Some approximate fixed point theorems without continuity of the operator using auxiliary functions

• Main Authors: Sumit Chandok, Arslan Hojjat Ansari, Tulsi Dass Narang
• Format: Article
• Language: English
• Published: Institute of Mathematics of the Czech Academy of Science 2019-10-01
• Series: Mathematica Bohemica
• Subjects: $\varepsilon$-fixed point; $\alpha$-admissible mapping; partial generalized convex contraction of order $4$ and rank $4$; $\alpha$-complete metric space
• Online Access: http://mb.math.cas.cz/full/144/3/mb144_3_3.pdf

We introduce partial generalized convex contractions of order $4$ and rank $4$ using some auxiliary functions. We present some results on approximate fixed points and fixed points for such class of mappings having no continuity condition in $\alpha$-complete metric spaces and $\mu$-complete metric spaces. Also, as an application, some fixed point results in a metric space endowed with a binary relation and some approximate fixed point results in a metric space endowed with a graph have been obtained. Some examples are also provided to illustrate the main results and to show the usability of the given hypotheses.