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Eigenvalues of Lidstone boundary value problems. (English) Zbl 0933.65089

The authors consider a two-point boundary value problem of the form \((-1)^ny^{(2n)}=\lambda F(t,y)\), \(t\in (0,1)\); \(y^{(2i)}(0)=y^{(2i)}(1)=0\), for \(i=0,\dots,n-1\), where \(\lambda >0\), under the assumption that there exist nonnegative continuous functions \(f\), \(\alpha\) \(\beta \) such that \(\alpha(t)f(x)\leq F(t,x)\leq \beta(t)f(x)\), where \(t\in(0,1)\), \(x\in(0,\infty)\). The aim of the paper is to characterize the set \(E\) of positive eigenvalues \(\lambda\) for which eigenfunctions \(y\) are positive solutions, i.e.: \(y\geq 0\) and \(y\not\equiv 0\) on \([0,1]\). It is proved that the set \(E\) is an interval and conditions under which this interval is bounded or unbounded are established. Moreover, explicit eigenvalue intervals are obtained in terms of some limits of the function \(f\).

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
34L30 Nonlinear ordinary differential operators
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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