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

Let $F(u)=h$ be an operator equation in a Banach space $X$, $\mathrm{\parallel }{F}^{\mathrm{\prime }}(u)-F^{\mathrm{\prime }}(v)\mathrm{\parallel }\le \omega (\mathrm{\parallel }u-v\mathrm{\parallel })$, where $\omega \in C([0,\mathrm{\infty }))$, $\omega (0)=0$, $\omega (r)>0$ if $r>0$, $\omega (r)$ is strictly growing on $[0,\mathrm{\infty })$. Denote $A(u):=F^{\mathrm{\prime }}(u)$, where $F^{\mathrm{\prime }}(u)$ is the Fr'{e}chet derivative of $F$, and $A_a:=A+aI\mathrm{.}$ Assume that () $|A^{-1}_a(u)|\leq \frac{c_1}{|a|^b}$,$|a|>0$,$b>0$,$a\in L$\mathrm{.}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, $0<\mathrm{\mid }a\mathrm{\mid }<{ϵ}_0$, where ${ϵ}_0>0$ 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 $t\to +\mathrm{\infty }$, where $a(t)\in L,$ $F(y)=f$, $r(t):=\mathrm{\mid }a(t)\mathrm{\mid }$, 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 $f\in X$, $a\in L$, and ${\lim }_{\mathrm{\mid }a\mathrm{\mid }\to 0,a\in L}\mathrm{\parallel }{w}_a-y\mathrm{\parallel }=0.$