Stationary Points of Set-valued Contractive and Nonexpansive Mappings on Ultrametric Spaces Meraj Hosseini 1 , Kourosh Nourouzi 1* , Donal O’Regan 2 August 21, 2015 1 Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran 2 Department of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Ireland Abstract In this paper we show that contractive set-valued mappings on spher- ically complete ultrametric spaces have stationary (or end) points if they have the approximate stationary point property. We also extend some known fixed point results to nonexpansive set-valued mappings. 1 Introduction and Preliminaries An ultrametric space (X, d) is said to be spherically complete if every shrinking collection of balls in X has nonempty intersection. Let CB(X) be the set of all closed and bounded subsets of X, T : X → CB(X) be a set-valued mapping, and x, y ∈ X be two distinct points. If H(Tx,Ty) <d(x, y), then T is called contractive, and if H(Tx,Ty) ≤ d(x, y), then T is called nonexpansive; here H(·, ·) is Hausdorff distance. A point x ∈ X is said to be a fixed point of T if x ∈ Tx and a stationary point (or end point) * Corresponding author E-mail addresses: meraj.hy@gmail.com (Meraj Hosseini); nourouzi@kntu.ac.ir (Kourosh Nourouzi); donal.oregan@nuigalway.ie (Donal O’Regan) 2010 Mathematics Subject Classification. 47H09, 54C60 Key words and Phrases: Stationary point, Fixed point, Approximate stationary point property, Ultrametric spaces. 1