On the two-point problem for implicit second-order ordinary differential equations

Given a nonempty set Y subset of R-n. Rn and a function f : [a, b] x R-n x R-n xY -> R, we are interested in the problem of finding u. W-2,W-p([a, b], R-n) such that {f(t, u(t), u'(t), u"(t)) = 0 for a.e.t. [a, b], u(a) = u(b) = 0(Rn). We prove an existence result where, for any fixed (t, y) is an element of [a, b] x Y, the function f (t, center dot, center dot, y) can be discontinuous even at all points (x, z) is an element of R-n x R-n. The function f (t, x, z, center dot) is only assumed to be continuous and locally nonconstant. We also show how the same approach can be applied to the implicit integral equation f(t, integral(b)(a) g(t, z)u(z) dz, u(t))=0. We prove an existence result (with f (t, x, y) discontinuous in x and continuous and locally nonconstant in y) which extends and improves in several directions some recent results in the field.