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: A NOTE ON WELL-POSED NULL AND FIXED POINT PROBLEMS | A NOTE ON WELL-POSED NULL AND FIXED POINT PROBLEMS SIMEON REICH AND ALEXANDER J. ZASLAVSKI Received 16 October 2004 We establish generic well-posedness of certain null and fixed point problems for ordered Banach space-valued continuous mappings. The notion of well-posedness is of great importance in many areas of mathematics and its applications. In this note we consider two complete metric spaces of continuous mappings and establish generic well-posedness of certain null and fixed point problems Theorems 1 and 2 resp. . Our results are a consequence of the variational principle established in 2 . For other recent results concerning the well-posedness of fixed point problems see 1 3 . Let X II II be a Banach space ordered by a closed convex cone X x e X x 0 such that x IIy II for each pair of points x y e X satisfying x y. Let K p be a complete metric space. Denote by M the set of all continuous mappings A K - X. We equip the set M with the uniformity determined by the following base E e A B e M X M Ax- Bx e Vx e K 1 where e 0. It is not difficult to see that this uniform space is metrizable by a metric d and complete. Denote by Mp the set of all A e M such that Ax e X Vx e K inf Ax x e K 0. 2 It is not difficult to see that Mp is a closed subset of M d . We can now state and prove our first result. Theorem 1. There exists an everywhere dense Gg subset c Mp such that for each A e the following properties hold. 1 There is a unique x e K such that Ax 0. 2 For any e 0 there exist g 0 and a neighborhood U of A in Mp such that if B e U and ifx e K satisfies Bxh g then p x x e. Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005 2 2005 207-211 DOI 208 Well-posed problems Proof. We obtain this theorem as a realization of the variational principle established in 2 Theorem with fA x Ax x e K .In order to prove our theorem by using this variational principle we need to prove the following assertion. A For each A e
