A new extension to the controlled metric type spaces endowed with a graph
Nabil Mlaiki, Nizar Souayah, Thabet Abdeljawad, Hassen Aydi
Abstract
Abstract In this paper, we initiate a new extension of b -metric spaces, called controlled metric-like spaces, by changing the condition $$ \bigl[\wp (s,r)=0 \Leftrightarrow s=r\bigr]\quad \text{by } \bigl[\wp (s,r)=0 \Rightarrow s=r\bigr] $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>℘</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>⇔</mml:mo> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mi>r</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mspace/> <mml:mtext>by </mml:mtext> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>℘</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>⇒</mml:mo> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mi>r</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> 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.