xin. Theory and Application of Stability for Dynamical Systems[M]. Beijing:Defence Industrial Press. 2000.(in Chinese)
[6]魏俊杰,黄启昌.关于具有限时滞Lienard方程周期解的存在性[J].科学通报,42, 1997, 42(9):906-909.
[7]陈红斌,李开泰,李东升.Lienard方程周期解的存在唯一与唯二性问题[J].数学学报,2004, 47(3):417-424.
[8]陈世哲,陈仕洲.具有两个偏差变元的Lienard型方程的周期解的存在唯一性[J].科技通报,2012,28(11):11-15.
[9]陈仕洲.一类Lienard型p-Laplacian方程周期解的存在唯一性[J].数学的实践与认识,2013, 43(8):244-253.
[10]陈月红·具有两个偏差变元的Lienard型方程周期解的存在性[J].数学的实践与认识,2014, 44(18):315-320.
[11]黄勇,黄燕革.同时带有强迫项的有限时滞Lienard方程周期解的存在性[J].数学的实践与认识,2016,46(16):213-220.
[12]刘炳文.一类具偏差变元的Lienard型方程周期解的存在与唯一性[J].系统科学与数学,2009, 29(3):360-366.
[13] liu B, Huang L, Existence and uniqueness of periodic solutions for a kind of Lienard equation with a deviating argument[J], Applied Mathematics Letters, 2008, 21(1):56-62.
[14] Lu S, Ge W, Sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument[J]. J Math Anal Appl, 2005, 308:393-419.
[15]黄勇,黄燕革.具有强迫项的有限时滞Lienard方程周期解的存在性[J].四川师范大学学报(自然科学版),2016, 39(1):111-116.
[16] Cheung W S, Ren Jingli, On the existence of periodic solutions for p-Laplacian generalized Lienard equation[J], Nonlinear Anal, 2005, 60(1):65-75.
[17] Omari P, Villari G, and Zanolin F. Periodic solutions of the Lienard equation with one-side growth restrictions[J]. J Diff Equs, 1987, 67:278-293.
[18] Gaines R E. and Mawhin J L. Coincidence Degree and Nonlinear Differential Equations[M].LMN586 Berlin Springer-Verlag, 1977.
[19] Deimling K. Nonlinear functional analysis[M]. Berlin Springer-Verlag, 1985.
[20] JIN Shan, LU Shi-ping. Periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument[J]. Nonlinear Anal TMA, 2008, 69(5/6):1710-1718.
[21] Mawhin. J. Periodic solutions of some vector retarded functional differential equations[J]. Math Anal Appl, 1974, 45:588-603.