A third order boundary value problem with jumping nonlinearities (Q690934)

This work concerns the existence of solutions to the third-order boundary value problem \[ -(pu'')'(x)=f(x,u(x))-t\phi(x)+h(x),\quad x\in(0,1), \] \[ \begin{aligned} (0)=au'(0)-bp(0)u''(0)&=0, \\ cu'(1)+dp(1)u''(1)&=0, \end{aligned} \] where \(a,b,c,d\in\mathbb{R}^+\) with \(ac+ad+bc>0\), \(p\in C^1([0,1],(0,+\infty))\), \(\phi,h\in C([0,1],\mathbb{R})\) and \(f\in C^1([0,1]\times\mathbb{R},\mathbb{R}).\) By tackling the corresponding linear eigenvalue problem, the authors establish existence conditions for solutions to the above problem by means of Schauder's fixed point theorem and fixed point index properties.