Asymptotic analysis of nonlinear ordinary differential equations and its applications

1.考虑了二阶拟线性常微分方程，它可以看作是Emden-Fowler方程的推广。我们确定了每一个正解的渐近形式。我们还建立了几类正解的唯一性。2.考虑Emden-Fowler型椭圆方程组。我们建立了系统所有解振动的充分条件。当系数函数的行为类似于|X| ^a，我们的条件在某种意义下是最好的。3.到目前为止，在对非线性项施加几个附加假设的情况下，二阶拟线性椭圆型方程的振动性得到了研究。然而，我们可以建立有效的振动准则，而不这样做。特别地，对于自治方程，我们可以建立所有解振动的充分必要条件。4.研究了二阶半近似常微分方程和椭圆型偏微分方程的特征值问题。得到了变分特征函数和特征值的渐近形式。5.研究了n阶拟线性常微分方程正解的渐近性态。建立了在无穷远点附近存在具有特定渐近行为的最终正解的充分必要条件。6.研究了四阶拟线性常微分方程。我们可以建立这样的方程没有最终正解的必要和/或充分条件。

1.We consider second-order quasilinear ordinary differential equations which can be regarded as generalization of the Emden-Fowler equation. We determine asymptotic forms of every positive solutions. We also establish uniqueness of several types of positive solutions.2.We consider elliptic systems of Emden-Fowler type. We establish sufficient conditions for the oscillation of all solutions to the system. When the coefficient functions behave like positive constant multiples of |x|^a, our conditions are best possible in some sense.3.Until now oscillatory properties of second order quasilinear elliptic equations have been studied under several additional assumptions imposed on the nonlinear terms. However, we can establish effective oscillation criteria without doing so. In particular, for autonomous equations we can establish necessary and sufficient conditions for the oscillation of all solutions.4.Eigenvalue problems are studied for second-order semilnear ordinary differential equations, as well as partial differential equations of elliptic type. Asymptotic forms are obtained for variational eigenfunctions and eigenvalues.5.Asymptotic behavior of evetually positive solutions of n-th order quasilinear ordinary differential equations is studied. We establish necessary and sufficient conditions for the existence of eventually positive solutions with specified asymptotic behavior near the infinity.6.Fourth-order quasilinear ordinary differential equations are studied. We can establish necessary and/or sufficient conditions for such equations to have no eventually positive solutions.