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

Let I be a compact interval of real axis ℝ, 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 Pn = (x0, x1, ..., xn) is a return trajectory of f and that p ∈ [minPn, maxPn] with p ∈ f(p). In this paper, we show that if there exist k (≥ 1) centripetal point pairs of f (relative to p) in {(xi; 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.