NONLINEAR INITIAL VALUE PROBLEMS WITH p-LAPLACIAN

We study the nonlinear initial value problem consisting of the equation − [ p(t) φ(y′ ) ]′ + q(t) φ(y) = w(t) f(y) with φ(y) = |y|^r−1y for r > 0 and the initial conditions y(t₀) = y₀, (p^1/r y′ ) (t₀) = z₀. By establishing nonlinear integral inequalities and applying a generalized energy function and a generalized Pr¨ufer 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.