Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-23T12:55:49.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Upper and lower bounds for solutions of linear operator problems with unilateral constraints*

Published online by Cambridge University Press:  14 February 2012

W. D. Collins
W. D. Collins
Affiliation:
Mathematics Research Centre, University of Wisconsin, and Department of Applied Mathematics and Computing Science, University of Sheffield
Get access Rights & Permissions [Opens in a new window]

Synopsis

Dual extremum principles characterising the solutions of problems for a positive-definite self-adjoint operator on a Hilbert space which involve unilateral constraints are formulated using a Hilbert space decomposition theorem due to Moreau. Various upper and lower bounds to these solutions are then obtained, these bounds involving the solutions to subsidiary problems with less restrictive conditions than the solution to the original problem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 76 , Issue 2 , 1977 , pp. 95 - 105
Copyright
Copyright © Royal Society of Edinburgh 1977

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

1Kato, T.. On some approximate methods concerning the operators T*T. Math. Ann. 126 (1953), 253–262.CrossRefGoogle Scholar
2Kato, T.. Perturbation theory for linear operators, 275–276 (New York: Springer Verlag, 1966).Google Scholar
3Fujita, H.. Contribution to the theory of upper and lower bounds in boundary value problems. J. Phys. Soc. Japan 10 (1955), 1–8.CrossRefGoogle Scholar
4Noble, B. and Sewell, M. J.. On dual extremum principles in applied mathematics. Inst. Math. Appl. 9 (1972), 123–193.CrossRefGoogle Scholar
5Courant, R. and Hilbert, D.. Methods of mathematical physics, I, 252–257 (New York: Interscience, 1953).Google Scholar
6Moreau, J. J.. Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires. C. R. Acad. Sci. Paris 255 (1962), 238–240.Google Scholar
7Moreau, J. J.. Convexity and duality. In Functional analysis and optimization (Ed. Caianiello, E. R.), 145–169 (New York: Academic Press, 1966).Google Scholar
8Moreau, J. J.. One-sided constraints in hydrodynamics. In Nonlinear programming (Ed. Abadie, J.), 259–279 (New York: Wiley, 1967).Google Scholar
9Luenberger, D. G.. Optimization by vector space methods (New York: Wiley, 1969).Google Scholar
10Fichera, G.. Boundary value problems of elasticity with unilateral constraints. In Handbuch der Physik VIa/2, 391–424 (Berlin: Springer Verlag, 1972).Google Scholar
11Sewell, M. J.. On dual approximation principles and optimization in continuum mechanics. Philos. Trans. Roy. Soc. London Ser. A 265 (1969), 319–351.Google Scholar
12Sewell, M. J.. The governing equations and extremum principles of elasticity and plasticity generated from a single functional. Struct. Mech. 2 (1973), 1–32, 135.Google Scholar
13Barnsley, M. F. and Robinson, P. D.. Bivariational bounds. Proc. Roy. Soc. London Ser. A 338 (1974), 527–533.Google Scholar
14Cole, R. J. and Pack, D. C.. Some complementary bivariational principles for linear integral equations of Fredholm type. Proc. Roy. Soc. London Ser. A 347 (1975), 239–252.Google Scholar