A Fixed Point Result with a Contractive Iterate at a Point

Badr Alqahtani, Andreea Fulga and Erdal Karapınar
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Badr Alqahtani: Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia
Andreea Fulga: Department of Mathematics and Computer Sciences, Universitatea Transilvania Brasov, 500036 Brasov, Romania
Erdal Karapınar: Department of Medical Research, China Medical University Hospital, China Medical University, 40402 Taichung, Taiwan

Mathematics, 2019, vol. 7, issue 7, 1-12

Abstract: In this manuscript, we define generalized Kincses-Totik type contractions within the context of metric space and consider the existence of a fixed point for such operators. Kincses-Totik type contractions extends the renowned Banach contraction mapping principle in different aspects. First, the continuity condition for the considered mapping is not required. Second, the contraction inequality contains all possible geometrical distances. Third, the contraction inequality is formulated for some iteration of the considered operator, instead of the dealing with the given operator. Fourth and last, the iteration number may vary for each point in the domain of the operator for which we look for a fixed point. Consequently, the proved results generalize the acknowledged results in the field, including the well-known theorems of Seghal, Kincses-Totik, and Banach-Caccioppoli. We present two illustrative examples to support our results. As an application, we consider an Ulam-stability of one of our results.

Keywords: contractive iterate at a point; fixed point; Ulam stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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