非线性全连续算子的正不动点和正固有元及其运用

Positive Fixed Points and Eigenvectors of Nonlinear Compact Operators and Applications

本文,先将[1]P326的引理4.9和引理4.10的条件,用一个简单而又便于应用的等价条件来代替,得到用算子A的导算子A′_+(θ)及A′_+(∞)的谱半径ρ(A′_+(θ)),ρ(A′_+(∞))的值判定不动点指数是否为0的基本结果,并给出它的一个新的证明;然后将它们与其他已知的一种计算不动点指数的方法结合起来,便可产生新的不动点定理;最后,用这些不动点定理来证明一个微分方程的两点边值问题的多解定理及固有值存在性定理。

In the present paper, first, we give the condition eguivalent with the condition of Lemma 4.9 and 4.10 in the book [1], and then obtain Theorem 1.1 and 1.2 in this paper which are more convenient for the application, and that give a new proof of this result. Second, we prove a number of fixed point theorems using the above results for the nonlinear compact operator. Finally, tve give the application of these results in the boundary value problem of the eguation x' + f(x) =0, and so obtain the multi-solutions theorems for this problem and the exist- ence theorems of the eigenvalue prblem for this boundary value problem.