Abstract
A general theory is developed for the estimation of linear functionals, in three distinct classes of nonlinear problems. The functional is linear in the solution vector
x
0
of the problem, an example being (
x
)
0
,
p
, where
p
is assignable. The considered problems are all generated via the gradients of some given quadratic or non-quadratic Lagrangian functional over two inner product spaces. This may be a saddle functional, or it may be constructed by embedding a given nonlinear problem with the aid of a Lagrange multiplier. Many different problems in applied mathematics are thereby included. In some cases the assignable coefficient can be chosen in such a way that the bounds calculated for the linear functional are pointwise bounds on the solution vector. In general this requires further investigation, but estimation of the deflexion at a point on a cantilever beam is illustrated in § 6.
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