Dynamical Systems Method for solving nonlinear operator equations in Banach spaces

2012/06/24 by А. Г. Рамм, Ramm, A. G.
Mathematics · #47J05 #47J06 #47J35 #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.1206.5518

openalex publication_date 2012/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F(u)=h be a solvable operator equation in a Banach space X with a Gateaux differentiable norm. Under minimal smoothness assumptions on F, sufficient conditions are given for the validity of the Dynamical Systems Method (DSM) for solving the above operator equation. It is proved that the DSM (Dynamical Systems Method) \bee u(t)=-A^-1_a(t)(u(t))[F(u(t))+a(t)u(t)-f], u(0)=u₀, %u=(d u)/(dt), \eee converges to y as t→ +∞, for a(t) properly chosen. Here F(y)=f, and u denotes the time derivative.

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