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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Dynamical systems method (DSM) for unbounded operators


Author: A. G. Ramm
Journal: Proc. Amer. Math. Soc. 134 (2006), 1059-1063
MSC (2000): Primary 35R25, 35R30, 37B55, 47H20, 47J05, 49N45, 65M32, 65R30
Published electronically: July 20, 2005
MathSciNet review: 2196039
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Abstract: Let $L$ be an unbounded linear operator in a real Hilbert space $H$, a generator of a $C_0$ semigroup, and let $g:H\to H$ be a $C^2_{loc}$nonlinear map. The DSM (dynamical systems method) for solving equation $F(v):=Lv+g(v)=0$ consists of solving the Cauchy problem $\dot {u}=\Phi(t,u)$, $u(0)=u_0$, where $\Phi$ is a suitable operator, and proving that i) $\exists u(t) \quad \forall t>0$, ii) $\exists u(\infty)$, and iii) $F(u(\infty))=0$.

Conditions on $L$ and $g$ are given which allow one to choose $\Phi$ such that i), ii), and iii) hold.


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Additional Information

A. G. Ramm
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602
Email: ramm@math.ksu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08076-7
PII: S 0002-9939(05)08076-7
Keywords: Dynamical systems method, nonlinear operator equations, ill-posed problems
Received by editor(s): February 18, 2004
Received by editor(s) in revised form: October 26, 2004
Published electronically: July 20, 2005
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society