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: The fixed point theorems of 1-set-contractive operators in Banach space | Wang Fixed Point Theory and Applications 2011 2011 15 http content 2011 1 15 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access The fixed point theorems of 1-set-contractive operators in Banach space Shuang Wang Correspondence wangshuang19841119@ School of Mathematical Sciences Yancheng Teachers University Yancheng 224051 Jiangsu PR China SpringerOpen0 Abstract In this paper we obtain some new fixed point theorems and existence theorems of solutions for the equation Ax ụx using properties of strictly convex concave function and theories of topological degree. Our results and methods are different from the corresponding ones announced by many others. MSC 47H09 47H10 Keywords 1-Set-contractive operator Topological degree Convex function Concave function Fixed point theorems 1 Introduction For convenience we first recall the topological degree of 1-set-contractive fields due to Petryshyn 1 . Let E be a real Banach space p e E Q be a bounded open subset of E. Suppose that A E is a 1-set-contractive operator such that I - A x - p s 0 Vx e do. In addition if there exists a k-set-contractive operator k 1 w D E such that Ax - Wx 3 Vx e dD then I - W x p Vx e dD and so it is easy to see that deg I - W D p is well defined and independent of W. Therefore we are led to define the topological degree as follows deg I - A D p deg I - w D p . Without loss of generality we set p 0 in the above definition. Let A E be a 1-set-contractive operator. A is said to be a semi-closed 1-set- contractive operator if I -A is closed operator see 2 . It should be noted that this class of operators as special cases includes completely continuous operators strict set-contractive operators condensing operators semi-compact 1-set-contractive operators and others see 2 . Petryshyn 1 and Nussbaum 3 first introduced the topological degree of 1-set-con-tractive fields studied its basic properties and obtained fixed point .
