Extrema of Differentiable Functions
- 585 Downloads
In this chapter we would like to investigate how one can calculate, for a variational problem, a minimising point whose existence and possibly also whose uniqueness (in the sense of Chap. 1) have been established. It is not possible, with the theorems we proved in Chap. 1 for the existence of an extremum of a functional f: M → ℝ, (where M is an open subset of a Banach space E), to find those points at which this functional attains, for example, its minimum. However, it is possible to do this with the aid of differential calculus in Banach spaces, which we introduced in Chap. 2, taking the approach used in the well-known case of differentiable functions on the real axis.
KeywordsBanach Space Open Subset Extremal Point Differentiable Function Real Hilbert Space
Unable to display preview. Download preview PDF.
- Courant, R., Hilbert, D.: Methods of mathematical physics. Wiley-Interscience, New York 1966Google Scholar
- Young, L.C.: Calculus of variation and optimal control theory. Saunders, Philadelphia 1969Google Scholar
- Gelfand, I.M., Fomin, S.V.: Calculus of variations. Prentice-Hall, Englewood Cliffs 1963Google Scholar
- Morse, M.: The calculus of variations in the large. Am. Math. Soc., Providence 1934Google Scholar