Báo cáo sinh học: Maximal and minimal point theorems and Caristi's fixed point theorem

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: Maximal and minimal point theorems and Caristis fixed point theorem
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Báo cáo sinh học: "Maximal and minimal point theorems and Caristis fixed point theorem"Fixed Point Theory andApplications This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Maximal and minimal point theorems and Caristis fixed point theorem Fixed Point Theory and Applications 2011, 2011:103 doi:10.1186/1687-1812-2011-103 Zhilong Li (lzl771218@sina.com) Shujun Jiang (jiangshujun_s@yahoo.com.cn) ISSN 1687-1812 Article type Research Submission date 8 August 2011 Acceptance date 21 December 2011 Publication date 21 December 2011 Article URL http://www.fixedpointtheoryandapplications.com/content/2011/1/103 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Fixed Point Theory and Applications go to http://www.fixedpointtheoryandapplications.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Li and Jiang ; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Maximal and minimal pointtheorems and Caristi’s fixed point theorem Zhilong Li∗ and Shujun JiangDepartment of Mathematics, Jiangxi University of Finance and Economics, Nanchang, Jiangxi 330013, China ∗ Corresponding author: lzl771218@sina.com E-mail address: SJ: jiangshujun s@yahoo.com.cn 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 1 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 IntroductionIn the past decades, Caristi’s fixed point theorem has been generalizedand extended in several directions, and the proofs given for Caristi’sresult 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] pro-vided that there exists a function η : [0, +∞) → [0, +∞) and a functionϕ : X → (−∞, +∞) such that η (d(x, T x)) ≤ ϕ(x) − ϕ(T x), ∀ x ∈ X,where (X, d) is a complete metric space. Let be a relationship definedon X as follows(1) x y ⇐⇒ η (d(x, y )) ≤ ϕ(x) − ϕ(y ), ∀ x, y ∈ X.Clearly, x T x for each x ∈ X provided that T is a Caristi-typemapping. Therefore, the existence of fixed points of Caristi-type map- 2pings is equivalent to the existence of maximal point of (X, ). Assumethat η is a continuous, nondecreasing, and subadditive function withη −1 ({0}) = {0}, then the relationship defined by (1) is a partial orderon X . Feng and Liu [12] proved each Caristi-type mapping has a fixedpoint by investigating the existence of maximal point of (X, ) providedthat ϕ is lower semicontinuous and bounded below. The additivity of ηappearing in [12] guarantees that the relationship defined by (1) is apartial order on X . However, if η is not subadditive, then the relation-ship defined by (1) may not be a partial order on X , and consequentlythe method used there becomes invalid. Recently, Khamsi [13] removedthe additivity of η by introducing a partial order on Q as follows x y ⇐⇒ cd(x, y ) ≤ ϕ(x) − ϕ(y ), ∀ x, y ∈ Q, ∗where Q = {x ∈ X : ϕ(x) ≤ inf ϕ(t) + ε} for some ε > 0. Assume ...