Fixed Point Results and Their Applications to Markov Processes

A. Tarski proved in his fundamental paper [18] that the set Fix(G) of fixed points of any increasing self-mappingG of a complete lattice is also a complete lattice. Davis completed this work by showing in [3] that a lattice is complete if each of its increasing self-mappings has a fixed point. As a generalization of this result Markowsky proved in [16] that each self-mapping G of a partially ordered set (poset) X has the least fixed point if and only if each chain of X , also the empty chain, has the supremum, and that in such a case each chain of Fix(G) has the supremum in Fix(G) (see also [2]). 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 , then 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]. 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. Results on the structure of the fixed point set are also derived. The proofs are based on a recursion principle introduced in [11]. In [18] existence of common fixed points is also proved for commutative families of increasing self-mappings of a complete lattice X . As for generalizations of these results, see, for example [4, 16, 19]. In Section 3 we derive existence results for common fixed points of a family of mappingsGt : X → X , t ∈ S, where S is a nonempty set, in cases when for some t0 ∈ S results of Section 2 are applicable to G= Gt0 , when (i) GtGt0 = Gt0Gt for each t ∈ S, and when (ii) either Gt0x ≤ Gtx or Gtx ≤ Gt0x for all t ∈ S and x ∈ X . For