p（t）-拉普拉斯系统中等价Hartman型的周期解
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摘要
首先,本文通过临界点理论中的极大极小原理,得到如下的p(t)拉普拉斯系统的周期解
接着,根据以上的结果,若存在R>0,对任意的t∈R1,(?)u∈RN有|u|=R,使得<▽F(t,u),u)≤0,则可以得出此问题满足Hartman型结果,也即是,至少有一个解u,对t∈[0,T]使|u(t)|≤R.
In this paper, firstly, by using ininimax methods in critical point theory, we get the existence of the periodic solutions for the following p(t)-Laplacian system
Then, according to the above result, if (?)R＞0, such that <▽F(t, u), u>≤0 for t∈R1, (?)u∈RN with|u|= R, we show that the problem satisfies the Hartman-type result, that is, there is at least a solution u such that |u(t)|≤R for t∈[0,T].
接着,根据以上的结果,若存在R>0,对任意的t∈R1,(?)u∈RN有|u|=R,使得<▽F(t,u),u)≤0,则可以得出此问题满足Hartman型结果,也即是,至少有一个解u,对t∈[0,T]使|u(t)|≤R.
In this paper, firstly, by using ininimax methods in critical point theory, we get the existence of the periodic solutions for the following p(t)-Laplacian system
Then, according to the above result, if (?)R＞0, such that <▽F(t, u), u>≤0 for t∈R1, (?)u∈RN with|u|= R, we show that the problem satisfies the Hartman-type result, that is, there is at least a solution u such that |u(t)|≤R for t∈[0,T].
引文
[1]W.Orlicz, Orlicz, Uber konjugierte exponentenfolgen, Studia Math.,3(1931),200-212.
[2]J.Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin,1993.
[3]H.Nakano, Modularde Semi-ordered Linear Spaces, Maruzen Co., Ltd., Tikyo,1950
[4]v.v.Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math.USSR Izv,29(1987),33-66
[5]O.Kavacik,J.Rakosnik, On spaces Lp(x)(Ω) and W1,P(x)(Ω) Czechoslovak Math.J.41(1991) 592-618
[6]M.Ruzicka, Electororheological fluids:modeling and mathematical theory, Springer Verlag, Berlin,2000.
[7]Ph.Hartman, On boundary value problems for systems of ordinary nonliear secord order differential equations, Trans. Amer. Math. Soc.96 (1960)493-509.
[8]H.W.knobloch, On the existence of periodic solutions for second order vactor differen-tial equations, J.Differential Equations 9 (1971)67-85.
[9]N.Rouche, J.Mawhin, Equations Differentielles Ordinaries, Vol.2, Masson, paris,1973; English transl.:Ordinary Differential Equations, Stability and periodic Solutions, Pit-man, Boston,1980.
[10]J.Mawhin, Some boundary value problems for Hartman-type perturbations of the or-dinary vector p-Laplacian, Nonliear Anal.40 (2000)497-503.
[11]X.L.Fan, X.Fan, A Knobloch-type result for p(t)-Laplacian systems, J.Math. Anal.Appl., 282 (2003)453-464.
[12]V. Benci, Some critical point theorems and applicationss, Comm. Pure Appl. Math., 33(2) (1980)147-172.
[13]X.b.Shu, Y.T.Xu, L.H.Huang, Infinite periodic solutions to a class of second-order Sturm-Liouville neutral differential equations, Monliear Anal., (2007),68 (2008) 905-911.
[14]C.L.Tang, X.P.Wu, Notes on periodic solutions of subquadradic second order systems, J.Math. Anal. Appl.,285(1)(2003) 8-16.
[15]B.Xu, C.L.Tang, Some existence results on periodic solutions of ordinary p-Laplacian systems, J.Math. Anal. Appl.,333 (2007) 1228-1236.
[16]X.L.Fan, Q.H.Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Non-liear Anal.,52 (2003)1843-1852.
[17]J.Mawhin, M.Willem, Critical point Theory and Hamiltonian systems, Appl. Math. Sci.vol.vol.74, Springer, New YorK,1989.
[18]X.J.Wang, R.Yuan, Existence of Periodic solutions for p(t)-Laplacian systems, Non-linear Anal.,70 (2009) 866-180.
[19]C.k.Chang, Critical point Theory and its Applications, Shanghai science and Technol-ogy Press, Shanghai 1986(in Chinese).
[20]P.H.Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Dif-ferential Equations, American Mathematical Society,1986.
[21]X.L.Fan, D.Zhao, On the space L P(X)(Ω) and W m,p(x)(Ω), J. Math. Anal. Appl.,263 (2001)424-446.
[22]J.T.Oden, Qualitative Methods in Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, NJ,1986.
[23]I.P.Natanson, Theory of Functions of a Real Variable, Nauka, Mosow,1950.
[24]张恭庆,临界点理论及其应用,上海科学技术出版社,1986.
[2]J.Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin,1993.
[3]H.Nakano, Modularde Semi-ordered Linear Spaces, Maruzen Co., Ltd., Tikyo,1950
[4]v.v.Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math.USSR Izv,29(1987),33-66
[5]O.Kavacik,J.Rakosnik, On spaces Lp(x)(Ω) and W1,P(x)(Ω) Czechoslovak Math.J.41(1991) 592-618
[6]M.Ruzicka, Electororheological fluids:modeling and mathematical theory, Springer Verlag, Berlin,2000.
[7]Ph.Hartman, On boundary value problems for systems of ordinary nonliear secord order differential equations, Trans. Amer. Math. Soc.96 (1960)493-509.
[8]H.W.knobloch, On the existence of periodic solutions for second order vactor differen-tial equations, J.Differential Equations 9 (1971)67-85.
[9]N.Rouche, J.Mawhin, Equations Differentielles Ordinaries, Vol.2, Masson, paris,1973; English transl.:Ordinary Differential Equations, Stability and periodic Solutions, Pit-man, Boston,1980.
[10]J.Mawhin, Some boundary value problems for Hartman-type perturbations of the or-dinary vector p-Laplacian, Nonliear Anal.40 (2000)497-503.
[11]X.L.Fan, X.Fan, A Knobloch-type result for p(t)-Laplacian systems, J.Math. Anal.Appl., 282 (2003)453-464.
[12]V. Benci, Some critical point theorems and applicationss, Comm. Pure Appl. Math., 33(2) (1980)147-172.
[13]X.b.Shu, Y.T.Xu, L.H.Huang, Infinite periodic solutions to a class of second-order Sturm-Liouville neutral differential equations, Monliear Anal., (2007),68 (2008) 905-911.
[14]C.L.Tang, X.P.Wu, Notes on periodic solutions of subquadradic second order systems, J.Math. Anal. Appl.,285(1)(2003) 8-16.
[15]B.Xu, C.L.Tang, Some existence results on periodic solutions of ordinary p-Laplacian systems, J.Math. Anal. Appl.,333 (2007) 1228-1236.
[16]X.L.Fan, Q.H.Zhang, Existence of solutions for p(x)-Laplacian Dirichlet problem, Non-liear Anal.,52 (2003)1843-1852.
[17]J.Mawhin, M.Willem, Critical point Theory and Hamiltonian systems, Appl. Math. Sci.vol.vol.74, Springer, New YorK,1989.
[18]X.J.Wang, R.Yuan, Existence of Periodic solutions for p(t)-Laplacian systems, Non-linear Anal.,70 (2009) 866-180.
[19]C.k.Chang, Critical point Theory and its Applications, Shanghai science and Technol-ogy Press, Shanghai 1986(in Chinese).
[20]P.H.Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Dif-ferential Equations, American Mathematical Society,1986.
[21]X.L.Fan, D.Zhao, On the space L P(X)(Ω) and W m,p(x)(Ω), J. Math. Anal. Appl.,263 (2001)424-446.
[22]J.T.Oden, Qualitative Methods in Nonlinear Mechanics, Prentice-Hall, Englewood Cliffs, NJ,1986.
[23]I.P.Natanson, Theory of Functions of a Real Variable, Nauka, Mosow,1950.
[24]张恭庆,临界点理论及其应用,上海科学技术出版社,1986.