Qualitative Theory on Oscillation of Solutions of Differential Equations

对偏泛函微分方程解的振动性定性理论，即振动性理论，得到了各种推广的结果。建立了中立型抛物型方程每一解振动的充分条件。此外，还得到了推广的经典Rosenau-Burgers型方程的振动性定理，从而建立了向量抛物型和双曲型方程的振动性理论。对离散型中立型双曲型方程已有研究，但对于具有连续分布偏差变元的中立型双曲型方程，我们可以建立振动性理论。关于这一主题的三篇论文已提交出版。其中两个是与中国东华大学陶友山教授合作的作品。一篇关于肿瘤的论文被批准发表在《非线性》杂志上。此外，还与东华大学严立坦教授合作发表了一篇研究双曲型双曲型方程特征初值问题的振动性理论的论文。我在斯洛伐克举行的国际会议(微分方程式及应用会议)上作了一次特邀演讲。然后我可以和斯洛伐克的数学家讨论，我可以和Jaros教授(夸美纽斯大学)合作，建立Picone恒等式，并建立带一阶项的半线性椭圆型方程的Sturmian比较定理。我可以在富山大学举办“富山微分方程式会议-2003”，并与嘉宾讲者千叶大学的E.Minchev教授就数学问题进行富有成效的讨论。

Various extended results were obtained for qualitative theory on oscillation of solutions of partial functional-differential equations, that is, oscillation theory. Sufficient conditions for every solution of parabolic equations of neutral type to be oscillatory were established. Moreover, oscillation theorems for a generalization of the classical Rosenau-Burgers equation were obtained^Oscillation theory for vector parabolic and hyperbolic equations could be constructed. Neutral hyperbolic equations of discrete type have been treated, but we could make oscillation theory for hyperbolic equations with continuous distributed deviating arguments. Three papers on this subject were submitted for publication. Two of them are joint works with Professor Youshan Tao of Dong Hua University in P.R.China. A paper on tumor which is a joint work was accpted for publication in the journal "Ndnlinearity". Furthermore, a joint paper with Professor Litan Yan of Dong Hua University which treat oscillation theory of characteristic initial value problems for hyperbolic equations of hyperbolic type was submitted for publication. I have given an invited lecture at the international conference(Conference on Differential Equations and Applications) held in Slovakia. Then I could discuss with Slovakian Mathematicians, and I could cooperate with Professor Jaros(Comenius University) to establish Picone identities and make Sturmian comparison theorems for half-linear elliptic equations with first order terms. I could hold "Toyama Conference on Differential Equations-2003" at Toyama University, and have fruitful discussions about Mathematics with the guest speaker, Professor E. Minchev of Chiba University.