On the Directional Differentiability of the Solution Mapping for a Class of Variational Inequalities of the Second Kind

2017 ◽  
Vol 26 (3) ◽  
pp. 631-642 ◽  
Author(s):  
M. Hintermüller ◽  
T. M. Surowiec
Keyword(s):  
Variational Inequalities ◽  
Directional Differentiability ◽  
Solution Mapping

scholarly journals From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability

2022 ◽  
Vol Volume 3 (Original research articles) ◽  
Author(s):  
Gerd Wachsmuth
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Generalized Equation ◽  
Generalized Equations ◽  
Directional Differentiability ◽  
Solution Map ◽  
Strongly Monotone ◽  
Set Valued Mapping ◽  
Maximally Monotone Set

We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.

Variational and Quasi-Variational Inequalities in Mechanics

2007 ◽  
Author(s):  
Alexander S. Kravchuk ◽  
Pekka J. Neittaanmäki
Keyword(s):  
Variational Inequalities

scholarly journals On a Class of Differential Variational Inequalities in Infinite-Dimensional Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 266 ◽  
Author(s):  
Savin Treanţă
Keyword(s):  
Optimal Control ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Optimal Control Theory ◽  
Infinite Dimensional ◽  
Nash Games ◽  
New Class ◽  
Set Valued Mappings ◽  
Differential Variational Inequalities ◽  
Theoretical Developments

A new class of differential variational inequalities (DVIs), governed by a variational inequality and an evolution equation formulated in infinite-dimensional spaces, is investigated in this paper. More precisely, based on Browder’s result, optimal control theory, measurability of set-valued mappings and the theory of semigroups, we establish that the solution set of DVI is nonempty and compact. In addition, the theoretical developments are accompanied by an application to differential Nash games.

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