function weighted $mathcal{g}$-metric spaces and hausdorff $delta$-distances; an application to fixed point theory

In this paper, we introduce a new space which is a generalization of function weighted metric space introduced by jleli and samet [on a new generalization of metric spaces, j. fixed point theory appl. 2018, 20:128] where namely function weighted $mathcal{g}$-metric space. also, a hausdorff $delta$-distance is introduced in these spaces. then several fixed point results for both single-valued and multi-valued mappings in such spaces are proved. we also construct some examples for the validity of the given results and present an application to the existence of a solution of the volterra-type integral equation.