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 P_n=(x₀, x₁,..., x_n) is a return trajectory of f and that p ∈min P_n, max P_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; x_i+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.