Decreasing flow invariant sets and fixed points of quasimonotone increasing operators (Q2716110)

This article deals with quasimonotone increasing and potential operators in a Hilbert space \(E\) ordered by a solid cone \(P\); an operator \(A: D(\subseteq E)\to E\) is called quasimonotone increasing if \(x,y\in D\), \(x\leq y\), \(\varphi\in P^*\), \(\varphi(x)= \varphi(y)\) implies \(\varphi(Ax)\leq \varphi(Ay)\). The problem of fixed points for the quasimonotone increasing and potential operator \(A\) is reduced to the analysis of the Cauchy problem NEWLINE\[NEWLINE{dx\over dt}= Ax- x,\quad x(0)= x_0NEWLINE\]NEWLINE with a suitable \(x_0\); in this analysis, decreasing flow invariant sets play the basic rôle. The basis results are a series of theorems of existence of at least one, and five fixed points of \(A\).