Roger D. Nussbaum

Some remarks on a conjecture in parameter adaptive control

Abstract A.S. Morse has raised the following question: Do there exist differentiable functions f:R 2 → R and g:R 2 → R with the property that for every nonzero real number λ and every (x0, y0) ∈ R 2 the solution (x(t),y(t)) of x (t) = x(t) + λf(x(t),y(t)) , y (t) = g(x(t),y(t)) , x(0) = x 0 , y(0) = y 0 , is defined for all t ⩾ 0 and satisfies lim t → + ∞ and y(t) is bounded on [0,∞)? We prove that the answer is yes, and we give explicit real analytic functions f and g which work. However, we prove that if f and g are restricted to be rational functions, the answer is no.

The fixed point index for local condensing maps

SummaryWe define below a fixed point index for local condensing maps f defined on open subset of «nice» metricANR’s. We prove that all the properties of classical fixed point index for continuous maps defined in compact polyhedra have appropriate generalizations. If our map is compact (a special case of a condensing map) and defined on an open subset of a Banach space, we prove that our fixed point index agrees with Leray-Schauder degree.

Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation

SummaryThe singularly perturbed differential-delay equation

A differential-delay equation arising in optics and physiology

\varepsilon \dot x(t) = - x(t) + f(x(t - 1))

Convexity and log convexity for the spectral radius

is studied. Existence of periodic solutions is shown using a global continuation technique based on degree theory. For small ɛ these solutions are proved to have a square-wave shape, and are related to periodic points of the mappingf:R→R.Whenfis not monotone the convergence of x(t) to the square-wave typically is not uniform, and resembles the Gibbs phenomenon of Fourier series.

A limit set trichotomy for self-mappings of normal cones in banach spaces

Abstract We are interested here in studying the closure S of the set {( x , α ): F ( x , α ) = x and x ≠ 0} for a certain class of nonlinear operators F such that F (0, α ) = 0 for all α in an interval of real numbers. Our main abstract theorem establishes under certain conditions on F the existence of a closed, connected unbounded subset S 0 of S which contains a point (0, α 0 ) and no points of the form (0, α) for α ≠ α 0 . The novelty of this result is that it is obtained under hypotheses of the type used in asymptotic fixed point theorems; in our applications F is not, in general, Frechet differentiable at points (0, α) and may not even be continuous at such points. The abstract theorem is then applied to study the structure of the set of periodic solutions of the equation x ′( t ) = − αf ( x ( t − 1)) and to obtain sharp results on the range of periods as α varies.

Cranberry proanthocyanidins are cytotoxic to human cancer cells and sensitize platinum-resistant ovarian cancer cells to paraplatin.

In recent papers the authors have studied differential-delay equations

Positive solutions of nonlinear elliptic boundary value problems

E_\varepsilon