T. A. Burton

A FIXED-POINT THEOREM OF KRASNOSELSKII

Abstract Krasnoselskiis fixed-point theorem asks for a convex set M and a mapping Pz = Bz + Az such that: 1. (i) Bx + Ay ∈ M for each x , y ∈ M 2. (ii) A is continuous and compact 3. (iii) B is a contraction. Then P has a fixed point. A careful reading of the proof reveals that (i) need only ask that Bx + Ay ∈ M when x = Bx + Ay . The proof also yields a technique for showing that such x is in M .

Krasnoselskii's fixed point theorem and stability

In this paper we use a fixed point theorem of Krasnoselskii to prove that the zero solution of a nonlinear ordinary differential equation is asymptotically stable. The result is applied to an equation x + f(x)x + g(x) = Kh(t, x, x). Although the discussion concerns ordinary differential equations, it can be applied equally well to functional differential equations.

Integral equations, implicit functions, and fixed points

The problem is to show that (1) V(t, x) = S(t, fo H(t, s, x(s)) ds) has a solution, where V defines a contraction, V, and S defines a compact map, S. A fixed point of Pfp = SW + (I V)p would solve the problem. Such equations arise naturally in the search for a solution of f (t, x) = 0 where f (0, 0) = 0, but 9f (0, 0)/63x = 0 so that the standard conditions of the implicit function theorem fail. Now Pp = Sp + (I V)(p would be in the form for a classical fixed point theorem of Krasnoselskii if I V were a contraction. But I V fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that I V has enough properties that an extension of Krasnoselskiis theorem still holds and, hence, (1) has a solution. This substantially improves the classical implicit function theorem and proves that a general class of integral equations has a solution.

Fixed Points and Stability of an Integral Equation: Nonuniqueness

We consider a paper of Banaś and Rzepka which deals with existence and asymptotic stability of an integral equation by means of fixed-point theory and measures of noncompactness. By choosing a different fixed-point theorem, we show that the measures of noncompactness can be avoided and the existence and stability can be proved under weaker conditions. Moreover, we show that this is actually a problem about a bound on the behavior of a nonunique solution. In fact, without nonuniqueness, the theorems of stability are vacuous.

Unified boundedness, periodicity, and stability in ordinary and functional differential equations

SummaryWe discuss a unified theory of periodicity of dissipative ordinary and functional differential equations in terms of uniform boundedness. Sufficient conditions for the uniform boundedness are given by means of Liapunov functionals having a weighted norm as an upper bound. The theory is developed for ordinary differential equations, equations with bounded delay, and equations with infinite delay.

Original Contribution: Averaged neural networks

Cohen and Grossberg studied an almost gradient system of ordinary differential equations with application to neural networks and used a Liapunov function, together with an invariance principle, to show that some equilibrium points attract solutions. Independently, Hopfield modeled a neural network by means of a system of ordinary differential equations which turn out to be a special case of the Cohen-Grossberg system, as pointed out by Cohen. In the Hopfield model it is clear that the functions involved in the equations are averages and that current will flow through a synapse only if a certain threshold is reached; however, none of the models take into account an averaging technique. Investigators have been interested in sustained oscillations in neural networks and have produced them in computer simulations when there is a pointwise delay. The linearized systems with a delay have also exhibited sustained oscillations. But our conjecture is that oscillations are not caused by a delay. This paper is intended to put substance to that conjecture by examining models with both pointwise and distributed delays. None of the models have solutions with sustained oscillations.

Perron-type stability theorems for neutral equations

Abstract In this paper, we present two Perron-type asymptotic stability results for a neutral functional differential equation of the form x′(t)=Sx(t)+Px(t−r)+ d d t Q(t,x t )+G(t,x t ), when the linear part (x′(t)=Sx(t)+Px(t−r)) is asymptotically stable. In particular, Q and G are allowed to be unbounded functions of t and Q need not be differentiable. The results are based on Krasnoselskiis fixed point theorem. It is to be emphasized that, unlike Perron, we obtain only asymptotic stability because of the unboundedness of Q and G.

Repellers in systems with infinite delay

Abstract One of the most important questions concerning interacting species is whether all the species survive in the long term. Here, models governed by a class of autonomous functional differential equations with infinite delay are considered, long term survival being interpreted in terms of the criterion of “permanent coexistence” which appears to provide a biologically realistic test. This criterion is based on the natural idea that it is enough if the boundary (corresponding to extinction of at least one species) is a repeller in a strong sense, and the purpose of the present investigation is to present a mathematical theory which often resolves this problem even when the detailed asymptotics of the system are inaccessible to analysis. Section 3 contains a result which may be of some independent interest. It is shown how to construct a space of initial functions so that when solutions are uniformly ultimately bounded, then there is a compact forward invariant set absorbing all solutions.

Stability, fixed points and inverses of delays

Abstract. The scalar equation (1) x′(t) = − ∫ t t−r(t) a(t, s)g(x(s)) ds with variable delay r(t) ≥ 0 is investigated, where t− r(t) is increasing and xg(x) > 0 (x 6= 0) in a neighborhood of x = 0. We find conditions for r, a, and g so that for a given continuous initial function ψ a mapping P for (1) can be defined on a complete metric space Cψ and in which P has a unique fixed point. The end result is not only conditions for the existence and uniqueness of solutions of (1) but also for the stability of the zero solution. We also find conditions ensuring the zero solution is asymptotically stable by changing to an exponentially weighted metric on a closed subset of Cψ . Finally, we parlay the methods for (1) into results for (2) x′(t) = − ∫ t t−r(t) a(t, s)g(s, x(s)) ds and (3) x ′(t) = −a(t)g(x(t − r(t))).

Periodic solutions of abstract differential equations with infinite delay

where L and C are linear operators (at least L is unbounded, L is sectorial) in a real Banach space B, f and F are quite smooth, F (t+T ) = F (t) and C(t+T, s+T ) = C(t, s) for some T > 0. The object is to give conditions to ensure that (E) has a T -periodic solution. When u is in a certain space, convergence properties for the integral will be required later. The work proceeds as follows. First, (E) is written as a functional differential equation with a parameter λ, together with an associated homotopy h; if the homotopy has a fixed point for λ = 1, it is a periodic solution of (E). This is the content of Section 2. In Section 3 the degree-theoretic work of Granas is summarized. This will enable us to show that if the homotopy h is compact and admissible, then the existence of an a priori bound on all possible T -periodic fixed points of h for 0 ≤ λ ≤ 1 implies the existence of a T -periodic fixed point of h for λ = 1.