摘要
对任给的一个定义在无限维Banach空间上具有无限维值域的全连续算子T,我们分析了Leray-Schauder拓扑度和不动点存在性之间的关系。如果T有一个不动点,那么可建立一个具有有限维值域的近似连续算了Te,使Te至少有一个不动点。如果T有一个孤立不动点,则存在一个开有界集D使Leray-Schauder拓扑度deg(I—T,D,0)不为零。对[0,1]区间上的一个两点边值问题,对应的积分算子T_(Q,A)可以被建立。
Given an arbitrary non-degenerate (with infinite dimensional range) completely continuous operator T defined on an infinite dimensional Banach space, we consider the relation between the Leray-Schauder degree of I-T and the existence of the fixed point of T. If T has afixed points one may alwaysconstruct an approximated continuous operator T, with finite rank which hasat least one fixed point. Furthermore, if T has an isolated fixed point at x_0; there exists an open bounded subset D containing x_0 such that the Leray-Schauder degree deg(I-T, D, 0) is non-zero. For a two-point-boundary-value problem (1) on [0, 1], an integral operator T_(Q,A) is obtained by integrating the equation withthe boundary conditions. T_(Q,A) is compact on bounded subsets of C[0, 1]. The existence of the fixed point of T_(Q,A) is proved.
出处
《应用数学与计算数学学报》
1992年第2期69-75,共7页
Communication on Applied Mathematics and Computation
