Abstract
In this paper, we initiate a new extension of
b
-metric spaces, called controlled metric-like spaces, by changing the condition
[
℘
(
s
,
r
)
=
0
⇔
s
=
r
]
by
[
℘
(
s
,
r
)
=
0
⇒
s
=
r
]
and that means basically we may have a non-zero self-distance. We prove some fixed point theorems which generalize many results in the literature. Also, we present an interesting application to illustrate our results by considering controlled metric-like spaces endowed with a graph.