Appl Math Optim 24:35-54 (1991) Applied Mathematics and Optimization © 1991 Springer-Verlag New York Inc. On Implicit Function Theorems for Set-Valued Maps and Their Application to Mathematical Programming under Inclusion Constraints Pham Huy Dien and Nguyen Dong Yen Institute of Mathematics, P.O. Box 631, Bo Ho, 10,000 Hanoi, Vietnam Communicated by J. Stoer Abstract. In this paper we establish some implicit function theorems for a class of locally Lipschitz set-valued maps and then apply them to investigate some questions concerning the stability of optimization problems with inclu- sion constraints. In consequence we have an extension of some of the corresponding results of Robinson, Aubin, and others. O. Introduction Implicit and inverse function theorems form an effective tool for solving many problems of optimization theory. In recent years, optimization problems involving set-valued maps have risen to become a subject that attracts much attention, and it is natural to raise the problem of extending these theorems to the case of set-valued maps. Following the studies of Methlouthi [10], Aubin [2], Aubin and Ekeland [3], Aubin and Frankowska I-4], Pschenichnyi [11] and others, the present paper gives several new results for this and some closely related problems. In the classical setting, implicit function theorems state sufficient conditions for deriving a function from the equation F(x, p)= 0 in some neighborhood of a solution point (x0, Po), i.e., F(xo, Po)= 0. In the case of set-valued maps the problem is restated in an alternative form. Namely, let there be given three linear topological spaces X, Y, P, a set-valued map F from X x P into Y, and closed subsets C c X, D c P. For a point Po ~ D satisfying 0 ~ F(x o, Po) for some Xo e C, we investigate under what condition the set G(p) := {x ~ C/O E F(x, p)} (0.1)