Acta Math. Hungar. 105 (1–2) (2004), 139–143. A FIXED POINT RESULT IN STRICTLY CONVEX BANACH SPACES A. AMINI-HARANDI (Isfahan) Abstract. We define an alternate convexically nonexpansive map T on a bounded, closed, convex subset C of a Banach space X and prove that if X is a strictly convex Banach space and C is a nonempty weakly compact convex subset of X, then every alternate convexically nonexpansive map T : C → C has a fixed point. As its application, we give an existence result for the solution of an integral equation. 1. Introduction Let C be a weakly compact convex subset of a Banach space X . A map- ping T : C → C is said to be nonexpansive if   T (x) − T (y)   ≦ ‖x − y‖ for all x, y ∈ C . By a famous counterexample of Alspach [2], there exists a weakly compact convex subset of L 1 [0, 1] and an isometry T : C → C such that T is fixed point free. The restrictions that must be added on T to assure the existence of at least one fixed point, is one of the most basic questions asked in the study of nonexpansive mappings [1], [5]. Here we introduce a class of nonexpansive mappings which is fundamental in proving our main theorem. Definition 1.1. Let C be a bounded, closed, convex subset of a Banach space X . A mapping T : C → C is called alternate convexically nonexpan- sive if     n  i=1 (−1) i+1 n Tx i − Ty     ≦     n  i=1 (−1) i+1 n x i − y     , for each n ∈ N,x i ,y ∈ C. Definition 1.2. Let (Ω, Σ,μ) be a finite measure space and X a Banach space. We say that a mapping f :Ω × X → X is alternate convexically k- Key words and phrases: fixed point, alternate convexically nonexpansive map, integral equa- tion, strictly convex space. 2000 Mathematics Subject Classification: 47H10, 45E99. 0236–5294/4/$ 20.00 c  2004 Akad´ emiai Kiad´ o, Budapest