Volume 72 , issue 3 ( 2020 ) | back |

ON A GENERALIZED FIXED POINT THEOREM IN INCOMPLETE SOFT METRIC SPACES | 187$-$195 |

**Abstract**

In this paper, we introduce the notion of orthogonal relation on a soft set $(F,A)$ and some related concepts. This notion allows us to consider fixed point theorem in SO-complete instead of complete soft metric spaces introduced by Yazar et.al. (Filomat 30:2 (2016), 269--279). Then, the existence and uniqueness of soft fixed points for a generalized soft contractive mapping are proved. Also, some examples are given to support that our main theorem is a real extension of Yazar et.al.

**Keywords:** Soft metric space; soft fixed point; orthogonal set; Picard operator.

**MSC:** 34A12, 65R10, 65R20

ON A CLASS OF ELLIPTIC NAVIER BOUNDARY VALUE PROBLEMS INVOLVING THE $\boldsymbol{(p_{1}(\cdot),p_{2}(\cdot))}$-BIHARMONIC OPERATOR | 196$-$206 |

**Abstract**

In this article, we study the existence and multiplicity of weak solutions for a class of elliptic Navier boundary value problems involving the $(p_{1}(\cdot),p_{2}(\cdot))$-biharmonic operator. Our technical approach is based on variational methods and the theory of the variable exponent Lebesgue spaces. We establish the existence of at least one solution and infinitely many solutions of this problem, respectively.

**Keywords:** $p_{1}(\cdot)$-Laplacian; mountain pass theorem; multiple solutions; critical point theory.

**MSC:** 39A05, 34B15

ON THE PARTIAL NORMALITY OF A CLASS OF BOUNDED OPERATORS | 207$-$214 |

**Abstract**

In this paper, some various partial normality classes of weighted conditional expectation type operators on $L^{2}(\Sigma)$ are investigated. For a weakly hyponormal weighted conditional expectation type operator $M_wEM_u$, we show that the conditional Cauchy-Schwartz inequality for u and w becomes an equality. Assuming this equality, we then show that the joint point spectrum is equal to the point spectrum of $M_wEM_u$. Also, we compute the approximate point spectrum of $M_wEM_u$ and we prove that under a mild condition the approximate point spectrum and the spectrum of $M_wEM_u$ are the same.

**Keywords:** Conditional expectation; hyponorma;, weakly hyponorma operators; spectrum.

**MSC:** 47B47

CONFORMAL CURVATURE TENSOR ON PARACONTACT METRIC MANIFOLDS | 215$-$225 |

**Abstract**

In this paper, we consider paracontact metric manifolds satisfying certain flatness conditions on the conformal curvature tensor. Specifically, we study $\xi$-conformally flat $K$-paracontact manifolds and $\varphi$-conformally flat $K$-paracontact and paraSasakian manifolds. Also we discuss $\varphi$-conformally flat compact regular $K$-paracontact manifolds. Finally, we study conformally flat paracontact metric manifolds..

**Keywords:** Conformal curvature tensor; paracontact metric manifold; $K$-paracontact manifold; paraSasakian manifold; regular $K$-paracontact manifold.

**MSC:** 53C15, 53C25, 53D10

CERTAIN RESULTS ON A CLASS OF INTEGRAL FUNCTIONS REPRESENTED BY MULTIPLE DIRICHLET SERIES | 226$-$231 |

**Abstract**

In the present paper we obtain a condition on vector valued coefficients of multiple Dirichlet series for when the series converges in the whole complex plane. We also prove some results related to Banach algebraic structure, topological divisor of zero and more on a class of such series satisfying certain condition.

**Keywords:** Dirichlet series; Banach algebra; topological divisor of zero; continous linear functional and total set.

**MSC:** 30B50, 46J15, 17A35

COMMON FIXED POINTS OF GENERALIZED CONTRACTIVE MAPPINGS IN UNIFORM SPACES | 232$-$242 |

**Abstract**

In order to establish some common fixed point theorems on Hausdorff uniform spaces endowed with a graph we will define a new kind of generalized contraction for self-mappings. A few related examples are also provided to support our main results. Finally an application of our results in $b$-metric spaces is exhibited.

**Keywords:** Fixed points; uniform spaces; graph; connectivity.

**MSC:** 47H10, 54E15, 05C40, 54H25

ON PRIME STRONG IDEALS OF A SEMINEARRING | 243$-$256 |

**Abstract**

The concepts of prime ideals and corresponding radicals play an important role in the study of nearrings. In this paper, we define different prime strong ideals of a seminearring $S$ and study the corresponding prime radicals. In particular, we prove that $P_e=S\mid P_e(S)=S $ is a Kurosh-Amitsur radical class where $P_{e}(S)$ denotes the intersection of equiprime strong ideals of $S$.

**Keywords:** Seminearring; strong ideal; radical.

**MSC:** 16Y30, 16Y60

INFINITE SERIES OF COMPACT HYPERBOLIC MANIFOLDS, AS POSSIBLE CRYSTAL STRUCTURES | 257$-$272 |

**Abstract**

Previous discoveries of the first author (1984-88) on so-called hyperbolic football manifolds and our recent works (2016-17) on locally extremal ball packing and covering hyperbolic space ${H}^3$ with congruent balls had led us to the idea that our ``experience space in small size'' could be of hyperbolic structure. In this paper we construct a new infinite series of oriented hyperbolic space forms so-called cobweb (or tube) manifolds $Cw(2z, 2z, 2z)=Cw(2z)$, $3\le z$ odd, which can describe nanotubes, very probably. So we get a structure of rotational order $z=5,7\dots$, as new phenomena. Although the theoretical basis of compact manifolds of constant curvature (i.e.\ space forms) are well-known (100 years old), we are far from an overview. So our new very natural hyperbolic infinite series $Cw(2z)$ seems to be very timely also for crystallographic applications. Mathematical novelties are foreseen as well, for future investigations.

**Keywords:** Hyperbolic space form; cobweb manifold; fullerene and nanotube.

**MSC:** 57M07, 57M60,52C17

SEMILATTICE DECOMPOSITION OF LOCALLY ASSOCIATIVE $\Gamma$-AG$^{**}$-GROUPOIDS | 273$-$280 |

**Abstract**

In this paper, we have shown that a locally associative $\Gamma$-AG$^{**}$-groupoid $S$ has associative powers and $S/\rho_{\Gamma}$ is a maximal separative homomorphic image of $S$, where $a\rho_{\Gamma}b$ implies that $a\Gamma b_{\Gamma}^{n}=b_{\Gamma}^{n+1}, b\Gamma a_{\Gamma}^{n}=a_{\Gamma}^{n+1}, \forall a, b\in S$. The relation $\eta_{\Gamma}$ is the least left zero semilattice congruence on $S$, where $\eta_{\Gamma}$ is defined on $S$ as $a\eta_{\Gamma}b$ if and only if there exist some positive integers $m, n$ such that $b_{\Gamma}^{m}\subseteq a\Gamma S$ and $a_{\Gamma}^{n}\subseteq b\Gamma S$.

**Keywords:** $\Gamma$-AG-groupoid; $\Gamma$-left invertive law; $\Gamma$-medial law; $\Gamma$-congruences.

**MSC:** 20M10, 20N99