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: GENERIC CONVERGENCE OF ITERATES FOR A CLASS OF NONLINEAR MAPPINGS | GENERIC CONVERGENCE OF ITERATES FOR A CLASS OF NONLINEAR MAPPINGS SIMEON REICH AND ALEXANDER J. ZASLAVSKI Received 4 March 2004 Let K be a nonempty bounded closed and convex subset of a Banach space. We show that the iterates of a typical element in the sense of Baire s categories of a class of continuous self-mappings of K converge uniformly on K to the unique fixed point of this typical element. 1. Introduction Let K be a nonempty bounded closed and convex subset of a Banach space X II II . We consider the topological subspace K c X with the relative topology induced by the norm II II. Set diam K sup x - y II x y G K . Denote by A the set of all continuous mappings A K - K which have the following property P1 for each e 0 there exists xe G K such that 11 Ax xe I1 I x xe I1 E V x G K. For each A B G set d A B sup Ax Bx x G K . Clearly the metric space d is complete. In this paper we use the concept of porosity 1 2 3 4 5 6 which we now recall. Let Y p be a complete metric space. We denote by B y r the closed ball of center y G Y and radius r 0. A subset E c Y is called porous in Y p if there exist a G 0 1 and r0 0 such that for each r G 0 r0 and each y G Y there exists z G Y for which B z ar c B y r E. Copyright 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004 3 2004 211-220 2000 Mathematics Subject Classification 47H09 47H10 54E50 54E52 URL http S1687182004403015 212 Generic convergence A subset of the space Y is called Ơ-porous in Y p if it is a countable union of porous subsets in Y p . Since porous sets are nowhere dense all Ơ-porous sets are of the first category. If Y is a finite-dimensional Euclidean space R then Ơ-porous sets are of Lebesgue measure 0. To point out the difference between porous and nowhere dense sets note that if E c Y is nowhere dense y e Y and r 0 then there are a point z e Y and a number s 0 such that B z s c B y r E. If however E is also porous then for small enough r we can
