Triple solutions for nonlinear singular m-point boundary value problem

Abstract

In this paper, we study the existence of three solutions to the following nonlinear m-point boundary value problem \[ \begin{cases} u''(t) + \beta^2u(t) = h(t)f(t, u(t)),\,\,\,\,\, 0 < t < 1,\\ u'(0) = 0, u(1) =\Sigma^{m-2}_{i=1}\alpha_i u(\eta_i), \end{cases} \] where \(0<\beta<\frac{\pi}{2}, f\in C([0,1]\times \mathbb{R}^+, \mathbb{R}^+). h(t)\) is allowed to be singular at \(t = 0\) and \(t = 1\). The arguments are based only upon the Leggett-Williams fixed point theorem. We also prove nonexist results.