In this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.

We consider nonlinear Sturm-Liouville problems with spectral parameter in the boundary condition. We investigate the structure of the set of bifurcation points, and study the behavior of two families of continua of nontrivial solutions of this problem contained in the classes of functions having oscillation properties of the eigenfunctions of the corresponding linear problem, and bifurcating from the points and intervals of the line of trivial solutions.

In this paper we consider the problem $\begin{array}{c}{y}^{iv}+p_2\left(x\right)y^{\text{'}\text{'}}+p_1\left(x\right)y^{\text{'}}+p_0\left(x\right)y=\lambda y,0<x<1,\\ y^{\left(s\right)}\left(1\right)-{(-1)}^{\sigma }{y}^{\left(s\right)}\left(0\right)+\sum _{l=0}^{s-1}{\alpha }_{s,l}{y}^{\left(l\right)}\left(0\right)=0,s=1,2,3,\\ y\left(1\right)-{(-1)}^{\sigma }y\left(0\right)=0,\end{array}$ where λ is a spectral parameter; p j (x) ∈ L 1(0, 1), j = 0, 1, 2, are complex-valued functions; α s;l, s = 1, 2, 3, $l=\overline{0,s-1}$, are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α 3,2 + α 1,0 ≠ α 2,1. It is proved that the system of root functions of this spectral problem...