The Prederivatives Of γ-paraconvex Set-valued Mappings And Its Applications In Set-valued Optimization

Prederivatives play an important role in the research of nonconvex optimiza-tion problems. In this paper, we mainly consider the existence of Holder order pred-erivatives and applications in set-valued optimization problems. Firstly, we intro-duce several kinds of Holder prederivatives. Then we establish existence theorems of prederivatives for γ-paraconvex set-valued mappings in Banach spaces(γ>0), generalized the result of Gaydu, Geoffroy and Marcelin[18] from convex set-valued mappings to γ-paraconvex set-valued mappings, and from finite dimensional s-paces to infinite dimensional spaces. Compared with weak minimal solution and strong minimal solution, minimal solution has more application value in the practi-cal problems. In the final of this paper, in terms of Holder order prederivatives, we discuss the necessary or sufficient conditions of the existence of minimal solution for set-valued optimization problems.