For the problem of minimizing a suitable type of integral functionalJ[y] in the class\(\mathfrak{M}\)k of real, monotone nonincreasing functionsy which are Lipschitzian on a compact interval [a, b] with Lipschitz constantk, there is presented an existence theorem and a characterization of minimizing functions as solutions, in the sense of Filippov, of associated differential equations whose members involve discontinuities. For the problem of minimizingJ[y] in the class\(\mathfrak{M}\) of all real, monotone nonincreasing functions for whichJ[y] exists, there is established an existence theorem and proof that, under suitable hypotheses, a solution of this second problem is the limit of solutions of the aforementioned problem ask → ∞. For the particular case in whichJ[y] is the integral of 1/2[y −h(t)]2, whereh(t) is measurable and bounded on [a, b], it is shown that the minimizing function forJ[y] in the class\(\mathfrak{M}\) is the derivative almost everywhere of the least concave majorant of the functionH(t)=∫0th(s)ds, t e [a,b].