Let f (z1, . . . ,zk)∈ C(Ik,I) be a given function, where I is (bounded or unbounded) subinterval of R, and k ∈ N. Assume that f (y1, y2, . . . , yk) ≥ f (y2, . . . , yk, y1) if y1 ≥ max{y2, . . . , yk}, f (y1, y2, . . . , yk) ≤ f (y2, . . . , yk, y1) if y1 ≤ min{y2, . . . , yk}, and f is nondecreasing in the last variable zk. We then prove that every bounded solution of an autonomous differenc...

Let G be a strict RS-set (resp. an RS-set) in X and let F be a bounded (resp. totally bounded) subset of X satisfying rG(F )> rX(F ), where rG(F ) is the restricted Chebyshev radius of F with respect to G. It is shown that the restricted Chebyshev center of F with respect to G is strongly unique in the case when X is a real Banach space, and that, under some additional convexity assumptions, th...

B. Preliminaries B.1. Functions of Bounded Variation. We begin by discussing the concept of bounded variation. This concept will be used to formulate our assumptions on the distribution function F in the subsection immediately following this one, namely Section B.2. More importantly, it plays an essential role in our proof of sufficiency in a much later section, namely Section P. The simplest d...

We prove that the statement “there is a k such that for every f there is a k-bounded diagonally non-recursive function relative to f” does not imply weak König’s lemma over RCA0+BΣ 0 2. This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every k-bounded diagonally non-recursive function computes a 2-bounded diagonally nonrecursive function m...

This paper deals with obtaing necessary and sufficient conditions for the existence of at least one Ψ-bounded solution for the non-homogeneous matrix difference equation X(n+1) = A(n)X(n)B(n)+F (n), where F (n) is a Ψ-bounded matrix valued function on Z+. Finally, we prove a result relating to the asymptotic behavior of the Ψ-bounded solutions of this equation on Z+.

Using methods of higher algebraic K-theory, van der Kallen proved that SLn(F [x]) does not have bounded word length with respect to elementary matrices if the field F has infinite transcendence degree over its prime subfield. A short explicit proof of this result is exhibited by constructing a sequence of matrices with infinitely growing word length. This construction is also used to show that ...

We consider bounded solutions of the semilinear heat equation ut = uxx+f(u) on R, where f is of a bistable type. We show that there always exist bounded solutions whose ω-limit set with respect to the locally uniform convergence contains functions which are not steady states. For balanced bistable nonlinearities, there are examples of such solutions with initial values u(x, 0) converging to 0 a...

For a polynomial P (ξ) in ξ in R with constant complex coefficients, the operator defined by R(z)f = F((P (·)−z)f̂), where ∧ denotes the Fourier transform and F denotes its inverse, is not bounded from L to L when z is in the spectrum of P (D). What are suitable spaces B and C so that R(z) is bounded from B to C? When P (ξ) is simply characteristic, we prove that the operator R(z) is bounded fro...

The order of positive and copositive spline approximation in the L p-norm, 1 p < 1, is studied, the main results are (1) the error of positive approximation by splines is bounded by C! 2 (f; 1=n) p if f has a nonnegative extension; (2) the order deteriorates to ! 1 if f does not have such an extension; (3) the error of copositive spline approximation is bounded by C!(f; 1=n) p ; (4) if f is als...

It is proved that if 0 < q ≤ (1− 3 √ 2 + 3 √ 4)/3, then the zero function is the only solution f : IR → IR of (1) satisfying (2) and bounded in a neighbourhood of at least one point of the set (3). The paper concerns bounded solutions f : IR → IR of the functional equation f(qx) = 1 4q [f(x − 1) + f(x + 1) + 2f(x)] (1) such that f(x) = 0 for | x |> Q (2) where q is a fixed number from the open ...