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This paper is concerned with
the problem of characterizing sub-(L) functions, where L is the Euler-Lagrange
operator for the functional Icd[y] = ∫
cd[∑
j=0npj(Djy)2], with n a positive integer,
[c,d] a subinterval of a fixed interval [a,b], and p0,p1,⋯,pn continuous real-valued
functions on [a,b] with pn(x) > 0 on this interval. Under certain hypotheses on the
operator L, it is shown that if f is a function in the domain of L on a subinterval
[c,d] of [a,b], then the statement that f is sub-(L) on [c,d] is equivalent to each of
the following conditions: (i) (−1)nLf(x) ≦ 0 on [c,d] (ii) Icd[y] ≧ Icd[f] whenever y is
a function having continuous derivatives of the first n − 1 orders with Dn−1y
having a piecewise continuous derivative on [c,d] such that Dj−1y and Dj−1f
have the same value at c and at d for j in {1,⋯,n}, and y(x) − f(x) ≦ 0 on
[c,d].
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