Suppose that L(x) is a
differential operator and R(t) a continuous function, and consider the differential
equation (∗) L(αj) = R(t). Then a problem in approximation theory is whether we
can approximate a solution x(t) of (∗) uniformly with a sequence of polynomials Pn
for which we have ∥R(t) − L(Pn)∥≦ ηn, where ∥⋅∥ is a certain norm and ηn a
specific sequence of nonnegative constants. This is done here for a first order
nonlinear differential operator L and for two different norms, the uniform norm and
the Lp norm (1 ≦ p < +∞).