Dynamical Systems Method (DSM) for general nonlinear equations

• Published in: Nonlinear analysis Vol. 69; no. 7; pp. 1934 - 1940
• Main Author: Ramm, A.G.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 01-10-2008; Elsevier
• Subjects: Approximation; Distributed memory; DSM—dynamical systems method; Dynamical systems; Exact sciences and technology; Global analysis, analysis on manifolds; Iterative methods; Mathematical analysis; Mathematics; Nonlinear equations; Nonlinear operator equations; Nonlinearity; Numerical analysis; Numerical analysis in abstract spaces; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; 47H20; 65H10; DSM—dynamical systems method; Nonlinear operator equations; Iterative methods; 65J15; 47H15; 65N20; Nonlinear analysis; Iterative method; Dynamical system; Non linear equation; DSM-dynamical systems method; Mathematical model
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2007.07.034

If
F
:
H
→
H
is a map in a Hilbert space
H
,
F
∈
C
loc
2
, and there exists
y
such that
F
(
y
)
=
0
,
F
′
(
y
)
≠
0
, then equation
F
(
u
)
=
0
can be solved by a DSM (dynamical systems method). This method yields also a convergent iterative method for finding
y
, and this method converges at the rate of a geometric series. It is not assumed that
y
is the only solution to
F
(
u
)
=
0
. A stable approximation to a solution of the equation
F
(
u
)
=
f
is constructed by a DSM when
f
is unknown but
f
δ
is known, where
‖
f
δ
−
f
‖
≤
δ
.