Unbounded solutions of third order three-point boundary value problems on a half-line

Abstract

Abstract We consider the following third order three-point boundary value problem on a half-line: x'''(t)+q(t)f(t,x(t),x'(t),x''(t)) = 0, t ∈ (0,+∞), x'(0) = A, x(η) = B, x''(+∞) = C, where η ∈ (0,+∞), but fixed, and f : [0,+∞) × ℝ3 → ℝ satisfies Nagumo's condition. We apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory, to establish existence theory for at least one unbounded solution, and at least three unbounded solutions. To demonstrate the usefulness of our results, we illustrate two examples.