A New Concept of Fixed Point in Metric and Normed Interval Spaces

• Main Author: Hsien-Chung Wu
• Format: Article
• Language: English
• Published: MDPI AG 2018-10-01
• Series: Mathematics
• Subjects: metric interval space; normed interval space; near fixed point; null set; triangle inequality
• Online Access: https://www.mdpi.com/2227-7390/6/11/219

The main aim of this paper is to propose the concept of so-called near fixed point and establish many types of near fixed point theorems in the set of all bounded and closed intervals in <inline-formula> <math display="inline"> <semantics> <mi mathvariant="double-struck">R</mi> </semantics> </math> </inline-formula>. The concept of null set will be proposed in order to interpret the additive inverse element in the set of all bounded closed intervals. Based on the null set, the concepts of metric interval space and normed interval space are proposed, which are not the conventional metric and normed spaces. The concept of near fixed point is also defined based on the null set. In this case, we shall establish many types of near fixed point theorems in the metric and normed interval spaces.