We study the problem of placing effective upper bounds for the number of zeroes of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension $n$ having $m$ singular points. As a function of $n,m$, this bound turns out to be double exponential in the precise sense explained in the paper.As a corollary, we obtain a solution of the so-called restricted infinitesimal...

We are concerned with the uniqueness problem for solutions to the second order ODE of the form $x^{\text{'}\text{'}}+f(x,t)=0$, subject to appropriate initial conditions, under the sole assumption that $f$ is non-decreasing with respect to $x$, for each $t$ fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among other, from the study of oscillations of lumped parameter systems with implicit...

The main purpose of this paper is to consider the analytic solutions of the non-homogeneous linear differential equation $f^{\left(k\right)}+a_{k-1}\left(z\right)f^{(k-1)}+\cdots +a₁\left(z\right)f^{\text{'}}+a₀\left(z\right)f=F\left(z\right)$, where all coefficients $a₀,a₁,...,a_{k-1}$, F ≢ 0 are analytic functions in the unit disc = z∈ℂ: |z|<1. We obtain some results classifying the growth of finite iterated order solutions in terms of the coefficients with finite iterated type. The convergence exponents of zeros and fixed points of solutions are also investigated.

This paper is devoted to considering the complex oscillation of differential polynomials generated by meromorphic solutions of the differential equation $$f^{\left(k\right)}+A_{k-1}\left(z\right)f^{(k-1)}+\cdots +A_1\left(z\right)f^{\text{'}}+A_0\left(z\right)f=0,$$ where $A_i\left(z\right)$$(i=0,1...$

This paper is devoted to considering the iterated order and the fixed points of some differential polynomials generated by solutions of the differential equation $$f^{{}^{\text{'}\text{'}}}+A_1\left(z\right)f^{{}^{\text{'}}}+A_0\left(z\right)f=F,$$ where $A_1\left(z\right)$, $A_0\left(z\right)$$(\neg \equiv 0)...$

This paper is devoted to studying the growth and oscillation of solutions and their derivatives of higher order non-homogeneous linear differential equations with finite order meromorphic coefficients. Illustrative examples are also treated.