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Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem. (English) Zbl 1200.34023
Consider the fourth-order boundary value problem
$u^{(4)}(t)=f(t,u(t),u''(t)), t \in (0,1),$
$u(0) =u(1)=u''(0)=u''(1)=0,$
where $$f:[0,1] \times [0,+\infty) \times (-\infty,0] \to [0,+\infty)$$ is continuous, such that $$f(t,0,0)=0$$ and satisfies a technical condition ensuring that, roughly speaking, $$f$$ is not necessarily linearizable at $$(0,0)$$ and $$(+\infty,-\infty).$$ Moreover, it is assumed that there exist a non-negative function $$c_1$$ and a non-negative constant $$c_2$$ such that $$c_1(t)+c_2>0$$ and $$f(t,u,p) \geq c_1(t)u-c_2 p$$ for all $$t,u,p.$$ The authors give a sufficient condition, expressed in terms of the generalized eigenvalues of the associated BVP $$u^{(4)}=\lambda(A(t)u-B(t)u''),$$ for the existence of at least one positive solution to the given BVP. The proof is performed by applying the global P. H. Rabinowitz bifurcation theorem [Rocky Mountain J. Math. 3, 161–202 (1973; Zbl 0255.47069)].

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations
##### Keywords:
Fourth-order ODE; positive solution; eigenvalue; bifurcation
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##### References:
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