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The starting point of this paper is a theorem by J. F. C. Kingman which asserts that if the entries of a nonnegative matrix are log convex functions of a variable then so is the spectral radius of the matrix. A related result of J. Cohen asserts that the spectral radius of a nonnegative matrix is a convex function of the diagonal elements. The first section(More)
We investigate the iterative behaviour of continuous order preserving subhomo-geneous 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 β N = max q +r +s=N N ! q!r!s! = N ! N 3 ! N + 1 3 ! N(More)
a r t i c l e i n f o a b s t r a c t We study the singularly perturbed state-dependent delay-differential equation ε˙ x(t) = −x(t) − kx(t − r), r = r x(t) = 1 + x(t), (*) which is a special case of the equation ε˙ x(t) = g x(t), x(t − r) , r = r x(t). One knows that for every sufficiently small ε > 0, Eq. (*) possesses at least one so-called slowly(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 that the operator A have a strictly positive periodic point of minimal period m. Under mild compactness conditions on the operator A, we prove that these necessary conditions are also sufficient(More)
Let K be a closed, normal cone with nonempty interior int(K) in a Banach space X. Let Σ = {x ∈ int(K) : q(x) = 1} where q: int(K) → (0, ∞) is continuous and homogeneous of degree 1 and it is usually assumed that Σ is bounded in norm. In this framework there is a complete metric d, Hilbert's projective metric, defined on Σ and a complete metric d,(More)
We consider the equation ˙ x(t) = f (t, x(t), x(η(t))) with a variable time shift η(t). Both the nonlinearity f and the shift function η are given, and are assumed to be analytic (that is, holomorphic) functions of their arguments. Typically the time shift represents a delay, namely, that η(t) = t − r(t) with r(t) ≥ 0. The main problem considered is to(More)
Maps / defined on the interior of the standard non-negative cone K in R. N which are both homogeneous of degree 1 and order-preserving arise naturally in the study of certain classes of Discrete Event Systems. Such maps are non-expanding in Thompson's part metric and continuous on the interior of the cone. It follows from more general results presented here(More)
We consider a class of singularly perturbed delay-differential equations of the form e ' xðtÞ ¼ f ðxðtÞ; xðt À rÞÞ; where r ¼ 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)