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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

Citations:

Zbl 0255.47069
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References:

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