Extended partial $b$-metric spaces and some fixed point results

Abstract

In this paper, we introduce the concept of extended partial
$b$-metric space. We demonstrate a fundamental lemma for the
convergence of sequences in such spaces. Then we prove some fixed
point results for weakly contractive mappings in the setup of
ordered extended partial $b$-metric spaces. An example is given to
verify the effectiveness and applicability of our main results. An
application of these results to Volterra-type integral equations
is provided at the end.