Comparioson Theory and Asymptotic Theory for Differential Equations

通过建立半线性椭圆型方程的Picone恒等式，得到了方程每一解振动的充分条件。在非线性项为超线性项与次线性项之和的情形下，利用Picone型不等式得到了振动性结果。通过导出一个Picone恒等式，建立了四阶微分算子的Sturmian比较定理.对于主部比p-Laplacin更一般的拟线性椭圆方程，通过建立半线性情形的Picone恒等式和非线性项为超线性项与次线性项之和的Picone型不等式，得到了振动性结果.进一步得到了带强迫项的超线性椭圆型方程的振动性定理。利用拟线性抛物型方程的Picone型不等式，得到了解的无界性和有界解振动的充分条件。以上结果对指导我国农业生产具有重要意义。 ...更多信息 因为Picone恒等式或Picone型不等式的重要性再次被认识到。通过将泛函微分方程化为泛函微分方程，建立了一类带函数变元的抛物型方程和带函数变元的双曲型方程边值问题解振动的充分条件.得到了二阶中立型正、负系数微分方程的每一个解振动的充分条件。这些结果对偏微分方程的振动理论和一元泛函微分方程的振动理论都有很大的贡献。我在国际会议“EQUADIFF 11”，“微分和差分方程及其应用会议”，“微分和差分方程学术讨论会”，“EQUADIFF 2007”上发表演讲，能够与国外研究人员交流并获得最新信息，这对我来说非常富有成效。我也可以参加在日本的各种研讨会，并提出研究成果，而且我可以做研究安排。我相信微分方程的比较理论和渐近理论会有很大的发展。少

Sufficient conditions for every solution to oscillate were obtained by establishing a Picone identity for half-linear elliptic equation. In the case where nonlinear term is the sum of superlinear term and sublinear term oscillation results were obtained via Picone-type inequalities. We established Sturmian comparison theorems for fourth order differential operators by deriving a Picone identity. For quasilinear elliptic equations where principal parts are more general than p-Laplacin we obtained oscillation results by establishing Picone identity for half-linear case and Picone-type inequality for the case where nonlinear term is the sum of superlinear term and sublinear term. Moreover, oscillation theorems were obtained for superlinear elliptic equations with forcing terms. Using Picone-type inequality applicable to quasilinear parabolic equations, we derived unboundedness of solutions and sufficient conditions for bounded solution to oscillate. All of above results are very significa … More nt because importance of Picone identity or Picone-type inequality was realized again. Sufficient conditions for solution of boundary value problems of parabolic equations with functional arguments and hyperbolic equations with functional arguments to oscillate were established by reducing to functional differential equations. We obtained sufficient conditions for every solution of second order neutral differential equations with positive and negative coefficients to oscillate. These results seem to contribute greatly to oscillation theory for partial differential equations and oscillation theory for functional differential equations with one variable. I gave lectures on international conferences "EQUADIFF 11", "Conference on Differential and Difference Equations and Applications", "Colloquium on Differential and Difference Equations", "EQUADIFF 2007" and it was very fruitful for me to be able to communicate with foreign researchers and get newest informations. Also I could participate in various workshops in Japan, and presented research results, and moreover I could do research arrangements. I believe that comparison theory and asymptotic theory for differential equations could be developed greatly. Less