Abstract
New existence and comparison results are proved for fixed points of increasing operators and for common fixed points of operator families in partially ordered sets. These results are then applied to derive existence and comparison results for invariant measures of Markov processes in a partially ordered Polish space.
Highlights
To meet the demands of our applications to Markov processes we will prove in Section 2 similar fixed point results when the existence of supremums or infimums of {c,x}, x ∈ X, is replaced by weaker hypotheses
If S is a poset if G : X × S → X is increasing with respect to the product ordering of X × S, it follows from Corollary 2.7 that Gt = G(·, t) has the greatest fixed point xt∗ in
Assume there exists a t0 ∈ S such that (P0) x ≤ y in E implies P(t0, x, ·) P(t0, y, ·) in ᏹ; (P1) P(t, x, ·) P(t0, x, ·) for all t ∈ S and x ∈ E; (P2) there exists a closed subset B of E whose monotone sequences converge such that P(t0, x, B) = 1 for each x ∈ E
Summary
In [9, 10] it is shown that if G : X → X is increasing, if nonempty well-ordered (w.o.) and inversely well-ordered (i.w.o.) subsets of G[X] have supremums and infimums in X, and if for some c ∈ X either supremums or infimums of {c,x} exist for each x ∈ X, G has maximal or minimal fixed points, and least or greatest fixed points in certain order intervals of X Applications of these results to operator equations, as well as various types of explicit and implicit differential equations are presented, for example, in [1, 8, 9, 10]. Such results have applications in ergodic theory, in economics and in statistics (see, e.g., [13, 17, 20]).
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