Multiple positive solutions of nonlinear singular boundary value problem for fourth-order equations. (English) Zbl 1073.34018

The existence of at least two positive solutions of the following fourth-order boundary value problem \[ x^{(4)}(t)=f(t,x(t),x''(t)),\quad 0<t<1;\qquad x(0)=x(1)=x''(0)=x''(1)=0, \] is discussed. Here, \(f\) may be singular at \(t=0\), \(t=1\), \(x=0\) and \(x''=0\). The proofs are based on Krasnoselskii’s fixed-point theorem on cone expansion and compression, a priori estimates on positive solutions and some properties of the spectral radius of a positive completely continuous operator.


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