### A boundary value problem for non-linear differential equations with a retarded argument

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We consider a large class of impulsive retarded functional differential equations (IRFDEs) and prove a result concerning uniqueness of solutions of impulsive FDEs. Also, we present a new result on continuous dependence of solutions on parameters for this class of equations. More precisely, we consider a sequence of initial value problems for impulsive RFDEs in the above setting, with convergent right-hand sides, convergent impulse operators and uniformly convergent initial data. We assume that the...

Existence and uniqueness theorem for state-dependent delay-differential equations of neutral type is given. This theorem generalizes previous results by Grimm and the author.

It is proved that under some conditions the set of solutions to initial value problem for second order functional differential system on an unbounded interval is a compact ${R}_{\delta}$-set and hence nonvoid, compact and connected set in a Fréchet space. The proof is based on a Kubáček’s theorem.

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...

We study existence of analytic solutions of a second-order iterative functional differential equation ${x}^{\text{'}\text{'}}\left(z\right)={\sum}_{j=0}^{k}{\sum}_{t=1}^{\infty}{C}_{t,j}\left(z\right){\left({x}^{\left[j\right]}\left(z\right)\right)}^{t}+G\left(z\right)$ in the complex field ℂ. By constructing an invertible analytic solution y(z) of an auxiliary equation of the form $\alpha \xb2{y}^{\text{'}\text{'}}\left(\alpha z\right){y}^{\text{'}}\left(z\right)=\alpha {y}^{\text{'}}\left(\alpha z\right){y}^{\text{'}\text{'}}\left(z\right)+\left[{y}^{\text{'}}\left(z\right)\right]\xb3[{\sum}_{j=0}^{k}{\sum}_{t=1}^{\infty}{C}_{t,j}\left(y\left(z\right)\right){\left(y\left({\alpha}^{j}z\right)\right)}^{t}+G\left(y\left(z\right)\right)]$ invertible analytic solutions of the form $y\left(\alpha {y}^{-1}\left(z\right)\right)$ for the original equation are obtained. Besides the hyperbolic case 0 < |α| < 1, we focus on α on the unit circle S¹, i.e., |α|=1. We discuss not only those α at resonance, i.e. at a root of unity, but also near resonance under the...