NONLINEAR INITIAL VALUE PROBLEMS WITH p-LAPLACIAN

We study the nonlinear initial value problem consisting of the equation -[p(t)phi)(y')]'+ q(t)phi(y) = w(t)f(y) with phi(y) = vertical bar y vertical bar(r-1)y for r > 0 and the initial conditions y(t(0)) = y(0), (P(1/r)y')(t(0)) = z(0). By establishing nonlinear integral inequalities and applying a generalized energy function and a generalized Prufer transformation, we prove that the solution of this initial value problem exists on the whole domain and is unique. This paper provides a foundation for a forthcoming paper on the existence of nodal solutioins of second order nonlinear boundary value problems with p-Laplacian.