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Sufficient conditions for a problem of Mayer in the calculus of variations

1933; American Mathematical Society; Volume: 35; Issue: 1 Linguagem: Inglês

10.1090/s0002-9947-1933-1501685-1

1088-6850

Gilbert Ames Bliss, Magnus R. Hestenes,

Analytic and geometric function theory

and it is clear at once that the problem of Mayer is a problem of Bolza having f_=O. Sufficient conditions for the problem of Bolza have been established by Morse (XI, p. 528) and Bliss (XII, p. 271). However the hypotheses which they make, in particular that of normality on every sub-interval, imply that the function f is not identically zero, and the sets of sufficient conditions established by them are therefore not applicable to the problem of Mayer without further modification. In view of this fact it is the purpose of the authors of the present paper to establish a set of sufficient conditions for the problem of Mayer with variable end points. This will be done in two parts, the first of which is the paper here presented, dealing only with the special case in which the number of end conditions t', = 0 is exactly 2n + 1. By methods similar to those used by Bliss for the problem of Bolza (XII, pp. 261-274) the results obtained will be extended to the general case in a second paper by Hestenes.