Minimax solutions of ordinary differential systems

The classical existence theorem of Peano states that there is at least one solution1 yi(x), • ■ • , yn(x) of (S) through £ existing in2 £ over some interval Xo — h^x^x0+k, where h, k>0. If for some such interval this solution through £ is not unique, then there are infinitely many solutions through £ and the existence of critical type solutions is a possibility. W. Osgood,3 P. Montel,4 and O. Perron,6 using different methods, considered the case » = 1 and proved the existence of a maximum and a minimum solution. E. Kamke6 gave an example to show that for n > 1 there will not in general be a maximum and a minimum solution through £, but that such solutions do exist provided /?(x, yu • • • , yn), i = l, • • ■ , n, satisfy certain monotone properties with respect to y1( ■ • • , yn. This paper is devoted to establishing the existence of other types of critical solutions and to a consideration of some of their properties.