A New Solution Operator of One-Dimensionalp-Laplacian with a Sign-Changing Weight and Its Application

We establish a new solution operator for the following problem -phi(p)(u')' = g, t is an element of (0, 1), u(0) = 0 = u(1), where phi(p)(x) = vertical bar x vertical bar(p-2)x, p > 1. g may be singular at the boundary or change signs or may not be in L-1(0, 1) so that this solution operator can cover larger class of g than previously known ones. As an application, by checking complete continuity of the solution operator, we show the existence of nontrivial solutions for p-Laplacian phi p(u')' + lambda h(t)f(u(t)) = 0, t is an element of (0, 1), u(0) = 0 = u(1), where lambda > 0 a parameter and f is an element of C(R, R) and f(0) > 0 and h may change signs or may be beyond of L-1(0, 1).