On Fundamental Solution for Autonomous Linear Retarded Functional Differential Equations

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.

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.

An identification problem is studied for linear retarded functional differential equations (FDEs) in Banach spaces. The unique identification of solution states and scalar forcing functions from the given data is proved.

Nonoscillations for odd-dimensional systems of linear retarded functional differential equations

The authors present some results on the numerical approximations for linear retarded functional differential equations with delays in control and observation. Specifically, they consider the averaging approximation scheme for such systems treated as evolution equations with unbounded input and output operators in the state space set-up developed by A.J. Pritchard and D. Salamon (Technical Report no.2624, Mathematics Research Center, University of Wisconsin, Madison, WI, 1984).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The paper gives new results onF-controllability of linear retarded functional differential equations. The concept ofF-controllability, weaker than the approximate controllability in the state space, has been introduced by this author in 1976. In the present paper new conditions are presented which, in the case of systems with one delay, become easily verifiable criteria ofF-controllability. A duality betweenF-controllability and observability is discussed, and several examples are given.

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.

Ważewski's Principle for retarded 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.

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.

&lt;p style='text-indent:20px;'&gt;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.&lt;/p&gt;

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).

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]