tailieunhanh - Báo cáo hóa học: " A NOTE ON WELL-POSED NULL AND FIXED POINT PROBLEMS"

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