Stability of Periodic Orbits and Return Trajectories of Continuous Multi-valued Maps on Intervals

Let I be a compact interval of real axis ., and (I, H) be the metric space of all nonempty closed subintervals of I with the Hausdorff metric H and f : I . I be a continuous multi-valued map. Assume that P n = (x0, x1, ..., x n ) is a return trajectory of f and that p . [minP n , maxP n ] with p . f(p). In this paper, we show that if there exist k (. 1) centripetal point pairs of f (relative to p) in {(x i ; xi+1): 0 . i . n . 1} and n = sk + r (0 . r . k . 1), then f has an R-periodic orbit, where R = s + 1 if s is even, and R = s if s is odd and r = 0, and R = s + 2 if s is odd and r > 0. Besides, we also study stability of periodic orbits of continuous multi-valued maps from I to I.