# zbMATH — the first resource for mathematics

On a fourth-order eigenvalue problem. (English) Zbl 1089.34511
Summary: We consider the existence of positive solutions for the equation $\frac{d^4y} {x^4}-\lambda f\bigl(x,y(x)\bigr)=0,\tag{1}$ with one of the following set of boundary value conditions $y(0)=y(1)=y''(0)=y''(1)= 0,\tag{2}$
$y(0)=y'(1)= y''(0)=y'''(1)=0,\tag{3}$ and prove the existence of a positive solution for sufficiently small $$\lambda$$. BVP (1)–(2) describes the deformations of an elastic beam both of whose ends simply supported, and BVP (1)–(3) describes the deformations of an elastic beam with one end simply supported and the other end clamped by sliding clamps. Several results on the existence of solutions for nonlinear beam equations have been established by several authors by using the Leray-Schauder continuation theorem.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B24 Sturm-Liouville theory