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: Research Article Remarks on Separation of Convex Sets, Fixed-Point Theorem, and Applications in Theory of Linear Operators | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 80987 14 pages doi 2007 80987 Research Article Remarks on Separation of Convex Sets Fixed-Point Theorem and Applications in Theory of Linear Operators Kamal N. Soltanov Received 20 February 2007 Accepted 2 May 2007 Recommended by Simeon Reich Some properties of the linear continuous operator and separation of convex subsets are investigated in this paper and a dual space for a subspace of a reflexive Banach space with a strictly convex norm is constructed. Here also an existence theorem and fixed-point theorem for general mappings are obtained. Moreover certain remarks on the problem of existence of invariant subspaces of a linear continuous operator are given. Copyright 2007 Kamal N. Soltanov. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In this paper the separation of convex sets in a real reflexive Banach space are investigated existence of a fixed-point theorem for a general mapping acting in a Banach space and the obtained results are applied to study certain properties of continuous linear operators. Furthermore here is proved the solvability theorem for an inclusion with sufficiently general mapping. Fixed-point theorems obtained here are some generalizations of results obtained earlier in 1 2 see also 3 . It is known that see 4-6 sufficiently general results about the separation of convex sets are available for the case when the space considered is a finite-dimensional Euclidean space. But if X is infinite-dimensional it is not possible to prove such results since the geometrical characteristics of an infinite-dimensional space essentially differ from those of a finite-dimensional space. Here we prove results about the separation of convex sets in an infinite-dimensional space which
