Results 1 
2 of
2
The Dynamical Systems Method for solving nonlinear equations with monotone operators
"... A review of the authors’s results is given. Several methods are discussed for solving nonlinear equations F(u) = f, where F is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy principle for solving the equation is formulated and justified. V ..."
Abstract

Cited by 15 (12 self)
 Add to MetaCart
A review of the authors’s results is given. Several methods are discussed for solving nonlinear equations F(u) = f, where F is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy principle for solving the equation is formulated and justified. Various versions of the Dynamical Systems Method (DSM) for solving the equation are formulated. These methods consist of a regularized Newtontype method, a gradienttype method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation F(u) = f is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to equation F(u) = f is justified. New nonlinear differential inequalities are derived and applied to a study of largetime behavior of solutions to evolution equations. Discrete versions of these inequalities are established.
Stability of Solutions to Evolution Problems
, 2013
"... Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t → ∞, ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t → ∞, in particular, sufficient conditions for this limit to be zero. The evolution problem is: ˙u = A(t)u + F (t, u) + b(t), t ≥ 0; u(0) = u0. (∗) Here ˙u: = du, u = u(t) ∈ H, H is a Hilbert space, t ∈ R+:= dt [0, ∞), A(t) is a linear dissipative operator: Re(A(t)u, u) ≤ −γ(t)(u, u), where F (t, u) is a nonlinear operator, ‖F (t, u) ‖ ≤ c0‖u‖p, p> 1, c0 and p are positive constants, ‖b(t) ‖ ≤ β(t), and β(t) ≥ 0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The nonclassical case γ(t) ≤ 0 is also treated.