Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces

zbMATH Open1266.47097arXiv1001.0368MaRDI QIDQ5300408

A. G. Ramm

Publication date: 27 June 2013

Abstract: Let $F(u)=h$ be an operator equation in a Banach space $X$, $|F^{\prime }(u)-F^{\prime }(v)|leqomega(|u-v|)$, where $omegainC([0,infty))$, $omega(0)=0$, $omega(r)>0$ if $r>0$, $omega(r)$ is strictly growing on $[0,infty)$. Denote $A(u):=F^{\prime }(u)$, where $F^{\prime }(u)$ is the Fr'{e}chet derivative of $F$, and $A_a:=A+aI.$ Assume that (*) $|A_a^{-1}(u)|leqfrac{c}_1|a{|}^b$, $|a|>0$, $b>0$, $ainL$. Here $a$ may be a complex number, and $L$ is a smooth path on the complex $a$-plane, joining the origin and some point on the complex $a-$plane, $0<|a|<epsilo{n}_0$, where $epsilo{n}_0>0$ is a small fixed number, such that for any $ainL$ estimate (*) holds. It is proved that the DSM (Dynamical Systems Method) �ee dot{u}(t)=-A^{-1}_{a(t)}(u(t))[F(u(t))+a(t)u(t)-f],quad u(0)=u_0, dot{u}=frac{d u}{dt}, eee converges to $y$ as $to+infty$, where $a(t)inL,$ $F(y)=f$, $r(t):=|a(t)|$, and $r(t)=c_4(t+c_2{)}^{-c_3}$, where $c_j>0$ are some suitably chosen constants, $j=2,3,4.$ Existence of a solution $y$ to the equation $F(u)=f$ is assumed. It is also assumed that the equation $F(w_a)+a{w}_a-f=0$ is uniquely solvable for any $finX$, $ainL$, and $li{m}_{|a|o0,ainL}|w_a-y|=0.$

Full work available at URL: https://arxiv.org/abs/1001.0368

zbMATH Keywords

Banach spacesnonlinear operator equationsDSM (Dynamical Systems Method)

Mathematics Subject Classification ID

Iterative procedures involving nonlinear operators (47J25) Nonlinear ill-posed problems (47J06) Nonlinear evolution equations (47J35)

Related Items (2)

Title not available (Why is that?) ⋮ Title not available (Why is that?)

This page was built for publication: Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q5300408)