Volume 64 , issue 4 ( 2012 ) | back |

Fixed point theorems for occasionally weakly compatible mappings in Menger spaces | 267$-$274 |

**Abstract**

In this paper, we prove common fixed point theorems for occasionally weakly compatible maps in Menger spaces. Our results never require the completeness of the whole space, continuity of the involved maps and containment of ranges amongst involved maps.

**Keywords:** Menger space; occasionally weakly compatible; common fixed point.

**MSC:** 54H25, 47H10

Asymptotic distribution of robust k-nearest neighbour estimator for functional nonparametric models | 275$-$285 |

**Abstract**

We propose a family of robust nonparametric estimators for robust regression function based on k-nearest neighbour (k-NN) method. We establish the asymptotic normality of the estimator under the concentration properties on small balls of the probability measure of the functional explanatory variables.

**Keywords:** Asymptotic distribution; functional data; k-nearest neighbour; robust estimation; small balls probability.

**MSC:** 62G05, 62G08, 62G20, 62G35

Score lists in bipartite multi hypertournaments | 286$-$296 |

**Abstract**

Given non-negative integers $m$, $n$, $h$ and $k$ with $m\ge h\ge 1$ and $n\ge k\ge 1$, an $[h,k]$-bipartite multi hypertournament (or briefly $[h,k]$-BMHT) on $m+n$ vertices is a triple $(U,V,\bold A)$, where $U$ and $V$ are two sets of vertices with $|U|=m$ and $|V|=n$ and $\bold A$ is a set of $(h+k$)-tuples of vertices, called arcs with exactly $h$ vertices from $U$ and exactly $k$ vertices from $V$, such that for any $h+k$ subset $U_{1}\cup V_{1}$ of $U\cup V$, $\bold A$ contains at least one and at most $(h+k)!$ $(h+k)$-tuples whose entries belong to $U_{1}\cup V_{1}$. If $\bold A$ is a set of $(r+s)$-tuples of vertices, called arcs for $r$ ($1\leq r\leq h$) vertices from $U$ and $s$ ($1\leq s\leq k$) vertices from $V$ such that $\bold A$ contains at least one and at most $(r+s)!$ $(r+s)$-tuples, then the bipartite multi hypertournament is called an $(h,k)$-bipartite multi hypertournament (or briefly $(h,k)$-BMHT). We obtain necessary and sufficient conditions for a pair of sequences of non-negative integers in non-decreasing order to be losing score lists and score lists of $[h,k]$-BMHT and $(h,k)$-BMHT.

**Keywords:** Hypertournaments; bipartite hypertournaments; score; losing score.

**MSC:** 05C65

On $\pi$-images of separable metric spaces and a problem of Shou Lin | 297$-$302 |

**Abstract**

In this paper, we give some characterizations of images of separable metric spaces under certain $\pi$-maps, and give an affirmative answer to the problem posed by Shou Lin in [Point-Countable Covers and Sequence-Covering Mappings, Chinese Science Press, Beijing, 2002].

**Keywords:** $cs^*$-network; $cs^*$-cover; $cs$-cover; $sn$-cover; separable metric space; Cauchy $sn$-symmetric;
$\sigma$-strong network; compact map; $\pi$-map.

**MSC:** 54C10, 54D55, 54E40, 54E99

Global smoothness preservation by some nonlinear max-product operators | 303$-$315 |

**Abstract**

In this paper we study the problem of partial global smoothness preservation in the cases of max-product Bernstein approximation operators, max-product Hermite-Féjer interpolation operators based on the Chebyshev nodes of first kind and max-product Lagrange interpolation operators based on the Chebyshev nodes of second kind.

**Keywords:** Max-product Bernstein approximation operator; max-product Hermite-Féjer interpolation operator; max-product
Lagrange interpolation operator; Chebyshev nodes of the first and second kind; global smoothness preservation.

**MSC:** 41A05, 41A20, 41A17

Harmonic starlike functions of complex order involving hypergeometric functions | 316$-$325 |

**Abstract**

A family of harmonic starlike functions of complex order in the unit disc has been introduced and investigated by S.A. Halim and A. Janteng [Harmonic functions starlike of complex order, Proc. Int. Symp. on New Development of Geometric function Theory and its Applications, (2008), 132--140]. In this paper we consider a subclass consisting of harmonic parabolic starlike functions of complex order involving special functions and obtain coefficient conditions, extreme points and a growth result.

**Keywords:** Harmonic functions; harmonic starlike functions; hypergeometric functions; Dziok-Srivastava operator.

**MSC:** 30C45, 30C50

A note on sequence-covering $\pi$-images of metric spaces | 326$-$329 |

**Abstract**

In this paper, we prove that a space is a sequence-covering $\pi$-image of a metric space if and only if it has a $\sigma$-strong network consisting of $cs$-covers (or $sn$-covers) if and only if it is a Cauchy $sn$-symmetric space.

**Keywords:** Sequence-covering mappings; $\pi$-mappings; $cs$-covers; $sn$-covers; $\sigma$-strong networks; Cauchy $sn$-symmetric spaces.

**MSC:** 54D55, 54E40, 54E99

Factorization of weakly compact operators between Banach spaces and Fréchet or (LB)-spaces | 330$-$335 |

**Abstract**

In this note we show that weakly compact operators from a Banach space $X$ into a complete (LB)-space $E$ need not factorize through a reflexive Banach space. If $E$ is a Fréchet space, then weakly compact operators from a Banach space $X$ into $E$ factorize through a reflexive Banach space. The factorization of operators from a Fréchet or a complete (LB)-space into a Banach space mapping bounded sets into relatively weakly compact sets is also investigated.

**Keywords:** Weakly compact operators; reflexive operators; factorization; reflexive Banach spaces; (LB)-spaces; Fréchet spaces.

**MSC:** 46A04, 46A03, 46A25, 46B10, 47B07

Ostrowski inequalities for cosine and sine operator functions | 336$-$346 |

**Abstract**

Here we present Ostrowski type inequalities on Cosine and Sine Operator Functions for various norms. At the end we give some applications.

**Keywords:** Ostrowski inequality; Cosine operator function; Sine operator function.

**MSC:** 26D99, 47D09, 47D99

Finite dimensions defined by means of $m$-coverings | 347$-$360 |

**Abstract**

We introduce and investigate finite dimensions
$(m,n)\text{-}\tx{\rm dim}$ defined by means of $m$-coverings. These
dimensions generalize the Lebesgue dimension: $\tx{\rm
dim}=(2,1)\text{-}\tx{\rm dim}$. If $n

**Keywords:** Dimension; dimension $(m,n)\text{-}\tx{\rm dim}$; metrizable space; hereditarily normal space.

**MSC:** 54F45