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...

The paper investigates singular nonlinear problems arising in hydrodynamics. In particular, it deals with the problem on the half-line of the form $${\left(p\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}=p\left(t\right)f\left(u\left(t\right)\right),$$$$u^{\text{'}}(...$$

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 $p>0$, $\lambda >0$, $p\ne \lambda$).

We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $\alpha$-Gevrey vector field with an hyperbolic linear part admits a smooth $\beta$-Gevrey transformation to a smooth $\beta$-Gevrey normal form. The Gevrey order $\beta$ depends on...

It is known that in two-dimensional systems of ODEs of the form ${}^i=x^i{f}^i\left(x\right)$ with $\partial f^i/\partial x^j<0$ (strongly competitive systems), boundaries of the basins of repulsion of equilibria consist of invariant Lipschitz curves, unordered with respect to the coordinatewise (partial) order. We prove that such curves are in fact of class $C^1$.

Strata of bifurcation sets related to the nature of the singular points or to connections between hyperbolic saddles in smooth families of planar vector fields, are smoothly equivalent to subanalytic sets. But it is no longer true when the bifurcation is related to transition near singular points, for instance for a line of double limit cycles in a generic 2-parameter family at its end point which is a codimension 2 saddle connection bifurcation point. This line has a flat contact with the line...