In this paper, we introduce the concept of $$\alpha $$ -fractal function and fractal approximation for a set-valued continuous map defined on closed bounded interval real numbers. Also, study some properties such functions. Further, estimate perturbation error between given its function. Additionally, define new graph different from standard introduced in literature establish bounds dimension n...

In this paper, we first present a new important property for Bouligand tangent cone (contingent cone) of a star-shaped set. We then establish optimality conditions for Pareto minima and proper ideal efficiencies in nonsmooth vector optimization problems by means of Bouligand tangent cone of image set, where the objective is generalized cone convex set-valued map, in general real normed spaces.