Mean-Periodic Solutions of Retarded Functional Differential Equations

In this paper we present a spectral criterion for existence of mean-periodic solutions of retarded functional differential equations with a time-independent main part.

Smoothness Properties of Solutions of Caputo-Type Fractional Differential Equations

Mathematics Subject Classification: 26A33, 34A25, 45D05, 45E10

Anti-Periodic Boundary Value Problem for Impulsive Fractional Integro Differential Equations

MSC 2010: 34A37, 34B15, 26A33, 34C25, 34K37

On the Operational Solution of a System of Fractional Differential Equations

MSC 2010: 26A33, 44A45, 44A40, 65J10

Noninear Integral Inequalities Involving Maxima

Снежана Христова, Кремена Стефанова, Лозанка Тренкова - В статията се изучават някои интегрални неравенства, които съдържат макси-мума на неизвестната функция на една променлива. Разглежданите неравенства са обобщения на класическото неравенство на Бихари. Значимостта на тези интегрални неравенства се дълже на широкото им приложение при качественото изследванене на различни свойства на решенията на диференциални уравнения с “максимум” и е илюстрирано с някои директни приложения.

Discrete Generalization of Gronwall-Bellman Inequality with Maxima and Application

Снежана Христова, Кремена Стефанова, Лиляна Ванкова - В работата са решени няколко нови видове линейни дискретни неравенства, които съдържат максимума на неизвестната функция в отминал интервал от време. Някои от тези неравенства са приложени за изучаване непрекъснатата зависимост от смущения при дискретни уравнения с максимуми.

Existence of Solutions for a Class of Quasi-Linear Singular Integro-Differential Equations

2000 Mathematics Subject Classification: 45F15, 45G10, 46B38.

Modeling some real phenomena by fractional differential equations

This paper deals with fractional differential equations, with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional differential equations may model more efficiently certain problems than ordinary differential equations. A numerical optimization approach based on least squares approximation is used to determine the order of the fractional operator that better describes real data, as well as other related parameters.

Instabilities in threshold-diffusion equations with delay

The introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio-temporal chaos. In this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria, periodic orbits and travelling waves. Importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function. Using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback. Direct numerical simulations are used to confirm our bifurcation analysis, and to probe some of the rich behaviour possible for mixed feedback.

Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling points t(i) is an element of [t(0), t(J)] for i = 0, 1, . . . , J of the solution. Two examples are provided.

Discontinuous local semiflows for Kurzweil equations leading to LaSalle`s invariance principle for differential systems with impulses at variable times

In this paper, we consider an initial value problem for a class of generalized ODEs, also known as Kurzweil equations, and we prove the existence of a local semidynamical system there. Under certain perturbation conditions, we also show that this class of generalized ODEs admits a discontinuous semiflow which we shall refer to as an impulsive semidynamical system. As a consequence, we obtain LaSalle`s invariance principle for such a class of generalized ODEs. Due to the importance of LaSalle`s invariance principle in studying stability of differential systems, we include an application to autonomous ordinary differential systems with impulse action at variable times. (C) 2011 Elsevier Inc. All rights reserved.