Qualitative Theory on Oscillation of Solutions of Differential Equations

得到了关于偏泛函微分方程解的振动定性理论的各种推广结果，即振动理论。建立了中立型抛物型方程各解振动的充分条件。此外，还得到了推广经典Rosenau-Burgers方程的振动定理，并构造了矢量抛物型和双曲型方程的振动理论。对于离散型中立型双曲方程，我们已经讨论过了，但是对于具有连续分布偏差参数的双曲方程，我们可以建立振荡理论。关于这个问题的三篇论文已提交出版。其中两篇是与中国东华大学陶友山教授合作完成的。在《非线性》杂志上发表了一篇关于肿瘤的合作论文。并与东华大学阎立潭教授共同发表了一篇关于双曲型双曲方程特征初值问题的振荡理论的论文。我曾在斯洛伐克举行的国际会议（微分方程与应用会议）上做过特邀演讲。然后我可以和斯洛伐克的数学家讨论，我可以和夸美纽斯大学的Jaros教授合作建立Picone恒等式，并给出一阶半线性椭圆方程的Sturmian比较定理。我可以在富山大学举办“富山微分方程会议-2003”，并与演讲嘉宾千叶大学的E. Minchev教授就数学进行了富有成果的讨论。

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.