On the continuity and differentiability of the implicit functions for generalized equations
On the continuity and differentiability of the implicit functions for generalized equations
Summary: The aim of this paper is to show that existence, continuity and differentiability of the implicit functions can be proved at the same time, using a sequence of successive approximations of a mapping of two variables. The proof from this paper unifies methods used in the study of local stability and sensitivity of the solutions of integral equations [see the paper by \textit{I. A. Rus}, Proceedings of the International Conference and Numerical Computation of Solutions of Nonlinear Systems Modeling Physical Phenomena, Timişoara, 256-270 (1997)], variational inequalities and nonsmooth generalized equations [see the papers by \textit{A.L. Dontchev} and \textit{W.W. Hager}, Math. Oper. Res. 19, 753-768 (1994; Zbl 0835.49019)], \textit{A.L. Dontchev} [Math. Program. 70A, 91-106 (1995; Zbl 0843.49010)], \textit{S.M. Robinson} [Math. Oper. Res. 5, 43-62 (1980; Zbl 0437.90094) and Math. Oper. Res. 16, 292-309 (1991; Zbl 0746.46039)]. We prove the continuous differentiability of the solution mapping in a neighborhood of a fixed parameter \(\lambda_0\).
Derivatives of functions in infinite-dimensional spaces, Variational and other types of inequalities involving nonlinear operators (general), existence, successive approximations, Set-valued operators, implicit functions, continuous differentiability, Continuous and differentiable maps in nonlinear functional analysis, Implicit function theorems; global Newton methods on manifolds
Derivatives of functions in infinite-dimensional spaces, Variational and other types of inequalities involving nonlinear operators (general), existence, successive approximations, Set-valued operators, implicit functions, continuous differentiability, Continuous and differentiable maps in nonlinear functional analysis, Implicit function theorems; global Newton methods on manifolds