Two Positive Solutions for a Fourth-Order Three-Point BVP with Sign-Changing Green's Function

Abstract

This paper concerns the fourth-order three-point boundary value problem (BVP) \[ u^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right)\right),\quad t\in\left[0,1\right], \] \[ u'\left(0\right)=u''\left(0\right)=u\left(1\right)=0,\;\alpha u''\left(1\right)-u'''\left(\eta\right)=0, \] where $f\in C\left(\left[0,1\right]\times\left[0,+\infty\right),\left[0,+\infty\right)\right)$, $\alpha\in\left[0,1\right)$ and $\eta\in\left[\frac{2\alpha+10}{15-2\alpha},1\right)$. Although the corresponding Green\textquoteright s function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions on $f$ by applying the two-fixed-point theorem due to Avery and Henderson. An example is also given to illustrate the main results.

Keywords

• two positive solutions
• completely continuous
• fourth-order boundary value problem
• green\textquoteright s function
• two positive solutions