On the DSM Newton-type method
نویسنده
چکیده
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 u̇(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 proof is new. This method is based on a novel version of the abstract inverse function theorem.
منابع مشابه
How large is the class of operator equations solvable by a DSM Newton-type method?
It is proved that the class of operator equations F(y) = f solvable by a DSM (dynamical systems method) Newton-type method, u̇ = −[F (u) + a(t)I]−1[Fu(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 assumpti...
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