Asymptotic Periodicity of the Iterates of Positivity Preserving Operators
M. Miklavcic · Transactions of the American Mathematical Society · 1988
Assume that (AI) X is a real Banach space.(A2) X+ is a closed subset of X with the following properties:(i) if xe X+, y € X+, a e [0, oo) then x + y & X+ and ax € X+;(ii) there exists Mo € (0, oo) such that for each x € X there exist x+ e X+ and x_ 6 X+ which satisfy _=_+-__, ||_+|| < Mo||x||, ||x_||<Mo||x|| and if x = j/+ -j/_ for some y+ 6 X+, j/_ _ X+ then y+ -x.|-6 X+;(iii) if x e X+, y e X+ then ||x|| < ||x + y\\.(A3) B is a bounded linear operator on X.(A4) BX+ C X+.(A5) Fq is a nonempty compact subset of X and limn-oo dist(_J"x, Fq) = 0 whenever x 6 X+ and ||x|| = 1.Then Bnx is asymptotically periodic for every x 6 X.This, and other properties of B, are proven in the paper.