Theory Of Functional Differential Equations Research Articles

LMI-based approach for asymptotically stability analysis of delayed neural networks

This paper derives some sufficient conditions for asymptotic stability of neural networks with constant or time-varying delays. The Lyapunov-Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) approach are employed to investigate the problem. It shows how some well-known results can be refined and generalized in a straightforward manner. For the case of constant time delays, the stability criteria are delay-independent; for the case of time-varying delays, the stability criteria are delay-dependent. The results obtained in this paper are less conservative than the ones reported so far in the literature and provides one more set of criteria for determining the stability of delayed neural networks.

Global Attractivity of Nonautonomous Inshore-Offshore Fishing Models with Stage-structure

The article deals with a nonautonomous fishing model with stage-structure in inshore and offshore. By using the theory of monotone of functional differential equation and the concave operator theory of functional differential equation, the stability of the positive periodic solution of a dynamical system representing the fishery are derived, and their globally attracting behaviour is discussed. A briefly biology meaning of the mathematical results is given.

Functional Differential Equations in Sustainable Forest Harvesting

This article deals with the dynamics of the volume control method in forest regulation. The sustainability of harvesting a given constant volume in a simplified forest model is studied. Depending on the initial age distribution, the cut can lead to the forest exhaustion, or to an asymptotic steady-state uniform age distribution. A continuous model is formulated in terms of various kinds of partial differential equations, delay differential equations, and non-linear integral equations. Equilibrium solutions and their stability properties are determined. Discrete models are also obtained, both by direct reasoning and as approximations to the continuous case. These are used for simulation and graphical exploration of the system behavior. In addition, contrasting various discrete and continuous versions was found useful in clarifying some issues, in particular, ambiguity/redundancy problems in the relation between integral equations and delay differential equations derived from them. The basic problem of evaluating sustainability for an initial distribution remains unsolved, however. Further progress is linked to the asymptotic properties of a second-order recurrence relationship. It is hoped that the interplay between the theory of functional differential equations and this concrete and easily interpretable problem in forest management might prove fruitful in both fields.

Normal forms for semilinear functional differential equations in Banach spaces and applications. Part II

A normal form theory for functional differential equations in Banach spaces of retarded type is addressed. The theory is based on a formal adjoint theory for the linearized equation at an equilibrium and on the existence of center manifolds for perturbed inhomogeneous equations, established in the first part of this work under weaker hypotheses than those that usually appear in the literature. Based on these results, an algorithm to compute normal forms on finite dimensional invariant manifolds of the origin is presented. Such normal forms are important in obtaining the ordinary differential equation giving the flow on center manifolds explicitly in terms of the original functional differential equation. Applications to Bogdanov-Takens and Hopf bifurcations are presented.

Global Asymptotic Stability of Traveling Waves in Delayed Reaction-Diffusion Equations

The existence and comparison theorem of solutions is first established for the quasi-monotone delayed reaction-diffusion equations on R by appealing to the theory of abstract functional differential equations. The global asymptotic stability, Liapunov stability, and uniqueness of traveling wave solutions are then proved by the elementary super- and subsolution comparison and squeezing methods.

The advancement of oscillation theory of functional differential equations

The oscillation theory of functional differential equations (FDEs) is surveyed including (I) formulation of oscillation problems; (II) a brief history of the oscillation theory of FDEs; (III) main topics in the oscillation theory of FDEs; (IV) open problems in the oscillation theory of FDEs.

Annulus arguments in the stability theory for functional differential equations

An annulus argument is a method of proof which can detect that a curve in $\mathbb{R}^n$ crosses an annulus around the origin infinitely many times. In this paper we give annulus arguments not requiring the boundedness of the derivatives of the functions involved. Using these results we establish Lyapunov type theorems for the attractivity, asymptotic stability, and partial stability properties of the zero solution of nonautonomous functional differential equations whose right hand sides are not bounded with respect to the time. We apply these results to the scalar equation $$ x'(t) = -c(t)x(t) + b(t)x(t-h)\qquad(c(t) \ge 0), $$ the scalar equation with several delays $$ x'(t) = -c(t)x(t) + \sum^n_{i=1} b_i(t)x(t - h_i)\qquad(c(t) \ge 0), $$ as well as to the system $$ x'(t) = B(t)x(t-h)-C(t)x(t), $$ where $B (t)$ and $C(t)$ are continuous matrix functions.

Oscillation Theory for Functional Differential Equations (L. H. Erbe, Q. K. Kong, and B. G. Zhang)

Oscillation Theory for Functional Differential Equations (L. H. Erbe, Q. K. Kong, and B. G. Zhang)

C∗-Algebra methods in the theory of functional differential equations

The paper presents a number of fundamental ideas and C ∗-algebraic methods related to the problems of solvability of functional differential equations.

Linear Theory of Abstract Functional Differential Equations of Retarded Type

The purpose of this paper is to provide an extension of the linear theory of functional differential equations of retarded type to abstract equations. Such equations include examples borrowed from population dynamics to which the theory applies. An application will be given elsewhere. Our main effort in this work consists in providing a suitable extension of the formal adjoint equation and the formal duality. The solutions of the linear autonomous retarded functional differential equation x′(t) = L(xt), (1) where L is a bounded linear operator mapping the space C([−r, 0]; E) into the Banach space E, define a strongly continuous translation semigroup. We show the existence of a direct sum decomposition of C([−r, 0]; E) into two subspaces which are semigroup invariants. The flow induced by the solutions of Eq. (1) can be interpreted as the flow induced by an ordinary differential equation in a finite-dimensional space. We explicitly characterize this decomposition by an orthogonality relation associated to a certain definition of formal duality. The existence of an integral representation for the operator L leads to an equation formally adjoint to (1) characterizing the projection operator defined by the above decomposition of C([−r, 0]; E).

A structured population model and a related functional differential equation: Global attractors and uniform persistence

A structured population model of a single population having two distinct life stages is considered. The model equations, consisting of a hyperbolic partial differential equation coupled to an ordinary differential equation, can be reduced to a single, scalar functional differential equation. This allows us to use the well-developed dynamical systems theory for functional differential equations in order to study the dynamical system generated by the more complicated coupled system. A precise relation is established between the dynamical systems generated by each system of equations and a correspondence between their respective global attractors is made. The two systems are topologically equivalent on their respective attractors. These relationships are used to determine sharp sufficient conditions for the uniform persistence of the population.

Liapunov Theory for Functional Differential Equations

Liapunov Theory for Functional Differential Equations

S1-degree and global Hopf bifurcation theory of functional differential equations

The recently developed S 1-degree and bifrucation theory are applied to provide a purely topological argument of a global Hopf bifurcation theory for functional differential equations of mixed type. In the special case where the equation is of retarded type, the established result represents an analog of Alexander and Yorke's global Hopf bifurcation theorem which has been obtained by Chow, Fiedler, Mallet-Paret, and Nussbaum, using different approaches.

On the stability of index of nonlocal elliptic problems

One of the first publications devoted to nonlocal elliptic problems was the paper of T. Carleman [S]. Together with this paper are closely associated further investigations of singular integral equations with shifts and elliptic boundary value problems with shifts which map the boundary onto itself [9]. Bibliographies of publications and the theory of abstract nonlocal elliptic problems are contained in [4, 71. In [2] the Laplace equation was considered with boundary conditions connecting the values of the unknown function on a manifold r, c aQ (Q c R”) with its values on some manifold o(T, ) c Q and Dirichlet conditions on 13Q\f,. The solvability of this problem in various functional spaces was studied in [16-18,201 and elsewhere. In [ 17,201 it was shown that in the case o(F,) n (r,\r,) # 0 the Dirichlet problem with arbitrary small coefficients in nonlocal terms can have a negative index. In [lo] the formula of index for some nonlocal elliptic boundary value problems of second order in a plane domain was obtained. In the present paper we consider elliptic equations of order 2m in a bounded domain Q c R with nonlocal conditions connecting the traces of the unknown function and its derivatives on the manifolds rj (IJ, F, = 8Q) with its traces on some compacturn a c Q. In the case a c Q (Section 2) and in the case Sz n K = 0 (Section 3) we prove the stability of index of nonlocal problems in Sobolev spaces and weighted spaces, where Xi = Uj (T,\r,). In particular the set D can consist of one or several manifolds (see [2, 163). In Section 4 it is established that in the case Q n Xi = 0 index of the equation of second order, with Dirichlet conditions containing nonlocal perturbations, is equal to zero. The elliptic problems with nonlocal conditions are closely associated with boundary value problems for elliptic functional differential equations [ 171. The theory of functional differential equations for the functions of one variable was reflected in the publications most completely [IS, 131. The 323 0022-247X/91 $3.00

J-nonexpansive mappings in uniform spaces and applications

The purpose of the paper is to introduce a class of “j-nonexpansive” mappings and to prove fixed point theorems for such mappings. They naturally arise in the existence theory of functional differential equations. These mappings act in spaces without specific geometric properties as, for instance, uniform convexity. Critical examples are given.

Stabilization of large-scale non-linear infinite delay systems

The problem of stabilizing a class of large-scale non-linear infinite delay systems is considered. It is shown that exponential stability can be achieved by a suitable choice of state feedback controls of large-scale systems with infinite delay, and that a Lyapunov functional can be constructed by using certain results from J. K. Hale's Theory of Functional Differential Equations (1977).

Semigroups of translations and functional differential equations

where x(t) E R” and x,, cp are elements of a space of functions c-r, O] -+ R”, called the state space. We will define a very general state space in the next section and discuss its properties. A solution of (l.l), if it exists, is a continuous extention to the right and satisfies Eq. (1.1). In case r = co we put [-r, 0] = (co, 0] and call (1.1) a functional differential equation with infinite delay. For r = 0, (1.1) becomes an ordinary differential equation. If the function f in (1.1) does not explicitly depend on t we call the functional differential equation (1.1) autonomous. In this paper we study the nonautonomous case with infinite delay. So let f(t, cp) be defined for t E R and cp in a space X of functions (-co, 0] + R”. The choice of the state space X depends, of course, on the special situation one wants to describe by (1.1). The fundamental theory of functional differential equations, however, (e.g., existence and uniqueness of solutions, connection with the theory of semigroups) should be formulated in most general state spaces [S, 7, 12, lo]. In this paper we study a state space of classes of functions. We start with a linear space 3 of Lebesgue-measurable functions (00, 0] -+ R” with a 417 0022-247X/85 $3.00

Stability Theory for Functional Differential Equations

Stability Theory for Functional Differential Equations

Theory of Functional Differential Equations (Jack Hale)

Theory of Functional Differential Equations (Jack Hale)