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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.
On the DSM version of Newton’s method
 Eurasian Math. Journ (EMJ
"... Abstract. The DSM (dynamical systems method) version of the Newton’s method is for solving operator equation F (u) = f in Banach spaces is discussed. If F is a global homeomorphism of a Banach space X onto X, that is continuously Fréchet differentiable, and the DSM version of the Newton’s method i ..."
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Abstract. The DSM (dynamical systems method) version of the Newton’s method is for solving operator equation F (u) = f in Banach spaces is discussed. If F is a global homeomorphism of a Banach space X onto X, that is continuously Fréchet differentiable, and the DSM version of the Newton’s method is u ̇ = −[F ′(u)]−1(F (u)− f), u(0) = u0, then it is proved that u(t) exists for all t ≥ 0 and is unique, that there exists u(∞): = limt→ ∞ u(t), and that F (u(∞)) = f. These results are obtained for an arbitrary initial approximation u0. This means that convergence of the DSM version of the Newton’s method is global. The proof is simple, short, and is based on a new idea. If F is not a global homeomorphism, then a similar result is obtained for u0 sufficiently close to y, where F (y) = f and F is a local homeomorphism of a neighborhood of y onto a neighborhood of f. These neighborhoods are specified. 1 Introduction and