【作者】 刘娜；

【导师】 钟晓珠；

【作者基本信息】 燕山大学 ， 运筹学与控制论， 2010， 硕士

【摘要】 近年来,由于生物学、经济学、物理学、航天卫星、计算机技术、控制理论等自然学科的不断发展,在科学研究和社会实践中不断提出大量新的中立型差分方程描述的具体的数学模型。由于应用的广泛性和它本身涉及到大量的数学问题,因而对中立型差分方程定性理论的研究吸引了大批学者的关注。中立型差分方程的振动性与渐近性理论是中立型差分方程定性理论中的重要内容。因此,对其进行研究不仅具有重要的理论意义,而且也具有重要的实际应用价值。论文分别研究了具连续变量的三阶非线性中立型差分方程、三阶非线性中立型差分方程和带有极大值项的高阶差分方程的定性问题。所得结论对已有文献的相关结论做了推广和改进。并分别给出了其解的振动性、渐近性的一些充分条件。首先,论文利用求和法和Riccati技巧对一类具连续变量的三阶多时滞非线性中立型差分方程的有界振动进行了研究,并在不同条件下给出了该方程振动的两个较简单的充分条件。其次,对于一类三阶非线性中立型非线性差分方程,论文利用Riccati分部差分法以及差分不等式的技巧,讨论了该类差分方程的振动性,所讨论的方程将已有文献中的结果推广到更高阶且方程较文献中更为复杂,最终获得了方程振动的几个充分条件。最后,论文研究了带有极大值项的高阶中立型差分方程的振动性与渐近性。运用反证法,将已有文献中的结论推广到高阶方程上,获得了该类方程振动的充分条件。更多还原

【Abstract】 In recent years, with the development of biology, economics, physics, space satellite, computer science, control theory, ect., People have put forward a great many new mathematical models that described by neutral difference equations in scientific research and social practice. Because of applied universality and a mass of mathematics problems relate to itself, qualitative analysis of the difference equations with neutral has attracted the attention of large number of scholars. Moreover, oscillation and asymptotic theory being the key content of qualitative study of the neutral difference equations; it is of great theoretical significance and practical value to research the oscillatory properties, asymptotic behavior and existence of positive solution.This paper is devoted to the problems of the third order nonlinear neutral delay difference equations with continuous arguments, the third-order nonli- near neutral delay difference equations and the high order neutral difference equations with maxima. The relative topics extend the problems studied in the literatures, and give separately some sufficient conditions for oscillation, asymptotic behavior of solutions. Some results which expand and improved the related conclusions in previous literatures are obtained.Firstly, the oscillation for third order nonlinear neutral difference equatio- ns with continuous arguments is discussed by using Summation Method and Riccati technique and two simple oscillatory criteria of this equation is obtained.Secondly, for the third order nonlinear neutral delay difference equations, this thesis studies the oscillation and asymptotic for a class of third order nonlinear neutral delay difference equations by using the methods of Riccati technique , segment difference and difference inequality .the equation which is discussed by us is expand and improved more higher than before and complica- ted than the ones in the previous literatures, some sufficient conditions of this equation to be oscillatory are obtained finally.Finally this thesis researches the oscillation and asymptotic for high order neutral difference equations with maxima. By using reduction to absurdity, these results about oscillation in the prevention literatures were expanded to the high order case. Some new sufficient conditions are obtained.更多还原