Smoothing Procedure for Lipschitzian Equations and Continuity of Solutions

The paper is devoted to the question of existence of continuous implicit functions for non-smooth equations. The equation f ( x , σ ) = 0 is studied with the unknown x and the parameter σ . We assume that the mapping which defines the equation is locally Lipschitzian in the unknown variable, the parameter belongs to a topological space, and both the unknown variable and the value of the mapping belong to finite-dimensional spaces. Sufficient conditions for the existence of a continuous implicit function in a neighbourhood of a given value of the parameter and conditions for the existence of a continuous implicit function on a given subset of the parameter space are obtained in terms of the Clarke generalized Jacobian. The key tool of this development is the smoothing of the initial equation and application of recent results on solvability of smooth nonlinear equations and a priori solution estimates. Applications to control and optimization are discussed.