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We study the arithmetic three-spheres theorems for subsolutions of subelliptic PDEs of p p -harmonic type in Carnot groups of Heisenberg type for 1 > p > ∞ 1>p>\infty . In the presentation we exhibit the special cases of sub-Laplace equations ( p = 2 p=2 ) and the case p p is equal to the homogeneous dimension of a Carnot group. Corollaries include asymptotic behavior of subsolutions for small and large radii and the Liouville-type theorems.

In this paper, we are concerned with the existence and multiplicity of positive periodic solutions for first-order vector differential equations. By using the Leray-Schauder alternative theorem and the Kransnosel’skii fixed point theorem, we show that the differential equations under the periodic boundary value conditions have at least two positive periodic solutions.

We prove a fixed point theorem that is a very simple generalization of the Banach contraction principle and characterizes the metric completeness of the underlying space. We also discuss the Meir-Keeler fixed point theorem.

The Kato spectrum of an operator is deployed to give necessary and sufficient conditions for Browder’s theorem to hold.

We study quasiregular mappings from a punctured Euclidean ball into n n -manifolds with many ends and prove, by using Harnack’s inequality, a version of the big Picard theorem.

W. T. Gowers' c 0 theorem asserts that for every Lipschitz function F : S c0 → R and e > 0, there exists an infinite-dimensional subspace Y of c 0 such that the oscillation of F on Sy is at most e. The proof of this theorem has been reduced by W. T. Cowers to the proof of a new Ramsey type theorem. Our aim is to present a proof of the last result.

By a well-known theorem of Lohwater and Piranian, for any set E E on | z | = 1 |z|=1 of type F σ F_\sigma and of measure zero there exists a bounded analytic function in | z | > 1 |z|>1 which fails to have radial limits exactly at the points of E E . We show that this theorem is an immediate corollary of Fatou’s interpolation theorem of 1906.

The Bartle-Graves theorem extends the Banach open mapping principle to a family of linear and bounded mappings, thus showing that surjectivity of each member of the family is equivalent to the openness of the whole family. In this paper we place this theorem in the perspective of recent concepts and results, and present a general Bartle-Graves theorem for set-valued mappings. As application, we obtain versions of this theorem for mappings defined by systems of inequalities, and for monotone variational inequalities.

This paper presents a generalization of the invariant subspace theorem of Helson and Lowdenslager along the lines of de Branges’ generalization of Beurling’s theorem.

We show that a theorem of Kisyński on the generation of Banach-algebra homomorphisms of certain convolution algebras is equivalent to the Hille-Yosida theorem on the generation of operator-valued one-parameter semigroups.

Let d μ ≥ 0 d\mu \geq 0 be a Borel measure on [ 0 , ∞ ) [0,\infty ) and A n = ∫ 0 ∞ t n d μ ( t ) > ∞ ( n = 0 , 1 , 2 , . . . ) A_{n}=\int \limits _{0}^{\infty }t^{n}d\mu (t) > \infty ~~(n=0,1,2,...) be its moments. T. Carleman found sharp conditions on the magnitude of { A n } 0 ∞ \{A_{n}\}_{0}^{\infty } for d μ d\mu to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if A n ′ = ∫ 0 ∞ t n d μ 1 ( t ) A_{n}’ =\int \limits _{0}^{\infty }t^{n} d\mu _{1} (t) are the moments of another measure, d μ 1 ≥ 0 , d\mu _{1} \geq 0, with lim sup n → ∞ | A n − A n ′ | 1 n = ρ > ∞ , \limsup \limits _{n\to \infty } |A_{n}-A_{n}’|^{\frac {1}{n}}=\rho >\infty , then the measure d μ − d μ 1 d\mu -d\mu _{1} is supported on the interval [ 0 , ρ ] . [0,\rho ]. This result generalizes both the Carleman theorem and a theorem of J. Mikusiński. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem.

We consider a linear sytem in a Banach lattice and provide a simple theorem of the alternative (or Farkas lemma) without the usual closure condition.

This paper establishes a generalization of the Lusin-Privalov radial uniqueness theorem which applies to subharmonic functions in all dimensions. In particular, it answers a question of Rippon by showing that no subharmonic function on the upper half-space can have normal limit − ∞ -\infty at every boundary point.

We study the a.e. convergence of orthogonal series defined over a general measure space. We give sufficient conditions which contain the Menshov-Rademacher theorem as an endpoint case. These conditions turn out to be necessary in the particular case where the measure space is the unit interval [0, 1] and the moduli of the coefficients form a nonincreasing sequence. We also prove a new version of the Menshov-Rademacher inequality.

We generalize the classical Wolff-Denjoy theorem to certain infinitely connected Riemann surfaces. Let X X be a non-parabolic Riemann surface with Martin boundary Δ \Delta . Suppose each Martin function k y k_{y} , y ∈ Δ y\in \Delta , extends continuously to Δ ∖ { y } \Delta \setminus \{y\} and vanishes there. We show that if f f is an endomorphism of X X and the iterates of f f converge to the point at infinity, then the iterates converge locally uniformly to a point in Δ \Delta . As an application, we extend the Wolff-Denjoy theorem to non-elementary Gromov hyperbolic covering spaces of compact Riemann surfaces. Such covering surfaces are of independent interest. Finally, we use the theory of non-tangential boundary limits to give a version of the Wolff-Denjoy theorem that imposes certain mild restrictions on f f but none on X X itself.

Let A A be a semisimple Banach algebra, and let Ω A \Omega _A be its spectral unit ball. We show that every holomorphic map G : Ω A → Ω A G\colon \Omega _A\to \Omega _A satisfying G ( 0 ) = 0 G(0)=0 and G ′ ( 0 ) = I G’(0)=I fixes those elements of Ω A \Omega _A which belong to the centre of A A , but not necessarily any others. Using this, we deduce that the automorphisms of Ω A \Omega _A all leave the centre invariant. As a further application, we give a new proof of Nagasawa’s generalization of the Banach-Stone theorem.

In 1961, Simon and Ando wrote a classical paper describing the convergence properties of nearly completely decomposable matrices. Basically, their work concerned a partitioned stochastic matrix e.g. \[ A = [ A 1 a m p ; E 1 E 2 a m p ; A 2 ] A = \begin {bmatrix} A_1&E_1\ E_2&A_2\end {bmatrix} \] where A 1 A_1 and A 2 A_2 are square blocks whose entries are all larger than those of E 1 E_1 and E 2 E_2 respectively. Setting \[ A k = [ A 1 ( k ) a m p ; E 1 ( k ) E 2 ( k ) a m p ; A 2 ( k ) ] , A^k=\begin {bmatrix} A^{(k)}_1&E^{(k)}_1\ E^{(k)}_2&A^{(k)}_2\end {bmatrix}, \] partitioned as in A A , they observed that for some, rather short, initial sequence of iterates the main diagonal blocks tended to matrices all of whose rows are identical, e.g. A 1 ( k ) A^{(k)}_1 to F 1 F_1 and A 2 ( k ) A^{(k)}_2 to F 2 F_2 . After this initial sequence, subsequent iterations showed that all blocks lying in the same column as those matrices tended to a scalar multiple of them, e.g. \[ lim k → ∞ A k = [ α F 1 a m p ; β F 2 α F 1 a m p ; β F 2 ] \lim _{k\to \infty }A^k=\begin {bmatrix} \alpha F_1&\beta F_2\ \alpha F_1&\beta F_2\end {bmatrix} \] where α ≥ 0 \alpha \geq 0 and β ≥ 0 \beta \geq 0 . The purpose of this paper is to give a qualitative proof of the Simon-Ando theorem.

Recently, we introduced a space [ H μ ( A ) ; B ] [{H_\mu }(A);B] which consists of Banach space-valued distributions for which the Hankel transformation is an automorphism (The Hankel transformation of a Banach space-valued generalized function, Proc. Amer. Math. Soc. 119 (1993), 153-163). One of the cornerstones in distribution theory is the kernel theorem of Schwartz which characterizes continuous bilinear functionals as kernel operators. The object of this paper is to prove a kernel theorem which states that for an arbitrary element of [ H μ × A ; B ] [{H_\mu } \times A;B] , it can be uniquely represented by an element of [ H μ ( A ) ; B ] [{H_\mu }(A);B] and hence of [ H μ ; [ A ; B ] ] [{H_\mu };[A;B]] . This is motivated by a generalization of Zemanian (Realizability theory for continuous linear systems, Academic Press, New York, 1972) for the product space D R n × V {D_{{R^n}}} \times V where V is a Fréchet space. His work is based on the facts that the space D R n {D_{{R^n}}} is an inductive limit space and the convolution product is well defined in D K j {D_{{K_j}}} . This is not possible here since the space H μ ( A ) {H_\mu }(A) is not an inductive limit space. Furthermore, D ( A ) D(A) is not dense in H μ ( A ) {H_\mu }(A) . To overcome this, it is necessary to apply some results from our aforementioned paper. We close this paper with some applications to integral transformations by a suitable choice of A.

In an earlier paper, we began a study of linear random operators which have a certain probability of behaving as continuous operators. In this paper we study the pointwise limit in probability of a sequence of such operators, extending the Banach-Steinhaus theorem in a stochastical sense.

Let C be a nonempty closed convex subset of a uniformly convex Banach space, and let S be a semitopological semigroup such that RUC ( S ) {\text {RUC}}(S) has a left invariant submean. Then we prove a fixed point theorem for a continuous representation of S as nonexpansive mappings on C.

In this note we give a very elementary proof of Picard’s Theorem and Rickman’s Theorem which uses only Harnack’s inequality.

We study the existence of positive solutions of the equation u + a ( t ) f ( u ) = 0 {u^{}} + a(t)f(u) = 0 with linear boundary conditions. We show the existence of at least one positive solution if f f is either superlinear or sublinear by a simple application of a Fixed Point Theorem in cones.

That Antosik's Lemma is not a special case of the Antosik-Mikusinski Basic Matrix Theorem will be shown and, an equivalent form of the Antosik-Mikusinski Basic Matrix Theorem will also be presented in this paper.

By applying a transplantation theorem of Kanjin, a multiplier theorem and a Cesáro summability result are proved for multiple Laguerre expansions. In the one-dimensional case an improved version of the multiplier theorem is obtained.

By using the paradifferential calculus, Hormander's classical uniqueness theorem for the Cauchy problem is shown to hold for operators with F2 coefficients in the principal part, instead of WI , under a special normality assumption. This work is devoted to the local uniqueness of the solutions to the Cauchy problem for partial differential operators. This subject has been studied widely during the last ten years, and we refer to [1, 7] for a comprehensive bibliography. Our main aim is to give a partial answer to a question asked by Hormander [5, Chapter XXVIII], specifically, after having proved his uniqueness theorem for principally normal operators with F?o coefficients in the principal part, under a convexity condition, Hormander remarks that it is not clear how smooth the coefficients must be for this theorem to hold. We think that the requirement on the smoothness of the coefficients in the principal part essentially depends on the strength of the normality assumption. Indeed, it was already observed by Hormander [4, Chapter 8] that WI coefficients are sufficient in the case of elliptic or real principal part. In this note, we show that the assumption of W2 coefficients will work under a strong normality assumption. In order to give a precise statement, let us set the problem. Let Q be an open set of RI and let P(x, D) be the operator (1) P(x, D) = E a. (x)D'X, IaI

Based on an extension of Aumann’s measurable selection theorem due to Leese, it is shown that each fixed point theorem for F ( ω , ⋅ ) F(\omega , \cdot ) produces a random fixed point theorem for F F provided the σ \sigma -algebra Σ \Sigma for Ω \Omega is a Suslin family and F F has a measurable graph (in particular, when F F is random continuous with closed values and X X is a separable metric space). As applications and illustrations, some random fixed points in the literature are obtained or extended.

We prove a conjecture about finite fields that arose in Péterfalvi’s study of the Feit-Thompson Theorem.

Univalent logharmonic ring mappings are characterized in terms of univalent starlike mappings. An existence and uniqueness theorem is also given.

Let X X be a uniformly convex Banach space with a Frechet differentiable norm, C C a bounded closed convex subset of X X , and T : C → C T:C \to C an asymptotically nonexpansive mapping. It is shown that for each x x in C C , the sequence { T n x } \{ {T^n}x\} is weakly almost-convergent to a fixed point y y of T T , i.e., ( 1 / n ) ∑ i = 0 n − 1 T k + i x → y (1/n)\sum \nolimits _{i = 0}^{n - 1} {{T^{k + i}}x \to y} weakly as n n tends to infinity uniformly in k = 0 , 1 , 2 , … k = 0,1,2, \ldots

The n n -skeleton of a ( 2 n + 2 ) (2n + 2) -simplex does not embed in R 2 n {{\mathbf {R}}^{2n}} . This well-known result is due (independently) to van Kampen, 1932, and Flores, 1933, who proved the case p = 2 p = 2 of the following: Theorem. Let p p be a prime, and let s s and l l be positive integers such that l ( p − 1 ) ≤ p ( s − 1 ) l(p - 1) \leq p(s - 1) . Then, for any continuous map f f from a ( p s + p − 2 ) (ps + p - 2) -dimensional simplex into R l {{\mathbf {R}}^l} , there must exist p p points { x 1 , … , x p } \{ {x_1}, \ldots ,{x_p}\} , lying in pairwise disjoint faces of dimensions ≤ s − 1 \leq s - 1 of this simplex, such that f ( x 1 ) = ⋯ = f ( x p ) f({x_1}) = \cdots = f({x_p}) .

In 1979 Eric K. van Douwen announced a powerful theorem about the Stone-Čech compactification of a discrete semigroup which he called The Right Ideal Theorem. Its proof, however, was lost with his untimely death. In this paper we present a proof of the theorem and a derivation of some of its corollaries.

We show that for a class of operators which properly contains the normal operators on L 2 {L_2} , \[ 1 n ∑ i = 0 n − 1 T i f → a . e . iff 1 2 n ∑ i = 0 2 n − 1 T i f → a . e . \frac {1}{n}\sum \limits _{i = 0}^{n - 1} {{T^i}f \to a.e.} {\text {iff}}\frac {1}{{{2^n}}}\sum \limits _{i = 0}^{{2^n} - 1} {{T^i}f \to a.e.} \] This theorem is used to give an alternate form of a theorem of Gaposhkin concerning the pointwise ergodic theorem for normal operators.

We prove a stability theorem for frames. Our result is a generalization of a classical result of Paley and Wiener about Riesz bases; it is also related to the Perturbation Theorem of Kato. The classical Paley-Wiener Theorem states the following: Let {fil}ol be a basis for the Banach space B, and let {gi} =1 be a family of vectors in B. If there exists a constant A E [O; 1 [ such that

By separating the algebraic and analytic aspect of Frobenius' theorem on involutive distributions, we are able to give a simplified proof. The Frobenius theorem may be stated as follows [B, p. 161]: Theorem. An r-distribution A on an m-manifold M is involutive if and only if A is completely integrable. We recall that A is involutive if in the neighborhood of each point of M there is a local basis of vector fields {X1, X2, ... , Xr} such that r [Xi, Xj] = c jXk for ? < i, j < r, k=1 and A is completely integrable if for each point p e M there is a coordinate system u: (U, p) -* (Rmw, 0) such that A is spanned by {&/0u I, ..., 0,/0uI. Clearly, if A is completely integrable then A is involutive. Thus we assume that A is involutive. The following proof separates the algebraic and analytic aspects of the theorem. The algebraic part is completely elementary; we give a reference for the analytic part. Lemma 1. There is a coordinate system v: (V, p) -* (Rm, 0) such that A has a local basis {Xl, ... , Xr} of the form Xi= 0?+ Sb!00 for l

In this paper, by means of the technique of measures of noncompactness, we establish a generalized form of the fixed point theorem for the sum of T+S, where S is noncompact, I -T may not be injective, and T is not necessarily continuous. The obtained results unify and significantly extend a number of previously known generalizations of the Krasnosel'skii fixed point theorem. The analysis presented here reveals the essential characteristics of the Krasnosel'skii type fixed point theorem in strong topology setups. Further, the results are used to prove the existence of periodic solutions of a nonlinear neutral differential equation with delay in the critical case.

We use bordism theory to extend Lefschetz-Nielsen theory to a family of manifolds and endomorphisms. In particular, we define an invariant, and prove a parametrized fixed point theorem and its converse.

An upper bound for the length of a commutative Artinian ring in terms of the length of a faithful module is given. This generalizes a theorem of Schur and Jacobson. Long ago, in days of yore, Schur [3] proved that commutative subalgebras of M, (C) have dimension less than or equal to [n2/4] + 1 . In 1945 Jacobson [2] proved the same inequality for any field K. His proof consists of putting the matrices in the subalgebra in a "standard form" by conjugation. Later many proofs by manipulation of matrices were given. In 1976 Gustafson [1] proved the theorem by representation theoretic methods. In his paper Gustafson asked whether the theorem could be proved for Artinian rings (which are not algebras) in terms of length. Here we prove the following theorem (the proof of which is essentially Gustafson's proof of his theorem) answering the question. We thank W. Brown for bringing back to mind the problem. Theorem 1. Let A be a commutative Artinian ring and let M be a faithful A-module of length n. Then l(A) < [n2/4] + 1. (Here 1 denotes the length function and [x] denotes the integral part of x.) Proof. We may assume by going to the components of A that A is a local ring with maximal ideal m. Lemma 2. Let M and N be A-modules, N having finite length. Then, l(Hom(M, N)) < u(M) * l(N). Proof. Let u(M) = m. We have an exact sequence Am -, M -* 0. Homing this into N we get: Hom(M, N) is a submodule of Nm that has length mm l(N). So the lemma is proved. [Aliter: This can also be proved by induction on length of N.] To continue the. proof of the theorem, as M is a faithful A-module, m injects into Hom(M, mM). Applying the lemma we get 1(m) < l(Hom(M, mM)) < (M) * 1(mM) = u(M)* {l(M) u(M)} < [(l(M))2/4]. Whence, l(A) = 1(m) + 1 < [(l(M))2/4] + 1. Received by the editors November 14, 1991. 1991 Mathematics Subject Classification. Primary 13E1 0; Secondary 16G1 0. ( 1993 American Mathematical Society 0002-9939/93 $1.00 + $.25 per page

A sort of the Lyapounov central limit theorem for row-wise factorizable triangular arrays is obtained. Also a new version of the classical Lyapounov theorem for independent random variables, being a tool in the proof of the main result, seems to be of independent interest.

We should show that we may use a lemma of Day to prove, among others, a generalization of a common fixed point theorem of Brodskii and Milman when restricted to normed linear spaces that are uniformly convex in every direction.

A short proof is given of a result due to Avakumović. More specifically the asymptotic behavior of the solution y ( x ) → 0 ( x → ∞ ) y\left ( x \right ) \to 0\left ( {x \to \infty } \right ) of the differential equation y = ϕ ( x ) y λ ( λ > 1 ) y = \phi \left ( x \right ){y^\lambda }\left ( {\lambda > 1} \right ) in case ϕ ( t x ) / ϕ ( x ) → t σ ( x → ∞ ) , σ > − 2 \phi \left ( {tx} \right )/\phi \left ( x \right ) \to {t^\sigma }\left ( {x \to \infty } \right ),\sigma > - 2 is given.

A theorem is proved which generalizes both the Vietoris-Begle theorem and the cell-like theorem for spaces of finite defomation dimension. The proof is geometric and uses a double mapping cylinder trick.

Let H be a real Hilbert space, and let B 1 ( H ) {B_1}(H) denote the space of symmetric, bounded operators on H which have numerical range in [0, 1], topologized by the strong operator topology, and let L be a strongly continuous function on H into B 1 ( H ) {B_1}(H) . In this paper, methods are given to locate all z ∈ H z \in H which are fixed points of L in the sense that L ( z ) z = z L(z)z = z . In particular, if w ∈ H w \in H and if α \alpha and β \beta are fixed positive rational numbers with α ∈ [ 1 2 , ∞ ) \alpha \in [\tfrac {1}{2},\infty ) , a decreasing sequence of elements of B 1 ( H ) {B_1}(H) is recursively defined, and converges to Q ∈ B 1 ( H ) Q \in {B_1}(H) . If α > 1 2 \alpha > \tfrac {1}{2} , then Q is idempotent and z = Q w z = Qw is a fixed point of L, and if α = 1 2 , β ⩾ 1 2 \alpha = \tfrac {1}{2},\beta \geqslant \tfrac {1}{2} , then z = Q β w z = {Q^\beta }w is a fixed point of L.

In this paper it is proved that the two-point boundary value problem, namely ( d ( 4 ) / d x 4 + f ) y = g , y ( 0 ) − A 1 = y ( 1 ) − A 2 = y ( 0 ) − B 1 = y ( 1 ) − B 2 = 0 ({d^{(4)}}/d{x^4} + f)y = g,y(0) - {A_1} = y(1) - {A_2} = y(0) - {B_1} = y(1) - {B_2} = 0 , has a unique solution provided inf x f ( x ) = − η > − π 4 {\inf _x}f(x) = - \eta > - {\pi ^4} . The given boundary value problem is discretized by a finite difference scheme. This numerical approximation is proved to be a second order convergent process by establishing an error bound using the L 2 {L_2} -norm of a vector.

In this paper, we investigate the validity of an interesting theorem of Fan [3, Theorem 2] in cones. We prove that it is true for a continuous condensing map defined on a closed ball in cones. A more interesting case is that we prove that it is true on an annulus if suitable inner boundary conditions are posed. As applications of our theorems, some new fixed point theorems in the norm form are derived.

We show how D. Voiculescu’s proof of the Berger-Shaw trace inequality for rationally cyclic nearly hyponormal operators can be presented using only elementary operator-theoretic concepts. In addition we show that if T T is a hyponormal operator whose essential spectrum has zero area, then the question of whether [ T ∗ , T ] [{T^ * },T] is trace class depends only on the spectral picture of T T . We also show how a special case of results of Helton-Howe can be derived from the BDF theory.

The purpose of this paper is to prove individual and dominated ergodic theorems for Abel means of semigroups of positive L p {L_p} contractions, 1 > p > ∞ 1 > p > \infty .

A generalization of a fixed point theorem of Rzepecki is proved and it is shown that in a paranormed space E E this result yields, under certain circumstances, solutions to the equation x = T x + S x x = Tx + Sx for T : E → E T:E \to E either continuous and affine or a generalized contraction, and S : K ⊆ E → E S:K \subseteq E \to E compact.

We shall prove a fixed point theorem for a system of multivalued mappings which generalizes the result obtained by the author [1, Theorem 1]. For n = 1 n = 1 we obtain a generalization of results of Reich [5, Theorem 5] and Nadler [3, Theorem 5], [4, Theorem 1].