Solvability of a nonlinear four-point boundary value problem for a fourth order differential equation (Q1428960)

The authors consider the four-point boundary value problem for a fourth-order ordinary differential equations of the form \[ (\phi_p(u''(t)))''=a(t)f(u(t)),\quad t\in (0,1),\tag \(E\) \] with one of the following boundary conditions \[ u(0)-\lambda u'(\eta) =u'(1) =0, \;u'''(1)=\alpha_1 u'''(\xi ),\;u''(1)=\beta_1 u''(\xi ),\tag \(B_1\) \] or \[ u(1)+\lambda u'(\eta )=u'(0)=0,\;u'''(0)=\alpha_1 u'''(\xi ),\;u''(1)=\beta_1 u''(\xi ).\tag{\(B_2\)} \] They impose growth conditions on \( f\) which guarantee the existence of at least two positive solutions for problems \((E),(B_1)\) and \((E),(B_2).\) From the Introduction: ``Our arguments involve the use of the concavity and integral representation of solutions and the Avery-Henderson fixed-point theorem.'' Related literature: \textit{R. I. Avery} and \textit{C. J. Chyan} [Comput. Math. Appl. 42, 695--704 (2001; Zbl 1006.34022)].