Volume 70 , issue 4 ( 2018 ) | back |

On viability result for first-order functional differential inclusions | 283$-$291 |

**Abstract**

We prove the existence of solutions, in separable Banach spaces, for the following differential inclusion: \begin{align*} \left\{ \begin{array}{ll} \dot{x}(t) ın F(t,T(t)x),\quad &\mbox{a.e. on }[0,\tau];\\ x(s)=\varphi(s),\quad &\forall sın [-a,0]; \\ x(t) ın C(t),\quad &\forall tın [0,\tau]; \end{array} \right. \end{align*} We consider weaker hypotheses on the constraint.

**Keywords:** Multifunction; measurability; selection; functional differential inclusion.

**MSC:** 34A60, 49J52

A new extension of Bessel-Maitland function and its properties | 292$-$302 |

**Abstract**

This paper deals with a new extended Bessel-Maitland function. The $m^{th}$ differentiation, Beta transform, Laplace transform, Whittaker transform and various other transforms for our new extended Bessel-Maitland function are presented here. Further, the Riemann-Liouville fractional integration and differentiation for the function introduced here are also indicated.

**Keywords:** Generalized Bessel-Maitland function; Wright hypergeometric function; fractional calculus.

**MSC:** 42C05, 33C45

The Zariski topology on the graded classical prime spectrum of a graded module over a graded commutative ring | 303$-$313 |

**Abstract**

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. A proper graded submodule $N$ of $M$ is called a graded classical prime if whenever $r,sın h(R)$ and $mın h(M)$ with $rsmın N$, then either $rmın N$ or $smın N$. The graded classical prime spectrum $Cl.Spec^{g}(M)$ is defined to be the set of all graded classical prime submodules of $M$. In this paper, we introduce and study a topology on $Cl.Spec^{g}(M)$, which generalizes the Zariski topology of graded ring $R$ to graded module $M$, called Zariski topology of $M$, and investigate several properties of the topology.

**Keywords:** Graded classical prime spectrum; graded classical prime submodule; Zariski topology.

**MSC:** 13A02, 16W50

Existence and uniqueness results for three-point nonlinear fractional (arbitrary order) boundary value problem | 314$-$325 |

**Abstract**

We present here a new type of three-point nonlinear fractional boundary value problem of arbitrary order of the form \begin{align*} &^{c}D^{q}u(t) = f(t,u(t)),\ \ t ın [0,1],\\ &u(\eta) = u^{\prime}(0)= u^{\prime\prime}(0) = \dots = u^{n-2}(0) = 0,\ I^{p}u(1) = 0,\ \ \ \ 0 < \eta < 1, \end{align*} where $n-1 < q \leq n$, $n ın \mathbb{N}$, $n \geq 3$ and $^{c}D^{q}$ denotes the Caputo fractional derivative of order $q$, $I^{p}$ is the Riemann-Liouville fractional integral of order $p$, $f : [0,1] \times \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function and $\eta^{n-1} \neq \frac{\Gamma(n)}{(p+n-1)(p+n-2)\dots(p+1)}$. We give new existence and uniqueness results using Banach contraction principle, Krasnoselskii, Scheafer's fixed point theorem and Leray-Schauder degree theory. To justify the results, we give some illustrative examples.

**Keywords:** Filtered Lagrangian Floer homology; Künneth formula; PSS isomorphism.

**MSC:** 26A33, 34B15

Generalized Hölder's inequality in Morrey spaces | 326$-$337 |

**Abstract**

The aim of this paper is to present sufficient and necessary conditions for generalized Hölder's inequality in Morrey spaces and generalized Morrey spaces. We also obtain similar results in weak Morrey spaces and generalized weak Morrey spaces. The sufficient and necessary conditions for the generalized Hölder's inequality in these spaces are obtained through estimates for characteristic functions of balls in $R^d$.

**Keywords:** Hölder's inequality; generalized Hölder's inequality; Morrey spaces; weak Morrey spaces; generalized Morrey spaces;
generalized weak Morrey spaces.

**MSC:** 26D15, 46B25, 46E30

Non-normal $p$-bicirculants, $p$ a prime | 338$-$343 |

**Abstract**

A graph $\Gamma$ is called a semi-Cayley graph over a group $G$, if there exists a semiregular subgroup $R_G$ of $Aut(\Gamma)$ isomorphic to $G$ with two orbits (of equal size). We say that $\Gamma$ is normal if $R_G$ is a normal subgroup of $Aut(\Gamma)$. Semi-Cayley graphs over cyclic groups are called bicirculants. In this paper, we determine all non-normal bicirculants over a group of prime order.

**Keywords:** Semi-Cayley graph; bicirculant; normal semi-Cayley graph.

**MSC:** 05C25, 20B25

On some multivariate summatory functions of the Euler phi-function | 344$-$349 |

**Abstract**

In this note we obtain an asymptotic formula with a power saving error term for the summation function of Euler phi-function evaluated at iterated and generalized least common multiples of four integer variables.

**Keywords:** Euler phi-function; multiplicative functions; least common multiple; greatest common divisor; asymptotic formula.

**MSC:** 11A25, 11N37,11N60,11A05

Coarser compact topologies | 350$-$363 |

**Abstract**

The concept of a quasi-king space is introduced, which is a natural generalisation of a king space. In the realm of suborderable spaces, king spaces are precisely the compact spaces, so are the quasi-king spaces. In contrast, quasi-king spaces are more flexible in handling coarser selection topologies. The main purpose of this paper is to show that a weakly orderable space is quasi-king if and only if all of its coarser selection topologies are compact.

**Keywords:** Vietoris topology; separately continuous weak selection; coarser topology; weakly orderable space; quasi-king space; pseudocompact space.

**MSC:** 54B20, 54C65, 54D30, 54F05

On graded 2-absorbing submodules over $Gr$-multiplication modules | 364$-$376 |

**Abstract**

Let $G$ be a multiplicative group with identity $e$, $R$ be a $G$-graded commutative ring and $M$ be a graded $R$-module. The aim of this article is some investigations of graded $2$-absorbing submodules over $Gr$-multiplication modules. A graded submodule $N$ of $R$-module $M$ is called graded $2$-absorbing if whenever $a,bın h(R)$ and $mın h(M)$ with $abmın N$, then either $abın (N :_R M)$ or $amın N$ or $bmın N$. We also introduce the concept of graded classical $2$-absorbing submodule as a generalization of graded classical prime submodules and show a number of results in this class.

**Keywords:** Graded multiplication modules; graded prime submodules; graded 2-absorbing submodules; graded classical submodules; graded classical 2-absorbing submodules.

**MSC:** 13A02, 16W50