On fixed point results in $\mathcal{F}$-metric spaces with applications

The aim of this research article is to define locally rational contractions concerning control functions of one variable in the background of $\mathcal{F}$-metric spaces and establish common fixed point results. We also introduce ( ${\alpha }^{\ast }$- $\psi )$-contractions and generalized ( ${\alpha }^{\ast }$, $\psi ,{\delta }_{\mathcal{F}})$-contractions in $\mathcal{F}$-metric spaces and obtain fixed points of multifunctions. A non trivial example is also furnished to manifest the originality of the fundamental result. As application, we investigate the solution of nonlinear neutral differential equation.