Roger D. Nussbaum

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We investigate the iterative behaviour of continuous order preserving subhomogeneous maps f :K→K, where K is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of f converges to a periodic orbit and, moreover, the period of each periodic point of f is bounded by βN = max q+r+s=N N ! q!r!s! = N ! ⌊ N 3 ⌋ ! ⌊ N + 1 3 ⌋ ! ⌊(More)
In this paper we begin a study of the differential-delay equation ex ' ( t ) = x ( t ) + f ( x ( t r ) ) , r = r ( x ( t ) ) . We prove the existence of periodic solutions for 0 < e < e0, where e0 is an optimal positive number. We investigate regularity and monotonicity properties of solutions x ( t ) which are defined for all t and of associated functions(More)
(Here, we use the notation x > y and x 2 y to mean, respectively, x y E k” and x y E K”.) This result extends an earlier theorem of Smith [2] concerning “discrete dynamics of monotone, concave maps”; some interesting applications to differential equations can be found in [l, 21. Another extension of Smith’s theorem has been given by TakaE in [3]. In this(More)
We consider a class of singularly perturbed delay-differential equations of the form e ’ xðtÞ 1⁄4 f ðxðtÞ; xðt rÞÞ; where r 1⁄4 rðxðtÞÞ is a state-dependent delay. We study the asymptotic shape, as e-0; of slowly oscillating periodic solutions. In particular, we show that the limiting shape of such solutions can be explicitly described by the solution of a(More)
We study the nonlinear eigenvalue problem f(x) = λx for a class of maps f : K → K which are homogeneous of degree one and order-preserving, where K ⊆ X is a closed convex cone in a Banach space X. Solutions are obtained, in part, using a theory of the “cone spectral radius” which we develop. Principal technical tools are the generalized measure of(More)
By an asymptotic fixed point theorem we mean a theorem in functional analysis in which the existence of fixed points of a map y is established with the aid of assumptions on the iterates fn of /. We prove below some new theorems of this type, and we obtain as corollaries results of F. E. Browder, G. Darbo, R. L. Frum-Ketkov, W. A. Horn and others. We also(More)
Let C(S) denote the Banach space of continuous, real-valued maps f : S -+ IR and let A denote a positive linear map of C(S) into itself. We give necessary conditions tha t the operator A have a s tr ict ly positive periodic point of minimal period m. Under mild compactness conditions on the operator A, we prove tha t these necessary conditions are also(More)
Polyphenolic extracts of the principal flavonoid classes present in cranberry were screened in vitro for cytotoxicity against solid tumor cells lines, identifying two fractions composed principally of proanthocyanidins (PACs) with potential anticancer activity. Matrix-Assisted Laser Desorption/Ionization Time-Of-Flight Mass Spectrometry (MALDI-TOF-MS)(More)