Rotation Numbers And ?-bouned Solutions For Quasi-Periocally Forced Monotone Recurrence Relations

We study the existence of solutions with rotation number p for quasi-periodically forced monotone recurrence relations,and the existence of p-bounded orbits for quasi-periodically forced continuous circle maps with degree 1.We firstly explore the theory of supersolutions and subsolutions for quasi-periodically forced monotone recurrence relations.Then using the theory of supersolutions and subsotions,we prove that if?0<?1,and a quasi-periodically forced monotone recurrence relation has a strict supersolution x and a strict subsolution x with exchanging rotation numbers ?0 and?1,then(?)??(?0,?1),there is a solution with rotation number ? and it is ?-bounded In particular,if p is in the interior of the rotation interval of a quasi-periodically forced continious circle map with degree 1,then there exists a ?-bounded orbit with rotation number ?.