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.     

Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed, convex sets in a Banach space

We show that any fixed point of a Lipschitzian, strictly pseudocontractive mapping on a closed, convex subset of a Banach space is necessarily unique, and may be norm approximated by an iterative procedure. Our argument provides a convergence rate estimate and removes the boundedness assumption on , generalizing theorems of Liu.

     

On differential properties of mappings into a Banach space

We prove that the Rieffel sharpness condition for a Banach space E is necessary and sufficient for an arbitrary Lipschitz function f: [a, b]→E to be differentiable almost everywhere on a segment [a, b]. We establish that, in the case where the sharpness condition is not satisfied, the major part (in the category sense) of Lipschitz functions has no derivatives at any point of the segment [a, b].     

Extending bounded type holomorphic mappings on a Banach space

We show that a Riemann domain Ω over a symmetrically regular Banach space E admits holomorphic extension to pseudo-convex domains over E″ with respect to two natural spaces of holomorphic functions of bounded type on Ω.     

Continuous bijective mappings in topological and Banach manifolds

Continuity and monotonicity of the inverse operator is investigated for the case when the original operator is a bijective mapping acting in topological manifolds or in Banach spaces. For finite-dimensional topological manifolds, the inverse operator is always continuous; for Banach spaces, continuity of the inverse operator is guaranteed only if the space is finite-dimensional. For an arbitrary differentiable bijective mapping acting in the Euclidean space, the Jacobian preserves its sign.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 117–124, 1985;     

Absolute continuity for Banach space valued mappings

We consider the notion ofp, λ, δ-absolute continuity for Banach space valued mappings introduced in [2] for real valued functions and for λ=1. We investigate the validity of some basic properties that are shared byn, λ-absolutely continuous functions in the sense of Maly and hencl. We introduce the class δ-BV _λ,loc ^p and we give a characterization of the functions belonging to this class.     

Absolute continuity for Banach space valued mappings

We consider the notion ofp, λ, δ-absolute continuity for Banach space valued mappings introduced in [2] for real valued functions and for λ=1. We investigate the validity of some basic properties that are shared byn, λ-absolutely continuous functions in the sense of Maly and hencl. We introduce the class δ-BV _λ,loc^p and we give a characterization of the functions belonging to this class.     

Normal families of holomorphic functions and mappings on a Banach space

The authors lay the foundations for the study of normal families of holomorphic functions and mappings on an infinite-dimensional normed linear space. Characterizations of normal families, in terms of value distribution, spherical derivatives, and other geometric properties are derived. Montel-type theorems are established. A number of different topologies on spaces of holomorphic mappings are considered. Theorems about normal families are formulated and proved in the language of these various topologies. Normal functions are also introduced. Characterizations in terms of automorphisms and also in terms of invariant derivatives are presented.     

A strong convergence theorem for relatively nonexpansive mappings in a Banach space

In this paper, we prove a strong convergence theorem for relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Using this result, we also discuss the problem of strong convergence concerning nonexpansive mappings in a Hilbert space and maximal monotone operators in a Banach space.     

Growth and distribution of zeros of holomorphic mappings of a finite-dimensional space into a Banach space

Khar'kov. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 1, pp. 161–171, January–February, 1990.     

Fixed points of non-expansive mappings associated with invariant means in a Banach space

In this paper, we study the fixed point set of the non-expansive mapping _Tμ$T_{\mu }$ for a Banach space with uniformly Gâteaux differentiable norm when μ$\mu$ is a multiplicative left invariant mean on l^∞(S)$l^{\infty }\left(S\right)$. As an application, we establish nonlinear ergodic properties for an extremely amenable semigroup of non-expansive mappings in a Banach space with uniformly Gâteaux differentiable norm. Furthermore, we improve a recent result of Atsushiba and Takahashi [S. Atsushiba, W. Takahashi, Weak and strong convergence theorems for non-expansive semigroups in a Banach spaces satisfying Opial’s condition, Sci. Math. Jpn. (in press)] on the fixed point set of non-expansive mappings associated with a left invariant mean on a left amenable semigroup.     

Expansion of open sets and decomposition of continuous mappings

The notion of expansionA of open sets is introduced. ThenA-expansion continuous mappingf:X→Y is defined. The main result of this note is that a mappingf is continuous if and only if it is bothA-expansion continuous andB-expansion continuous, whereA-expansion,B-expansion are two mutually dual expansions.     

On a hybrid method for a family of relatively nonexpansive mappings in a Banach space

We prove strong convergence theorems by the hybrid method given by Takahashi, Takeuchi, and Kubota for a family of relatively nonexpansive mappings under weaker conditions. The method of the proof is different from the original one and it shows that the type of projection used in the iterative method is independent of the properties of the mappings. We also deal with the problem of finding a zero of a maximal monotone operator and obtain a strong convergence theorem using this method.