The functional-differential equation $f^{\text{'}}\left(t\right)=1/f\left(f\left(t\right)\right)$ is closely related to Golomb’s self-described sequence $F$,$$\underset{1,}{\underbrace{1,}}\underset{2,}{\underbrace{2,2,}}\underset{2,}{\underbrace{3,3,}}\underset{3,}{\underbrace{4,4,4}}\underset{3,}{\underbrace{5,5,5,}}\underset{4,}{\underbrace{6,6,6,6,}}\cdots .$$We describe the increasing solutions of this equation. We show that such a solution must have a nonnegative fixed point, and that for every number $p\ge 0$ there is exactly one increasing solution with $p$ as a fixed point. We also show that in general an initial condition doesn’t determine a unique solution: indeed the graphs of two distinct increasing solutions cross each other infinitely many times. In fact...

In the present paper we study some basic qualitative properties of solutions of a nonlinear parabolic integrodifferential equation of Barbashin type which occurs frequently in applications. The fundamental integral inequality with explicit estimate is used to establish the results.

A relatively simple proof is given for Haimo's theorem that a meromorphic function with suitably controlled Schwarzian derivative is a concave mapping. More easily verified conditions are found to imply Haimo's criterion, which is now shown to be sharp. It is proved that Haimo's functions map the unit disk onto the outside of an asymptotically conformal Jordan curve, thus ruling out the presence of corners.

Two point boundary value problem for the linear system of ordinary differential equations with deviating arguments $$x^{\text{'}}\left(t\right)=A\left(t\right)x\left({\tau }_{11}\left(t\right)\right)+B\left(t\right)u\left({\tau }_{12}\left(t\right)\right)+q_1\left(t\right),u^{\text{'}}\left(t\right)=C\left(t\right)x\left({\tau }_{21}\left(t\right)\right)+D\left(t\right)u\left({\tau }_{22}\left(t\right)\right)+q_2\left(t\right),{\alpha }_{11}x\left(0\right)+{\alpha }_{12}u\left(0\right)=c_0,{\alpha }_{21}x\left(T\right)+{\alpha }_{22}u\left(T\right)=c_T$$ is considered. For this problem the sufficient condition for existence and uniqueness of solution is obtained. The same approach as in [2], [3] is applied.

Formulating the two-body problem of classical relativistic electrodynamics in terms of action at a distance and using retarded potential, the equations of one-dimensional motion are functional differential equations of the retarded type. For this kind of equations, in general it is not enough to specify instantaneous data to specify unique trajectories. Nevertheless, Driver (1969) has shown that under special conditions for these electrodynamic equations, there exists an unique solution for this...

An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in general it has rank 1 on a nonempty set which is generically a smooth curve. In this paper we provide a short introduction to 2-dimensional almost-Riemannian geometry highlighting its...