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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)

The purpose of this note is to indicate some applications of a new fixed point theorem to the question of periodic solutions of nonlinear autonomous functional differential equations. The techniques developed give the standard periodicity examples in the literature and some new results, notably for the neutral case, which do not seem accessible by previous… (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 like tl (t) = t-r(x(t)).… (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)

For Σ a compact subset of C symmetric with respect to conjugation and f : Σ → C a continuous function, we obtain sharp conditions on f and Σ that insure that f can be approximated uniformly on Σ by polynomials with nonnegative coefficients. For X a real Banach space, K ⊆ X a closed but not necessarily normal cone with K − K = X, and A : X → X a bounded… (More)

- Roger Nussbaum
- 2005

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)