Geodesic
Dynamical systems method (DSM) for solving nonlinear operator equations in Banach spaces
Eurasian mathematical journal, Tome 3 (2012) no. 1, pp. 86-96

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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) $$ \dot u(t)=-A_{a(t)}^{-1}(u(t))[F(u(t))+a(t)u(t)-f)],\quad u(0)=u_0, $$ converges to $y$ as $t\to+\infty$, for $a(t)$ properly chosen. Here $F(y)=f$, and $\dot u$ denotes the time derivative. 
@article{EMJ_2012_3_1_a6,
     author = {A. G. Ramm},
     title = {Dynamical systems method {(DSM)} for solving nonlinear operator equations in {Banach} spaces},
     journal = {Eurasian mathematical journal},
     pages = {86--96},
     publisher = {mathdoc},
     volume = {3},
     number = {1},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2012_3_1_a6/}
}
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A. G. Ramm. Dynamical systems method (DSM) for solving nonlinear operator equations in Banach spaces. Eurasian mathematical journal, Tome 3 (2012) no. 1, pp. 86-96. http://geodesic.mathdoc.fr/item/EMJ_2012_3_1_a6/