二阶非线性奇异ф-Laplace算子方程的无穷多次调和解
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摘要
本文讨论二阶非线性奇异φ-Laplace算子方程(φ(x'))'+f(t,x)=0的无穷多个周期解(次调和解)的存在性问题,其中φ:(-α,α)→R(0<α<+∞)是单调递增同胚,满足φ(0)=0.f(t,x)是关于时间t为2π-周期的连续函数,满足一定的符号条件.
对于特殊的其典型的模型就是相对场中带电粒子的运动的描述.
本文通过改造与所考虑方程等价的Hamilton系统的方法,在所给条件下构造环域,使Hamilton系统的Poincare映射的若干次迭代在环域上满足扭转条件.然后应用Poincare-Birkhoff扭转定理得到新系统的次调和解的存在性.并且我们可以构造不同的环域并不断提高迭代次数得到无穷多个次调和解的存在性.
本文的结果是奇异φ-Laplace算子方程无穷多个周期解(次调和解)的存在性的第一个结果.
In this paper we deal with the existence of infinitely many subharmonic solutions for nonlinear second order equation involving singularφ-Laplacian (φ(x'))'+f(t, x)=0, whereφ:(-α,α)→R (0 For special which is a particular model for mechanical dynamics of particle moving under relativistic effect.
In this paper we modify the considered equation into a new plane Hamiltonian system and we can construct an annulus such that some iteration of the Poincare map for Hamiltonian system is twist on the annulus. By using Poincare-Birkhoff twist theorem, we can obtain the existence of the fixed points for the iteration of the Poincare map which corresponding to the subharmonic solutions for Hamiltonian system. The existence of infinitely many subharmonic solutions for nonlinear second order equation involving singularφ-Laplacian then is proved by considering more and more iterations on different annuli.
As we known, this is the first result on the existence of infinitely many subhar-monic solutions for nonlinear second order equation involving singularφ-Laplacian.
对于特殊的其典型的模型就是相对场中带电粒子的运动的描述.
本文通过改造与所考虑方程等价的Hamilton系统的方法,在所给条件下构造环域,使Hamilton系统的Poincare映射的若干次迭代在环域上满足扭转条件.然后应用Poincare-Birkhoff扭转定理得到新系统的次调和解的存在性.并且我们可以构造不同的环域并不断提高迭代次数得到无穷多个次调和解的存在性.
本文的结果是奇异φ-Laplace算子方程无穷多个周期解(次调和解)的存在性的第一个结果.
In this paper we deal with the existence of infinitely many subharmonic solutions for nonlinear second order equation involving singularφ-Laplacian (φ(x'))'+f(t, x)=0, whereφ:(-α,α)→R (0
In this paper we modify the considered equation into a new plane Hamiltonian system and we can construct an annulus such that some iteration of the Poincare map for Hamiltonian system is twist on the annulus. By using Poincare-Birkhoff twist theorem, we can obtain the existence of the fixed points for the iteration of the Poincare map which corresponding to the subharmonic solutions for Hamiltonian system. The existence of infinitely many subharmonic solutions for nonlinear second order equation involving singularφ-Laplacian then is proved by considering more and more iterations on different annuli.
As we known, this is the first result on the existence of infinitely many subhar-monic solutions for nonlinear second order equation involving singularφ-Laplacian.
引文
[1]J. Moser, E. Zehnder, Notes on dynamical systems, Amer Mathematical Society, New York,2005.
[2]C. Roberts, S. Buchsbaum, Motion of a charged particle in a constant magnetic field and a transverse electromagnetic wave propagating along the field, Phys. Rev. A 135 (1964),381-389.
[3]G. Mourou, T. Tajima, S. Bulanov, Optics in the relativistic regime, Rev. Mod. Phys.78 (2006),309-371.
[4]W. Allison, J. Cobb, Relativistic charged particle identification by energy loss, Ann. Rev. Nucl.& Part. Sci.30 (1980),253-298.
[5]J. Kirk, P. Schneider, On the acceleration of charged particles at relativistic shock fronts, Astrophys. J.315 (1987),425-433.
[6]R. Schlickeiser, I. Lerche, Nonlinear radiative cooling of relativistic particles under equipartition conditions, Astron. Astrophys.476 (2007),1-8.
[7]H. Goldstein, Classical mechanics, Addison-Wesley Press, Cambridge,1951.
[8]J. Kim, H. Lee, Relativistic chaos in the driven harmonic oscillator, Phys. Rev. E 51 (1995),1579-1581.
[9]P. Torres, Periodic oscillations of the relativistic pendulum eith friction, Phys. Lett. A 372 (2008) 6386-6387.
[10]C. Bereanu, J. Mawhin, Nonlinear Neumann boundary value problems with φ-Laplacian operators, An. Stiint. Univ. Ovidius Constanta Ser. Mat.12 (2004), 73-92.
[11]C. Bereanu, J. Mawhin, Boundary-value problems with non-surjective φ-Laplacian and one-side bounded nonlinearity, Adv. Differential Equations 11 (2006),35-60.
[12]C. Bereanu, J. Mawhin, Existence and multiplicity results for some nonlinear prob-lems with singular φ-Laplacian, J. Differential Equations 243 (2007),536-557.
[13]C. Bereanu, J. Mawhin, Multiple periodic solutions of ordinary differential equa-tions with bounded nonlinearities and φ-Laplacian, NoDEA:Nonlinear Differential Equations and Applications 15 (2008),159-168.
[14]C. Bereanu, J. Mawhin, Periodic solutions of nonlinear perburbations of φ-Lapla-cians with possibly bounded φ, Nonlinear Anal. TMA 68 (2008),1668-1681.
[15]C. Bereanu, J. Mawhin, Boundary value problems for second-order nonlinear dif-ference equations with discrete φ-Laplacian and singular φ, J. Differ. Equ. Appl. 14 (2008),1099-1118.
[16]C. Bereanu, J. Mawhin, Boundary value problems for some nonlinear systems with singular φ-Laplacian, J. Fixed Point Theory and Appl.4 (2008),57-75.
[17]C. Bereanu, J. Mawhin, Nonhomogeneous boundary value problems for some non-linear equations with singular φ-Laplacian, J. Math. Anal. Appl.352 (2009),218-233.
[18]C. Bereanu, P. Jebelean, J. Mawhin, Non-homogeneous boundary value problems for ordinary and partial differential equations involving singular φ-Laplacians, Matematica Contemporanea 36 (2009),51-65.
[19]J. Cid, P. Torres, Solvability for some boundary value problems with φ-Laplacian operators, Discrete contin. Dyn. Sys.23 (2009),727-732.
[20]J. Mawhin and M. Wilem, Critical Point Theory and Hamiltonian systems, Springer-Verlag, Berlin-Heidelberg-New York,1993.
[21]H. Jacobowitz, Periodic solutions of x"+f(x,t)=0 via the Poincare-Birkhoff Theorem, J. Differential Equations,20 (1976),37-52.
[22]W. Ding, A generalization of the Poincare-Birkhoff Theorem, Proceedings of the American Mathematical Society 88 (1983),341-346.
[23]T. Ding, F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J Differential Equations 97 (1992),328-378.
[24]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.
[25]D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach-Dancer condition, J. Differential Equations,171 (2001),233-250.
[26]A. Fonda, R. Manasevich and F. Zanolin, Subharmonic solutions for some second-order differential equations with singularities, SIAM J. Math. Anal.,24 (1993), 1294-1311.
[27]D. Qian, P. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM Journal on Mathematical Analysis 36 (2005),1707-1725.
[28]丁同仁,关于亚线性Duffing方程的研究,应用数学学报,12(1989),449-455.
[29]T. Ding and F. Zanolin, Subharmonic solution of second-order nonlinear equa-tions:a time-map approach, Nonlinear Analysis TMA 20 (1993),509-532.
[30]J. Franks, Generalizations of the Poincare-Birkhoff Theorem, Ann. of Math.128 (1998),139-151.
[31]丁伟岳,扭转映射的不动点和常微分方程的周期解,数学学报,25(1982),227-235.
[32]J. Ginocchio, Relativistic symmetries in nuclei and hadrons, Phys. Rep.414 (2005), 165-261.
[33]D. Kulikov, R. Tutik, Oscillator model for the relativistic fermion-boson system, Phys. Lette. A 372 (2008),7105-7108.
[34]N. Atakishiev, R. Mir-Kasimov, Generalized coherent states for relativistic model of a linear oscillator, Theor. Math. Phys.67 (1986),362-367.
[35]R. Ortega, Asymmertric oscillators and twist mapping, J. Lond. Math. Soc.,53 (1996),325-342.
[2]C. Roberts, S. Buchsbaum, Motion of a charged particle in a constant magnetic field and a transverse electromagnetic wave propagating along the field, Phys. Rev. A 135 (1964),381-389.
[3]G. Mourou, T. Tajima, S. Bulanov, Optics in the relativistic regime, Rev. Mod. Phys.78 (2006),309-371.
[4]W. Allison, J. Cobb, Relativistic charged particle identification by energy loss, Ann. Rev. Nucl.& Part. Sci.30 (1980),253-298.
[5]J. Kirk, P. Schneider, On the acceleration of charged particles at relativistic shock fronts, Astrophys. J.315 (1987),425-433.
[6]R. Schlickeiser, I. Lerche, Nonlinear radiative cooling of relativistic particles under equipartition conditions, Astron. Astrophys.476 (2007),1-8.
[7]H. Goldstein, Classical mechanics, Addison-Wesley Press, Cambridge,1951.
[8]J. Kim, H. Lee, Relativistic chaos in the driven harmonic oscillator, Phys. Rev. E 51 (1995),1579-1581.
[9]P. Torres, Periodic oscillations of the relativistic pendulum eith friction, Phys. Lett. A 372 (2008) 6386-6387.
[10]C. Bereanu, J. Mawhin, Nonlinear Neumann boundary value problems with φ-Laplacian operators, An. Stiint. Univ. Ovidius Constanta Ser. Mat.12 (2004), 73-92.
[11]C. Bereanu, J. Mawhin, Boundary-value problems with non-surjective φ-Laplacian and one-side bounded nonlinearity, Adv. Differential Equations 11 (2006),35-60.
[12]C. Bereanu, J. Mawhin, Existence and multiplicity results for some nonlinear prob-lems with singular φ-Laplacian, J. Differential Equations 243 (2007),536-557.
[13]C. Bereanu, J. Mawhin, Multiple periodic solutions of ordinary differential equa-tions with bounded nonlinearities and φ-Laplacian, NoDEA:Nonlinear Differential Equations and Applications 15 (2008),159-168.
[14]C. Bereanu, J. Mawhin, Periodic solutions of nonlinear perburbations of φ-Lapla-cians with possibly bounded φ, Nonlinear Anal. TMA 68 (2008),1668-1681.
[15]C. Bereanu, J. Mawhin, Boundary value problems for second-order nonlinear dif-ference equations with discrete φ-Laplacian and singular φ, J. Differ. Equ. Appl. 14 (2008),1099-1118.
[16]C. Bereanu, J. Mawhin, Boundary value problems for some nonlinear systems with singular φ-Laplacian, J. Fixed Point Theory and Appl.4 (2008),57-75.
[17]C. Bereanu, J. Mawhin, Nonhomogeneous boundary value problems for some non-linear equations with singular φ-Laplacian, J. Math. Anal. Appl.352 (2009),218-233.
[18]C. Bereanu, P. Jebelean, J. Mawhin, Non-homogeneous boundary value problems for ordinary and partial differential equations involving singular φ-Laplacians, Matematica Contemporanea 36 (2009),51-65.
[19]J. Cid, P. Torres, Solvability for some boundary value problems with φ-Laplacian operators, Discrete contin. Dyn. Sys.23 (2009),727-732.
[20]J. Mawhin and M. Wilem, Critical Point Theory and Hamiltonian systems, Springer-Verlag, Berlin-Heidelberg-New York,1993.
[21]H. Jacobowitz, Periodic solutions of x"+f(x,t)=0 via the Poincare-Birkhoff Theorem, J. Differential Equations,20 (1976),37-52.
[22]W. Ding, A generalization of the Poincare-Birkhoff Theorem, Proceedings of the American Mathematical Society 88 (1983),341-346.
[23]T. Ding, F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J Differential Equations 97 (1992),328-378.
[24]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.
[25]D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach-Dancer condition, J. Differential Equations,171 (2001),233-250.
[26]A. Fonda, R. Manasevich and F. Zanolin, Subharmonic solutions for some second-order differential equations with singularities, SIAM J. Math. Anal.,24 (1993), 1294-1311.
[27]D. Qian, P. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM Journal on Mathematical Analysis 36 (2005),1707-1725.
[28]丁同仁,关于亚线性Duffing方程的研究,应用数学学报,12(1989),449-455.
[29]T. Ding and F. Zanolin, Subharmonic solution of second-order nonlinear equa-tions:a time-map approach, Nonlinear Analysis TMA 20 (1993),509-532.
[30]J. Franks, Generalizations of the Poincare-Birkhoff Theorem, Ann. of Math.128 (1998),139-151.
[31]丁伟岳,扭转映射的不动点和常微分方程的周期解,数学学报,25(1982),227-235.
[32]J. Ginocchio, Relativistic symmetries in nuclei and hadrons, Phys. Rep.414 (2005), 165-261.
[33]D. Kulikov, R. Tutik, Oscillator model for the relativistic fermion-boson system, Phys. Lette. A 372 (2008),7105-7108.
[34]N. Atakishiev, R. Mir-Kasimov, Generalized coherent states for relativistic model of a linear oscillator, Theor. Math. Phys.67 (1986),362-367.
[35]R. Ortega, Asymmertric oscillators and twist mapping, J. Lond. Math. Soc.,53 (1996),325-342.