On the existence of unique eigensets of monotone processes
Wei Shen Hsia, Balakrish R. Natarajan · Rocky Mountain Journal of Mathematics · 1982
A Sufficient condition is given to guarantee the existence of a unique eigenset of a monotone process.Then, a special class of monotone processes is proved to have unique eigensets through this condition and the Perron-Frobenius Theorem. Introduction.Rockafellar [4, p. 69, Theorem 4] proved a theorem which provides necessary and sufficient conditions for the existence of unique eigensets of monotone processes.Since those necessary and sufficient conditions must be satisfied by every pair of non-singular monotone sets in P n and P*, it is almost impossible to verify that a certain monotone process actually satisfies these conditions.In this paper, a sufficient condition in a simpler form is given to guarantee the existence of a unique eigenset.This sufficient condition in fact is a modification of Rockafellar's conditions.Then, a special class of monotone processes is proved to have unique eigensets through this modified condition and the Perron-Frobenius Theorem [2].We shall only give the definitions of monotone sets, monotone processes, and eigensets of a monotone process.For more detailed definitions (e.g., positively homogeneous, sub-additive, non-singular, etc.), examples, and properties of monotone processes see [3], [4], and the references therein.DEFINITION 1.1.[4, p. 11].A monotone set of concave type in P n , the nonnegative orthant of R n , is a non-empty closed bounded convex set C such that O^^^eC implies ^eC.A monotone set of convex type is a non-empty closed convex set such that y x ^ y 2 e C implies y 1 e C. DEFINITION 1.2.[4, p. 9].A monotone process of concave type from P n to P m is a nonnegative process T which is positively homogeneous, sub-additive, closed, and satisfies (a) T(x) is a monotone set of concave type for all xeP n , and (b) 0 ^ x 1 ^ x 2 implies T(x x ) g T(x 2 ).