The Kakutani's precompactness lemma

In this paper we establish a theorem that extends and sharpens an old precompactness lemma due to Kakutani. We use this theorem to derive the classical Arzelà-Ascoli theorem and a theorem of Defant and Floret for families of linear operators. We also use this theorem to derive a theorem for composition operators which yields as immediate corollaries a theorem of Geue and a locally convex version of a theorem of Aron and Schottenloher.     

An algebraic certificate for Budan’s theorem

The existing algorithms to construct the real closure of an ordered field involve very high complexities. These algorithms are based on Sturm’s theorem which we suspect to be one reason for the complexities since all known proofs of Sturm’s theorem use Rolle’s theorem which is problematic in a constructive context.Therefore we propose to replace the use of Sturm’s theorem by Budan’s theorem. In this paper we present as a first step in this direction an algebraic certificate for Budan’s theorem. An algebraic certificate is a certain kind of proof of a statement. In particular, it is an algorithm which produces, from an arbitrary data in the premise of the statement, explicit (in)equalities which express the conclusion.     

An extension of Suffridge's convolution theorem

We present an extension of Suffridge's convolution theorem for polynomials with restricted zeros on the unit circle. We also discuss a possible extension of the theorem of Laguerre for those polynomials and give an answer to a long-standing open question by Suffridge regarding an extension of the theorem of Gauß-Lucas.     

An Extension of Set-Valued Contraction Principle for Mappings on a Metric Space with a Graph and Application

In this paper, we obtain an existence theorem for fixed points of contractive set-valued mappings on a metric space endowed with a graph. This theorem unifies and extends several fixed point theorems for mappings on metric spaces and for mappings on metric spaces endowed with a graph. As an application, we obtain a theorem on the convergence of successive approximations for some linear operators on an arbitrary Banach space. This result yields the well-known Kelisky–Rivlin theorem on iterates of the Bernstein operators on C[0,1].     

基于广义逆的矩阵Padé逼近的De Montessus-De Ballore型收敛性定理

基于广义逆的矩阵Padé 逼近^[4,5]的一个行收敛性定理,即著名的De Montessus-De Ballore回收敛定理在本文首次得以建立,根据这一结果,唯一性定理被简洁地证明,并获得一个实用的存在性定理,     

Implicit Multifunction Theorems

We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson–Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson–Ursescu theorem, we also establish a refined version of the multidirectional mean value inequality which is of independent interest.     

A variant of Newton's method and its application

This paper details an existence and uniqueness theorem for solving an operator equation of the form F(x)=0, where F is a Gateaux differentiable operator defined on an open convex subset of a Banach space proved. From the main theorem, an earlier theorem of Argyros follows as a consequence. Other corollaries constitute the semilocal versions of the theorems due to Ozban and Weerakoon and Fernando in a general Banach space. Our main theorem leads to the existence of solutions for a class of nonlinear Urysohn-type integral equations in the n-dimensional Euclidean space.     

Analytic proofs of a network feasibility theorem and a theorem of Fulkerson

We provide an elementary proff of Fulkerson's theorem which gives the permutation matrices as extreme points of a certain unbounded convex polyhedron. An adaptation of the proof also establishes an analogous feasibility theorem for network flows which has Fulkerson's theorem as a corollary.     

A relative interpolation theorem for infinitary universal Horn logic and its applications

In this paper we deal with infinitary universal Horn logic both with and without equality. First, we obtain a relative Lyndon-style interpolation theorem. Using this result, we prove a non-standard preservation theorem which contains, as a particular case, a Lyndon-style theorem on surjective homomorphisms in its Makkai-style formulation. Another consequence of the preservation theorem is a theorem on bimorphisms, which, in particular, provides a tool for immediate obtaining characterizations of infinitary universal Horn classes without equality from those with equality. From the theorem on surjective homomorphisms we also derive a non-standard Beth-style preservation theorem that yields a non-standard Beth-style definability theorem, according to which implicit definability of a relation symbol in an infinitary universal Horn theory implies its explicit definability by a conjunction of atomic formulas. We also apply our theorem on surjective homomorphisms, theorem on bimorphisms and definability theorem to algebraic logic for general propositional logic.     

A Newton-like method and its application

In this paper we prove an existence and uniqueness theorem for solving the operator equation F(x)+G(x)=0, where F is a Gateaux differentiable continuous operator while the operator G satisfies a Lipschitz-condition on an open convex subset of a Banach space. As corollaries, a theorem of Tapia on a weak Newton's method and the classical convergence theorem for modified Newton-iterates are deduced. An existence theorem for a generalized Euler-Lagrange equation in the setting of Sobolev space is obtained as a consequence of the main theorem. We also obtain a class of Gateaux differentiable operators which are nowhere Frechet differentiable. Illustrative examples are also provided.     

On solutions of generalized complementarity and eigenvector problems

From a new Fan–Browder type fixed point theorem due to the second author, we deduce an existence theorem for a solution of an equilibrium problem in Section 3. This theorem is applied to generalized complementarity problems in Section 4 and to eigenvector problems in Section 5.     

Two extension theorems. Modular functions on complemented lattices

We prove an extension theorem for modular functions on arbitrary lattices and an extension theorem for measures on orthomodular lattices. The first is used to obtain a representation of modular vector-valued functions defined on complemented lattices by measures on Boolean algebras. With the aid of this representation theorem we transfer control measure theorems, Vitali-Hahn-Saks and Nikodým theorems and the Liapunoff theorem about the range of measures to the setting of modular functions on complemented lattices.     

Induced partition properties of combinatorial cubes

An induced version of the partition theorem for parameter-sets of R. L. Graham and B. L. Rothschild (Trans. Amer. Math. Soc.159 (1971), 257–291) is proven. This theorem generalizes the Graham-Rothschild theorem in the same way as the partition theorem for finite hypergraphs (F. G. Abramson and L. A. Harrington, J. Symblic Logic43 (1978), 572–600 and J. Ne?et?il and V. Rödl; J. Combin. Theory Ser. A22 (1977), 289–312; 34 (1983), 183–201) generalizes Ramsey's theorem. Some applications are given, e.g., an induced version of the Rado-Folkman-Sanders theorem and an induced version of the partition theorem for finite Boolean lattices. Also it turns out that the partition theorem for finite hypergraphs is an easy consequence of the induced partition theorem for parameter-sets.     

A generalization of Dirichlet's unit theorem

In this paper, we obtain a unit theorem for algebraic tori defined over an algebraic number field, which generalizes Dirichlet's unit theorem as well as the S-unit theorem due to Hasse and Chevalley.     

Asymptotic existence theorems for frames and group divisible designs

In this paper, we establish an asymptotic existence theorem for group divisible designs of type mn with block sizes in any given set K of integers greater than 1. As consequences, we will prove an asymptotic existence theorem for frames and derive a partial asymptotic existence theorem for resolvable group divisible designs.     

Cauchy problems for Hilfer fractional evolution equations on an infinite interval

It is well known that the classical Ascoli-Arzelà theorem is powerful technique to give a necessary and sufficient condition for investigating the relative compactness of a family of abstract continuous functions, while it is limited to finite compact interval. In this paper, we shall generalize the Ascoli-Arzelà theorem on an infinite interval. As its application, we investigate an initial value problem for fractional evolution equations on infinite interval in the sense of Hilfer type, which is a generalization of both Riemann-Liuoville and Caputo fractional derivatives. Our methods are based on the Hausdorff theorem, classical/generalized Ascoli-Arzelà theorem, Schauder fixed point theorem, Wright function, and Kuratowski measure of noncompactness. We obtain the existence of mild solutions on an infinite interval when the semigroup is compact as well as noncompact.     

A remark on Barth’s connectivity theorem

It has been remarked by Hartshorne, that Barth’s theorem for a smooth projective X follows from the strong Lefschetz theorem for the cohomology of X. Using the strong Lefschetz theorem for intersection cohomology, we give an extension of Barth’s theorem to singular X. This naturally raises several questions concerning possible Barth theorems on the level of intersection cohomology.     

An inequality for the group chromatic number of a graph

We give an inequality for the group chromatic number of a graph as an extension of Brooks’ Theorem. Moreover, we obtain a structural theorem for graphs satisfying the equality and discuss applications of the theorem.     

On estimates for integral solutions of linear inequalities

Recently, Bombieri and Vaaler obtained an interesting adelic formulation of the first and the second theorems of Minkowski in the Geometry of Numbers and derived an effective formulation of the well-known “Siegel’s lemma” on the size of integral solutions of linear equations. In a similar context involving linearinequalities, this paper is concerned with an analogue of a theorem of Khintchine on integral solutions for inequalities arising from systems of linear forms and also with an analogue of a Kronecker-type theorem with regard to euclidean frames of integral vectors. The proof of the former theorem invokes Bombieri-Vaaler’s adelic formulation of Minkowski’s theorem.     

Uncertainty principles for the Cherednik transform

We shall investigate two uncertainty principles for the Cherednik transform on the Euclidean space $\mathfrak a$ ; Miyachi??s theorem and Beurling??s theorem. We give an analogue of Miyachi?? theorem for the Cherednik transform and under the assumption that $\mathfrak a$ has a hypergroup structure, an analogue of Beurling??s theorem for the Cherednik transform.