Eigenvalues of Lidstone boundary value problems (Q1294229)

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 \(f\).