非连续混合单调算子的耦合不动点定理

THEOREMS FOR COUPLED FIXED POINTS OF DISCONTINUOUS MIXED MONOTONE OPERATORS

在半序拓扑空间内获得了非连续混合单调算子的耦合不动点定理：定理２设Ｘ是半序拓扑空间，Ｍ是Ｘ中的闭集，Ａ：Ｍ×Ｍ→Ｘ是混合单调算子，又设（ⅰ）Ｍ的每一个全序子集都是相对紧的；（ⅱ）存在（ｘ０，ｙ０）∈Ｍ×Ｍ使得ｘ０≤Ａ（ｘ０，ｙ０），Ａ（ｙ０，ｘ０）≤ｙ０；则Ａ在Ｍ×Ｍ中必有耦合不动点．还给出了它在Ｂａｎａｃｈ空间常微分方程中的应用．

The coupled fixed point theorems of discontinuous mixed monotone operators are obtained:Theorem 2 Let M be a closed subset of a partially ordered topological space X. Suppose that A:M ×M→X is a mixed monotone operator and the follwing conditions hold:(ⅰ) Each totally ordered subset of M is relatively compact;(ⅱ) There exists (x0, y0) ∈ M × M suchthat x0≤A(x0, y0), and A(y0, x0)≤y0 ;Then A has the coupled fixed point in M × M.The application to differential equation in a real Banach space is also given out.