Optimal Lipschitz Thresholds for Weighted Dirichlet Problems, with an Explicit Bessel-Zero Bound for Power Weights

We study the two-point problem −u′′(t)=w(t)f(t,u(t)), a<t<b, u(a)=M1, u(b)=M2, where the weight w is positive and integrable on (a,b) but may be unbounded near one or both endpoints. We prove that the problem has exactly one solution whenever f satisfies a one-sided Lipschitz condition with constant K<λ1(w) and the first eigenvalue of −φ′′=λwφ with Dirichlet conditions, and no condition depending only on K and w can do better: for K=λ1(w), there are nonlinearities for which the problem has no solution and others for which it has infinitely many. For globally Lipschitz f, the successive approximations converge geometrically in a weighted L2 metric, in which the Green operator is self-adjoint, with a ratio of at most K/λ1(w). For power weights w(t)=(t−a)γ−2, γ>1, the threshold is λ1=γ2j1/γ,12/(4(b−a)γ), where j1/γ,1 is the first positive zero of the Bessel function J1/γ, that is, jν,1 with ν=1/γ. Spectral sums of the Green operator give elementary two-sided bounds, and λ1 strictly exceeds the supremum-norm, Hilbert–Schmidt and Lyapunov-type thresholds. As an application, we obtain sharp uniqueness thresholds, within criteria depending only on the Lipschitz constant, the order and the interval length, for conformable boundary value problems of order γ∈(1,2] and for sequential problems of every order δ∈(0,1], enlarging the admissible Lipschitz constant for the problem of order γ by up to 35% relative to the supremum-norm criterion and by up to 22% relative to the Hilbert–Schmidt criterion, with the largest gains for orders close to 1. Numerical experiments confirm these results.