October 15, 2004 13:26 Journal of Difference Equations and Applications FOP Journal of Difference Equations and Applications Vol. , No. , , 1–13 The antipodal mapping theorem and difference equations in Banach spaces † Daniel Franco ‡, Donal O’Regan § and Juan Peran *‡ ‡ Departamento de Matem´ atica Aplicada, Universidad Nacional de Educaci´on a Distancia, Apartado 60149, Madrid 28080, Spain. § Department of Mathematics. National University of Ireland. Galway. Ireland. () We employ the Borsuk-Krasnoselskii antipodal theorem to prove a new fixed point theorem in ordered Banach spaces. Then, the applicability of the result is shown by presenting sufficient conditions for the existence of solutions to initial value problems for first-order difference equations in Banach spaces. To prove that result we shall employ set valued analysis techniques. Keywords : difference equations; antipodal theorem; set valued analysis AMS Subject Classification (2000): 39A05; 47H10 1 Introduction Let E be a Banach space and f a compact map from B = {x ∈E : ‖x‖≤ 1} to E . Theorem 1.1 Borsuk-Krasnoselskii antipodal theorem. Suppose that f has no fixed points on ∂B and that the antipodal condition f (x) − x ‖f (x) − x‖ = f (−x)+ x ‖f (−x)+ x‖ ,x ∈ ∂B is satisfied. Then f has a fixed point in B. The above result has been taken from section 16.3 in [22] and it was first † This research has been supported in part by Ministerio de Ciencia y Tecnolog´ ıa (Spain), project MTM2004-06652-C03-03 * Corresponding author. E-mail: jperan@ind.uned.es