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Almost periodic solutions of neutral impulsive systems with periodic time-dependent perturbed delays

Valéry Covachev, Zlatinka Covacheva, Haydar Akça, Eada Al-Zahrani (2003)

Open Mathematics

A neutral impulsive system with a small delay of the argument of the derivative and another delay which differs from a constant by a periodic perturbation of a small amplitude is considered. If the corresponding system with constant delay has an isolated ω-periodic solution and the period of the delay is not rationally dependent on ω, then under a nondegeneracy assumption it is proved that in any sufficiently small neighbourhood of this orbit the perturbed system has a unique almost periodic solution....

An asymptotic theorem for a class of nonlinear neutral differential equations

Manabu Naito (1998)

Czechoslovak Mathematical Journal

The neutral differential equation (1.1) d n d t n [ x ( t ) + x ( t - τ ) ] + σ F ( t , x ( g ( t ) ) ) = 0 , is considered under the following conditions: n 2 , τ > 0 , σ = ± 1 , F ( t , u ) is nonnegative on [ t 0 , ) × ( 0 , ) and is nondecreasing in u ( 0 , ) , and lim g ( t ) = as t . It is shown that equation (1.1) has a solution x ( t ) such that (1.2) lim t x ( t ) t k exists and is a positive finite value if and only if t 0 t n - k - 1 F ( t , c [ g ( t ) ] k ) d t < for some c > 0 . Here, k is an integer with 0 k n - 1 . To prove the existence of a solution x ( t ) satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.

Approximate solutions for integrodifferential equations of the neutral type

B. G. Pachpatte (2010)

Commentationes Mathematicae Universitatis Carolinae

The main objective of the present paper is to study the approximate solutions for integrodifferential equations of the neutral type with given initial condition. A variant of a certain fundamental integral inequality with explicit estimate is used to establish the results. The discrete analogues of the main results are also given.

Approximation of solutions of a difference-differential equation

B. G. Pachpatte (2010)

Archivum Mathematicum

In the present paper we study the approximate solutions of a certain difference-differential equation under the given initial conditions. The well known Gronwall-Bellman integral inequality is used to establish the results. Applications to a Volterra type difference-integral equation are also given.

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