Volume 68 , issue 3 ( 2016 ) | back |

On a generalization of a result of Zhang and Yang | 155$-$163 |

**Abstract**

In this paper we find out the specific form of a meromorphic function when a generalized linear expression of the function share a small function with its $k$-th derivative counterpart. Our result will improve and generalize a few existing results, especially that of Zhang and Yang [J. L. Zhang and L. Z. Yang, A power of a meromorphic function sharing a small function with its derivative, Ann. Acad. Sci. Fenn. Math., 34 (2009), 249-260].

**Keywords:** Meromorphic function; small function; weighted sharing.

**MSC:** 30D35

An Engel condition of generalized derivations with annihilator on Lie ideal in prime rings | 164$-$174 |

**Abstract**

Let $R$ be a prime ring with its Utumi ring of quotients $U$, $C=Z(U)$ extended centroid of $R$, $F$ a nonzero generalized derivation of $R$, $L$ a noncentral Lie ideal of $R$ and $k\geq 2$ a fixed integer. Suppose that there exists $0\neq a\in R$ such that $a[F(u^{n_1}),u^{n_2},\ldots,u^{n_k}]=0$ for all $u \in L$, where $n_1, n_2, \ldots, n_k\geq 1$ are fixed integers. Then either there exists $\lambda\in C$ such that $F(x)=\lambda x$ for all $x\in R$, or $R$ satisfies $s_4$, the standard identity in four variables.

**Keywords:** Prime ring; generalized derivation; extended centroid; Utumi quotient ring; Engel condition.

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

Parity results for 13-core partitions | 175$-$181 |

**Keywords:** $t$-core partition; theta function; dissection; congruence.

**MSC:** 11P83, 05A17

Non-null helicoidal surfaces as non-null Bonnet surfaces | 182$-$191 |

**Abstract**

In this study, we obtain an equivalent of the Codazzi-Mainardi equations for spacelike and timelike surfaces in three dimensional Lorentz space $\Bbb{R}_1^3$. Also, we find necessary and sufficient conditions for spacelike and timelike helicoidal surfaces with non-null axis to be Bonnet surfaces.

**Keywords:** Bonnet surface; helicoidal surface; Lorentz space form.

**MSC:** 53A10, 53B30

$S$-paracompactness in ideal topological spaces | 192$-$203 |

**Abstract**

In this paper, we study the notion of $S$-paracompact spaces in ideal topological spaces. We provide some characterizations of these spaces and investigate relationships to other classes of spaces. Moreover, we study the invariance of such spaces under some special types of functions.

**Keywords:** $S$-paracompact; almost-paracompact; ideal;
$\Cal{I}$-para\-compact; countably $\Cal{I}$-compact; perfect functions.

**MSC:** 54A05, 54D20, 54F65, 54G05, 54C10

Common fixed points of commuting mappings in ultrametric spaces | 204$-$214 |

**Abstract**

In this paper, we will use implicit functions to obtain a general result about the existence of a unique common fixed point for commuting mappings in ultrametric spaces. This result enables us to improve some known fixed point theorems and enables us to obtain a relation between completeness and the existence of a unique fixed point for self-mappings in non-Archimedean metric spaces. By presenting some counterexamples, we will show that our results cannot be extended to general metric spaces.

**Keywords:** Contraction mapping; fixed point; non-Archimedean metric space.

**MSC:** 47H10, 47H09, 54E35

An existence result for a Kirchhoff $p(x)$-Laplacian equation | 215$-$224 |

**Abstract**

In this article, using Mountain Pass Theorem, we investigate the existence of a nontrivial weak solution for nonlocal equations driven by $p(x)$-Laplacian, under Dirichlet boundary condition.

**Keywords:** $p(x)$-Kirchhoff type operator; mountain pass theorem; nonlocal problem.

**MSC:** 35J20, 35J60, 34B27

On monotonicity of ratios of some $q$-hypergeometric functions | 225$-$231 |

**Abstract**

In this paper, we prove monotonicity of some ratios of $q$-Kummer confluent hypergeometric and $q$-hypergeometric functions. The results are also closely connected with Turán type inequalities. In order to obtain main results we apply methods developed for the case of classical Kummer and Gauss hypergeometric functions in [S.M. Sitnik, Inequalities for the exponential remainder, preprint, Institute of Automation and Control Process, Far Eastern Branch of the Russian Academy of Sciences, Vladivostok 1993 (in Russian)] and [S.M. Sitnik, Conjectures on Monotonicity of Ratios of Kummer and Gauss Hypergeometric Functions, RGMIA Research Report Collection 17 (2014), Article 107].

**Keywords:** Kummer functions; Gauss
hypergeometric functions; $q$-Kummer confluent hypergeometric
functions; $q$-hypergeometric functions; Turán type inequalities.

**MSC:** 33D15