Volume 20, Issue 2

Abstract. In [Numer. Funct. Anal. Optim. 32 (2011), 243–253], Azam, Fisher, and Khan introduced a notion of complex-valued metric spaces and obtained common fixed point results for mappings satisfying rational inequalities. In the present paper, employing the concept of asymptotically regular mappings, the existence of a fixed point is obtained in a complex-valued metric space.

Abstract. In this paper we prove some results on the covering morphisms of internal groupoids. We also give a result on the coverings of the crossed modules of groups with operations.

Abstract. Applying the concepts of almost increasing and -quasi-monotone sequences, the paper presents a generalization of a known result dealing with the -summability to the -summability factors. We also obtain some new results.

Abstract. We show that the fact that X has a compact resolution swallowing the compact sets characterizes those C c ( X ) spaces which have the so-called 𝔊-base. So, if X has a compact resolution which swallows all compact sets, then C c ( X ) belongs to the class 𝔊 of Cascales and Orihuela (a large class of locally convex spaces which includes the ( LM ) and ( DF )-spaces) for which all precompact sets are metrizable and, conversely, if C c ( X ) belongs to the class 𝔊 and X satisfies an additional mild condition, then X has a compact resolution which swallows all compact sets. This fully applicable result extends the classification of locally convex properties (due to Nachbin, Shirota, Warner and others) of the space C c ( X ) in terms of topological properties of X and leads to a nice theorem of Cascales and Orihuela stating that for X containing a dense subspace with a compact resolution, every compact set in C c ( X ) is metrizable.

Abstract. This paper addresses the issue of sharp maximal function estimates of Toeplitz type operators. As an application, the boundedness of the operator on weighted Lebesgue and Triebel–Lizorkin spaces is established.

Abstract. Assuming Martin's Axiom, it is proved that every function acting from the real line ℝ into itself is representable as a sum of two absolutely nonmeasurable injective functions. A similar result is obtained for any endomorphism of the additive group ℝ.

Abstract. We investigate an integral equation with Bessel potential in ℝ n + and prove that every positive solution of the integral equation is rotationally symmetric about some line parallel to the -axis. Furthermore, we consider an overdetermined problem of the same integral equation in a cylinder type domain and obtain the symmetry of both domains and solutions, where is a bounded domain with .

Abstract. In [Topology Appl. 158 (2011), 397–408], Fischer et al. introduced the Spanier group of a based space ( X , x ) which is denoted by . By a Spanier space we mean a space X such that for every . In this paper, first we give an example of Spanier spaces. Then we study the influence of the Spanier group on the covering theory and introduce Spanier coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of Spanier coverings for non-homotopically path Hausdorff spaces. Finally, we study the topological properties of Spanier groups and establish criteria for the Hausdorffness of topological fundamental groups.

Abstract. In 2009, Goginava showed that , where is the th Marcinkiewicz–Fejér mean of the Walsh–Kaczmarz–Fourier series, is not bounded from the Hardy space to the space . The main aim of this paper is to show that the maximal operator is bounded from the Hardy space to the space . Moreover, it is proved that the order of deviant behavior of the th Walsh–Kacmarz–Marcinkiewicz–Fejér mean is exactly .

Abstract. The aim of the present paper is to study a Hofer-like geometry on the space of Legendrian submanifolds of a contact manifold. Using contact Hofer energy we define a pseudometric on this space and find sufficient conditions that this be nondegenerate. It is proved that an ample class of distances reduces to multiples of that metric. Examples of subsets with positive contact displacement energy are given.

Abstract. The Cauchy problem for hyperbolic functional differential systems is considered. The Volterra and Fredholm dependence are considered. A theorem on the global existence of classical solutions defined on the Haar pyramid is proved. A result on the differentiability of a solution with respect to the initial function is obtained.

Abstract. In this paper we introduce a class of complex q -Bernstein–Schurer operators and study the approximation properties of these operators. We obtain the order of simultaneous approximation and a Voronovskaja-type result with a quantitative estimate for these complex q -Bernstein–Schurer operators attached to analytic functions in compact disks. More important, our results show the overconvergence phenomenon for these complex operators.

Abstract. A theorem of Ferenc Lukács states that the partial sums of conjugate Fourier series of a periodic Lebesgue integrable function f diverge at the logarithmic rate at the points of first kind discontinuity of f . The aim of this paper is to prove analogous theorems in terms of generalized Cesáro means, introduced by Akhobadze, as well as regular linear means. We also formulate analogous theorems in terms of Abel–Poisson means for functions and series introduced by Taberski.