Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-02-06T13:07:14.454Z Has data issue: false hasContentIssue false
  • English
  • Français

9.—Bivariational Bounds associated with Non-self-adjoint Linear Operators

Published online by Cambridge University Press:  14 February 2012

Michael F. Barnsley  and
Peter D. Robinson
Michael F. Barnsley
Affiliation:
School of Mathematics, University of Bradford.
Peter D. Robinson
Affiliation:
School of Mathematics, University of Bradford.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let A be a closed linear transformation from a real Hilbert space ℋ, with symmetric inner product 〈, 〉, into itself; and let f ∈ ℋ be given such that the problem Aø = f has a solution ø ∈ D(A), the domain of A. Then bivariational upper and lower bounds on 〈g, ø〉 for any g ∈ ℋ are exhibited when there exists a positive constant a such that 〈AΦ, AΦ⊖ ≧ a2〈Φ, Φ〉 for all Φ ∈ D(A). The applicability of the theory both to Fredholm integral equations and also to time-dependent diffusion equations is demonstrated.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 75 , Issue 2 , 1976 , pp. 109 - 118
Copyright
Copyright © Royal Society of Edinburgh 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1Barnsley, M. F. and Robinson, P. D.. Bivariational bounds. Proc. Roy. Soc. London Ser. A, 338 (1974), 527–533.Google Scholar
2Barnsley, M. F.. Correction terms for Padé approximants. J. Mathematical Phys. 16 (1975), 918–928.CrossRefGoogle Scholar
3Barnsley, M. F. and Robinson, P. D.. Bivariational bounds on solutions of two-point boundaryvalue problems. J. Math. Anal. & Appl., in press.Google Scholar
4Barnsley, M. F. and Robinson, P. D.. Pointwise accuracy for approximate solutions to integral equations, submitted for publication.Google Scholar
5Barnsley, M. F.. Pointwise bounds for eigenfunctions of one-electron systems. Phys. Lett. A 53 (1975), 124–126.CrossRefGoogle Scholar
6Riesz, F. and Sz-Nagy, B.. Functional Analysis (New York: Ungar, 1965).Google Scholar
7Grant, P. J.. Elementary Reactor Physics (Oxford: Pergamon, 1966).Google Scholar
8Murray, J. C.. A note on a general linear initial-boundary value problem. J. Inst. Maths Appl. 10 (1972), 305–311.Google Scholar