Uniqueness for a class of nonlinear initial value problems

AbstractThis article is concerned with nonnegative solutions of the nonlinear initial value problem x′ = p(t) xx + q(t), t > 0; x(0) = 0. Here, α is a constant, α ϵ (0, 1); p and q are real-valued functions that are integrable and not identically zero near 0, with p(t)⩾0. The function q may assume positive, as well as negative values. The following result is shown. Let q1 denote the integral of q, q1(t) = ∝01q(s) ds, t⩾0. Let x be a solution of the initial value problem which is positive on (0, T). Then x is unique if (i) q1(t)⩾0 for all t ϵ [0, T] and (ii) there exists a sequence {tj}j ϵ N of points tj ϵ (0, T) which converges to 0, such that q1(tj)>0 for j = 1, 2,…