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Complementary bounds for inner products associated with non-linear equations

Published online by Cambridge University Press:  14 November 2011

R. J. Cole ,
J. Mika  and
D. C. Pack
R. J. Cole
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow, Scotland
J. Mika
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow, Scotland
D. C. Pack
Affiliation:
Department of Mathematics, University of Strathclyde, Glasgow, Scotland
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Synopsis

Functionals are found that give upper and lower bounds to the inner product 〈g0, f〉 involving the unknown solution f of a non-linear equation T[f] = f0, with f∈H, a real Hilbert space, g0 a given function in H and f0 a given function in the range of the non-linear operator T. The method depends upon a re-ordering of terms in the expansion of T[f] about a trial function so as to transfer the non-linearity to a secondary problem that requires its own particular treatment and to enable earlier results obtained for linear operators to be used for the main part. First, bivariational bounds due to Barnsley and Robinson are re-derived. The new and more accurate bounds are given under relaxed assumptions on the operator T by introducing a third approximating function. The results are obtained from identities, thus avoiding some of the conditions imposed by the use of variational methods. The accuracy of the new method is illustrated by applying it to the problem of the heat contained in a bar.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 96 , Issue 1-2 , 1984 , pp. 135 - 142
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

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