EXISTENCE THEOREMS OF POSITIVE SOLUTIONS FOR A FOURTH-ORDER THREE-POINT BOUNDARY VALUE PROBLEM

Abstract

In this paper, the following fourth-order three-point boundary value problem with $p$-Laplacian operator is studied: $$ \left\{ \begin{array}{ll} (\phi_p(u''(t)))'' = a(t) f(u(t)), \quad t \in (0,1), \\ u(0) = \xi u(1), \; u'(1) = \eta u'(0), \\ u''(0) = \alpha_1 u''(\delta), \; u''(1) = \beta_1 u''(\delta), \end{array} \right. $$ where $\alpha_1, \; \beta_1 \ge 0$, $\xi \neq 1$, $\eta \neq 1$, $0 \lt \delta \lt 1$ and $\phi_p(z) = |z|^{p-2} z$ for $p \gt 1$. We impose growth conditions on $f$ which guarantee the existence of at least three positive solutions for the problem.