Topological degree for \((S)_{+}\)-mappings with maximal monotone perturbations and its applications to variational inequalities
This article deals with some generalization of the Browder degree theory for mappings of the type \(T + A: \;X \to X^*\), where \(T\) is a bounded demicontinuous \((S)_{(+)}\)-mapping and \(A\) a maximal monotone operator (\(X\) denoting a reflexive Banach space). The main part of the article is devoted to some revision of the original Browder theory. In particular, the authors study pseudo-monotone homotopies preserving the Browder degree, give a more general formulation for the normalization property, and show that the local degree for the generalized gradient of a nonsmooth functional at the local minimizer is equal to one. As applications, the variational inequalities \[ \int_\Omega | \nabla u| ^{p-2} \nabla u \cdot \nabla(v - u) \, dx \geq \int_\Omega f(x,u)(v - u) \, dx \;\;\forall v \in K, \] \[ K = \{u \in W_0^{1,p}(\Omega): \;| \nabla v| \leq 1 \;\text{ a.e. in } \Omega\}, \] and \[ \int_\Omega \nabla u \cdot \nabla(v - u) \, dx \geq \int_\Omega f(v - u) \, dx \;\;\forall \;v \in K, \qquad K = \{v \in H_0^1(\Omega: \;v \geq 0\;\text{ a.e. in } \Omega\}, \] are considered.
local minimizer of non-smooth functional
scientific article; zbMATH DE number 2117567
scientific article