A closed convex subset X of a Banach space E is said to have (i) asymptotic normal structure if for each bounded closed convex subset C of X containing more than one point and for each sequence in C satisfying ‖x n − x n + 1‖ → 0 as n → ∞, there is a point x ∈ C such that ; (ii) close-to-normal structure if for each bounded closed convex subset C of X containing more than one point, there is a point x ∈ C such that ‖x − y‖ < diam‖ ‖(C) for all y ∈ C While asymptotic normal structure and close-to-normal structure are both implied by normal structure, they are not related. The example that a reflexive Banach space which has asymptotic normal structure but not close-to normal structure provides us a non-empty weakly compact convex set which does not have close-to-normal structure. This answers an open question posed by Wong in [9] and hence also provides us a Kannan map defined on a weakly compact convex set which does not have a fixed point.

Let (X, , μ) be a σ-finite measure space, p fixed, 1 < p < ∞, T a linear operator of L p(X,μ), {αi } a sequence of complex numbers. If

exists and is finite a.e. we say the individual weighted ergodic theorem holds for T with the weights {αi }

In this paper we show that if {αi } is a bounded Besicovitch sequence and T is a Dunford-Schwartz operator (i.e.: ||T||1≤1, ||T||∞≤1) then the individual weighted ergodic theorem holds for T with the weights {αi }.

Let X and Y be Banach spaces, L(X, Y) the space of bounded linear operators from X to Y and C(X, Y) its subspace of the compact operators. A sequence {T i} in C(X, Y) is said to be an unconditional compact expansion of T ∈ L (X, Y) if ∑ Tix converges unconditionally to Tx for every x ∈ X. We prove: (1) If there exists a non-compact T ∈ L(X, Y) admitting an unconditional compact expansion then C(X, Y) is not complemented in L(X, Y), and (2) Let X and Y be classical Banach spaces (i.e. spaces whose duals are some LP(μ) spaces) then either L(X, Y) = C(X, Y) or C(X, Y) is not complemented in L(X, Y).

Let K be a closed, bounded, convex, nonempty subset of a Hilbert Space . It is shown that if is a left reversible, uniformly k-lipschitzian semigroup of mappings of K into itself, with k < √2, then has a common fixed point in K.

If X is a topological space then exp X denotes the space of non-empty closed subsets of X with the Vietoris topology and λX denotes the superextension of X Using Martin's axiom together with the negation of the continuum hypothesis the following is proved: If every discrete subset of exp X is countable the X is compact and metrizable. As a corollary, if λX contains no uncountable discrete subsets then X is compact and metrizable. A similar argument establishes the metrizability of any compact space X whose square X × X contains no uncountable discrete subsets.

The purpose of the note is to calculate the group K 0(A) where A is a ring all of whose Pierce stalks are simple artinian. This generalizes known results for A self injective of type I n. For A regular, with injective hull, Â, of type I n, a characterization is given for when A → Â it induces an isomorphism K 0(A) → K 0(Â).

Let Λn be the kernel of the map from GL(n, ℤ) to GL(n, ℤ2) induced by the quotient map ℤ → ℤ2. We give a presentation for Λn .

Comparison theorems are developed for the pair of first order Riccati equations (1) and (2) . The comparisons are of an integral type and involve an auxiliary function μ. Applications are given to disconjugacy theory for self-adjoint equations of the second and fourth order.

Let D be a division algebra whose class [D] is in B(K), the Brauer group of an algebraic number field K. If [D⊗K L] is the trivial class in B(L), then we say that L is a splitting field for D or L splits D. The splitting fields in D of smallest dimension are the maximal subfields of D. Although there are infinitely many maximal subfields of D which are cyclic extensions of K; from the perspective of the Schur Subgroup S(K) of B(K) the natural splitting fields are the cyclotomic ones. In (Cyclotomic Splitting Fields, Proc. Amer. Math. Soc. 25 (1970), 630-633) there are errors which have led to the main result of this paper, namely to provide necessary and sufficient conditions for (D) in S(K) to have a maximal subfield which is a cyclic cyclotomic extension of K, a finite abelian extension of Q. A similar result is provided for quaternion division algebras in B(K).

Consider the nth order (n ≥ 1) delay differential inequalities and and the delay differential equation , where q(t) ≥ 0 is a continuous function and p, τ are positive constants. Under the condition pτe ≥ 1 we prove that when n is odd (1) has no eventually positive solutions, (2) has no eventually negative solutions, and (3) has only oscillatory solutions and when n is even (1) has no eventually negative bounded solutions, (2) has no eventually positive bounded solutions, and every bounded solution of (3) is oscillatory. The condition pτe > 1 is sharp. The above results, which generalize previous results by Ladas and by Ladas and Stavroulakis for first order delay differential inequalities, are caused by the retarded argument and do not hold when τ = 0.

If an automorphism a on a ring R (with 1) leaves no non-trivial proper ideals of R invariant then we say that R is σ-simple. We construct examples of σ-simple rings and prove that finitely generated σ-simple algebras over fields are regular. A geometric interpretation of these concepts is also discussed.

In this paper we continue our study of the solvability of nonlinear equations involving uniform limits of A-proper and pseudo A-proper maps under a new growth condition (1) that we began in [14,15]. Applications of our results to quasimonotone, ball-condensing pertubations of c -accretive maps and maps of semibounded variation and of type (M) are also given.

Let Ĥ denote the class of functions analytic in |z| < 1 which are in every H p class, 0 < p < ∞. The class Ĥ strictly contains H ∞ and consists of those functions that are ‘almost in H ∞’ in the sense of integration. L. Hansen and W. Hayman have given simple geometric conditions for a function to belong to Ĥ. The purpose of this note is to show that Hansen and Hayman's conditions are far from necessary. Using techniques from normal functions, gap series, characterizations of BMOA, subordination, Bloch functions, and VMOA, six completely different examples of functions in Ĥ are given which ‘fill the plane’.

This paper gives a new characterization of the dimension of a normal Hausdorff space, which joins together the Eilenberg-Otto characterization and the characterization by finite coverings. The link is furnished by the notion of a system of faces of a certain type (N 1,..., N K ), where N 1,..., N K , K are natural numbers. It is shown that a space X contains a system of faces of type (N 1,..., N K ) if and only if dim(X) ≥ N 1 + … + N K . The two limit cases of the theorem, namely N k = 1 for 1 ≤ k ≤ K on the one hand, and K = 1 on the other hand, give the two known results mentioned above.

Dans un espace de Banach complexe (X, | |) on considère un opérateur linéaire borné A de spectre σ(A) et de rayon spectral r(A) = 1. On établit des conditions, en termes du spectre périphérique de A: σπ(A) = {λ ∊ σ(A): |λ| = 1}, qui garantissent l'existence d'une norme | |0, équivalente à | |, définie par un produit scalaire si | | l'est et telle que ‖A‖0 = Sup{|Ax|0: x|0 = 1} = 1. Si A est à itérés uniformément bornés (‖A n‖ ≤ M pour n = 1, 2, …) une telle norme peut ne pas exister.

In this note we use functional analysis arguments to prove the existence of families of analytic solutions for the singular system of complex ordinary differential equations zW′ = h(z, W).

In this note we show that the generic orthogonal stably free modules of type (2, 7) and (3, 8) have one free summand. This completes the work of other authors on free summands of orthogonal stably free modules.

It is known that if f is a real valued function on a rational vector space V, δ > 0,

1

and if f is unbounded then f(x + y) = f(x)f(y) for all x, y ∊ V. In response to a problem of E. Lukacs, in this paper we study the bounded solutions of (1). For example, it is shown that if f is a bounded solution of (1) then - δ ≤ f(x) ≤ (1 + (1 + 4δ)1/2)/2 for all x ∊ V and these bounds are optimal.

The Snake Theorem (terminology of Krein), due to Karlin in its original form, has been periodically improved. The theorem shows under appropriate conditions the existence of a function p* from a Tchebycheff space T, with a graph that alternately "touches" the graphs of functions f and g where f < g and f ≤ p * ≤ g on a compact interval [a, b]. The number of "touchings" depends upon the dimension of T. In this paper the conditions assumed are not the weakest known (see Gopinath and Kurshan, J. of Approximation Theory 21 (1977), 151–173), but the apparently new proof offered is elementary and fairly short. f and g are not assumed continuous.

Writing ϕl(n) = ϕ(ϕl−1(n)), the integers n which are solutions of n = kϕl (n) are considered. The set of all such n is completely characterized.