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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)

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)

The Kuratowski measure of noncompactness α on an infinite dimensional Banach space (X, ·) assigns to each bounded set S in X a non-negative real number α(S) by the formula In general a map β which assigns to each bounded set S in X a nonnegative real number and which shares most of the properties of α is called a homogeneous measure of noncompactness or… (More)