A third order boundary value problem on an unbounded domain

Abstract

We consider the third order nonlinear differential equation $$(p(t)u'(t))'' = f(t,u(t),u'(t),u''(t)), \quad a.\, e. \ \mathrm{in} \ (0,\infty), $$ satisfying the boundary conditions $$ u(0)=u'(0) = 0, \quad \lim_{t \to \infty} u(t) =\alpha u(\beta).$$ where $\alpha, \beta \in (0,\infty)$, $f: [0,\infty) \times \mathbb{R}^3 \to \mathbb{R}$ is Carath\'{e}odory with respect to $L_1[0,\infty)$, $p \in C^2(0,\infty)$ and $p(t) > 0$ for all $t > 0$. We obtain the existence of at least one solution using the Leray-Schauder Continuation Principle.