The existence of an extremum in problems of Mayer

Abstract

one which minimizes a function g [a, b, y(a), y(b) ]. All simple integral problems of the calculus of variations for which a fairly complete theory of the relative extremum problem has been developed may be transformed to a problem of the above type. The theorems we shall prove concerning the existence of an absolute minimum may be translated directly into corresponding existence theorems for the problem of Bolza or for the problem of Lagrange. The problem of Mayer is considered here because it seems notationally simpler than the problem of Bolza. An existence theorem for a complicated problem such as we are considering must naturally impose rather severe restrictions on the involved. In order to treat as wide a variety of cases as possible, the variable functions are divided into groups satisfying different types of conditions. We shall divide them notationally into two groups. The functions yi(x), defined for a ? x < b, will be regarded as independent functions, and the curve they determine in xy-space will be denoted by C. When the functions yi(x) are absolutely continuous, dependent functions z,(x) will be determined by differential equations and initial conditions of the special form