Volume 67 , issue 2 ( 2015 ) | back |

An iterative approximation of fixed points of strictly pseudocontractive mappings in Banach spaces | 79$-$91 |

**Abstract**

We prove strong convergence of an iterative scheme for approximation of fixed point of $\lambda$-strict pseudocontractive mapping in a uniformly smooth real Banach space (which is not necessarily uniformly convex). We apply our result to approximation of common fixed point of a finite family of strictly pseudocontractive mappings. Our result extends the results of Li and Yao [M. Li, Y. Yao, Strong convergence of an iterative algorithm for $\lambda$-strictly pseudocontractive mappings in Hilbert spaces, An. St. Univ. Ovidius Constanta 18 (2010), 219-228] and complements other new interesting results in the literature.

**Keywords:** Strong convergence; strictly pseudocontractive mappings; uniformly smooth Banach spaces; uniformly convex Banach spaces.

**MSC:** 47H06, 47H09, 47J05, 47J25

Generalized derivations as a generalization of Jordan homomorphisms acting on Lie ideals | 92$-$101 |

**Abstract**

Let $R$ be a prime ring with extended centroid $C$, $L$ a non-central Lie ideal of $R$ and $n\geq 1$ a fixed integer. If $R$ admits the generalized derivations $H$ and $G$ such that $H(u^2)^n=G(u)^{2n}$ for all $u\in L$, then one of the following holds: {(1)} $H(x)=ax$ and $G(x)=bx$ for all $x\in R$, with $a,b\in C$ and $a^n=b^{2n}$; {(2)} char$(R)\neq 2$, $R$ satisfies $s_4$, $H(x)=ax+[p,x]$ and $G(x)=bx$ for all $x\in R$, with $b\in C$ and $a^n=b^{2n}$; {(3)} char$(R)=2$ and $R$ satisfies $s_4$. As an application we also obtain some range inclusion results of continuous generalized derivations on Banach algebras.

**Keywords:** Prime ring; generalized derivation; extended centroid; Utumi quotient ring; Banach algebra.

**MSC:** 16W25, 16N60, 16R50, 16D60

The rainbow domination subdivision numbers of graph | 102$-$114 |

**Abstract**

{$2$-rainbow dominating function} (2RDF) of a graph $G$ is a function $f$ from the vertex set $V(G)$ to the set of all subsets of the set $\{1,2\}$ such that for any vertex $v\in V(G)$ with $f(v)=\emptyset$ the condition $\cup_{u\in N(v)}f(u)=\{1,2\}$ is fulfilled. The {weight} of a 2RDF $f$ is the value $\omega(f)=\Sigma_{v\in V}|f (v)|$. The {$2$-rainbow domination number} of a graph $G$, denoted by $\gamma_{r2}(G)$, is the minimum weight of a 2RDF of G. The {$2$-rainbow domination subdivision number} $\tx{\rm sd}_{\gamma_{r2}}(G)$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the $2$-rainbow domination number. In this paper, we initiate the study of $2$-rainbow domination subdivision number in graphs.

**Keywords:** domination number; $2$-rainbow domination number; $2$-rainbow domination subdivision number.

**MSC:** 05C69

Common fixed point theorems for expansive mappings satisfying an implicit relation | 115$-$122 |

**Abstract**

We prove common fixed point theorems in metric spaces for expansive mappings satisfying an implicit relation without non-decreasing assumption and surjectivity using the concept of weak compatibility which generalize some theorems appearing in the recent literature.

**Keywords:** Weakly compatible mappings; common fixed point; metric space.

**MSC:** 47H10, 54H25

Faber polynomial coefficient estimates for analytic bi-Bazilevič functions | 123$-$129 |

**Abstract**

A function is said to be bi-univalent in the open unit disk $\Bbb{D}$ if both the function and its inverse are univalent in $\Bbb{D} $. By the same token, a function is said to be bi-Bazilevič in $\Bbb{D}$ if both the function and its inverse are Bazilevič there. The behavior of these types of functions are unpredictable and not much is known about their coefficients. In this paper we use the Faber polynomial expansions to find upper bounds for the coefficients of classes of bi-Bazilevič functions. The coefficients bounds presented in this paper are better than those so far appeared in the literature. The technique used in this paper is also new and we hope that this will trigger further interest in applying our approach to other related problems.

**Keywords:** Faber polynomials; bi-Bazilevič functions; univalent functions.

**MSC:** 30C45, 30C50

Proximity structures and ideals | 130$-$142 |

**Abstract**

In this paper, we present a new approach to proximity structures based on the recognition of many of the entities important in the theory of ideals. So, we give a characterization of the basic proximity using ideals. Also, we introduce the concept of $g$-proximities and we show that for different choice of ``$g$'' one can obtain many of the known types of generalized proximities. Also, characterizations of some types of these proximities -- ($g_0,h_0$) -- are obtained.

**Keywords:** ideals; basic proximity; proximity space; g-proximities; nearness; topological space.

**MSC:** 54E05

Generalized relative lower order of entire functions | 143$-$154 |

**Abstract**

The basic properties of the generalized relative lower order of entire functions are discussed in this paper. In fact, we improve here some results of Datta, Biswas and Biswas [Casp. J. Appl. Math. Ecol. Econ., {\bf1}, 2 (2013), 3--18].

**Keywords:** Entire function; generalized relative lower order; Property~(A).

**MSC:** 30D20, 30D30, 30D35