非线性全特征型偏微分方程组的全纯解

本文在复空间C_t×C_x上讨论了一类非线性全特征型具正则奇性偏微分方程组的全纯解,在一定的条件下,证明了在复空C_t×C_x的原点附近的某个邻域内存在唯一的全纯解,并把结论推广到了高阶的情况.本文的结果实际上是经典的Cauchy-Kowalewski定理在非线性奇异方程组上的推广。     

On Biholomorphy in Infinite Dimensions

In this article, we consider the question of when a holomorphic mapping on a domain in a complex Hilbert space into itself has a holomorphic inverse. We give a strengthened version of a known result that involves a Fredholm condition on the mapping. We show that holomorphic mappings on certain domains that are biholomorphic near the boundary are biholomorphic on the domain itself.      

Open sets with the Runge property in Banach spaces

We show that every pseudoconvex open subset of a Banach space with an unconditional Schauder basis has the F-valued Runge property for every Banach space F.     

Existence and uniqueness of holomorphic solutions for fractional Cauchy problem

By employing majorant functions, the existence and uniqueness of holomorphic solutions to nonlinear fractional partial differential equations (the Cauchy problems) are introduced. Furthermore, the analytic continuation of solutions is studied.     

Angular and unrestricted limits of one-parameter semigroups in the unit disk

We study local boundary behaviour of one-parameter semigroups of holomorphic functions in the unit disk. Earlier, under some additional condition (the position of the Denjoy–Wolff point) it was shown in [13] that elements of one-parameter semigroups have angular limits everywhere on the unit circle and unrestricted limits at all boundary fixed points. We prove stronger versions of these statements with no assumption on the position of the Denjoy–Wolff point. In contrast to many other problems, in the question of existence for unrestricted limits it appears to be more complicated to deal with the boundary Denjoy–Wolff point (the case not covered in [13]) than with all the other boundary fixed points of the semigroup.     

Decomposition of $L^{p}(\partial D_{a})$ Space and Boundary Value of Holomorphic Functions

This paper deals with two topics mentioned in the title.First,it is proved that function f in Lp((O)Da) can be decomposed into a sum g + h,where Da is an angular domain in the complex plane,g and h are the non-tangential limits of functions in Hp(Da) and Hp((D)cn) in the sense of Lp(Da),respectively.Second,the sufficient and necessary conditions between boundary values of holomorphic functions and distributions in n-dimensional complex space are obtained.     

Periodicity of some classes of holomorphic difference equations

In this paper, we investigate the periodicity character of some classes of difference equations. Among other results, we give a simpler proof of an Open Problem addressed in Csörnyei and Laczkovich, 2001, “Some periodic and non-periodic recursions”, Monath. Math., 132, 215–236. We also consider Open Problem 2.8 in Grove and Ladas, 2005, Periodicities in Nonlinear Difference Equations, (Chapman and Hall/CRC), and the case when a difference equation has periodic solutions depending on arbitrary parameters.     

The Schwarz problem for analytic functions in torus-related domains

The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. It is also decisive for the establishment of connection between the Riemann–Hilbert problem and the Riemann problem. Thus, not only the Schwarz problem for holomorphic functions in poly-domains is solved, but also the basis is established for solving some other problems. The boundary values of functions, holomorphic in poly-domains, are classified in the Wiener algebra. The general integral representation formulas for these functions, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary poly-domains is given. At the end, well-posed formulations of the torus-related problems are considered.     

On compact holomorphically pseudosymmetric Kählerian manifolds

For compact Kählerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric Kählerian manifolds. These examples show that the compactness assumption cannot be omitted in the above stated theorem. Recently, the first examples of compact, simply connected essentially holomorphically pseudosymmetric Kählerian manifolds are discovered in [4]. In these examples, the structure functions change their signs on the manifold.