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

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

2010-01-03

arxiv.org/abs/1001.0368v1

Mathematics

Dynamical Systems

Scientific paper

Let $F(u)=h$ be an operator equation in a Banach space $X$, $\|F'(u)-F'(v)\|\leq \omega(\|u-v\|)$, where $\omega\in C([0,\infty))$, $\omega(0)=0$, $\omega(r)>0$ if $r>0$, $\omega(r)$ is strictly growing on $[0,\infty)$. Denote $A(u):=F'(u)$, where $F'(u)$ is the Fr\'{e}chet derivative of $F$, and $A_a:=A+aI.$ Assume that (*) $\|A^{-1}_a(u)\|\leq \frac{c_1}{|a|^b}$, $|a|>0$, $b>0$, $a\in L$. 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, $0<|a|<\epsilon_0$, where $\epsilon_0>0$ is a small fixed number, such that for any $a\in L$ estimate (*) holds. It is proved that the DSM (Dynamical Systems Method) \bee \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 $t\to +\infty$, where $a(t)\in L,$ $F(y)=f$, $r(t):=|a(t)|$, 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 $F(w_a)+aw_a-f=0$ is uniquely solvable for any $f\in X$, $a\in L$, and $\lim_{|a|\to 0,a\in L}\|w_a-y\|=0.$

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