An inverse map result and some applications to sensitivity of generalized equations

Abstract

a b s t r a c tThis work deals with non-global inverse map results for the sum of two maps. We prove two theorems which shed some new light on this aspect. Some implications in terms of sensitivity of parametric generalized equations are investigated. Finally, a class of wellconditioned operators is identified.

Newton's Method for Solving Inclusions Using Set-Valued Approximations

“…(iii) A result closely related to Theorem 3.2 was proved in [9], where the authors concentrate more on the size of the domain of the inverse of the sum. Under stronger assumptions, they obtained a stronger result: the Lipschitz continuity of (Φ + G) −1 .…”

“…(ii) The conclusion of Theorem 3.2 fails (see [12,Example 5E.6] and also [9]) when condition (ii) of Theorem 3.2 on the diameter is omitted.…”

Existence and continuity of solution trajectories of generalized equations with application in electronics

“…Clearly, in this framework for any fixed t ∈ [0, 1] one has a static case problem. This approach is well known and well studied in the literature, both as a pointwise study (see for instance [6,7,18,24] and references therein), or as a numerical method and for designing algorithms (see for example [3,4,11,14,15]). In this paper, instead of looking at the sets S(t), we focus on solution trajectories, functions like x : [0, 1] → R n such that x(t) ∈ S(t), for all t ∈ [0, 1], that is, x(•) is a selection for S over [0, 1].…”

More on the behaviors of fixed points sets of multifunctions and applications

“…Results about the behaviors of fixed points sets have brought in recent years the attention of several authors since they not only can be used to describe the dependence of solutions to differential inclusions or partial differential equations but, without claim of completeness, they reinforce the links among several purposes such as stability, optimal control, well-posedness, sensitivity analysis, generalized differentiation, generalized equations, differential inclusions and optimization, see for example, [8,9,14,15,17,18,25,28,30,32] and the references therein.…”