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Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces

Guillaume Vigeral (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider some discrete and continuous dynamics in a Banach space involving a non expansive operator J and a corresponding family of strictly contracting operators Φ (λ, x): = λ J( 1 - λ λ x) for λ ∈ ] 0,1] . Our motivation comes from the study of two-player zero-sum repeated games, where the value of the n-stage game (resp. the value of the λ-discounted game) satisfies the relation vn = Φ( 1 n , v n - 1 ) (resp.  v λ = Φ(λ, v λ )) where J is the Shapley operator of the game. We study the evolution equation u'(t) =...

Existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces with anti-periodic boundary conditions

Sahbi Boussandel (2018)

Applications of Mathematics

The paper is devoted to the study of the existence of solutions for nonlinear nonmonotone evolution equations in Banach spaces involving anti-periodic boundary conditions. Our approach in this study relies on the theory of monotone and maximal monotone operators combined with the Schaefer fixed-point theorem and the monotonicity method. We apply our abstract results in order to solve a diffusion equation of Kirchhoff type involving the Dirichlet p -Laplace operator.

Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation

Martin Heida (2015)

Applications of Mathematics

We show existence of solutions to two types of generalized anisotropic Cahn-Hilliard problems: In the first case, we assume the mobility to be dependent on the concentration and its gradient, where the system is supplied with dynamic boundary conditions. In the second case, we deal with classical no-flux boundary conditions where the mobility depends on concentration u , gradient of concentration ∇ u and the chemical potential Δ u - s ' ( u ) . The existence is shown using a newly developed generalization of gradient...

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