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

Abstract

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.

Keywords

• $\varepsilon$-fixed point
• $\alpha$-admissible mapping
• partial generalized convex contraction of order $4$ and rank $4$
• $\alpha$-complete metric space