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Dynamical systems method for . . .
, 2004
"... Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a ..."
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Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or nonlinear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of wellposed problems as well.
Inequalities for solutions to some nonlinear equations
"... Let F be a nonlinear Frechet differentiable map in a real Hilbert space. Condition sufficient for existence of a solution to the equation F(u) = 0 is given, and a method (dynamical systems method, DSM) to calculate the solution as the limit of the solution to a Cauchy problem is justified under suit ..."
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Let F be a nonlinear Frechet differentiable map in a real Hilbert space. Condition sufficient for existence of a solution to the equation F(u) = 0 is given, and a method (dynamical systems method, DSM) to calculate the solution as the limit of the solution to a Cauchy problem is justified under suitable assumptions.
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"... Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a ..."
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Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′ (u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or nonlinear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of wellposed problems as well. 1
Communications in Nonlinear Science and Numerical Simulation, 9, N2, (2003) Dynamical systems method for solving operator equations ∗
"... Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′(u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a ..."
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Consider an operator equation F (u) = 0 in a real Hilbert space. The problem of solving this equation is illposed if the operator F ′(u) is not boundedly invertible, and wellposed otherwise. A general method, dynamical systems method (DSM) for solving linear and nonlinear illposed problems in a Hilbert space is presented. This method consists of the construction of a nonlinear dynamical system, that is, a Cauchy problem, which has the following properties: 1) it has a global solution, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the original linear or nonlinear problem. New convergence and discretization theorems are obtained. Examples of the applications of this approach are given. The method works for a wide range of wellposed problems as well. 1
unknown title
, 2003
"... Inequalities for solutions to some nonlinear equations ∗† ..."
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