East Asian Math. J. 24 (2008), No. 4, pp. 369–375 FIXED POINT THEORY FOR INWARD SET VALUED MAPS IN HYPERCONVEX METRIC SPACES A. Amini-Harandi, A. P. Farajzadeh, D. O’Regan, and R. P. Agarwal Abstract. In this paper, we first introduce inwards set valued maps in hyperconvex metric spaces. Then we present fixed point theory for continuous condensing inward set valued maps. 1. Introduction and Preliminaries Let X and Y be topological spaces with A ⊆ X and B ⊆ Y . Let F : X ⊸ Y be a multimap with nonempty values. The image of A under F is the set F (A)= ∪ x∈A F (x) and the inverse image of B under F is F - (B)= {x ∈ X : F (x) ∩ B ̸= ∅}. Now F is said to be: (i) lower semicontinuous if, for each open set B ⊆ Y , F - (B)= {x ∈ X : F (x) ∩ B ̸= ∅} is open in X, (ii) upper semicontinuous, if for each closed set B ⊆ Y , F - (B)= {x ∈ X : F (x) ∩ B ̸= ∅} is closed in X, (iii) continuous if, F is both lower semicontinuous and upper semicontinu- ous. Let (M,d) be a metric space and B(x, r)= {y ∈ M : d(x, y) ≤ r}, denotes the closed ball with center x and radius r. Let co(A)= ∩ {B ⊆ M : B is a closed ball in M such that A ⊂ B}. If A = co(A), we say that A is admissible subset of M . Note, co(A) is admissible and the intersection of any family of admissible subsets of M is admissible. Note an admissible set is bounded. The following definition of a hyperconvex metric space is due to Aronszajn and Pantichpakdi [3]. Definition 1.1. A metric space (M,d) is said to be a hyperconvex metric space if for any collection of points x α of M and any collection r α of non-negative real numbers with d(x α ,x β ) ≤ r α + r β , we have ∩ α B(x α ,r α ) ̸= ∅. Received December 17, 2007; Accepted July 9, 2008. 2000 Mathematics Subject Classification. 47H10, 54H25. Key words and phrases. Fixed point, best approximation, inward map, hyperconvex met- ric space. c ⃝2008 The Busan Gyeongnam Mathematical Society 369