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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: FIXED POINT THEOREMS IN CAT(0) SPACES AND R-TREES | FIXED POINT THEOREMS IN CAT 0 SPACES AND R-TREES W. A. KIRK Received 10 June 2004 We showthat if U is abounded open set in a complete CAT 0 spaceX and if f U X is nonexpansive then f always has a fixed point if there exists p e U such that x e p f x for all x e dU. It is also shown that if K is a geodesically bounded closed convex subset of a complete R-tree with int K 0 and if f K X is a continuous mapping for which x e p f x for some p e int K and all x e dK then f has a fixed point. It is also noted that a geodesically bounded complete R-tree has the fixed point property for continuous mappings. These latter results are used to obtain variants of the classical fixed edge theorem in graph theory. 1. Introduction A metric space X is said to be a CAT 0 space the term is due to M. Gromov see e.g. 1 page 159 if it is geodesically connected and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. It is well known that any complete simply connected Riemannian manifold having nonpositive sectional curvature is a CAT 0 space. Other examples include the classical hyperbolic spaces Euclidean buildings see 2 the complex Hilbert ball with a hyperbolic metric see 6 also 12 inequality 4.3 and subsequent comments and many others. On the other hand if a Banach space is a CAT k space for some K e R then it is necessarily a Hilbert space and CAT 0 . For a thorough discussion of these spaces and of the fundamental role they play in geometry see Bridson and Haefliger 1 . Burago et al. 3 present a somewhat more elementary treatment and Gromov 8 a deeper study. In this paper it is shown that if U is a bounded open set in a complete CAT 0 space X and if f U X is nonexpansive then f always has a fixed point if there exists p e U such that x e p f x for all x e dU. In a Banach space this condition is equivalent to the classical Leray-Schauder boundary condition f x - p A x - p for x e dU and A 1. It is then shown that boundedness of U can
