Bolzano's Theorems for Holomorphic Mappings(To Ham Brezis,with friendship and admiration)

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions. A more complicated proof, using Cauchy's argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from C~n to C~n are obtained using Brouwer degree.     

Singular second order boundary value problems on purely discrete time scales

We study singular discrete second order boundary value problems with mixed boundary conditions over a finite interval. We prove the existence of a positive solution by means of the lower and upper solutions method and the Brouwer fixed point theorem in conjunction with perturbation methods to approximate regular problems.     

The full power of the half-power

We use the complex square root to define a very simple homotopic invariant over the non-vanishing functions defined on the circle. As a consequence we provide easy proofs of the plane Brouwer fixed point theorem and the Fundamental Theorem of Algebra. The relation of this new invariant with the winding number and the Brouwer degree will be fully unveiled.     

几个多复变数加权全纯函数空间的边界值与对偶性

该文将几个全纯函数空间的定义从单复变数推广到多复变数, 这些空间中全纯函数的增长性依赖于某个权函数. 作者研究了它们的增长性与边界值的关系以及这些空间相互之间的对偶关系, 所用方法和技巧与单复变数有不少区别. 所得到的结论是单复变数许多已知结果的推广.     

A boundary perturbation interior point homotopy method for solving fixed point problems

In this paper, a boundary perturbation interior point homotopy method is proposed to give a constructive proof of the general Brouwer fixed point theorem and thus solve fixed point problems in a class of nonconvex sets. Compared with the previous results, by using the newly proposed method, initial points can be chosen in the whole space of Rn, which may improve greatly the computational efficiency of reduced predictor-corrector algorithms resulted from that method. Some numerical examples are given to illustrate the results of this paper.     

~n中有界域上全纯函数的第Ⅰ型 C-F公式

本文得到Ｃｎ中有界域上全纯函数的一种其积分密度函数含有全纯函数导数的 Ｃａｕｃｈｙ－Ｆａｎｔａｐｐｉ　　公式，称之为第Ⅰ型 Ｃ－Ｆ 公式，利用这个公式，通过适当选择其中的向量函数，可以得到许多区域上全纯函数相应的第Ⅰ型积分表示式．     

~n中有界域上全纯函数的第Ⅰ型 C-F公式

本文得到Ｃｎ中有界域上全纯函数的一种其积分密度函数含有全纯函数导数的 Ｃａｕｃｈｙ－Ｆａｎｔａｐｐｉ　　公式，称之为第Ⅰ型 Ｃ－Ｆ 公式，利用这个公式，通过适当选择其中的向量函数，可以得到许多区域上全纯函数相应的第Ⅰ型积分表示式．     

On first-order discrete boundary value problems

This article analyzes a nonlinear system of first-order difference equations with periodic and non-periodic boundary conditions. Some sufficient conditions are presented under which: potential solutions to the equations will satisfy certain a priori bounds; and the equations will admit at least one solution. The methods involve new dynamic inequalities and use of Brouwer degree theory. The new results are compared with those featuring in the theory of solutions to boundary value problems for differential equations.     

Cauchy中值定理的推广及应用

讨论了Cauchy中值定理中分子分母上的导函数可以同时取零值的情况,获得了Cauchy中值定理的一种推广,改进了[1]和[2]中的结果.     

A Generalized Schwarz Lemma

In this paper,we establish a boundary Schwarz Lemma for holomorphic mapping on the generalized complex ellipsoid in Cn.     

Constructive Proofs of the Brouwer Type Coincidence Theorems

In 1 966,Schirmer[1 ] proved the following interesting Brouwer type coincidencetheorem for the n dimensional closed unit disc Dn.　 Theorem1　 Suppose that the continuous mapping f:Dn→ Dn maps the boundary Sn- 1essentially onto itself.Then for any continuous mapping g:Dn→Dn,the equationf(x) =g(x)has a solution in Dn.　 This result is closely related to Brouwer' s celebrated fixed point theorem. Someimportant generalizations and applications of it can be found in [2— 4 ] .　 In this not…     

Generic approximation of functions by their Padé approximants

In the present article we obtain generic approximations, under sharp conditions, of holomorphic functions on arbitrary open sets by sequences of their Padé approximants. Similar results hold for functions smooth on the boundary of their domain of definition. In addition, the approximation is valid simultaneously with respect to all centers of expansion.     

Generalized nonlinear models of suspension bridges

This paper generalizes some results established in [J. Malík, Nonlinear models of suspension bridges, J. Math. Anal. Appl., in press]. The geometric nonlinearity connected with the torsion of a road bed is included in the generalized model. The basic variational equations are derived from the principle of minimum energy. The existence of a solution to the generalized problem is proved. The existence is based on the Brouwer fixed-point theorem.     

Leray-Schauder不动点定理的一个新证明

给出Leray-Schauder不动点定理的一个新证明.我们首先给出集值映射的焊接引理,利用集值映射的焊接引理和Kakutani不动点定理证明Leray-Schauder不动点定理,并证明Leray-Schauder不动点定理与Brouwer不动点定理等价.     

Two applications of Brouwer’s fixed point theorem: in insurance and in biology models

In the first part of the article, a new interesting system of difference equations is introduced. It is developed for re-rating purposes in general insurance. A nonlinear transformation φ of a d-dimensional (d ≥ 2) Euclidean space is introduced that enables us to express the system in the form f^t+1:=φ( f^t), t = 0, 1, 2,. …. Under typical actuarial assumptions, existence of solutions of that system is proven by means of Brouwer’s fixed point theorem in normed spaces. In addition, conditions that guarantee uniqueness of a solution are given. The second, smaller part of the article is about Leslie–Gower’s system of d ≥ 2 difference equations. We focus on the system that satisfies conditions consistent with weak inter-specific competition. We prove existence and uniqueness of the equilibrium of the model under surprisingly simple and very general conditions. Even though the two parts of this article have applications in two different sciences, they are connected with similar mathematics, in particular by our use of Brouwer’s fixed point theorem.     

On Vector Variational Inequalities in Reflexive Banach Spaces

In this paper, we study the solvability for a class of vector variational inequalities in reflexive Banach spaces. By using Brouwer fixed point theorem, we prove the solvability for this class of vector variational inequalities without monotonicity assumption. The solvability results for this class of vector variational inequalities with monotone mappings are also presented by using the KKM-Fan lemma This paper is dedicated to Professor Franco Giannessi for his 68th birthday     

LINEAR SINGULAR INTEGRAL EQUATION ON DOMAINS COMPOSED BY BALLS

For domains composed by balls in Cn, this paper studies the boundary behaviour of Cauchy type integrals with discrete holomorphic kernels and the corresponding linear singular integral equation on each piece of smooth lower dimensional edges on the boundary of the domain.     

Maximal cluster sets along arbitrary curves

The existence of a dense linear manifold of holomorphic functions on a Jordan domain having except for zero maximal cluster set along any curve tending to the boundary with nontotal oscillation value set is shown.     

On a Generalized Dirichlet Problem for Analytic Family

In this paper, we give the Silov boundary for an analytic family on a bounded strictly pseudoconvex domain or an analytic polyhedron in Cn, and get a necessary and sufficient condition for a generalized Dirichlet problem to be solvable for an analytic family on a bounded holomorphic domain. Especially, we derive that this condition is just that the continuous real boundary value is prescribed on and only on the Silov boundary for an analytic family on a bounded strictly pseudoconvex domain or an analytic polyhedron.