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Zbl 1200.34023
Ma, Ruyun; Xu, Jia
Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 1, A, 113-122 (2010). ISSN 0362-546X

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 {\it P. H. Rabinowitz} bifurcation theorem [Rocky Mountain J. Math. 3, 161--202 (1973; Zbl 0255.47069)].
[Anna Capietto (Torino)]

MSC 2000:: *34B18 Positive solutions of nonlinear boundary value problems
34C23 Bifurcation (periodic solutions)

Keywords: Fourth-order ODE; positive solution; eigenvalue; bifurcation

Citations: Zbl 0255.47069

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