Positive Solutions and Nonlinear Multipoint Conjugate Eigenvalue Problems

Abstract

Values of λ are determined for which there exist solutions in a cone of the n^th order nonlinear differential equation, u⁽ⁿ⁾ = λα(t)ƒ(u), 0 < t < 1, satisfying the multipoint boundary conditions, u^(j) (α_i) = 0, 0 ≤ j ≤ n_i−1, 1 ≤ i ≤ k, where 0 =α₁ < α₂ < ∙∙∙ <a_k = 1, and Σ^k_i=1 n_i = n, where α and ƒ are nonnegative valued, and where both lim_|x|→0+ ƒ(x) / |x| and lim_|x|→∞ ƒ(x) / |x| exist.