Volume 66 , issue 1 ( 2014 ) | back |

On weak and strong convergence theorems for two nonexpansive mappings in Banach spaces | 1$-$8 |

**Abstract**

In this paper, we consider an iteration process for approximating common fixed points of two nonexpansive mappings and prove some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces.

**Keywords:** Common fixed point; two step modified iterative scheme; condition (B); nonexpansive mapping; Banach space.

**MSC:** 47H09, 47J25

A new class of meromorphic functions related to Cho-Kwon-Srivastava operator | 9$-$18 |

**Abstract**

In the present paper, we introduce a new class of meromorphic functions defined by means of the Hadamard product of Cho-Kwon-Srivastava operator and we define here a similar transformation by means of an operator introduced by Ghanim and Darus. We investigate a number of inclusion relationships of this class. We also derive some interesting properties of this class.

**Keywords:** Subordination; meromorphic function; Cho-Kwon-Srivastava operator; Choi-Saigo-Srivastava operator; Hadamard product; integral operator.

**MSC:** 30C45, 30C50

Generalized Hausdorff operators on weighted Herz spaces | 19$-$32 |

**Abstract**

In this paper, we introduce new generalized Hausdorff operators. They include many famous operators as special cases. We obtain necessary and sufficient conditions for these operators to be bounded on the weighted Herz spaces. The corresponding new operator norm inequalities are obtained. They are significant improvements and generalizations of many known results. Several open problems are formulated.

**Keywords:** Hausdorff operator; weighted Herz space; norm inequality.

**MSC:** 26D10, 47A30

$N(k)$-quasi Einstein manifolds satisfying certain curvature conditions | 33$-$45 |

**Abstract**

The object of the present paper is to study $N(k)$-quasi Einstein manifolds. Existence of $N(k)$-quasi Einstein manifolds are proved by two non-trivial examples. Also a physical example of an $N(k)$-quasi-Einstein manifold is given. We study an $N(k)$-quasi-Einstein manifold satisfying the curvature conditions $\tilde Z(\xi ,X)\cdot S=0$, $P(\xi ,X)\cdot\tilde Z=0$, $\tilde Z(\xi,X)\cdot P=0$, $\tilde Z(\xi,X)\cdot C=0$ and $P(\xi,X)\cdot C=0$. Finally, we study Ricci-pseudosymmetric $N(k)$-quasi-Einstein manifolds.

**Keywords:** Quasi Einstein manifold;
$N(k)$-quasi Einstein manifold; projective curvature tensor; concircular curvature tensor; conformal curvature tensor; Ricci-pseudosymmetric manifold.

**MSC:** 53C25

Functions of class $H(\alpha,p)$ and Taylor means | 46$-$57 |

**Abstract**

In this paper, we take up Taylor means to study the degree of approximation of $fın H(\alpha,p)$ space in the generalized Hölder metric and obtain a general theorem which is used to obtain a few more results that improve upon some earlier results obtained by Mohapatra, Holland and Sahney [J. Approx. Theory 45 (1985), 363--374] in $L_p$-norm, Mohapatra and Chandra [Math. Chronicle 11 (1982), 89--96] in Hölder metric and Chui and Holland [J. Approx. Theory 39 (1983), 24--38] in sup-norm.

**Keywords:** Generalized Hölder metric; Taylor mean; degree of approximation.

**MSC:** 41A25, 42A10, 40G10

Fixed points of multivalued Suzuki-Zamfirescu-$(f,g)$ contraction mappings | 58$-$72 |

**Abstract**

Coincidence point theorems for hybrid pairs of single valued and multivalued mappings on an arbitrary nonempty set have been proved. As an application of our main result, the existence of common solutions of functional equations arising in dynamic programming are discussed.

**Keywords:** Coincidence point; orbitally complete space; common fixed point.

**MSC:** 47H09, 47H10, 54H25

On \cal{I}-convergence of double sequences in the topology induced by random 2-norms | 73$-$83 |

**Abstract**

In this article we introduce the notion of $\cal{I}$-convergence and $\cal{I}$-Cauchyness of double sequences in the topology induced by random $2$-normed spaces and prove some important results.

**Keywords:** $t$-norm; random $2$-normed space; ideal convergence; ideal Cauchy sequences; $F$-topology.

**MSC:** 40A35, 46A70, 54E70

Some characterizations of spaces with locally countable networks | 84$-$90 |

**Abstract**

In this paper, we give some characterizations of spaces with locally countable network and some characterizations of $sn$-symmetric (or Cauchy $sn$-symmetric) spaces with locally countable $sn$-networks by compact images (or $\pi$-images) of locally separable metric spaces.

**Keywords:** Network; $cs^*$-network; $sn$-network; weak base; locally countable; sequence-covering; $1$-sequence-covering; weak-open; compact map; $\pi$-map.

**MSC:** 54C10, 54D65, 54E40, 54E99

Some spaces of double difference sequences of fuzzy numbers | 91$-$100 |

**Abstract**

In this paper, we introduce some spaces of double difference sequences of fuzzy numbers defined by a sequence of modulus functions $F=(f_{k,l})$. We also make an effort to study some topological properties and prove some inclusion relations between these spaces.

**Keywords:** Fuzzy numbers; modulus function; double sequence spaces

**MSC:** 40A05, 40D25

On the invertibility of $AA^{+}-A^{+}A$ in a Hilbert space | 101$-$108 |

**Abstract**

Let $H$ be a Hilbert space and $B(H)$ the algebra of all bounded linear operators on $H$. In this paper, we study the class of operators $Aın B(H)$ with closed range such that $AA^{+}-A^{+}A$ is invertible, where $A^{+}$ is the Moore-Penrose inverse of $A$. Also, we present new relations between $(AA^{*}+A^{*}A)^{-1}$ and $(A+A^{*})^{-1}$. The present paper is an extension of results from [J. Benítez and V. Rakočević, Appl. Math. Comput. 217 (2010) 3493--3503] to infinite-dimensional Hilbert space.

**Keywords:** Moore-Penrose inverse; idempotent; orthogonal projection; positive operator.

**MSC:** 47A05

Two infinite families of equivalences of the continuum hypothesis | 109$-$112 |

**Abstract**

In this brief note we present two infinite families of equivalences of the Continuum Hypothesis, as follows: $\bullet$ For every fixed $n \geq 2$, the Continuum Hypothesis is equivalent to the following statement: ``There is an $n$-dimensional real normed vector space $E$ including a subset $A$ of size $\aleph_1$ such that $E \setminus A$ is not path connected''. $\bullet$ For every fixed $T_1$ first-countable topological space $X$ with at least two points, the Continuum Hypothesis is equivalent to the following statement: ``There is a point of the Tychonoff product $X^{\mathbb{R}}$ with a fundamental system of open neighbourhoods $B$ of size $\aleph_1$''.

**Keywords:** Continuum Hypothesis; path connected subsets; normed spaces; $T_1$ spaces; product topology; function spaces.

**MSC:** 03E50, 54A35, 54B10, 54C30

Fixed point theorems on S-metric spaces | 113$-$124 |

**Abstract**

In this paper, we prove a general fixed point theorem in S-metric spaces which is a generalization of Theorem~3.1 from [S. Sedghi, N. Shobe, A. Aliouche, Mat. Vesnik 64 (2012), 258--266]. As applications, we get many analogues of fixed point theorems from metric spaces to S-metric spaces.

**Keywords:** S-metric space.

**MSC:** 54H25, 54E99