Abstract

The aim of this paper is to obtain new common fixed point theorems under strict contractive conditions for three and four maps without continuity.

Keywords: Coincidence point, fixed point, $(E\; A)$-property, tangent point, compatible maps, noncompatible maps, pointwise $R$-weak commutativity.

MSC: 54H25, 47H10

Abstract

The class of functions of $\Lambda BV^{(p)}$ shares many properties of functions of bounded variation. Here we have shown that $\Lambda BV^{(p)}$ is a Banach space with a suitable norm, the intersection of $\Lambda BV^{(p)}$, over all sequences $\Lambda$, is the class of functions of BV$^{(p)}$ and the union of $\Lambda BV^{(p)}$, over all sequences $\Lambda$, is the class of functions having right- and left-hand limits at every point.

Keywords: Banach space, right and left-hand limits, p-$\Lambda$-bounded variation.

MSC: 26A45

Abstract

The paper generalizes the well-known inequality of Littlewood-Paley in the polydisc. We establish a family of inequalities which are analogues and extensions of Littlewood-Paley type inequalities proved by Sh.\ Yamashita and D. Luecking in the unit disk. Some other generalizations of the Littlewood-Paley inequality are stated in terms of anisotropic Triebel-Lizorkin spaces. With the help of an extension of Hardy-Stein identity, we also obtain area inequalities and representations for quasi-norms in weighted spaces of holomorphic functions in the polydisc.

Keywords: Littlewood-Paley inequalities, polydisc, Triebel-Lizorkin spaces, weighted spaces.

MSC: 32A37, 32A36

Abstract

We characterize the representability of an interval order on a topological space through a pair of continuous real-valued functions which in addition represent two total preorders associated to the given interval order. Such a continuous representation is obtained by using the notion of a decreasing scale.

Keywords: Interval order; continuous numerical representation; decreasing scale.

MSC: 06A06, 91B16

Abstract

In the present paper, the authors introduce a new subclass ${\Cal K}^{(k)}_s(\alpha,\beta)$ of close-to-convex functions. The subordination and inclusion relationship, and some coefficient inequalities for this class are provided. The results presented here would provide extensions of those given in earlier works.

Keywords: Starlike functions, close-to-convex functions, differential subordination.

MSC: 30C45

Abstract

We characterize Ramsey theoretically two classes of spaces which are related to $\gamma$-sets.

Keywords: Selection principles, Ramsey theory, game theory, $\omega$-cover, $k$-cover, $\gamma$-cover, $\gamma_k$-cover, $\gamma$-set, $k$-$\gamma$-set, $\gamma_k^\prime$-set.

MSC: 54D20, 05C55, 03E02, 91A44

Abstract

We show that for metrizable topological groups %Tkachenko's notion of being a strictly o-bounded group is equivalent to being a Hurewicz group. In [5] Hernandez, Robbie and Tkachenko ask if there are strictly $o$-bounded groups $G$ and $H$ for which $G\times H$ is not strictly $o$-bounded. We show that for metrizable strictly $o$-bounded groups the answer is no. In the same paper the authors also ask if the product of an $o$-bounded group with a strictly $o$-bounded group is again an $o$-bounded group. We show that if the strictly $o$-bounded group is metrizable, then the answer is yes.

Keywords: Metrizable topological group, strict $o$-boundedness, product groups.

MSC: 54H11