Some Remarks on Perov Type Mappings in Cone Metric Spaces

Let \(\left( E,C,t\right) \) be a real ordered topological vector space and let (X, d) be a tvs-cone metric space over cone C. Using Proposition 19.9 of Deimling (Nonlinear functional analysis, Springer, Berlin, 1985), we show that E can be equipped with a norm such that C is a normal monotone solid cone. Hence, a tvs-cone metric space \(\left( X,d\right) \) over a solid cone C is a normal cone metric space over the same cone C. This assures that tvs-cone metric spaces are not a genuine generalization of cone metric spaces introduced by Huang and Zhang, recently. Further, if the cone C is solid then we have only cone metric spaces over normal solid cone (with coefficient of normality \(K=1\)). Here, we introduce also the notion of Sehgal–Guseman–Perov type mappings and we establish a result of existence and uniqueness of fixed points for this class of mappings.     

Mappings of higher order and nonlinear equations in some spaces of almost periodic functions

In the first part of the paper we examine mappings of higher order from a general point of view, that is, in normed spaces of bounded real-valued functions defined on R$\mathbb{R}$. Particular attention is paid to the relation of such mappings with the so-called autonomous superposition operators. Next we investigate mappings of higher order in Banach spaces of almost periodic functions and their perturbations. We also give necessary and sufficient conditions guaranteeing that a nonautonomous superposition operator acts in the space of almost periodic functions in the sense of Levitan and is uniformly continuous. In the Banach space of bounded almost periodic functions in the sense of Levitan we discuss mappings of higher order and a convolution operator. Some applications to nonlinear differential and integral equations are given.     

Approximating fixed points of mappings satisfying condition (E) in Busemann space

It is well-known that in a Banach space, using the Ishikawa iterative process, one can find fixed points of nonexpansive mappings via asymptotic center’s method. In this paper, we obtain the fixed points of mappings satisfying so-called condition (E) in a uniformly convex Busemann space. Many known results in CAT (0) spaces are improved and extended by our results.     

Mappings of Ruled Surfaces in Simply Isotropic Space <Emphasis Type="Italic">I</Emphasis><Subscript>3</Subscript><Superscript>1</Superscript> that Preserve the Generators

In this paper we study some mappings of skew ruled surfaces in simply isotropic space which preserve the generators. We study isometries, conformal mappings and mappings which preserve the area. Furthermore, we study mappings of surfaces in I ₃ ¹ which preserve the asymptotic lines.Received December 18, 2001; in revised form July 12, 2002 Published online April 4, 2003     

Weak fixed point property for nonexpansive mappings with respect to orbits in Banach spaces

In this paper, we first show that a Banach space X has weak normal structure if and only if X has the weak fixed point property for nonexpansive mappings with respect to (wrt) orbits. Then, we give a counterexample to show that the Goebel–Karlovitz lemma does not hold for minimal invariant sets of nonexpansive mappings wrt orbits, and we present a modified version of the Goebel–Karlovitz lemma.     

The space of -additive mappings on semigroups

In this paper, we introduce the concept of -pseudoadditive mappings from a semigroup into a Banach space, and we provide a generalized solution of Ulam's problem for approximately additive mappings.

     

Fixed point set of semigroups of non-expansive mappings and amenability

In this paper, we study the fixed point set of a strongly continuous non-expansive semigroup of a semi-topological semigroup S for which CB(S) is n-extremely left amenable. Also, we study the fixed point set of a strongly continuous semigroup of mappings when S is a semigroup which is a sub-semigroup of a locally convex topological vector space with addition. Some applications to harmonic analysis are also provided.     

Uniformly locally univalent harmonic mappings

The primary aim of this paper is to characterize the uniformly locally univalent harmonic mappings in the unit disk. Then, we obtain sharp distortion, growth and covering theorems for one parameter family \({{\mathcal {B}}}_{H}(\lambda )\) of uniformly locally univalent harmonic mappings. Finally, we show that the subclass of k-quasiconformal harmonic mappings in \({{\mathcal {B}}}_{H}(\lambda )\) and the class \({{\mathcal {B}}}_{H}(\lambda )\) are contained in the Hardy space of a specific exponent depending on \(\lambda \), respectively, and we also discuss the growth of coefficients for harmonic mappings in \({{\mathcal {B}}}_{H}(\lambda )\).     

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.     

The contraction principle for mappings on a metric space with a graph

We give some generalizations of the Banach Contraction Principle to mappings on a metric space endowed with a graph. This extends and subsumes many recent results of other authors which were obtained for mappings on a partially ordered metric space. As an application, we present a theorem on the convergence of successive approximations for some linear operators on a Banach space. In particular, the last result easily yields the Kelisky-Rivlin theorem on iterates of the Bernstein operators on the space .

     

Fekete and Szeg problem in one and higher dimensions

In this paper, we establish the Fekete and Szeg inequality for a class of holomorphic functions in the unit disk, and then we extend this result to a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in C~n.     

Involutions in Dual Split-Quaternions

Involutions and anti-involutions are self-inverse linear mappings. In three-dimensional Euclidean space \({\mathbb{R}^{3}}\), a reflection of a vector in a plane can be represented by an involution or anti-involution mapping obtained by real-quaternions. A reflection of a line about a line in \({\mathbb{R}^{3}}\) can also be represented by an involution or anti-involution mapping obtained by dual real-quaternions. In this paper, we will represent involution and anti-involution mappings obtaind by dual split-quaternions and a geometric interpretation of each as rigid-body (screw) motion in three-dimensional Lorentzian space \({\mathbb{R}_1^{3} }\).     

Fixed points of Mizoguchi–Takahashi contraction on a metric space with a graph and applications

In this paper we extend Mizoguchi–Takahashi's fixed point theorem for multi-valued mappings on a metric space endowed with a graph. As an application, we establish a fixed point theorem on an ε  -chainable metric space for mappings satisfying Mizoguchi–Takahashi contractive condition uniformly locally. Also, we establish a result on the convergence of successive approximations for certain operators (not necessarily linear) on a Banach space as another application. Consequently, this result yields the Kelisky–Rivlin theorem on iterates of the Bernstein operators on the space C[0,1]$C[0,1]$ and also enables us study the asymptotic behaviour of iterates of some nonlinear Bernstein type operators on C[0,1]$C[0,1]$.     

Cone monotone mappings: Continuity and differentiability

We generalize some results of Borwein, Burke, Lewis, and Wang to mappings with values in metric (resp. ordered normed linear) spaces, and we define two classes of monotone mappings between an ordered linear space and a metric space (resp. ordered linear space): K$K$-monotone dominated and cone-to-cone monotone mappings. K$K$-monotone dominated mappings naturally generalize mappings with finite variation (in the classical sense) and K$K$-monotone functions defined by Borwein, Burke and Lewis to mappings with domains and ranges of higher dimensions. First, using results of Veselý and Zají?ek, we show some relationships between these classes. Then, we show that every K$K$-monotone function f:X→R$f:X\to \mathbb{R}$, where X$X$ is any Banach space, is continuous outside of a set which can be covered by countably many Lipschitz hypersurfaces. This sharpens a result due to Borwein and Wang. As a consequence, we obtain a similar result for K$K$-monotone dominated and cone-to-cone monotone mappings. Finally, we prove several results concerning almost everywhere differentiability (also in metric and w^∗$w^{\ast }$-senses) of these mappings.     

Strong convergence theorems for finitely many nonexpansive mappings and applications

Let E$E$ be a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable. Recently, Takahashi and Shimoji [W. Takahashi, K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Modelling 32 (2000) 1463–1471] introduced an iterative scheme given by finitely many nonexpansive mappings in E$E$ and proved weak convergence theorems which are connected with the problem of image recovery. In this paper we introduce a new iterative scheme which includes their iterative scheme as a special case. Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gâteaux differentiable and which has a weakly continuous duality mapping, we prove strong convergence theorems which are connected with the problem of image recovery. Using the established results, we consider the problem of finding a common fixed point of finitely many nonexpansive mappings.     

On viscosity iterative methods for variational inequalities

Let be a commutative family of nonexpansive mappings of a closed convex subset C of a reflexive Banach space X such that the set of common fixed point is nonempty. In this paper, we suggest and analyze a new viscosity iterative method for a commutative family of nonexpansive mappings. We also prove that the approximate solution obtained by the proposed method converges to a solution of a variational inequality. Our method of proof is simple and different from the other methods. Results proved in this paper may be viewed as an improvement and refinement of the previously known results.     

On w-compatible mappings and common coupled coincidence point in cone metric spaces

Recently, Abbas et al. [M. Abbas, M.A. Khan, S. Radenovi?, Common coupled fixed point theorems in cone metric spaces for $w$-compatible mapping, Applied Mathematics and Computation (2010) doi:10.1016/j.amc.2010.05.042] introduced the concept of $w$-compatible mappings and obtained results on coupled coincidence point for nonlinear contractive mappings in a cone metric space. In the present paper, we introduce the concept of a common coupled coincidence point of the mappings $F,G:X\times X\to X$ and $f:X\to X$ and we prove some theorems for nonlinear contractive mappings in a cone metric space with a cone having nonempty interior. Our results generalize several well known comparable results in the literature.     

Zur Theorie Linearer Abbildungen I

The paper is concerned with a uniform geometric definition of linear mappings in a projective or grassmannian space into a projective space. We discuss sufficient conditions for the existence of a linear mapping in a finite dimensional pappian projective space which continues two given linear mappings in complementary subspaces.The subspace spanned by the image set of a linear mapping in the grassmannian of d-dimensional subspaces of an n-dimensional projective space has at most dimension –1.     

Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications

In this paper, we establish coincidence and common fixed point theorems for contractive mappings on a metric space endowed with an amorphous binary relation. The presented theorems extend the results of Samet and Turinici in [Commun. Math. Anal. 12 (2012), 82– 97] and generalize many existing results on metric and ordered metric spaces. We apply also our main results to derive coincidence and common fixed point theorems for cyclic contractive mappings.     

Distortion theorems for normalized biholomorphic quasi-convex mappings

In this paper, we will use the Schwarz lemma at the boundary to character the distortion theorems of determinant at the extreme points and distortion theorems of matrix on the complex tangent space at the extreme points for normalized locally biholomorphic quasi-convex mappings in the unit ball B ⁿ respectively.