Volume 65 , issue 2 ( 2013 ) | back |

Common fixed point result for two self-maps in $G$-metric spaces | 143$-$150 |

**Abstract**

In this paper, we prove a fixed point result for two maps in a generalized metric space $(X,G)$. Also, we prove the uniqueness of such fixed point. This result generalizes some well known results in the literature due to M. Abbas and B.E. Rhoades [Common fixed point results for noncommuting mapping without continuity in generalized metric spaces, Appl. Math. Computation 215 (2009), 262--269] and W. Shatanawi [Fixed point theory for contractive mappings satisfying $\Phi$-maps in $G$-metric spaces, Fixed Point Theory Appl. 2010 (2010), Article ID 181650, 9 pages].

**Keywords:** Common fixed point; weakly compatible maps; generalized metric spaces.

**MSC:** 54H25, 47H10, 54E50

Nonlinear differential polynomials sharing a small function | 151$-$165 |

**Abstract**

In the paper, we investigate the uniqueness problems on entire and meromorphic functions concerning nonlinear differential polynomials that share a small function and obtain some results which improve and generalize some previous results due to Zhang-Chen-Lin, Banerjee-Bhattacharjee and Xu-Han-Zhang.

**Keywords:** Uniqueness; meromorphic function; nonlinear differential polynomials.

**MSC:** 30D35

Exponential dichotomy and strongly stable vectors of Hilbert space contraction semigroups | 166$-$177 |

**Abstract**

The paper deals with exponential dichotomy and its relationship with strongly stable vectors associated with Hilbert space semigroups. Contraction semigroups are decomposed by using the strongly stability operator associated with the semigroup. Necessary and sufficient conditions for exponential stability and non-exponential stability are investigated in terms of norm inequalities---instead of a Lyapunov operator equation.

**Keywords:** Exponential stability; strong stability; exponential dichotomy; strong dichotomy

**MSC:** 47D03, 47A45

Sandwich-type results for a class of functions defined by a generalized differential operator | 178$-$186 |

**Abstract**

By making use of a generalized differential operator a new class of non-Bazilevič functions is introduced. Differential sandwich-type theorem for the above class is investigated. Relevant connections of the results, which are presented in this paper, with various other known results are also pointed out.

**Keywords:** Univalent function; derivative operator; differential subordination; differential superordination.

**MSC:** 30C45, 30C80

On convergence of $q$-Chlodovsky-type MKZD operators | 187$-$196 |

**Abstract**

In the present paper, we define a new kind of MKZD operators for functions defined on $[0,b_{n}]$, named q-Chlodovsky-type MKZD operators, and give some approximation properties.

**Keywords:** q-Chlodovsky-type MKZD operators; modulus of continuity; Peetre-K functional; Lipschitz space.

**MSC:** 41A25, 41A36

On slant submanifolds of $N(k)$-contact metric manifolds | 197$-$203 |

**Abstract**

The object of the present paper is to study slant submanifolds of an $N(k)$-contact metric manifold. We study the parallelism of $Q$, and find out necessary and sufficient conditions for the existence of proper slant submanifolds of $N(k)$-contact metric manifolds.

**Keywords:** Slant submanifold; $N(k)$-contact metric manifold; autoparallel.

**MSC:** 53C15, 53C40

Common fixed point results for non-linear contractions in $G$-metric spaces | 204$-$212 |

**Abstract**

We establish common fixed point results for three self-mappings on a $G$-metric space satisfying non linear contractions. Also, we prove the uniqueness of such common fixed point, as well as studying the $G$-continuity at such point. Our results extend some known works. Also, an example is given to illustrate our obtained results.

**Keywords:** $G$-metric space; common fixed point; non-linear contraction; $G$-continuity.

**MSC:** 47H10, 54H25, 54E35, 54E99

Convolution properties of a slanted right half-plane mapping | 213$-$221 |

**Abstract**

In the present paper, the authors identify some specific harmonic functions whose convolution with slanted right half-plane mapping is harmonic close-to-convex.

**Keywords:** Harmonic functions; univalent functions; convolution.

**MSC:** 30C45

Relative order of entire functions of several complex variables | 222$-$233 |

**Abstract**

In this paper we introduce the idea of relative order of entire functions of several complex variables. After proving some basic results, we observe that the relative order of a transcendental entire function with respect to an entire function is the same as that of its partial derivatives. Further we study the equality of relative order of two functions when they are asymptotically equivalent.

**Keywords:** Entire functions; relative order; several complex variables; polydisc; property (R).

**MSC:** 32A15

Eigenvalues of certain sparse matrices | 234$-$241 |

**Abstract**

We introduce a special class of matrices of arbitrary size, which we call C-matrices. We compute some exact values of the eigenvalues of the C-matrices.

**Keywords:** Matrices; eigenvalues; finite sequence.

**MSC:** 15A18, 11B83

The Euler theorem and Dupin indicatrix for surfaces at a constant distance from edge of regression on a surface in $E_1^3$ | 242$-$249 |

**Abstract**

In this paper we give the Euler theorem and Dupin indicatrix for surfaces at a constant distance from edge of regression on a surface in $E_1^3$.

**Keywords:** Euler theorem; Dupin indicatrix; edge of regression.

**MSC:** 51B20, 53B30

On Camina group and its generalizations | 250$-$260 |

**Abstract**

In this paper we present some new results on Camina groups. Infinite generalizations of Camina groups and generalized Camina groups are also discussed. We further define and study some new group structures which arise out of inter-relations between conjugacy classes, order classes, and cosets with respect to a normal subgroup.

**Keywords:** Camina group; infinite Camina group; generalized Camina group; infinite Frobenius group;
multiple conjugate type vector; order class; Or-Con group; Or-Cos group.

**MSC:** 20E45, 20D99

On right ideals and derivations in prime rings with Engel condition | 261$-$270 |

**Abstract**

Let $R$ be an associative ring with center $Z(R)$ and $d$ a nonzero derivation of $R$. The main object in this paper is to study the situation $[[d(x^r)x^n,x^r]_s,[y,d(y)]_t]^m\in Z(R)$ for all $x,y$ in some appropriate subset of $R$, where $n\geq 0$, $s\geq 0$, $t\geq 0$, $m\geq 1$, $r\geq 1$ are fixed integers and $R$ is a prime or semiprime ring.

**Keywords:** Prime ring; derivation; extended centroid; Martindale's quotient ring.

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

Fixed point techniques and stability in nonlinear neutral differential equations with variable delays | 271$-$284 |

**Abstract**

In this paper we use fixed point techniques to obtain asymptotic stability results of the zero solution of a nonlinear neutral differential equation with variable delays. This investigation uses new conditions which allow the coefficient functions to change sign and do not require the boundedness of delays. An asymptotic stability theorem with a necessary and sufficient condition is proved. The obtained results improve and extend those due to Burton, Zhang, Raffoul, Jin and Luo, Ardjouni and Djoudi, and Djoudi and Khemis. Two examples are also given to illustrate this work.

**Keywords:** Fixed points; stability; neutral differential equation; integral equation; variable delays.

**MSC:** 34K20, 34K30, 34K40