Stability analysis of partial differential variational inequalities in Banach spaces

Faming Guo,Yi-Bin Xiao,Wei Li,Stanisław Migórski

doi:10.15388/namc.2020.25.15730

Abstract

In this paper, we study a class of partial differential variational inequalities. A general stability result for the partial differential variational inequality is provided in the case the perturbed parameters are involved in both the nonlinear mapping and the set of constraints. The main tools are theory of semigroups, theory of monotone operators, and variational inequality techniques.

Highlights

Introduction and preliminariesLet X and Y be two real reflexive and separable Banach spaces, X∗ be the dual space of X, and K ⊂ X be a nonempty, compact, and convex set
A general stability result for the partial differential variational inequality is provided in the case the perturbed parameters are involved in both the nonlinear mapping and the set of constraints
We aim to study the following partial differential variational inequality (PDVI) in infinite dimensional space

Summary

Introduction and preliminaries

Let X and Y be two real reflexive and separable Banach spaces, X∗ be the dual space of X, and K ⊂ X be a nonempty, compact, and convex set. Wang et al in [33] studied the upper semicontinuity and continuity properties for the set of Carathéodory weak solution mapping for a differential mixed variational inequality when both the mapping and the constraint set are perturbed by different parameters in finite dimensional spaces. Gwinner in [7] studied the stability of the solution set to linear differential variational inequalities and gave a result on the upper convergence with respect to perturbations http://www.journals.vu.lt/nonlinear-analysis in the data, including perturbations in the associated linear maps and the constraint set in Hilbert spaces. Liu et al in [16] established the existence of solution for a class of partial differential variational inequalities involving nonlocal boundary conditions in infinite Banach spaces by using fixed point theorem for condensing set-valued operators, theory of measure of noncompactness, and the Filippov implicit function lemma.

Main results

An application