The paper is devoted to the study of the properties of the Fučík spectrum. In the first part, we analyse the Fučík spectra of the problems with one second order ordinary differential equation with Dirichlet, Neumann and mixed boundary conditions and we present the explicit form of nontrivial solutions. Then, we discuss the problem with two second order differential equations with mixed boundary conditions. We show the relation between the Dirichlet boundary value problem and mixed boundary value...

A special type of Jacobi matrices, discrete Schrödinger operators, is found to play an important role in quantum physics. In this paper, we show that given the spectrum of a discrete Schrödinger operator and the spectrum of the operator obtained by deleting the first row and the first column of it can determine the discrete Schrödinger operator uniquely, even though one eigenvalue of the latter is missing. Moreover, we find the forms of the discrete Schrödinger operators when their smallest and...

We consider the classical nonlinear fourth-order two-point boundary value problem $$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=\lambda h\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right)),0<t<1,\hfill \\ u\left(0\right)=u^{\text{'}}\left(1\right)=u^{\text{'}\text{'}}\left(0\right)=u^{\text{'}\text{'}\text{'}}\left(1\right)=0.\hfill \end{array}\right.$$ In this problem, the nonlinear term $h\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right))$ contains the first and second derivatives of the unknown function, and the function $h\left(t\right)f(t,x,y,z)$ may be singular at $t=0$, $t=1$ and at $x=0$, $y=0$, $z=0$. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.

We consider the existence of positive solutions of the equation $1/\lambda \left(t\right){\left(\lambda \left(t\right){\phi }_p\left(x^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right))=0$, where ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s$, p > 1, subject to some singular Sturm-Liouville boundary conditions. Using the Krasnosel’skiĭ fixed point theorem for operators on cones, we prove the existence of positive solutions under some structure conditions.

Values of $\lambda$ are determined for which there exist positive solutions of the system of three-point boundary value problems, $u^{\text{'}\text{'}}+\lambda a\left(t\right)f\left(v\right)=0$, $v^{\text{'}\text{'}}+\lambda b\left(t\right)g\left(u\right)=0$, for $0<t<1$, and satisfying, $u\left(0\right)=\beta u\left(\eta \right)$, $u\left(1\right)=\alpha u\left(\eta \right)$, $v\left(0\right)=\beta v\left(\eta \right)$, $v\left(1\right)=\alpha v\left(\eta \right)$. A Guo-Krasnosel’skii fixed point theorem is applied.

In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools...

In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools...

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...