Dynamical systems method (DSM) for unbounded operators
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- by A. G. Ramm PDF
- Proc. Amer. Math. Soc. 134 (2006), 1059-1063 Request permission
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.References
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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
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 1059-1063
- MSC (2000): Primary 35R25, 35R30, 37B55, 47H20, 47J05, 49N45, 65M32, 65R30
- DOI: https://doi.org/10.1090/S0002-9939-05-08076-7
- MathSciNet review: 2196039