测度链上p-Laplacian边值问题与Hamiltonian系统的周期解
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摘要
测度链上动力方程理论不但可以统一微分方程和差分方程、更好地洞察二者之间的本质差异,而且还可以更精确地描述那些有时在连续时间出现而有时在离散时间出现的现象。所以,研究测度链上动力方程既有理论意义,又有现实基础。
类似于微分方程和差分方程,非线性项变号的边值问题同样是一个困难且重要的问题。为此,我们研究了测度链上非线性项变号的p-Laplacian奇异多点边值问题正解的存在性。借助于上下解方法和Schauder不动点定理,获得了在广义Dirichlet,广义Robin及非线性Robin边值条件下非线性项变号的p-Laplacian奇异多点边值问题正解的一些新的存在性法则。
对于非奇异边值问题,我们首先考虑了测度链T上的p-Laplacian多点广义Neumann边值问题正解的存在性。利用Krasnosel'skii不动点定理、广义的AveryHenderson不动点定理以及Avery-Peterson不动点理论。获得了至少有一个、两个、三个和任意奇数个正解的新的充分条件,建立了p-Laplacian多点广义Neumann边值问题正解的存在性理论。为了进一步考虑解的特性,我们借助于对称技巧和五泛函不动点定理,给出了测度链T上一类p-Laplacian两点边值问题至少有三个正对称解的存在性条件。此外,利用伪对称技巧和五泛函不动点定理,得到了一类p-Laplacian三点边值问题三个正伪对称解的存在性准则。
我们知道,变分理论是非线性分析中非常有力的工具,能否在测度链分析中发挥类似的作用,是人们非常关注的问题。然而,正如美国数学评论员Ahlbrandt在评论(MR1962542)中指出,目前所使用的Hilger积分仅仅依赖于原函数,而所谓的“△积分”和“▽积分”又分别是一类Darboux积分和修改了的Riemann和的极限。这些积分的缺陷严重阻碍了测度链上变分理论进一步发展,从而使目前的积分理论很难将变分理论应用到测度链分析理论中去。鉴于这种理论背景,Rynne(JMAA,2007)引入了一种测度链T上的新积分。我们发现这种新积分克服了Hilger积分的缺陷,使得变分理论应用到测度链分析理论成为可能。因此,我们借助于临界点理论,研究了两类二阶Hamiltonian系统,并得到了周期解的存在性准则。这可能是第一次用临界点理论来研究测度链上二阶Hamiltonian系统周期解存在性的问题。
The theory of dynamic equations on time scales cannot only unify differential and difference equations and understand deeply the essential difference between them, but also provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. Hence, the studying of dynamic equations on time scales is worth with theoretical and practical values.
As considering the difference equations and differential equations, it is very difficult and important in studying the singular boundary value problems on time scales with the nonlinear term changing sign. Hence, we investigate singular m-point p-Laplacian boundary value problems on time scales with the sign changing nonlinearity. By using the well-known Schauder fixed point theorem and upper and lower solution method, we obtain some new existence criteria for positive solution of generalized Dirichlet, generalized Robin and nonlinear Robin boundary value problems.
For nonsingular boundary value problems, we firstly deal with p-Laplacian multi-point generalized Neumann boundary value problem on time scales T. By using Krasnosel'skii's fixed point theorem, the generalized Avery and Henderson fixed point theorem and Avery-Peterson fixed point theorem. Some new sufficient conditions are obtained for the existence of at least single, twin, triple and arbitrary odd positive solutions of the above generalized Neumann problem. For considering the character of solutions, we concern with a class of p-Laplacian two-point boundary value problem on time scales T. By using symmetry technique and a five functionals fixed-point theorem, we prove that the boundary value problem has at least three positive symmetric solutions. We also prove that another p-Laplacian three-point boundary value problem on time scales has at least three positive pseudo-symmetric solutions in view of pseudo-symmetric technique and a five functionals fixed-point theorem.
Variational theory is a very important method in nonlinear functional analysis. Does the variational theory exert analogous strength to study the problem in analysis on time scales? This problem arrests many people's attention. However, just as Ahlbrandt (MR1962542) reviewed for the reference (Buhner M., Peterson A., Advances in Dynamic Equation on Time Scales, Birkh(a|¨)user. Boston, 2003.), the Hilger's integral is based solely on antiderivatives. The so-called "delta integral" and "nabla integral" are defined by a sort of Darboux integral and as a limit of a modified Riemann sum, respectively. The absence of these integrals has hindered the development of variational theory on time scales. Therefore, Rynne defined a new integral on time scales T (JMAA, 2007). This new integral got over the absence of Hilger's integral. In view of variational methods and critical theory, we discuss two classes of second order Hamiltonian systems on time scales T and establish existence results for periodic solutions of the above-mentioned second order Hamiltonian systems on time scales T. This is probably the first time the existence of periodic solutions for second order Hamiltonian system on time scales has been studied by using variational methods and critical theory.
类似于微分方程和差分方程,非线性项变号的边值问题同样是一个困难且重要的问题。为此,我们研究了测度链上非线性项变号的p-Laplacian奇异多点边值问题正解的存在性。借助于上下解方法和Schauder不动点定理,获得了在广义Dirichlet,广义Robin及非线性Robin边值条件下非线性项变号的p-Laplacian奇异多点边值问题正解的一些新的存在性法则。
对于非奇异边值问题,我们首先考虑了测度链T上的p-Laplacian多点广义Neumann边值问题正解的存在性。利用Krasnosel'skii不动点定理、广义的AveryHenderson不动点定理以及Avery-Peterson不动点理论。获得了至少有一个、两个、三个和任意奇数个正解的新的充分条件,建立了p-Laplacian多点广义Neumann边值问题正解的存在性理论。为了进一步考虑解的特性,我们借助于对称技巧和五泛函不动点定理,给出了测度链T上一类p-Laplacian两点边值问题至少有三个正对称解的存在性条件。此外,利用伪对称技巧和五泛函不动点定理,得到了一类p-Laplacian三点边值问题三个正伪对称解的存在性准则。
我们知道,变分理论是非线性分析中非常有力的工具,能否在测度链分析中发挥类似的作用,是人们非常关注的问题。然而,正如美国数学评论员Ahlbrandt在评论(MR1962542)中指出,目前所使用的Hilger积分仅仅依赖于原函数,而所谓的“△积分”和“▽积分”又分别是一类Darboux积分和修改了的Riemann和的极限。这些积分的缺陷严重阻碍了测度链上变分理论进一步发展,从而使目前的积分理论很难将变分理论应用到测度链分析理论中去。鉴于这种理论背景,Rynne(JMAA,2007)引入了一种测度链T上的新积分。我们发现这种新积分克服了Hilger积分的缺陷,使得变分理论应用到测度链分析理论成为可能。因此,我们借助于临界点理论,研究了两类二阶Hamiltonian系统,并得到了周期解的存在性准则。这可能是第一次用临界点理论来研究测度链上二阶Hamiltonian系统周期解存在性的问题。
The theory of dynamic equations on time scales cannot only unify differential and difference equations and understand deeply the essential difference between them, but also provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time. Hence, the studying of dynamic equations on time scales is worth with theoretical and practical values.
As considering the difference equations and differential equations, it is very difficult and important in studying the singular boundary value problems on time scales with the nonlinear term changing sign. Hence, we investigate singular m-point p-Laplacian boundary value problems on time scales with the sign changing nonlinearity. By using the well-known Schauder fixed point theorem and upper and lower solution method, we obtain some new existence criteria for positive solution of generalized Dirichlet, generalized Robin and nonlinear Robin boundary value problems.
For nonsingular boundary value problems, we firstly deal with p-Laplacian multi-point generalized Neumann boundary value problem on time scales T. By using Krasnosel'skii's fixed point theorem, the generalized Avery and Henderson fixed point theorem and Avery-Peterson fixed point theorem. Some new sufficient conditions are obtained for the existence of at least single, twin, triple and arbitrary odd positive solutions of the above generalized Neumann problem. For considering the character of solutions, we concern with a class of p-Laplacian two-point boundary value problem on time scales T. By using symmetry technique and a five functionals fixed-point theorem, we prove that the boundary value problem has at least three positive symmetric solutions. We also prove that another p-Laplacian three-point boundary value problem on time scales has at least three positive pseudo-symmetric solutions in view of pseudo-symmetric technique and a five functionals fixed-point theorem.
Variational theory is a very important method in nonlinear functional analysis. Does the variational theory exert analogous strength to study the problem in analysis on time scales? This problem arrests many people's attention. However, just as Ahlbrandt (MR1962542) reviewed for the reference (Buhner M., Peterson A., Advances in Dynamic Equation on Time Scales, Birkh(a|¨)user. Boston, 2003.), the Hilger's integral is based solely on antiderivatives. The so-called "delta integral" and "nabla integral" are defined by a sort of Darboux integral and as a limit of a modified Riemann sum, respectively. The absence of these integrals has hindered the development of variational theory on time scales. Therefore, Rynne defined a new integral on time scales T (JMAA, 2007). This new integral got over the absence of Hilger's integral. In view of variational methods and critical theory, we discuss two classes of second order Hamiltonian systems on time scales T and establish existence results for periodic solutions of the above-mentioned second order Hamiltonian systems on time scales T. This is probably the first time the existence of periodic solutions for second order Hamiltonian system on time scales has been studied by using variational methods and critical theory.
引文
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