Introduction

The general ideas which have been developed in the first three chapters are now available to be used in a wider class of problems in applied mathematics than we have so far indicated. We can expect to go beyond the original objective, which was to find upper and lower bounds for the solution value of functional representing, for example, an overall energy expenditure in boundary value problems.

In this chapter we start to explore such extensions.

We begin §4.2 with a discussion of bounds on pre-assigned linear functionals. This is related to the question of pointwise bounds. Some rather different viewpoints in the literature are brought together and generalized in §§4.2(i)–(iii). We carry out some preliminary detailed calculations in §§4.2(iv) and (v). Then we discuss so-called ‘bivariational’ bounds, which require some new hypotheses.

In §§4.3 and 4.4 we give some discussion of bounds for initial value problems. In §4.5 we return briefly to comparison methods, in order to make contact with an approach which has been influential in solid mechanics, where information about a ‘hard’ problem is obtained by comparison with a notional ‘easy’ problem.

The idea of working in a pair of inner product spaces offers some fresh viewpoints in all these contexts, which already have their own substantial literature. The purpose in this chapter is to hint at what may be achieved by the systematic development of a body of detailed and substantial examples.