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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 Newton-type method, a gradient-type 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 large-time behavior of solutions to evolution equations. Discrete versions of these inequalities are established.

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A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed stopping rule is efficient. Equations with monotone operators are of interest in many applications.

An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A discrete version of the Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with monotone operators is proposed and its convergence is proved. A discrepancy principle is proposed and justified. A priori and a posteriori stopping rules for the iterative scheme are formulated and justified.

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Assume that Au = f (1) is a solvable linear equation in a Hilbert space H, A is a linear, closed, densely defined, unbounded operator in H, which is not boundedly invertible, so problem (1) is ill-posed. It is proved that the closure of the operator (A*A + ffI)-1A*, with the domain D(A*), where ff> 0 is a constant, is a linear bounded everywhere defined operator with norm<= 1 2pff. This result is applied to the variational problem F (u): = ||Au- f || 2 + ff||u||2 =

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It is proved that the class of operator equations F (y) = f solvable by a DSM (Dynamical Systems Method) Newton-type method ˙u = −[F ′ (u) + a(t)I] −1 [F u(t) + a(t)u − f], u(0) = u0, (∗) is large. Here F: X → X is a continuously Fréchet differentiable operator in a Banach space X, a(t) : [0, ∞) → C is a function, limt→ ∞ |a(t) | = 0, and there exists a y ∈ X such that F (y) = f. Under weak assumptions on F and a it is proved that This justifies the DSM (*). ∃!u(t) ∀t ≥ 0; ∃u(∞); F (u(∞)) = f.