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Existence of positive solutions of a nonlinear fourth-order boundary value problem. (English) Zbl 1195.34037

Summary: We study the existence of positive solutions of fourth-order boundary value problem

${u}^{\left(4\right)}\left(t\right)=f\left(t,u\left(t\right),{u}^{\text{'}\text{'}}\left(t\right)\right),t\in \left(0,1\right)·$
$u\left(0\right)=u\left(1\right)={u}^{\text{'}\text{'}}\left(0\right)={u}^{\text{'}\text{'}}\left(1\right)=0,$

where $f:\left[0,1\right]×\left[0,\infty \right)×\left(-\infty ,0\right]\to \left[0,\infty \right)$ is continuous. The proof of our main result is based upon the Krein-Rutman theorem and the global bifurcation techniques.

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations 47J15 Abstract bifurcation theory
##### References:
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