NONLINEAR INITIAL VALUE PROBLEMS WITH ρ-LAPLACIAN

摘要

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-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 ρ-Laplacian.