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.
Highlights
The existence of fixed points of Caristi-type mappings is equivalent to the existence of maximal point of (X, ≼)
The additivity of h appearing in [12] guarantees that the relationship ≼ defined by (1) is a partial order on X
Khamsi [13] removed the additivity of h by introducing a partial order on Q as follows x ∗y ⇔ cd(x, y) ≤ φ(x) − φ(y), ∀x, y ∈ Q, where
Summary
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 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. Assume that h is a continuous, nondecreasing, and subadditive function with h-1({0}) = {0}, 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, ≼) provided that is lower semicontinuous and bounded below. Assume that is lower semicontinuous and bounded below, h is continuous and nondecreasing, and there exists δ. We obtained some fixed point theorems of Caristi-type mappings in a partially ordered complete metric space without the lower semicontinuity of and the condition (H)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
