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

TL;DR

This paper investigates discrete and continuous-time evolution equations involving nonexpansive operators in Banach spaces, with applications to game theory, establishing new formulas and inequalities for their trajectories and asymptotic behaviors.

Contribution

It introduces a new exponential formula and Kobayashi-like inequality for evolution trajectories involving nonexpansive operators, linking continuous and discrete dynamics in game-theoretic contexts.

Findings

Established a new exponential formula for trajectories.

Proved asymptotic equivalence between evolution equations and game value sequences.

Derived a Kobayashi-like inequality for nonexpansive operator trajectories.

Abstract

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 $\Phi(\lambda,x):=\lambda J(\frac{1-\lambda}{\lambda}x)$ for $\lambda\in]0,1]$. 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 $\lambda$-discounted game) satisfies the relation $v_n=\Phi(\frac{1}{n},v_{n-1})$ (resp. $v_\lambda=\Phi(\lambda,v_\lambda)$) where $J$ is the Shapley operator of the game. We study the evolution equation $u'(t)=J(u(t))-u(t)$ as well as associated Eulerian schemes, establishing a new exponential formula and a Kobayashi-like inequality for such trajectories. We prove that the solution of the non-autonomous evolution equation $u'(t)=\Phi(\bm{\lambda}(t),u(t))-u(t)$ has the same asymptotic behavior (even when it diverges) as the sequence $v_n$ (resp. as the family $v_\lambda$) when $\bm{\lambda}(t)=1/t$ (resp. when $\bm{\lambda}(t)$ converges slowly enough to 0).