Ważewski's Principle for retarded functional differential equations

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix; and the Laplace transform. The fundamental solution is presented in the form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of an RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.

In this paper, we apply an extension of the abstract Hopf bifurcation theorem stated in Jacimovic (Non Anal TMA 73(8):2426---2432, 2010) to abstract integral equations (AIE) and retarded functional differential equations (RFDE). This yields sufficient conditions of what we refer to as extended Hopf bifurcation for AIE and RFDE, in which we have a relaxation of the non-resonance condition on the eigenvalues of the generator of corresponding semigroup. We illustrate our results with an explicit example of a system of two delay differential equations (DDE), undergoing extended Hopf bifurcation at the resonant eigenvalue.

An interesting problem that has been investigated over the past several years has been the study of asymptotic constancy of solutions of differential equations for which each constant function is a solution itself. For ordinary differential equations the only such equation is x'=0. A list of certain results and references for retarded functional differential equations (RFDEs) with finite or infinite delay can be found in Haddock [3]

In this note we establish a criterion to obtain classical solutions for a quasi-linear abstract retarded functional differential equation with infinite delay and we apply this result to characterize the infinitesimal generators of several strongly continuous semigroup of linear operators that arise in the theory of linear abstract retarded functional differential equations with infinite delay on a phase space defined axiomatically.

This paper deals with the approximation of the eigenvalues of evolution operators for linear retarded functional differential equations through the reduction to finite dimensional operators by a pseudospectral collocation. Fundamental applications such as determination of asymptotic stability of equilibria and periodic solutions of nonlinear autonomous retarded functional differential equations follow at once. Numerical tests are provided.

Imagine a biological population composed of adult and juvenile individuals. Let N(t) denote the density of adults at time t. Assume that the length of the juvenile period is exactly h units of time for each individual. Assume that adults produce offspring at a per capita rate α and that their probability per unit of time of dying is μ. Assume that a newborn survives the juvenile period with probability ρ and put r = αρ. Then the dynamics of N can be described by the differential equation$$\dot N\left( t \right) = - \mu N\left( t \right) + rN\left( {t - h} \right)$$ ((1.1)) which involves a nonlocal term, where N has argument t — h, since newborns become adults with some delay. So the rate of change of N involves the current as well as the past values of N. Such equations are called Retarded Functional Differential Equations (RFDE) or, alternatively, Delay Equations.KeywordsPeriodic OrbitHopf BifurcationSolution OperatorDelay EquationJuvenile PeriodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we study degenerate Hopf bifurcations in a class of parametrized retarded functional differential equations. Specifically, we are interested in the case where the eigenvalue crossing condition of the classical Hopf bifurcation theorem is violated. Our approach is based on center manifold reduction and Poincare-Birkhoff normal forms, and a singularity theoretical classification of this degenerate Hopf bifurcation. Our results are then applied to a recently developed SIS model incorporating a delayed behavioral response. We show that the phenomenon of endemic bubbles, which is characterized by a branch of periodic solutions which bifurcates from the endemic equilibrium at some value of the basic reproduction number R0, and then reconnects to the endemic equilibrium at a larger value of R0, originates in a codimension-two organizing center where the eigenvalue crossing condition for the Hopf bifurcation theorem is violated.

We present Lyapunov stability and boundedness results for a class of discontinuous dynamical systems (DDS) determined by retarded functional differential equations and we apply these results in the analysis of a class of DDS determined by differential-difference equations (DDS with time delays). DDS of the type considered herein arise in the modeling of a variety of (infinite dimensional) systems, including certain classes of hybrid system, discrete event systems, switched system, intelligent control systems, systems subjected to impulse effects, and the like.

A retarded functional differential equation (RFDE) consists of a rule to extend the “initial” condition. Starting from a given function defined on [—h, 0] one obtains, by solving the equation, a function defined on [—h,ω), with ω = ∞under mild conditions on the right hand side (such that blow up of solutions in finite time can be excluded).

<p style='text-indent:20px;'>In this paper we study nonlinear autonomous retarded functional differential equations; that is, functional equations where the time derivative may depend on the past values of the variables. When the nonlinearities in such equations are comprised of elementary functions, we give a constructive proof of the existence of an embedding of the original coordinates yielding a polynomial differential equation. This embedding is a topological conjugacy between the semi-flow of the original differential equation and the semi-flow of the auxiliary polynomial differential equation. Further dynamical features are investigated; notably, for an equilibrium or a periodic orbit and its embedded counterpart, the stable and unstable eigenvalues have the same algebraic and geometric multiplicity.</p>

By applying the Laplace transform to a certain Volterra convolution equation complex function theory can be made to bear on a linear autonomous retarded functional differential equation (RFDE) with finite delay.

Collocation based on piecewise polynomial approximation represents a powerful class of methods for the numerical solution of initial-value problems for functional differential and integral equations arising in a wide spectrum of applications, including biological and physical phenomena. The present book introduces the reader to the general principles underlying these methods and then describes in detail their convergence properties when applied to ordinary differential equations, functional equations with (Volterra type) memory terms, delay equations, and differential-algebraic and integral-algebraic equations. Each chapter starts with a self-contained introduction to the relevant theory of the class of equations under consideration. Numerous exercises and examples are supplied, along with extensive historical and bibliographical notes utilising the vast annotated reference list of over 1300 items. In sum, Hermann Brunner has written a treatise that can serve as an introduction for students, a guide for users, and a comprehensive resource for experts.

We introduce a new class of functional differential equations with functional response on piecewise constant argument, FDEPCA. It contains functional differential equations with continuous time as well as differential equations with piecewise constant argument which were introduced in Akhmet (On the integral manifolds of the differential equations with piecewise constant argument of generalized type. In: Proceedings of the conference on differential and difference equations at the Florida institute of technology, August 1–5, 2005, Melbourne, Florida, ed. by R.P. Agarval, K. Perera (Hindawi Publishing Corporation, London, 2006), pp. 11–20; Nonlinear Anal 66:367–383, 2007; Discontinuity Nonlinearity Complexity 1:1–6,2012; J Math Anal Appl 336:646–663, 2007; Nonlinear Anal Hybrid Syst 2:456–467, 2008), and then developed in many papers and books. This chapter considers equations which are introduced and developed in our papers. We concentrate only on retarded equations, but one can easily extend the discussion to any type of piecewise constant argument and functional differential equations. Nonlinear and quasilinear systems are under consideration. At the end of the chapter, we suggest how one can apply the systems for solution of real-world problems, provided more general systems for future investigations.

The existence of slowly and regularly varying solutions in the sense of Karamata implying nonoscillation is proved for a class of second order nonlinear retarded functional differential equations of Thomas-Fermi type. A motivation for such study is the extensively developed theory offering a number of properties of regularly and slowly varying functions ([2]) - consequently of such solutions of differential equations. As an illustration, the precise asymptotic behaviour for $t\rightarrow \infty$ of the slowly varying solutions for a subclass of considered equations is presented.

In Chap. 17, n-th-order functional differential equations are considered. In order to create a concept of nonoscillation for functional differential equations which will be an analogue of nonoscillation theory for ordinary differential equations, we define a nonoscillation interval for the n-th-order homogeneous functional differential equation as interval, where each nontrivial solution can have at most n−1 zeros, counting every zero with its multiplicity. The study of many classical questions of the qualitative theory of n-th-order equations such as existence and uniqueness of solutions of the interpolation boundary value problems, positivity or corresponding regular behavior of their Green’s functions, maximum principles and stability, was considered in the general framework based on the notion of nonoscillation intervals of corresponding linear functional differential equations. It is demonstrated that if the Wronskian of the fundamental system is not equal to zero at the initial point of the interval, then the nonoscillation interval exists for functional differential equations; this is true for a particular case of the n-th-order functional differential equations with Volterra operators (delay equations). For second-order functional differential equations, nonvanishing Wronskian is equivalent to the Sturm theorem on separation of zeros (between two adjacent zeros of any nontrivial solution there is a zero of every other solution). The analogue of the classical Mammana theorem, establishing necessary and sufficient nonoscillation conditions through nonvanishing normal chains of the Wronskians, is proven for n-th-order functional differential equations.