一类积分方程组解的存在性

Existence of solution of a class of integral equation system

某元有无原象等价于象集是否包含此元,等价于以此元为象的映射方程的解是否存在.若有两个映射,一个象集比另一个大,在一定条件下,可以确定此二映射之差包含一集合,则此集合的原象非空,且上述集合中的元为象的关于此二映射之差的映射方程的解存在.将上述想法运用到微分积分方程解的存在性问题中去,可以得到相应的结论,再将此结论用到Hammerstein型积分方程,Volterra型积分方程解的存在性问题中去,我们发现,我们的结论比起以往用半序理论,不动点指数理论,迭代逼近方法所得到的结论要好许多.

This paper proceeds according to following idea： whether an element has an image by inversion is equivalent to whether the mapping image set contains the element and equivalent to whether there exist solution of the mapping equation. If there are two mappings, and one set is larger than the other, in proper condition we can ascertain the difference of the two mappings contains a set, then original image of the set is not empty, and there exists a solution of the mapping equation of the difference. Applying the above idea to existence of solution of differential integral equation, we can obtain corresponding result. Applying the result to Hammerstein integral equation and Volterra integral equation, we obtain the result about existence of solution of these integral equations. We find our result is better than the results obtained by the partial ordering theory,fixed point index theory or iterative approximation theory.