We consider linear differential equations of the form $${\left(p\left(t\right)x^{\text{'}}\right)}^{\text{'}}+\lambda q\left(t\right)x=0(p\left(t\right)>0,q\left(t\right)>0)\left(\mathrm{A}\right)$$ on an infinite interval $[a,\infty )$ and study the problem of finding those values of $\lambda$ for which () has principal solutions $x_0(t;\lambda )$ vanishing at $t=a$. This problem may well be called a singular eigenvalue problem, since requiring $x_0(t;\lambda )$ to be a principal solution can be considered as a boundary condition at $t=\infty$. Similarly to the regular eigenvalue problems for () on compact intervals, we can prove a theorem asserting that there exists a sequence $\left\{{\lambda }_n\right\}$ of eigenvalues such...

Singular quadratic functionals of one dependent variable with nonseparated boundary conditions are investigated. Necessary and sufficient conditions for nonnegativity of these functionals are derived using the concept of coupled point and singularity condition. The paper also includes two comparison theorems for coupled points with respect to the various boundary conditions.

In the paper a sufficient condition for all solutions of the differential equation with $p$-Laplacian to be proper. Examples of super-half-linear and sub-half-linear equations $\left(\right|y^{\text{'}}{|}^{p-1}{y}^{\text{'}}{)}^{\text{'}}+r\left(t\right){\left|y\right|}^{\lambda }sgny=0$, $r>0$ are given for which singular solutions exist (for any $p>0$, $\lambda >0$, $p\ne \lambda$).

Soit $\varphi :x\mapsto \varphi \left(x\right),x\gg 0$ une solution à l’infini d’une équation différentielle algébrique d’ordre $n$, $P(x,y,y^{\prime },...,y^{\left(n\right)})=0$. Nous donnons un critère géométrique pour que les germes à l’infini de $\varphi$ et de la fonction identité sur $ℝ$ appartiennent à un même corps de Hardy. Ce critère repose sur le concept de non oscillation.

We establish an asymptotic formula for a pair of linearly independent solutions of the subcritical Riemann–Weber type half-linear differential equation. We also complement the results of the author and M. Ünal, Acta Math. Hungar. 120 (2008), 147–163, where the equation was considered in the critical case.

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.

Some new oscillation criteria are obtained for second order elliptic differential equations with damping ${\sum }_{i,j=1}^n{D}_i\left[A_{ij}\left(x\right)D_{j}y\right]+{\sum }_{i=1}^n{b}_i\left(x\right)D_{i}y+q\left(x\right)f\left(y\right)=0$, x ∈ Ω, where Ω is an exterior domain in ℝⁿ. These criteria are different from most known ones in the sense that they are based on the information only on a sequence of subdomains of Ω ⊂ ℝⁿ, rather than on the whole exterior domain Ω. Our results are more natural in view of the Sturm Separation Theorem.

Oscillatory properties of solutions to the system of first-order linear difference equations $$\begin{array}{cc}\hfill \Delta u_k& =q_k{v}_k\hfill \\ \hfill \Delta v_k& =-p_k{u}_{k+1},\hfill \end{array}$$ are studied. It can be regarded as a discrete analogy of the linear Hamiltonian system of differential equations. We establish some new conditions, which provide oscillation of the considered system. Obtained results extend and improve, in certain sense, results presented in Opluštil (2011).

In this paper we establish some oscillation or nonoscillation criteria for the second order half-linear differential equation $${\left(r\left(t\right)\Phi \left(u^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(u\left(t\right)\right)=0,$$ where (i) $r,c\in C([t_0,\infty )$, $ℝ:=(-\infty ,\infty ))$ and $r\left(t\right)>0$ on $[t_0,\infty )$ for some $t_0\ge 0$; (ii) $\Phi \left(u\right)={\left|u\right|}^{p-2}u$ for some fixed number $p>1$. We also generalize some results of Hille-Wintner, Leighton and Willet.