Results and estimates on multiple solutions of Lidstone boundary value problems. (English) Zbl 0966.34017

It is well known that in recent years Lidstone boundary value problems of the form \[ (-1)^{n}y^{(2n)}=F(t,y), \] with \(n\geq 1\), \(0<t<1,\) and \(y^{(2i)}(0)=y^{(2i)}(1)=0\), \(0\leq {i}\leq {n-1}\), have attracted the attention of many researchers. This is because of its importance in modeling several nonlinear phenomena. In this paper, after the preliminaries in section 2, the authors investigate when the above boundary value problem admits two positive solutions by using a fixed-point theorem applied on a suitable cone. This is done in section 3. In sections 4 and 5 they apply their first result on the problem \[ y''+a(t)(y^{\mu}+y^{\nu})=0,\quad 0<t<1, \;y(0)=y(1)=0, \] and \[ y''+a(t)e^{\sigma y}=0,\quad 0<t<1, \;y(0)=y(1)=0, \] respectively. Finally, in section 6 the authors provide sufficient conditions for the existence of positive solutions to the Lidstone boundary value problem \[ (-1)^{n}y^{(2n)}=p(t)h(y), \quad n\geq 1,\;0<t<1, \qquad y^{(2i)}(0)=y^{(2i)}(1)=0, \quad 0\leq {i}\leq {n-1}. \] In all previous cases the methods permit to have estimates of the norms of the solutions. Also the authors provide a sufficient number of illustrative examples.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
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