Dynamical systems method for solving nonlinear equations with locally Hölder continuous monotone operators (Q988532)

The author studies a new version of the dynamical systems method (DSM) for solving ill-posed nonlinear equations with monotone and locally Hölder continuous operators. The nonlinear equation is \[ F(u)=f \] with \(F\) nonlinear and monotone solvable in a real Hilbert space \(H\). In this paper the following version of the (DSM)is considered: \[ \dot{u}_{\delta}=-(F(u_{\delta})+\alpha(t)u_{\delta}-f_{\delta}),\quad u_{\delta}(0)=u_{0}, \] The main theorem of the paper asserts that if \(\alpha(t)\) fulfils some positivity conditions and if \[ \|F(u_{0})-f_{\delta}\|>C\delta^{\zeta}>\delta\quad C>0, 0>\zeta\leq 1 \] and \(\|f_\delta-f\|\leq\delta.\) Moreover let \(y\) be the minimal norm solution of our equation. Then the solution \(u_{\delta}\) exists globally and there exists a unique \(t_{\delta}\) such that \[ \| F(u_{\delta})(t_{\delta})-f_\delta\|=C\delta^{\zeta},\quad \|F(u_{\delta}(t))-f_\delta\|>C\delta^{\zeta},\quad \forall t\in [0,t_{\delta}). \] If \(\zeta\in (0,1)\) and \(\lim_{\delta\to 0}t_{\delta}=\infty,\) then \(\lim_{\delta\to 0}\|u(t_{\delta})-y\|=0.\) The second theorem yields conditions under which \(\lim_{\delta\to 0}t_{\delta}=\infty\).