Existence of solutions of non-autonomous second order functional differential equations with infinite delay

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.     

On the essential spectra of coupled operators matrix and application

We define an abstract setting to treat essential spectra of unbounded coupled operator matrix. We prove a well‐posedness result and develop a spectral theory which also allows us to prove an amelioration to many earlier works. We point out that a concrete example from integro‐differential equation fit into this abstract framework involving a general class of regular operator in L₁ spaces.     

Point initial value problem for linear functional differential equations in Banach spaces

Summary The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.     

Series expansions for functional differential equations

In this paper we prove convergence results for the series expansion of the solution to a linear functional differential equation. The results are consequences of an analysis of eigenfunction expansions for the generator of the solution map. This abstract approach unifies the treatment of retarded and neutral functional differential equations.The research of S.M. Verduyn Lunel has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences     

A system of non-local parabolic PDE and application to option pricing

This article includes a proof of well posedness of an initial-boundary value problem involving a system of non-local parabolic partial differential equation (PDE), which naturally arises in the study of derivative pricing in a generalized market model, which is known as a semi-Markov modulated geometric Brownian motion (GBM) model We study the well posedness of the problem via a Volterra integral equation of second kind. A probabilistic approach, in particular the method of conditioning on stopping times is used for showing the uniqueness.     

Approximate controllability of second-order distributed implicit functional systems

In this paper we study the approximate controllability of control systems with states and controls in Hilbert spaces, and described by a second-order semilinear abstract functional differential equation with infinite delay. Initially we establish a characterization for the approximate controllability of a second-order abstract linear system and, in the last section, we compare the approximate controllability of a semilinear abstract functional system with the approximate controllability of the associated linear system.     

An inverse problem for an immobilized enzyme model

A method for estimating unknown kinetic parameters in a mathematical model for catalysis by an immobilized enzyme is studied. The model consists of a semilinear parabolic partial differential equation modeling the reaction‐diffusion process coupled with an ordinary differential equation for the rate transport. The well posedness of the model is proven; a PDE‐constrained optimization approach is applied to the stated inverse problem; and finally, some numerical simulations are presented.     

Local asymptotic attractivity for nonlinear quadratic functional integral equations

In this paper, an existence result for local asymptotic attractivity of the solutions is proved for a nonlinear quadratic functional integral equation under certain growth conditions which in turn gives the existence as well as asymptotic stability of solutions. A couple of examples are provided for indicating the natural realizations of abstract theory presented in the paper.     

Rothe's method to strongly damped wave equations

A strongly damped wave equation is studied as an abstract differential equation in a Banach space. Existence and uniqueness of a strong solution is established with the help of Rothe's method.     

On a class of abstract neutral functional differential equations

By using the theory of semigroups of growth α, we discuss the existence of mild solutions for a class of abstract neutral functional differential equations. A concrete application to partial neutral functional differential equations is considered.     

Global solutions for abstract neutral differential equations

We study the existence of global solutions for a class of abstract neutral differential equation defined on the whole real axis. Some concrete applications related to ordinary and partial differential equations are considered.     

Attractivity,multistability, and bifurcation in delayed Hopfieldʼs model with non-monotonic feedback

For a system of delayed neural networks of Hopfield type, we deal with the study of global attractivity, multistability, and bifurcations. In general, we do not assume monotonicity conditions in the activation functions. For some architectures of the network and for some families of activation functions, we get optimal results on global attractivity. Our approach relies on a link between a system of functional differential equations and a finite-dimensional discrete dynamical system. For it, we introduce the notion of strong attractor for a discrete dynamical system, which is more restrictive than the usual concept of attractor when the dimension of the system is higher than one. Our principal result shows that a strong attractor of a discrete map gives a globally attractive equilibrium of a corresponding system of delay differential equations. Our abstract setting is not limited to applications in systems of neural networks; we illustrate its use in an equation with distributed delay motivated by biological models. We also obtain some results for neural systems with variable coefficients.     

Problems of Periodic Solutions for a Kind of Second Order Neutral Functional Differential Equation

By means of continuation theorem of coincidence degree theory, existence, nonexistence and unique existence criteria are established for the periodic solution of a kind of second order neutral functional differential equation.     

Well–posedness for an Abstract Semilinear Integro–differential Fractional Problem

We consider an abstract second order semilinear integrodifferential equation involving fractional time derivatives of order between 1 and 2. Well–posedness is established under appropriate conditions on the initial data and the nonlinearity which is itself a fractional integral. These conditions will determine the exact underlying space where to look for solutions.     

Construction of differential equation approximations to delay differential equations

A constructive method is presented for obtaining differential equation approximations to general functional delay differential equations. It is shown that the approximating differential equation systems can be used to determine the stability of the functional differential equations     

A Taylor method for numerical solution of generalized pantograph equations with linear functional argument

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this paper, we introduce a numerical method based on the Taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results.     

A second-order nonlinear problem with two-point and integral boundary conditions

The paper gives sufficient conditions for the existence and nonuniqueness of monotone solutions of a nonlinear ordinary differential equation of the second order subject to two nonlinear boundary conditions one of which is two-point and the other is integral. The proof is based on an existence result for a problem with functional boundary conditions obtained by the author in [6].     

Local well‐posedness and ill‐posedness for the fractal Burgers equation in homogeneous Sobolev spaces

In this paper, we consider local well‐posedness and ill‐posedness questions for the fractal Burgers equation. First, we obtain the well‐posedness result in the critical Sobolev space. We also present an unconditional uniqueness result. Second, we show the ill‐posedness from the point of view of the Picard iterations in the supercritical space. Copyright © 2008 John Wiley & Sons, Ltd.     

Hölder continuous solutions for fractional differential equations and maximal regularity

We characterize the existence and uniqueness of solutions of an abstract fractional differential equation with infinite delay in Hölder spaces.     

Well‐posedness and exact controllability of the mass balance equations for an extrusion process

In this paper, we study the well‐posedness and exact controllability of a physical model for an extrusion process in the isothermal case. The model expresses the mass balance in the extruder chamber and consists of a hyperbolic partial differential equation (PDE) and a nonlinear ordinary differential equation (ODE) whose dynamics describes the evolution of a moving interface. By suitable change of coordinates and fixed point arguments, we prove the existence, uniqueness, and regularity of the solution and finally, the exact controllability of the coupled system. Copyright © 2015 John Wiley & Sons, Ltd.