TRIPLE POSITIVE SOLUTIONS FOR A THIRD-ORDER THREE-POINT BOUNDARY VALUE PROBLEM WITH SIGN-CHANGING GREEN’S FUNCTION

Abstract

In this paper, we discuss the existence of triple positive solutions for a third-order three-point boundary value problem
     \begin{cases} u'''(t) = f(t,u(t)),  t ∈ (0, 1), \\ u'(0) = 0,    u(1) = αu(η),     u''(η) = 0, \end{cases}

where 0 < α < 1, max \(\left\{ \frac{1+2α}{1+24α}, \frac{1}{2-α} \right\}\).  We first study the associated Green’s function and obtain some useful properties. Our main tool is the fixed point theorem due to Avery and Peterson.  It is to be observed that although the associated Green’s function is sign-changing, the solution obtained is still positive.  The results of this paper are new.  An example demonstrates the main results.