Existence of a Continuous Solution of Parametric Nonlinear Equation with Constraints

Antoine and Zouaki ([1] and [16]) have given the existence of a continuous selection of (1.1) in the particular case where M − x0 and N − y0 are closed convex cones . In this paper, we extend this result when M (resp. N) approximates continuously its Clarke tangent cone at x0 (resp. y0) (see Definition 2.2). This property is verified at x0 for an important class of sets. We get among others: sets x0 + K, where K is a closed cone (not necessarily convex); graphs of strictly differentiable functions with respect to the first component of x0; sets locally convex at x0 in finite dimension.