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How large is the class of operator equations solvable by a DSM Newtontype method?
"... It is proved that the class of operator equations F (y) = f solvable by a DSM (Dynamical Systems Method) Newtontype 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, ∞) ..."
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It is proved that the class of operator equations F (y) = f solvable by a DSM (Dynamical Systems Method) Newtontype 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.
Dynamical systems gradient method for solving illconditioned linear algebraic systems
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EXISTENCE OF SOLUTION TO AN EVOLUTION EQUATION AND A JUSTIFICATION OF THE DSM FOR EQUATIONS WITH MONOTONE OPERATORS
"... An evolution equation, arising in the study of the Dynamical Systems Method (DSM) for solving equations with monotone operators, is studied in this paper. The evolution equation is a continuous analog of the regularized Newton method for solving illposed problems with monotone nonlinear operators ..."
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An evolution equation, arising in the study of the Dynamical Systems Method (DSM) for solving equations with monotone operators, is studied in this paper. The evolution equation is a continuous analog of the regularized Newton method for solving illposed problems with monotone nonlinear operators F. Local and global existence of the unique solution to this evolution equation are proved, apparently for the firs time, under the only assumption that F ′ (u) exists and is continuous with respect to u. The earlier published results required more smoothness of F. The Dynamical Systems method (DSM) for solving equations F (u) = 0 with monotone Fréchet differentiable operator F is justified under the above assumption apparently for the first time.
Stability of Solutions to Some Evolution Problems
, 2011
"... Abstract. Large time behavior of solutions to abstract differential equations is studied. The corresponding 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+: = [0, ∞), A(t) is a linear dt dissipative operator: Re(A( ..."
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Abstract. Large time behavior of solutions to abstract differential equations is studied. The corresponding 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+: = [0, ∞), A(t) is a linear dt dissipative operator: Re(A(t)u, u) ≤ −γ(t)(u, u), γ(t) ≥ 0, F (t, u) is a nonlinear operator, ‖F (t, u) ‖ ≤ c0‖u ‖ p, p> 1, c0, p are constants, ‖b(t) ‖ ≤ β(t), β(t) ≥ 0 is a continuous function. Sufficient conditions are given for the solution u(t) to problem (*) to exist for all t ≥ 0, to be bounded uniformly on R+, and a bound on ‖u(t) ‖ is given. This bound implies the relation limt→ ∞ ‖u(t) ‖ = 0 under suitable conditions on γ(t) and β(t). The basic technical tool in this work is the following nonlinear inequality: ˙g(t) ≤ −γ(t)g(t) + α(t, g(t)) + β(t), t ≥ 0; g(0) = g0, which holds on any interval [0, T) on which g(t) ≥ 0 exists and has bounded derivative from the right, ˙g(t): = lims→+0 g(t+s)−g(t) s. It is assumed that γ(t), and β(t) are realvalued, continuous functions of t, defined on R+: = [0, ∞), the function α(t, g) is defined for all t ∈ R+, locally Lipschitz with respect to g uniformly with respect to t on any compact subsets [0, T], T < ∞.If there exists a function µ(t)> 0, µ(t) ∈ C 1 (R+), such that α t, 1 + β(t) ≤ µ(t) 1 γ(t) − µ(t)
A justification of the Dynamical Systems Method (DSM) for global homeomorphisms
"... The Dynamical Systems Method (DSM) is justified for solving operator equations F (u) = f, where F is a nonlinear operator in a Hilbert space H. It is assumed that F is a global homeomorphism of H onto H, that F ∈ C 1 loc, that is, it has a continuous with respect to u Fréchet derivative F ′ (u), th ..."
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The Dynamical Systems Method (DSM) is justified for solving operator equations F (u) = f, where F is a nonlinear operator in a Hilbert space H. It is assumed that F is a global homeomorphism of H onto H, that F ∈ C 1 loc, that is, it has a continuous with respect to u Fréchet derivative F ′ (u), that the operator [F ′ (u)] −1 exists for all u ∈ H and is bounded, [F ′ (u)] −1   ≤ m(u), where m(u)> 0 is a constant, depending on u, and not necessarily uniformly bounded with respect to u. It is proved under these assumptions that the continuous analog of the Newton’s method ˙u = −[F ′ (u)] −1 (F (u) − f), u(0) = u0, converges strongly to the solution of the equation F (u) = f for any f ∈ H and any u0 ∈ H. The case when F is not a global homeomorphism but a monotone operator in H is also considered.
On the DSM Newtontype method
"... A wide class of the operator equations F (u) = h in a Hilbert space is studied. Convergence of a Dynamical Systems Method (DSM), based on the continuous analog of the Newton method, is proved without any smoothness assumptions on the F ′ (u). It is assumed that F ′ (u) depends on u continuously. Ex ..."
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A wide class of the operator equations F (u) = h in a Hilbert space is studied. Convergence of a Dynamical Systems Method (DSM), based on the continuous analog of the Newton method, is proved without any smoothness assumptions on the F ′ (u). It is assumed that F ′ (u) depends on u continuously. Existence and uniqueness of the solution to evolution equation ˙u(t) = −[F ′ (u(t))] −1 (F (u(t)) − h), u(0) = u0, is proved without assuming that F ′ (u) satisfies the Lipschitz condition. The method of the inverse function theorem.
Dynamical Systems Method (DSM) for solving equations with monotone operators without smoothness assumptions on F'(u)
 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
, 2010
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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 → ∞, ..."
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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.
Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces
"... Let F (u) = h be an operator equation in a Banach space X with Gateaux differentiable norm, ‖F ′ (u) − F ′ (v) ‖ ≤ ω(‖u − v‖), where ω ∈ C([0, ∞)), ω(0) = 0, ω(r) is strictly growing on [0, ∞). Denote A(u):= F ′ (u), where F ′ (u) is the Fréchet derivative of F, and Aa: = A + aI. Assume that (*) ..."
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Let F (u) = h be an operator equation in a Banach space X with Gateaux differentiable norm, ‖F ′ (u) − F ′ (v) ‖ ≤ ω(‖u − v‖), where ω ∈ C([0, ∞)), ω(0) = 0, ω(r) is strictly growing on [0, ∞). Denote A(u):= F ′ (u), where F ′ (u) is the Fréchet derivative of F, and Aa: = A + aI. Assume that (*) ‖A −1 a (u) ‖ ≤ c1 a  b, a > 0, b> 0, a ∈ L. Here a may be a complex number, and L is a smooth path on the complex aplane, joining the origin and some point on the complex a−plane, 0 < a  < ɛ0, where ɛ0> 0 is a small fixed number, such that for any a ∈ L estimate (*) holds. It is proved that the DSM (Dynamical Systems Method) ˙u(t) = −A −1 a(t) (u(t))[F (u(t)) + a(t)u(t) − f], du u(0) = u0, ˙u = dt, converges to y as t → +∞, where a(t) ∈ L, F (y) = f, r(t): = a(t), and r(t) = c4(t + c2) −c3, where cj> 0 are some suitably chosen constants, j = 2, 3, 4. Existence of a solution y to the equation F (u) = f is assumed. It is also assumed that the equation F (wa) + awa − f = 0 is uniquely solvable for any f ∈ X, a ∈ L, and lima→0,a∈L ‖wa − y ‖ = 0.