Positive solutions of singular fourth-order boundary-value problems. (English) Zbl 1096.34012

Cui, Yujun; Zou, Yumei

Electron. J. Differ. Equ. 2006, Paper No. 39, 10 p. (2006).

Summary: We present necessary and sufficient conditions for the existence of positive \(C^3[0,1]\cap C^4(0,1)\) solutions for the singular boundary value problem \[ x''''(t)=p(t)f(x(t)),\quad t\in(0,1);\quad x(0)=x(1)=x'(0)=x'(1)=0, \] where \(f(x)\) is either superlinear or sublinear and \(p:(0,1)\to [0,+\infty)\) may be singular at both ends \(t=0\) and \(t=1\). For this goal, we use fixed-point index results.

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34B16	Singular nonlinear boundary value problems for ordinary differential equations

Keywords:

Singular boundary value problem; fixed-point theorem; positive solution