集压缩算子的两点拉伸型不动点定理

A FIXED POINT THEOREM FOR K—SET—CONTRACTIONS OF THE TYPE OF TWO—POINT EXTENSION

著名的上下解方法使用的一个基本条件是方程的下解小于上解。但是,在一系列非线性问题中出现的下解及上解并不满足这一条件,而是满足相反的条件,即下解不小于上解。本文对于严格集压缩算子在下解不小于上解的基本条件下得到了非零不动点的存在性。

Theoreml. Let E be a Banach space and P be a solid cone in E. Let A: P→P be an increasing operator which is a k-set-contraction, k<1. Suppose that (i) there exist x_0,y_0 ∈P, such thatx_0≤Ax_0, Ay_0≤y_0, x_0≤y_0 (ii) there exist u~*∈P, c>0, Such that Ax≥c·‖Ax‖·u~*,(?) x∈P (iii) u~*∈IntP, y_0∈IntP, where IntP expresses the interior of P. Then A has at least one nontrivial fixed point in P.