# Article

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Keywords:
$\varepsilon$-fixed point; $\alpha$-admissible mapping; partial generalized convex contraction of order $4$ and rank $4$; $\alpha$-complete metric space
Summary:
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.
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