【作者】 侯小康；

【导师】 杨军；

【作者基本信息】 燕山大学 ， 运筹学与控制论， 2009， 硕士

【摘要】 近年来,时标上动力方程作为数学的一个新兴领域得到了人们的普遍重视。1988年,Stefan Hilger率先提出时标及时标上的微积分理论,并初步建立了时标上动力方程的基本理论。从本世纪初开始,这一理论受到关注,目前,这一理论正得到迅速的发展。一方面,它统一和推广了现有的微分和差分、微分方程和差分方程的理论,另一方面,时标上动力方程的研究对于刻画真实现象和过程的数学模型有着重要应用,例如:时标上的种群动力学、流行病模型、金融消费过程的数学模型等。总之,时标和时标上动力方程理论有着广阔的应用前景。在现有时标理论的基础上,论文分别就时标上动力方程解的振动性、解的存在性以及边值问题进行了研究。首先讨论了时标上一类二阶非线性中立型动力方程解的振动性,得到了解振动的充分条件,并且给出了实际应用的例子。其次给出了时标上一类二阶中立型时滞动力方程解振动的充分条件,同时给出了例子加以说明。然后运用Banach压缩映射原理研究了时标上一类高阶带强迫项中立型时滞动力方程非振动解的存在性,得到了方程非振动解存在的判别结果。进一步讨论了时标上一类具有非线性中立项的高阶变系数变时滞动力方程解的强迫振动与非振动解的存在性,所得结果推广了已知的一些结论。最后,研究了时标上一类二阶非线性边值问题及时标上的半直线上奇异m-点边值问题的正解存在性,得到了方程正解存在的判别结果。更多还原

【Abstract】 Recently, the study of dynamic equations on time scales is an area of mathematics that has received a lot of attention. Theory of dynamic equations on time scales, founded by Stefan Hilger in 1988, is undergoing a rapid development as it provides a powerful tool not only to unify the existing theories of both differential equations and difference equations but also to generalize the discussion to a wide class of equations on time scales. In addition, the study of dynamic equations on time scales has lead to several important applications related to mathematical modeling of real phenomena. For instance, investigation of population dynamics, financial assumption process, and epidemic models have been involved with the concept of dynamic equations on time scales. Clearly, theory of dynamic equations on time scales has been and are being proved to be of great importance and potential application.In this thesis, oscillation properties、existence properties of solutions of dynamic equations and boundary value problems on time scales are researched.Firstly, oscillation properties of solutions for a class of second order nonlinear neutral dynamic equations on time scales are researched, a series of sufficient conditions for oscillation of the solutions are established. These results are illustrated by examples.Secondly, some sufficient conditions for oscillation of a class of second order neutral delay dynamic equations on time scales are obtained. These results are explained by examples.The existence of nonoscillatory solutions for a class of high-order forced neutral delay dynamic equations on time scales are studied by using Banach contraction mapping principle. Some sufficient conditions to satisfy existence of nonoscillatory solutions of these equations are derived.A class of forced high-order nonlinear neutral variable delay dynamic equations with variable coefficient on time scales is studied. The sufficient conditions are given for the forced oscillation and existence of nonoscillatory solutions. These results generalize the corresponding known results.Finally, the existence of positive solutions for a class of second order nonlinear boundary value problems on time scales and nonlinear singular m-point boundary value problems on the half-line on time scales are considered. Some sufficient conditions to satisfy existence of positive solutions of these equations are derived.更多还原