The author studies the first-order equations
where is a Carathéodory function.
He proves that if there exist and such that and for a. e. then such problems have extremal solutions in for all and . Here, , for all , and () denotes the set of functions of bounded variation in with nondecreasing (nonincreasing) singular part.
Moreover, if and then, for every and , the problem for a.e. , , , has extremal solutions in .
These results are extended to the discontinuous problem , with a Carathéodory function and such that , .
Finally, some examples of functions that do not satisfy the Carathéodory conditions and for which there is no solution in are presented in the paper.