Fixed point results and their applications to Markov processes
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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. 1. Introduction A. Tarski proved in his fundamental paper [18] that the set Fix(G) of fixed points of any increasing self-mapping G 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 mappings Gt : 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) Gt Gt0 = Gt0 Gt for each t ∈ S, and when (ii) either Gt0 x ≤ Gt x or Gt x ≤ Gt0 x for all t ∈ S and x ∈ X. For Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 307–320 DOI: 10.1155/FPTA.2005.307
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Fixed points and Markov processes
instance, if X is a closed ball in Rm , ordered coordinatewise, a family {Gt }t∈S has a common fixed point if Gt0 is increasing and satisfies (i) and (ii) (cf. Example 3.4). The results of Section 2 can also be applied to prove the existence of increasing selectors for fixed points of an increasing family of increasing mappings (cf. Remarks 3.3 and Example 3.4). The obtained results are then applied in Section 4 to prove existence and comparison results for invariant measures of Markov processes in a partially ordered Polish space
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