Domains In Banach Spaces Research Articles

Neargeodesics in Gromov hyperbolic John domains in Banach spaces

In this paper, we prove that neargeodesics in Gromov hyperbolic John domains in Banach space are cone arcs. This result gives an improvement of a result of Li (2014).

The weak min-max property in Banach spaces

In this paper, we investigate the relationship between the weak min-max property and the diameter uniformity of domains in Banach spaces with dimension at least 2. As an application, we show that diameter uniform domains are invariant under relatively quasimöbius mappings.

On the Removability Property of φ‐Natural Domains in Real Banach Spaces

In this paper, we investigate the removability of φ‐natural domains in Banach spaces. Let E be a real Banach space with dimension at least 2, G a domain in E, and let be a countable subset of G which satisfies a quasi hyperbolic separation condition. Then, G is φ‐natural if and only if G is ψ‐natural, quantitatively.

Quasim

We study quasimöbius invariance of uniform domains in Banach spaces. We first investigate implications of certain geometric properties of domains in Banach spaces, such as (diameter) uniformity, $\delta $-uniformity and the min-max property. Then we show

Existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces

In this paper, the existence and uniqueness of solutions for a class of nonlinear integro-differential equations on unbounded domains in Banach spaces are established under more general conditions by constructing a special Banach space and using cone theory and the Banach contraction mapping principle. The results obtained herein improve and generalize some well-known results.

Holomorphic automorphisms of complex Banach spaces

We consider holomorphic automorphisms of infinite dimensional complex Banach spaces. First we look at automorphisms with an attracting fixed point to construct Fatou–Bieberbach domains in Banach spaces. Second, we look tame sets in Banach spaces. Recall that a discrete set in X is tame if it can be mapped onto an arithmetic progression via an automorphism of X. We show that bounded discrete sets of Banach spaces allowing a Schauder basis are tame. In contrast, $$l_\infty $$ has several bounded discrete sets which are not tame.

Hyperbolicity and Vitali properties of unbounded domains in Banach spaces

Let $\Om$ be a unbounded domain in a Banach space. In this work, we wish to impose {\it local conditions} on boundary point of $\Om$ (including the point at infinity) that guarantee complete hyperbolic of $\Om.$ We also search for local boundary conditions so that Vitali properties hold true for $\Om.$ These properties might be considered as analogues of the taut property in the finite dimensional case.

Monotone iterative method for a class of nonlinear fractional differential equations on unbounded domains in Banach spaces

In this paper, we investigate the existence of minimal nonnegative solution for a class of nonlinear fractional integro-differential equations on semi-infinite intervals in Banach spaces by applying the cone theory and the monotone iterative technique. An example is given for the illustration of main results.

Some Classes of Banach Analytic Spaces

In this paper we introduce three classes of Banach analytic spaces: hyperbolic Banach analytic spaces, weakly taut spaces and Banach analytic spaces having Vitali’s property. At the same time, we establish some relations between these three classes. Finally, we give some conditions under which balanced Hartogs domains in Banach spaces are weakly taut.

On John domains in Banach spaces

We study the stability of John domains in Banach spaces under removal of a countable set of points. In particular, we prove that the class of John domains is stable in the sense that removing a certain type of closed countable set from the domain yields a new domain which also is a John domain. We apply this result to prove the stability of the inner uniform domains. Finally, we consider a wider class of domains, so called -John domains and prove a similar result for this class.

Existence Results for Singular Boundary Value Problems on Unbounded Domains in Banach Spaces

In this paper, the Krasnosels’kii fixed point theorem in cones for strict set-contractions is used to investigate the existence of single and twin positive solutions for a class of a two-point boundary value problem of second-order nonlinear differential equations posed on an infinite interval. The nonlinearity, which may have a time-singularity, takes values in a general Banach space and has at most polynomial growth with respect to the unknown.

Measures of Weak Noncompactness and Fixed Point Theory for 1-Set Weakly Contractive Operators on Unbounded Domains

The main purpose of this paper is to prove a collection of new fixed point theorems and existence theorems for the nonlinear operator equation F(x) =αx (α ≥ 1) for so-called 1-set weakly contractive operators on unbounded domains in Banach spaces. We also introduce the concept of weakly semi-closed operator at the origin and obtain a series of new fixed point theorems and the existence theorems for the nonlinear operator equation F(x) = αx (α ≥ 1) for such class of operators. As consequences, the main results generalize and improve the relevant results, which are obtained by O’Regan and A. Ben Amar and M. Mnif in 1998 and 2009 respectively. In addition, we get the famous fixed point theorems of Leray-Schauder, Altman, Petryshyn and Rothe type in the case of weakly sequentially continuous, 1-set weakly contractive (µ-nonexpansive) and weakly semi-closed operators at the origin and their generalizations. The main condition in our results is formulated in terms of axiomatic measures of weak compactness.

Fréchet differential of a power series in Banach algebras

We present two new forms in which the Frechet differential of a power series in a unitary Banach algebra can be expressed in terms of absolutely convergent series involving the commutant \(C(T) : A \mapsto [A,T]\). Then we apply the results to study series of vector-valued functions on domains in Banach spaces and to the analytic functional calculus in a complex Banach space.

Weakly continuous holomorphic functions on pseudoconvex domains in Banach spaces

This paper is devoted to the study of weakly continuous holomorphic functions on pseudoconvex domains in Banach spaces with Schauder bases. We establish the identity of pseudoconvex domains and domains of existence of weakly continuous holomorphic functions. We show that holomorphic functions can be approximated by weakly continuous holomorphic functions. We study the spectra of certain algebras of weakly continuous holomorphic functions.

HOLOMORPHIC NEAR INVERSES

In this article we show that holomorphic Fredholm-valued mappings defined on connected pseudo-convex domains in Banach spaces with unconditional basis always have meromorphic generalised inverses. We show they have holomorphic generalised inverses if and only if the kernels have the same dimension at all points in Ω.

Fredholm-valued holomorphic mappings on a Banach space

In this article we show that the pointwise existence of a regulariser for holomorphic Fredhom-valued mappings defined on pseudo-convex domains in Banach spaces with an unconditional basis implies the existence of a holomorphic regulariser.

The extension operator in banach spaces for locally biholomorphic mappings

In this article, the generalized Roper—Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in C n are extended to Hilbert spaces or Banach spaces.

A unique positive solution for nth-order nonlinear impulsive singular integro-differential equations on unbounded domains in Banach spaces

In this paper, the fixed point theory combined with a monotone iterative technique is used to investigate the unique positive solution of a boundary problem for nth-order nonlinear impulsive singular integro-differential equations of mixed type on an infinite interval in a Banach space. The conditions for the existence of a unique positive solution are established. In addition, an explicit iterative sequence for approximating the solution of the boundary value problem is derived together with an error estimate. Furthermore, the conditions of the theorems can be easily verified.

A Schottky-type theorem for starlike domains in Banach spaces

We show that if $U$ is a starlike domain in a Banach space $E$ and $\mathscr {F}$ is a family of holomorphic functions on $U$ that omit two distinct values and is bounded at the origin, then $\mathscr {F}$ is uniformly bounded on each $U$-bounded set.

Geometrical properties of nonlinear maps and their application, Part I: The topological degree of quasiruled Fredholm maps on quasicylindrical domains

In this paper we introduce the notion of quasicylindrical domains in Banach spaces and develop a concept of a degree for quasiruled Fredholm mappings on quasicylindrical domains. Note that this quasicylindrical structure appears in a rather natural way, whenever “analytically” given nonlinear pseudodifferential operators are investigated in spaces of sufficiently smooth functions. Moreover, the class of quasiruled Fredholm mappings on quasicylindrical domains is sufficiently large, so that within this framework one can study a quite large class of nonlinear boundary value problems which are related to pseudodifferential operators.