Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học được đăng trên tạp chí toán học quốc tế đề tài: Maximal and minimal point theorems and Caristi’s fixed point theorem | Li and Jiang Fixed Point Theory and Applications 2011 2011 103 http content 2011 1 103 Fixed Point Theory and Applications a SpringerOpen Journal RESEARCH Open Access Maximal and minimal point theorems and Caristi s fixed point theorem Zhilong Li and Shujun Jiang Correspondence lzl771218@sina. com Department of Mathematics Jiangxi University of Finance and Economics Nanchang Jiangxi 330013 China Springer Abstract This study is concerned with the existence of fixed points of Caristi-type mappings motivated by a problem stated by Kirk. First several existence theorems of maximal and minimal points are established. By using them some generalized Caristi s fixed point theorems are proved which improve Caristi s fixed point theorem and the results in the studies of Jachymski Feng and Liu Khamsi and Li. MSC 2010 06A06 47H10. Keywords maximal and minimal point Caristi s fixed point theorem Caristi-type mapping partial order 1 Introduction In the past decades Caristi s fixed point theorem has been generalized and extended in several directions and the proofs given for Caristi s result varied and used different techniques we refer the readers to 1-15 . Recall that T X X is said to be a Caristi-type mapping 14 provided that there exists a function h 0 0 and a function ộ X - such that n d x Tx ự x p Tx V x e X where X d is a complete metric space. Let 4 be a relationship defined on X as follows x 4 y n d x y ự x ự y V x y e X. 1 Clearly x 4 Tx for each x e X provided that T is a Caristi-type mapping. Therefore the existence of fixed points of Caristi-type mappings is equivalent to the existence of maximal point of X 4 . Assume that h is a continuous nondecreasing and subadditive function with h-1 0 0 then the relationship defined by 1 is a partial order on X. Feng and Liu 12 proved each Caristi-type mapping has a fixed point by investigating the existence of maximal point of X 4 provided that Ộ is lower semicontinu-ous and bounded .
