Existence of solutions for fourth order three-point boundary value problems on a half-line

Abstract

In this paper, we apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory to establish the existence of unbounded solutions for the following fourth order three-point boundary value problem on a half- line x 0000 (t) + q(t) f(t, x(t), x 0 (t), x 00 (t), x 000 (t)) = 0, t2 (0,+¥), x 00 (0) = A, x(h) = B1, x 0 (h) = B2, x 000 (+¥) = C, where h2 (0,+¥), but fixed, and f : (0,+¥) R 4 ! R satisfies Nagumo's condition. We present easily verifiable sufficient conditions for the existence of at least one solu- tion, and at least three solutions of this problem. We also give two examples to illustrate the importance of our results.