Volume 65 , issue 3 ( 2013 ) | back |

Convergence theorems of two-step implicit iterative process with errors for a finite family of asymptotically nonexpansive mappings | 285$-$298 |

**Abstract**

The objective of this paper is to study the weak and strong convergence of two-step implicit iteration process with errors to a common fixed point for a finite family of asymptotically nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003), Xu and Ori (2001), Zhou and Chang (2002) and Gu and Lu (2006).

**Keywords:** Asymptotically nonexpansive mapping; implicit iteration process with errors; common fixed point; strong convergence; weak convergence.

**MSC:** 47H09, 47H10

On certain univalent class associated with functions of non-Bazilevič type | 299$-$305 |

**Abstract**

In this work, we study certain differential inequalities and first order differential subordinations. As their applications, we obtain some sufficient conditions for univalence, which generalize and refine some previous results.

**Keywords:** Univalent functions; starlike functions; convex functions; close-to-convex functions; differential subordination;
subordination; superordination; unit disk; $\Phi$-like functions; non-Bazilevič type; Dziok-Srivastava linear operator; sandwich theorem.

**MSC:** 30C45

Compact-like properties in hyperspaces | 306$-$318 |

**Abstract**

$\mathcal{CL}(X)$ and $\mathcal{K}(X)$ denote the hyperspaces of non-empty closed and non-empty compact subsets of $X$, respectively, with the Vietoris topology. For an infinite cardinal number $\alpha$, a space $X$ is $\alpha$-hyperbounded if for every family $\{S_{\xi}:\xi<\alpha\}$ of non-empty compact subsets of $X$, $Cl_X(\bigcup_{\xi<\alpha}S_{\xi})$ is a compact set, and a space $X$ is pseudo-$\omega$-bounded if for each countable family $\mathcal{U}$ of non-empty open subsets of $X$, there exists a compact set $K\subseteq X$ such that each element in $\mathcal{U}$ has a non-empty intersection with $K$. We prove that $X$ is $\alpha$-hyperbounded if and only if $\mathcal{K}(X)$ is $\alpha$-hyperbounded, if and only if $\mathcal{K}(X)$ is initially $\alpha$-compact. Moreover, $\mathcal{K}(X)$ is pseudocompact if and only if $X$ is pseudo-$\omega$-bounded. Also, we show than if $\mathcal{K}(X)$ is normal and $C^{*}$-embbeded in $\mathcal{CL}(X)$, then $X$ is $\omega$-hyperbounded, and $X$ is $\alpha$-bounded if and only if $X$ is $\alpha$-hyperbounded, for every infinite cardinal number $\alpha$.

**Keywords:** Hyperspaces; Vietoris topology; $\alpha$-hyperbounded spaces; pseudo-$\omega$-bounded spaces; normal and $C^*$-embedded spaces.

**MSC:** 54B20, 54D99, 54D15, 54C45

Mapping properties of some classes of analytic functions under a general integral operator defined by the Hadamard product | 319$-$325 |

**Abstract**

In this paper, we consider certain subclasses of analytic functions with bounded radius and bounded boundary rotation and study the mapping properties of these classes under a general integral operator defined by the Hadamard product.

**Keywords:** Bounded boundary and bounded radius rotations; integral operator; Hadamard product (convolution).

**MSC:** 30C45

On the rings on torsion-free groups | 326$-$333 |

**Abstract**

The typeset of a torsion-free group is one of the important concepts in the theory of abelian groups. We use the typeset of an abelian group to study the rings that exist over such groups. Moreover, we consider the types of rational groups belonging to an independent set of a group and obtain some results about their relation with the rings over the group.

**Keywords:** Typeset; ring on a group; nil group.

**MSC:** 20K15

On some new characterizations of near paracompactness and associated results | 334$-$345 |

**Abstract**

Near paracompactness is a concept, in Set Topology, which is weaker than paracompactness; in this paper, several characterizations of this concept have been enunciated and proved. In the process, several tools have been utilized. The main theorem uses the selection theory of Michael.

**Keywords:** Near paracompactness; almost regularity; regular open set; semiregularization; strongly lower semicontinuous carrier.

**MSC:** 54D20, 54C65, 54D99

Some results on trans-Sasakian manifolds | 346$-$352 |

**Abstract**

The object of the present paper is to study $\phi$-conformally (resp. conharmonically, projectively) flat trans-Sasakian manifolds.

**Keywords:** $\phi$-conformally flat; $\phi$-conharmonically flat; $\phi$-projectively flat and trans-Sasakian manifold

**MSC:** 53C50, 53C15

Weighted Hankel operators and matrices | 353$-$363 |

**Abstract**

In this paper, the notions of weighted Hankel matrix along with weighted Hankel operator $S_{\phi}^{\beta}$, with $\phi \in L^{\infty}({\beta})$ on the space $L^2(\beta)$, $\beta=\{\beta_n\}_{n\in \Bbb{Z}}$ being a sequence of positive numbers with $\beta_0=1$, are introduced. It is proved that an operator on $L^2(\beta)$ is a weighted Hankel operator on $L^2(\beta)$ if and only if its matrix is a weighted Hankel matrix. Various properties of the weighted Hankel operators $S_{\phi}^{\beta}$ on $L^2(\beta)$ are also discussed.

**Keywords:** Weighted Hankel matrix; weighted Hankel operator.

**MSC:** 47B35, 47B20

Relative EXT groups of Abelian categories | 364$-$372 |

**Abstract**

In this article, we characterize the relative Ext groups of abelian categories relative to a fixed precovering class $\Cal{F}$ and give some examples.

**Keywords:** Relative Ext group; precovering; $\mathcal{F}$-resolution.

**MSC:** 18G15, 18G25

Certain sufficient conditions for a subclass of analytic functions associated with Hohlov operator | 373$-$382 |

**Abstract**

Making use of the Hohlov operator, we obtain inclusion relations between the classes of certain normalized analytic functions. Relevant connections of our work with the earlier works are pointed out.

**Keywords:** Starlike function; convolution, negative coefficients; coefficient inequalities; growth and distortion theorems.

**MSC:** 33C45, 33A30, 30C45

Decomposition of an integer as a sum of two cubes to a fixed modulus | 383$-$386 |

**Abstract**

The representation of any integer as the sum of two cubes to a fixed modulus is always possible if and only if the modulus is not divisible by seven or nine. For a positive non-prime power there is given an inductive way to find its remainders that can be represented as the sum of two cubes to a fixed modulus $N$. Moreover, it is possible to find the components of this representation.

**Keywords:** Sum of two cubes; Diophantine equation.

**MSC:** 11A07, 11B50, 11D25

Signed total distance $k$-domatic numbers of graphs | 387$-$393 |

**Abstract**

In this paper we initiate the study of signed total distance $k$-domatic numbers in graphs and we present its sharp upper bounds.

**Keywords:** Signed total distance $k$-domatic number; signed total distance $k$-dominating function; signed total distance $k$-domination number.

**MSC:** 05C69

Univalence conditions of general integral operator | 394$-$402 |

**Abstract**

In this paper, we obtain new univalence conditions for the integral operator $$ I_{\xi}^{\alpha_{i},\beta_{i}}(f_{1},\dots,f_{n})(z)=\left[\xi\int_{0}^{z}t^{\xi-1}(f_{1}^{\prime}(t))^{\alpha_{1}} (\frac{f_{1}(t)}{t})^{\beta_{1}}\cdots(f_{n}^{\prime}(t))^{\alpha_{n}}(\frac{f_{n}(t)}{t})^{\beta_{n}}\,dt\right]^{\frac{1}{\xi}} $$ of analytic functions defined in the open unit disc.

**Keywords:** Analytic function; univalent function; integral operator.

**MSC:** 30C45

Majorization problem for a subclass of $p$-valently analytic functions defined by the Wright generalized hypergeometric function | 403$-$411 |

**Abstract**

In this paper we investigate the majorization problem for a subclass of $p$-valently analytic functions involving the Wright generalized hypergeometric function. Some useful consequences of the main result are mentioned and relevance with some of the earlier results are also pointed out.

**Keywords:** Analytic, $p$-valent, majorization, Wright generalized hypergeometric function.

**MSC:** 30C45

Ore type condition and Hamiltonian graphs | 412$-$418 |

**Abstract**

In 1960, Ore proved that if $G$ is a graph of order $n\geq3$ such that $d(x)+d(y)\geq n$ for each pair of nonadjacent vertices $x,y$ in $G$, then $G$ is Hamiltonian. In 1985, Ainouche and Christofides proved that if $G$ is a 2-connected graph of order $n\geq 3$ such that $d(x)+d(y)\geq n-1$ for each pair of nonadjacent vertices $x,y$ in $G$, then $G$ is Hamiltonian or $G$ belongs to two classes of exceptional graphs. In this paper, we prove that if $G$ is a connected graph of order $n\geq 3$ such that $d(x)+d(y)\geq n-2$ for each pair of nonadjacent vertices $x,y$ in $G$, then $G$ is Hamiltonian or $G$ belongs to one of several classes of well-structured graphs.

**Keywords:** Ore type condition; Hamiltonian graphs.

**MSC:** 05C38, 05C45

Totally bounded endomorphisms on a topological ring | 419$-$424 |

**Abstract**

Let $X$ be a topological ring. In this paper, we consider the three classes ($btb$-bounded, $tbtb$-bounded, and $tbb$-bounded) of endomorphisms defined on $X$ and denote these classes by $B_{btb}(X), B_{tbtb}(X)$, and $B_{tbb}(X)$, respectively. We equip them with an appropriate topology and we find some sufficient conditions under which, each class of these endomorphisms is complete.

**Keywords:** Totally bounded endomorphism; completeness; topological ring.

**MSC:** 54H13, 16W80