THE OSCILLATION OF SOLUTIONS OF VOLTERRA INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS WITH HIGHLY OSCILLATORY KERNELS
THE OSCILLATION OF SOLUTIONS OF VOLTERRA INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS WITH HIGHLY OSCILLATORY KERNELS
复制标题
具有高振荡核的Volterra积分和积分微分方程解的振荡
DOI：
10.1216/jie-2015-27-4-455
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发表时间：
2015-12-01
影响因子：
0.8
通讯作者：
Xu, Yuesheng
中科院分区：
文献类型：
--
作者：
Brunner, Hermann;Ma, Yunyun;Xu, Yuesheng
研究了具有高振动核的沃尔泰拉积分微分方程（VIE，VIDES）解的振动结构.基于Wang和Xu [28]中引入的结构振动空间，我们首先利用VIE解的预解式表示，分析了与属于某一结构振动空间的振动核相关联的VIE解的振动度.根据复平面上振荡积分的一种分解，证明了沃尔泰拉积分算子在与核的振荡结构相对应的结构振荡空间中降低了函数的振荡阶.类似的振动结构的解决方案的VIDES，然后分析表示的解决方案的VIDES的微分预解核，并利用VIDES和等效VIE之间的关系。我们的结论是，VIE和VIDES的解决方案保持振荡的核心组件。
We study the oscillatory structures of solutions of Volterra integral and integro-differential equations (VIEs, VIDEs) with highly oscillatory kernels. Based on the structured oscillatory spaces introduced in Wang and Xu [28], we first analyze the degree of oscillation of the solution of VIEs associated with the oscillatory kernels belonging to a certain structured oscillatory space by using the resolvent representation of the solution. According to a decomposition of the oscillatory integrals in the complex plane, we prove that the Volterra integral operator reduces the oscillatory order of the functions in the structured oscillatory spaces corresponding to the oscillatory structure of the kernel. The analogous oscillatory structure of solutions of VIDEs is then analyzed by representing the solution of the VIDEs by the differential resolvent kernel and by exploiting the relationship between the VIDEs and the equivalent VIE. We conclude that the solutions of the VIEs and VIDEs preserve the oscillatory components of the kernel.