Global Bifurcation from Zero in Some Fourth-Order Nonlinear Eigenvalue Problems

Abstract

In this paper, we study the nonlinear eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary condition. Global bifurcation of nontrivial solutions of this problem is investigated. We prove the existence of two families of unbounded continua of the set of solutions to this problem bifurcating from points and intervals of the line of trivial solutions. Moreover, it is shown that these continua are contained in classes of functions possessing oscillating properties of the eigenfunctions of the corresponding linear problem and their derivatives.