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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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