Non-periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses

We present a non-periodic averaging principle for measure functional differential equations and, using the correspondence between solutions of measure functional differential equations and solutions of functional dynamic equations on time scales (see Federson et al., 2012 [8]), we obtain a non-periodic averaging result for functional dynamic equations on time scales. Moreover, using the relation between measure functional differential equations and impulsive measure functional differential equations, we get a non-periodic averaging theorem for these equations. Also, it is a known fact that we can relate impulsive measure functional differential equations and impulsive functional dynamic equations on time scales (see Federson et al., 2013 [9]). Therefore, applying this correspondence to our averaging principle, we obtain a non-periodic averaging theorem for impulsive functional dynamic equations on time scales.     

Measure functional differential equations with infinite delay

We introduce measure functional differential equations with infinite delay and an axiomatically described phase space. We show how to transform these equations into generalized ordinary differential equations whose solutions take values in a suitable infinite-dimensional Banach space. Even in the special case of functional equations with finite delay, our result improves the existing one by imposing weaker conditions on the right-hand side.     

Linear measure functional differential equations with infinite delay

We use the theory of generalized linear ordinary differential equations in Banach spaces to study linear measure functional differential equations with infinite delay. We obtain new results concerning the existence, uniqueness, and continuous dependence of solutions. Even for equations with a finite delay, our results are stronger than the existing ones. Finally, we present an application to functional differential equations with impulses.     

Stochastic functional differential equations with infinite delay

The stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors, but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay. The main aim of this paper is to close this gap. We establish criteria of pth moment ψγ(t)-bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay, and we also illustrate the result with an example.     

Measure of noncompactness and semilinear nonlocal functional differential equations in Banach spaces

This paper is concerned with the measure of noncompactness in the spaces of continuous functions and semilinear functional differential equations with nonlocal conditions in Banach spaces.The relationship between the Hausdorff measure of noncompactness of intersections and the modulus of equicontinuity is studied for some subsets related to the semigroup of linear operators in Banach spaces.The existence of mild solutions is obtained for a class of nonlocal semilinear functional differential equations without the assumption of compactness or equicontinuity on the associated semigroups of linear operators.     

Embedding problem and functional equations

In this paper we give a method for solving the functional equations arising from the differential embedding problem. We also obtain the conditions for embedding one-dimensional diffeomorphisms into differential flows.     

Working with differential,functional and difference equations using functional networks

In this paper we first analyze the problem of equivalence of differential, functional and difference equations and give methods to move between them. We also introduce functional networks, a powerful alternative to neural networks, which allow neural functions to be different, multidimensional, multiargument and constrained by link connections, and use them for predicting values of magnitudes satisfying differential, functional and/or difference equations, and for obtaining the difference and differential equation associated with a set of data. The estimation of the differential or difference equation coefficients is done by simply solving systems of linear equations, in the cases of equally or unequally spaced or missing data points. Some examples of applications are given to illustrate the method.     

Averaging for retarded functional differential equations

We consider retarded functional differential equations in the setting of Kurzweil-Henstock integrable functions and we state an averaging result for these equations. Our result generalizes previous ones.     

Ergodic and weighted pseudo-almost periodic solutions for partial functional differential equations in fading memory spaces

We use a new concept of weighted ergodic function based on the measure theory to investigate the existence and uniqueness of weighted pseudo almost periodic solution for a class of partial functional differential equations with infinite delay in fading memory spaces. We illustrate our theoretical results by studying some Lotka-Voltera reaction-diffusion systems with infinite delay.     

Dissipativity of delay functional differential equations with bounded lag

Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if θ=1. One numerical example is given to confirm our theoretical result.     

Stability for retarded functional differential equations

It is known that retarded functional differential equations can be regarded as Banach-space-valued generalized ordinary differential equations (GODEs). In this paper, some stability concepts for retarded functional differential equations are introduced and they are discussed using known stability results for GODEs. Then the equivalence of the different concepts of stabilities considered here are proved and converse Lyapunov theorems for a very wide class of retarded functional differential equations are obtained by means of the correspondence of this class of equations with GODEs. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 107–126, January, 2008.     

Almost automorphic solutions for some partial functional differential equations

In this work, we study the existence of almost automorphic solutions for some partial functional differential equations. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution. Our results extend the classical known theorem by Bohr and Neugebauer on the existence of almost periodic solutions for inhomegeneous linear almost periodic differential equations. We give some applications to hyperbolic equations and Lotka-Volterra type equations used to describe the evolution of a single diffusive animal species.     

Theory of fractional functional differential equations

In this paper, the basic theory for the initial value problems for fractional functional differential equations is considered, extending the corresponding theory of ordinary functional differential equations.     

One class of singular linear functional differential equations

We consider one class of first-order functional differential equations with a singularity in the independent variable. We obtain conditions for the Fredholm property and the solvability of the mentioned equations.     

Pseudo almost automorphy of two-term fractional functional differential equations

In this paper, by measure theory, we introduce and investigate the concepts of (Stepanov-like) $(\mu,\nu)$-pseudo almost automorphic of class $r$ and class infinity, respectively. As applications, we establish some sufficient criteria for the existence, uniqueness of pseudo almost automorphic mild solutions to two-term fractional functional differential equations with finite or infinite delay. The working tools are based on the generalization of semigroup theory, Banach contraction mapping principle and Leray-Schauder alternative theorem. Finally, we explore the same topic for a fractional partial functional differential equation with delay.     

On a class of functional and differential functional equations

One class of functional and differential functional equations with deviating argument depending on the unknown function is considered in this study. The procedure of constructing and investigating solutions to such equations is also proposed.     

Positive periodic solutions of impulsive functional differential equations

We study the existence and nonexistence of positive periodic solutions of a non-autonomous functional differential equation with impulses. The equations we study may be of delay, advance or mixed type functional differential equations and the impulses may cause the existence of positive periodic solutions. The methods employed are fixed-point index theorem, Leray-Schauder degree, and upper and lower solutions. The results obtained are new, and some examples are given to illustrate our main results.     

中立型泛函微分方程的周期解

对于中立型泛函微分方程,证明了解的毕竟有界性蕴含周期解的存在性,把常微分方程中著名的Yoshizawa周期解存在定理推广到中立型泛函微分方程,然后利用所得结果给出一类产生于电力系统的中立型时滞泛函微分方程周期解存在惟一与吸引的条件。     

Bifurcation theory of functional differential equations: A survey

In this paper we survey the topic of bifurcation theory of functionaldifferential equations. We begin with a brief discussion of the position of bifurcationand functional differential equations in dynamical systems. We followwith a survey of the state of the art on the bifurcation theory of functionaldifferential equations, including results on Hopf bifurcation, center manifoldtheory, normal form theory, Lyapunov-Schmidt reduction, and degree theory.     

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.