J.R. Haddock

Liapunov-Razumikhin functions and an invariance principle for functional differential equations

The purpose of this paper is to present a general theory for Liapunov (Razumikhin) functions, and autonomous systems of functional differential equations. Fundamental to this theory is the introduction and development of a new “invariance principle.” During the past two decades, invariance principles have been established for various dynamical systems and, in particular, for several areas of differential equations (cf. [ 1 I] and references therein). For the most part, these principles have been motivated by the work of LaSalle in the 1960’s on Liapunov functions and ordinary differential equations (see, for example, [lOI)* The invariance principles that have previously been given, have two main ingredients in common. First of all, they exploit the fact that limit sets of solutions possess an invariance property. Secondly, they rely on the use of certain auxiliary (Liapunov) functions which are nonincreasing along solutions. The invariance principle and related theory that is developed in this paper also exploits, in addition to these, the same invariance property of limit sets. However, the auxiliary functions that we employ do not enjoy the

On the Location of Positive Limit Sets for Autonomous Functional Differential Equations with Infinite Delay

In 1983, the authors published a paper [lo] on an invariance principle for finite delay functional differential equations (FDEs). In [9], T. Krisztin and the authors extended many of the ideas of [lo] to include FDEs with infinite delay. The basic results in both [9] and [lo] centered around the use of Liapunov-Razumikhin techniques and the location of positive limit sets of precompact orbits. The main purpose of this paper is to extend the results of [9] and to provide several examples to illustrate the main theorems. Of particular interest will be to establish asymptotic constancy of solutions of equations for which each constant function is itself a solution. This idea was one of the main themes in [lo] but was not examined in [9]. Likewise, we will see how a family of spaces (as opposed to a single phase space) can be useful in examining certain equations (cf. Example 3.4). In considering the use of limit sets in conjunction with Razumikhin techniques and invariance principles for infinite delay equations, there are several difficulties that are encountered. These include: (i) choice of space(s); (ii) precompactness of orbits; and (iii) appropriate comparison principles. These topics are discussed individually in [ 11, [7], and [8],

Asymptotic constancy for pseudo monotone dynamical systems on function spaces

Abstract A pseudo monotone dynamical system is a dynamical system which preserves the order relation between initial points and equilibrium points. The purpose of this paper is to present some convergence, oscillation, and order stability criteria for pseudo monotone dynamical systems on function spaces for which each constant function is an equilibrium point. Some applications to neutral functional differential equations and semilinear parabolic partial differential equations with Neumann boundary condition are given.

Asymptotic Theory for a Class of Nonautonomous Delay Differential Equations

This paper deals with asymptotic behavior of solutions of the nonlinear nonautonomous delay differential equation x′(t) = −∝t − r(t)t f(t, x(s))dμ(t, s), (su∗) where xf(t, x) ⩾ 0, f(t, 0) = 0, t − r(t) nondecreasing, μ(t, s) is nondecreasing and of bounded variation. General sufficient conditions, which are easy to verify, are obtained for the solutions to be bounded and asymptotically stable (locally and globally). These results improve many existing ones principally by allowing: (i) r(t) to be unbounded, (ii) both discrete and distributed delays, and (iii) the equation to be strongly nonlinear and nonautonomous. Various examples are given in the form of corollaries with a highly flexible integrand.

Friendly Spaces for Functional Differential Equations with Infinite Delay

Publisher Summary This chapter analyzes friendly spaces for functional differential equations (FDE) with infinite delay. Several interesting theorems and techniques for FDEs with infinite delay have been developed. One of the purposes of the chapter is to provide an update of some of the results that have been obtained, and discusses certain phase spaces (that is, spaces of initial functions) that are “friendly” with respect to current research involving existence, comparison theorems, convergence of solutions, and periodic solutions. The choice of phase space for FDEs with finite delay (or for ordinary differential equations—ODES) is standard. However the situation for infinite delay equations is quite different. For this case, there are several possibilities, and an underlying space usually is chosen in connection with the particular equation at hand. The chapter discusses some results on admissible spaces. A convergence results is also presented, and existence of a periodic solution is described.

On Liapunov Functions for Nonautonomous Systems

During the last decade LaSalle developed a general theory for examining the behavior of solutions of systems of differential equations with respect to certain sets (see [4] and its references). His main tool was to employ properties of the derivative of a Liapunov function in order to locate limit sets of soiu- tions of a given system. Although the most significant results were given for autonomous systems for which an invariance principle can be applied, Yoshizawa and LaSalle have proved some useful results for nonautonomous systems (see [4, 61). Th e p u rp ose of this paper is to give an improved result for nonautonomous systems. Let x’ =f(t, x) (’ = dldt), (1) where f: [0, co)

A VARIATION OF RAZUMIKHIN'S METHOD FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

Publisher Summary This chapter discusses a variation of Razumikhins method for retarded functional differential equations. Although Liapunov functionals are theoretically perhaps the natural way to study various stability properties of functional differential equations, Liapunov functions have played a prominent role in such investigations. The idea of employing Liapunov functions to functional differential equations was apparently first conceived by Razumikhin in 1960. In 1962, Driver helped to clarify some of the ideas of the Razumikhin method and introduced them to the non-Russian reader. One of the basic techniques involved in this classical method has been to examine inequalities in connection with the derivative of a Liapunov function. This derivative is often examined with respect to a certain subset C0 of the general solution space C. This chapter presents a variation of the Razumikhin method to gain information concerning the behavior of solutions of functional differential equations.

Stability Theory for Nonautonomous Systems

This chapter describes the stability theory for nonautonomous systems. The nonautonomous system of differential equations x′ = ƒ ( t, x ), where ƒ: [0 , ∞ ) × R n → R n is continuous. The Liapunov theory is employed to determine behavior of solutions relative to some closed set H ⊆ R n . In general, no invariance principle is readily available. The chapter presents results that improve two of the classical Liapunov theorems. It also presents an improvement of the LaSalle–Yoshizawa theorem for nonautonomous systems for solutions approaching a set and also an improvement of the Marachkoff asymptotic stability theorem. It is evident that one relies heavily on a close relationship between the derivative of a Liapunov function and the norm of the right-hand side equation. It is found that all bounded solutions tend to H or all solutions in the stability region tend to zero.

Fundamental inequalities and applications to neutral equations

Abstract Two fundamental inequalities are derived for neutral functional differential equations with stable D-operator. These inequalities provide some exponential estimates about the relation between solution operators and D-operators, an essential characterization of neutral functional differential equations, and an effective tool for the application of Liapunovs direct method and Razumikhin techniques to boundedness, stability, and convergence of solutions.