**Abstract**Employing sequential generalized Caputo fractional left and right vectorial Taylor formulae we establish mixed sequential generalized fractional Ostrowski and Gruss ¨ type inequalities for several Banach algebra valued functions. The estimates are with respect to all norms k·kp , 1 ≤ p ≤ ∞. We finish with applications.

Volume 4, Number 3 (2021)-Table of Contents

## Sequential Generalized Fractional Ostrowski and Grüss type inequalities for several Banach algebra valued functions

Annals of Communications in Mathematics 2021

, 4 (3)

, 207-225

DOI: https://doi.org/10.62072/acm.2021.040301

## Positive solutions for nonlinear Caputo-Hadamard fractional relaxation differential equations

Annals of Communications in Mathematics 2021

, 4 (3)

, 226-236

DOI: https://doi.org/10.62072/acm.2021.040302

**Abstract**We study the existence and uniqueness of positive solutions of the nonlinearfractional relaxation differential equation where Dα/1 is the Caputo-Hadamard fractional derivative of order 0 < α ≤ 1. In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct an appropriate mapping and employ the Schauder fixed point theorem and the method of upper and lower solutions to show the existence of a positive solution of this equation. We also use the Banach fixed point theorem to show the existence of a unique positive solution. Finally, an example is given to illustrate our results.

## On New Approach Towards Cubic Vague Subbisemirings in Bisemirings

Annals of Communications in Mathematics 2021

, 4 (3)

, 237-248

DOI: https://doi.org/10.62072/acm.2021.040303

**Abstract**From the nature of subbisemiring, we develop a new generalized hybrid structure of vague subbisemiring known as cubic vague subbisemiring (shortly CVSBS). We talk about the CVSBS and level sets CVSBS of bisemiring. At first we define some basic operation such as intersection, cartesian product on them and use these to obtain some of its basic properties under CVSBS. Let L = hA¯L, VLi be the cubic vague subset of S. It is shown that L is a CVSBS if and only if all non empty level set L(α,β) (α, β ∈ D[0, 1]) is a SBS. Let L be the CVSBS and W be the strongest vague relation of S. We show that L is a CVSBS if and only if W is a CVSBS of S × S. After we define homomorphic image and preimage of bisemiring. It will be shown that the homomorphic image and preimage of CVSBS is a CVSBS. To strengthen our results with examples are indicated.

## Somewhat Fuzzy Completely e-irresolute Mappings

Annals of Communications in Mathematics 2021

, 4 (3)

, 249-253

DOI: https://doi.org/10.62072/acm.2021.040304

**Abstract**The aim of this paper is to introduce and study the concept of somewhat fuzzy completely e-irresolute mapping and somewhat fuzzy irresolute e-open mapping. Further, some interesting properties of those mappings are given and some comparative results discussed.

## Neutrosophic Regular Semi Continuous Functions

Annals of Communications in Mathematics 2021

, 4 (3)

, 254-260

DOI: https://doi.org/10.62072/acm.2021.040305

**Abstract**In this paper, we introduce and study the concept of regular semi continuous, regular semi irresolute, regular semi-T1/2 space, regular semi homeomorphisms and regular semi c-homeomorphisms in neutrosophic topological spaces. Moreover, we investigate the relationship among neutrosophic regular semi continuous, neutrosophic regular semi irresolute, neutrosophic regular semi homeomorphism and neutrosophic regular semi Chomeomorphisms mappings. Finally, we have given some counter examples to show that these types of mappings are not equivalent.

## Generalized Spherical Fuzzy Soft Sets in Medical Diagnosis for a Decision

Annals of Communications in Mathematics 2021

, 4 (3)

, 261-277

DOI: https://doi.org/10.62072/acm.2021.040306

**Abstract**In the present communication, we introduce the theory of generalized spherical fuzzy soft set and define some operations such as complement, union, intersection, AND and OR. Notably, we tend to showed De Morgan’s laws, associate laws and distributive laws that are holds in generalized spherical fuzzy soft set. Also, we advocate an algorithm to solve the decision making problem based on generalized soft set model. We introduce a similarity measure of two generalized spherical fuzzy soft sets and discuss its application in a medical diagnosis problem. Suppose that there are five patients P1, P2, P3, P4 and P5 in a hospital with certain symptoms of dengue hemorrhagic fever. Let the universal set contain only three elements. That is X = {x1 : severe, x2: mild, x3 : no}, the set of parameters E is the set of certain symptoms of dengue hemorrhagic fever is represented by E = {e1 : severe abdominal pain, e2: persistent vomiting, e3 : rapid breathing, e4 : bleeding gums, e5: restlessness and blood in vomit}. An illustrative examples are mentioned to show that they can be successfully used to solve problems with uncertainties.

## On basic properties of relative Γ-ideals in Γ-near rings

Annals of Communications in Mathematics 2021

, 4 (3)

, 278-283

DOI: https://doi.org/10.62072/acm.2021.040307

**Abstract**The algebraic system Γ-near rings was introduced by Satyanarayana. Tamizh and Ganesan introduced the concept of bi-ideals in near-rings [On bi-ideals of near-rings, Indian J. Pure Appl. Math., 18(11), 1002-1005(1987)]. Tamizh and Meenakumari defined the concept of bi-ideals in Γ-near-rings and characterized Γ-near-fields [Bi-Ideals of Gamma Near-Rings, Southeast Asian Bulletin of Mathematics(2004), 27: 983-988]. Satyanarayana, Yahya, Basar and Kuncham studied abstract affine Γ-nearrings [Some Results on Abstract Affine Gamma-Near-Rings, International Journal of Pure and Applied Mathematical Sciences, 7(1) (2014), 43-49]. Recently, Basar, Satyanarayana, Kuncham, Kumar and Yahya studied some relative ideals in Γ-nearrings [A note on relative Γ-ideals in abstract affine Γ-nearrings, GIS Science Journal, 8(10)(2021), 9-13]. In this paper, we study relative quasi-Γ-ideals and relative bi-Γ-ideals in Γ-near rings. We also proved nice characterizations of Γ-near rings by these basic relative Γ-ideals.

## On Dynamic Multisets and their Operations

Annals of Communications in Mathematics 2021

, 4 (3)

, 284-292

DOI: https://doi.org/10.62072/acm.2021.040308

**Abstract**The concept of multisets emerged by violating a principle of distinct collection of object in crisp sets. In some practical situations, multisets with multiplicity of their objects varying overtime are frequently encountered, such multisets are called dynamic multisets. However, there has been no formal mathematical study on dynamic multisets. Dynamic multiset is a special kind of multiset with time varying multiplicity of elements. The importance of dynamic multisets stems from their potential usefulness in resolving a task of finding duplicate records within large databases. In this paper, we vividly explore the concept of dynamic multisets and present some of its properties. We observe that, the operations on dynamic multisets are the same as that of static multisets, with the time parameter as the only distinction. Finally, some application-driven examples of dynamic multisets are presented.

## A study of ordered quasi-hyperideals and ordered bi-hyperideals in regular ordered semihypergoups

Annals of Communications in Mathematics 2021

, 4 (3)

, 293-306

DOI: https://doi.org/10.62072/acm.2021.040309

**Abstract**In this paper, we introduce the concept of ordered quasi-hyperideals of regular ordered semihypergroups, and study the basic results on ordered quasi-hyperideals of ordered semihypergroups. We also investigate regular ordered semihypergroups in terms of its ordered quasi-hyperideals, ordered right hyperideals and ordered left hyperideals. We prove that: (i) A partially ordered semihypergroup S is regular if and only if for every ordered bi-hyperideal B, every ordered hyperideal I and every ordered quasi-hyperideal Q, we have B ∩ I ∩ Q ⊆ (B ◦ I ◦ Q], and (ii) A partially ordered semihypergroup S is regular if and only if for every ordered quasi-hyperideal Q, every ordered left hyperideal L and every ordered right-hyperideal R, we have R ∩ Q ∩ L ⊆ (R ◦ Q ◦ L].

## m-polar cubic set theory applied to BCK/BCI-algebras

Annals of Communications in Mathematics 2021

, 4 (3)

, 307-319

DOI: https://doi.org/10.62072/acm.2021.0403010

**Abstract**In this paper, by combinig the notions of m-polar fuzzy structures and interval valued m-polar fuzzy structures, the notion of m-polar cubic structures is introduced and applied on the ideal theory of BCK/BCI-algebras. In this respect, the notions of m-polar cubic subalgebras and m-polar cubic (commutative) ideals are introduced and some essential properties are discussed. Characterizations of m-polar cubic subalgebras and m-polar cubic (commutative) ideals are considered. Moreover, the relations among m-polar cubic subalgebras, m-polar cubic ideals and m-polar cubic commutative ideals are obtained.