Fixed points of non-linear set-valued $\left({\alpha }_{\ast },{\varphi }_M\right)$-contraction mappings and related applications

The aim of this manuscript is to prove some fixed point results for non-linear set-valued maps with new approach of $\left({\alpha }_{\ast },{\varphi }_M\right)$-contraction in complete $M$-metric space. Also, we prove some fixed point results in ordered $M$-metric space. As an presented work which are the extension and improves the current study of set-valued mappings. Finally, we also give an non-trivial extensive examples and application to homotopy theory and the existence solution of functional equations to show that our concepts are meaningful and to support our results.