## Existence of positive solutions of a fourth-order boundary value problem.(English)Zbl 1082.34023

Summary: We consider the fourth-order boundary value problem $u''''= f(t,u,u''),\;0<t<1 \quad u(0)=u(1)=u''(0)=u''(1)=0,$ where $$f(t,u,p)= au-bp+o(|(u,p)|)$$ near $$(0,0)$$, and $$f(t,u,p)=cu-dp+o(|(u,p)|)$$ near $$\infty$$. We give conditions on the constants $$a,b,c,d$$ that guarantee the existence of positive solutions. The proof of our main result is based upon global bifurcation techniques.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations
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### References:

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