The linear differential equation $\left(q\right):y^{\text{'}\text{'}}=q\left(t\right)y$ with the uniformly almost-periodic function $q$ is considered. Necessary and sufficient conditions which guarantee that all bounded (on $ℝ$) solutions of $\left(q\right)$ are uniformly almost-periodic functions are presented. The conditions are stated by a phase of $\left(q\right)$. Next, a class of equations of the type $\left(q\right)$ whose all non-trivial solutions are bounded and not uniformly almost-periodic is given. Finally, uniformly almost-periodic solutions of the non-homogeneous differential equations...

This paper deals with basic stability properties of a two-term linear autonomous fractional difference system involving the Riemann-Liouville difference. In particular, we focus on the case when eigenvalues of the system matrix lie on a boundary curve separating asymptotic stability and unstability regions. This issue was posed as an open problem in the paper J. Čermák, T. Kisela, and L. Nechvátal (2013). Thus, the paper completes the stability analysis of the corresponding fractional difference...

In this paper, we present an analysis for the stability of a differential equation with state-dependent delay. We establish existence and uniqueness of solutions of differential equation with delay term [...] τ(u(t))=a+bu(t)c+bu(t).$\tau \left(u\left(t\right)\right)=\frac{a+bu\left(t\right)}{c+bu\left(t\right)}.$ Moreover, we put the some restrictions for the positivity of delay term τ(u(t)) Based on the boundedness of delay term, we obtain stability criterion in terms of the parameters of the equation.

In questo articolo si investigano le proprietà di stabilità asintotica dei metodi numerici per equazioni differenziali con ritardo, prendendo in esame l'equazione test: $$U^{\prime }\left(t\right)=aU\left(t\right)+bU\left(t-\tau \right)$$ dove $a$, $b\in \mathbb{R}$, $\tau >0$ e $g\left(t\right)$ è una funzione a valori reali e continua. In particolare, viene analizzata la dipendenza dal ritardo della stabilità numerica dei metodi di collocazione Gaussiana. Nel recente lavoro [GH99], la stabilità di questi metodi è stata dimostrata facendo uso di un approccio geometrico, basato sul legame tra la proprietà...

A general existence and uniqueness result of Picard-Lindelöf type is proved for ordinary differential equations in Fréchet spaces as an application of a generalized Nash-Moser implicit function theorem. Many examples show that the assumptions of the main result are natural. Applications are given for the Fréchet spaces $C^{\infty }\left(K\right)$, $S\left({ℝ}^N\right)$, $B\left(ℝR^N\right)$, $D_{L_1}\left({ℝ}^N\right)$, for Köthe sequence spaces, and for the general class of subbinomic Fréchet algebras.

The paper presents overview of applications of A. M. Lyapunov’s direct method to stability investigation of systems with argument delay. Methods of building Lyapunov-Krasovskiy funcionals for linear systems with constant coefficients are considered. Lyapunov quadratic forms are used to obtain applicable methods for stability investigation and estimation of solution convergence for linear stationary systems, as well as non-linear control systems and systems with quadratic and rational right hand...

We establish a Hartman type asymptotic formula for nonoscillatory solutions of the half-linear second order differential equation $${\left(r\left(t\right)\Phi \left(y^{\text{'}}\right)\right)}^{\text{'}}+c\left(t\right)\Phi \left(y\right)=0,\Phi \left(y\right):={\left|y\right|}^{p-2}y,p>1.$$