Asian-European Journal of Mathematics

In this paper, we create tripled fixed point outcomes via a subjective measure of noncompactness in the sense of Banas and Goebel. Furthermore, we introduce some applications of the measure of noncompactness notion to functional equations including nonlinear integral equations as well as local fractional integral equations.

In this paper, we introduce the concept of an $a$-pseudo complementation $x\mapsto x\ast a$ on an Almost Distributive Lattice $A$ for an arbitrary fixed element $a$ and any $x$ of $A$ and discuss several properties of this. Here, we obtain a bijection correspondence between the $a$-pseudo complementations on $A$ and maximal elements of $A.$ We prove that the set $A\ast a=\{x\ast a:a\in A\}$ is a Boolean algebra which is independent(upto isomorphism) of the $a$-pseudo complementation $x\mapsto x\ast a$.

A set $S$ of vertices in a graph $G=(V,E)$ is a $2$-dominating set of $G$ if every vertex of $V-S$ is adjacent to at least two vertices in $S.$ The $2$-domination number is the minimum cardinality of a $2$-dominating set of $G$. A graph is $2$-domination dot-stable if the contraction of any arbitrary edge has no effect on the $2$-domination number. On the other hand, a graph is $2$-domination dot-critical if the contraction of any arbitrary edge decreases the $2$-domination number. We present some properties of these graphs and we provide a characterization of all trees that are $2$-domination dot-stable or $2$-domination dot-critical.

In this paper, we investigate the notion of dc-po-flat $S$-posets as the ones for which the associated tensor functors transfer merely down-closed embeddings (embeddings with down-closed images in codomains) to embeddings. We investigate derived flatness notions in regard to dc-po-flatness in parallel with po-flatness notions and give examples to clarify new notions and their implications. As the characterization of flat acts by means of embeddings into cyclic acts, stated by Fleischer, is not valid for $S$-posets, it eventuates in introducing the new notion of cyclical po-flatness, situated strictly between weak po-flatness and po-flatness, though, we express a counterpart characterization for dc-po-flatness. At the end, we expose relationships between some po-flatness properties and regular injectivity.

In this paper, we study Johnson pseudo-contractibility of second dual of some Banach algebras. We show that the semigroup algebra ${\ell }^1{(S)}^{\ast \ast }$ is Johnson pseudo-contractible if and only if $S$ is a finite amenable group, where $S$ is an archimedean semigroup. We also show that the matrix algebra $M_I{(ℂ)}^{\ast \ast }$ is Johnson pseudo-contractible if and only if $I$ is finite. We study Johnson pseudo-contractibility of certain projective tensor product second duals Banach algebras.

In this paper, some results on representations of Hom-Leibniz algebras are found. Specifically the adjoint representation and trivial representation of Hom-Leibniz algebras are studied in detail. Deformations and central extensions of Hom-Leibniz algebras are also studied as applications.

The notion of $(\alpha ,\psi )$ G-contractive mappings is defined and the existence and uniqueness of fixed points of such mappings on quasi-partial b-metric-like spaces endowed with a graph are obtained. An application and examples are provided to illustrate the results.

There are two different concepts for hypersubstitutions for algebraic systems [K. Denecke and D. Phusanga, Hyperformulas and solid algebraic systems, Studia Logica 90(2) (2008) 263–286; J. Koppitz and D. Phusanga, The monoid of hypersubstitutions for algebraic systems, J. Announcements Union Sci. Sliven 33(1) (2018) 120–127]. In this paper, we follow the more natural and practicable one given in [J. Koppitz and D. Phusanga, The monoid of hypersubstitutions for algebraic systems, J. Announcements Union Sci. Sliven 33(1) (2018) 120–127]. On the other hand, in [S. Leeratanavalee and K. Denecke, Generalized hypersubstitutions and strongly solid varieties, General Algebra and Applications, Proc. of 59th Workshop on General Algebra; 15th Conf. for Young Algebraists Potsdam 2000 (Shaker Verlag, 2000), pp. 135–145], the concept of the monoid of generalized hypersubstitutions was introduced. Following both ideas, one obtains the concept of a monoid of generalized hypersubstitutions for algebraic systems in a canonical way. The purpose of this paper is the study of the monoid of generalized hypersubstitutions for algebraic systems. We characterize the idempotent as well as regular elements in this monoid.

Due to their wide range of applications in many areas of modern analysis, the splendid Hermite–Hadamard inequalities for convex function are very attractive for researcher community. Many authors have been working on their variants, refinements and extensions. The aim of this research is to establish a novel identity and Hermite–Hadamard-type inequalities for convex functions linked with the renowned conformable integral operators. We have pointed some particular cases of the results obtained here in order to show that they reduce some well-known results to relatively simple ones.

In this paper, we prove some weak and strong convergence results for generalized $\alpha$-nonexpansive mappings using $M$ iteration process in the framework of Banach spaces. This generalizes former results proved by Ullah and Arshad [Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat 32(1) (2018) 187–196].

In this paper, we introduce and study a new class of rings which we call semireversible rings. A ring $R$ is called semireversible if for any $a,b\in R,ab=0$ implies there exists a positive integer $n$ such that $b^{n}a=0$. We observe that the class of semireversible rings strictly lies between the class of central reversible rings and weakly reversible rings. Some relations are provided between semireversible rings and many other known classes of rings. Some extensions of semireversible rings such as ring of fractions, Dorroh extension, subrings of matrix rings are investigated. Finally, we study semireversible rings via modules over them wherein among other results, we prove that a semireversible left (right) SF-ring is strongly regular.

Let $G$ be a simple connected graph and $c:V(G)\to \mathbb{N}$ a coloring of the vertices in $G.$ For any $v\in V(G)$, let $\sigma (v)$ be the sum of colors of the vertices adjacent to $v$. Then $c$ is called a sigma coloring of $G$ if for any two adjacent vertices $u,v\in V(G),\sigma (v)\ne \sigma (u)$. The minimum number of colors needed in a sigma coloring of $G$ is the sigma chromatic number of $G$, denoted by $\sigma (G).$

In this paper, we prescribe a sigma coloring of the join of paths and cycles. As a consequence, we determine the sigma chromatic number of the join of a finite number of paths and cycles. In particular, let $G={\sum }_{i=1}^{\ell }{H}_i$, where $H_i=P_{n_i}$ or $H_i=C_{n_i}$ with $6\le n_1\le \cdots \le n_{\ell }.$ If $n_{i+2}-n_i\ge 2$, where $1\le i\le \ell -2$ and $(H_1,H_2)\ne (C_6,C_6)$, then $\sigma (G)=3$ if $H_i$ is an odd cycle, for some $i,$ and $\sigma (G)=2$ otherwise.

In this paper, we investigate the commutativity of a prime Banach algebra $𝔄$ which admits a nonzero continuous linear generalized derivation $𝔉:𝔄\to 𝔄$ associated with continuous linear derivation $\delta$ such that either $𝔉([{𝔵}^m,{𝔶}^n])-[𝔉({𝔵}^m),{𝔶}^n]\in \mathcal{𝒵}(𝔄)$ or $𝔉([{𝔵}^m,{𝔶}^n])+[𝔉({𝔵}^m),{𝔶}^n]\in \mathcal{𝒵}(𝔄)$ for intergers $m=m(𝔵,𝔶)>1$ and $n=n(𝔵,𝔶)>1$ and sufficiently many $𝔵,𝔶\in 𝔄$. Further, similar results are also obtained for unital prime Banach algebra $𝔄$ which admits a nonzero continuous linear generalized derivations $𝔉$ satisfying either $𝔉{(𝔵𝔶)}^n-𝔉({𝔵}^n){𝔶}^n\in \mathcal{𝒵}(𝔄)$ or $𝔉{(𝔵𝔶)}^n-{𝔶}^n𝔉({𝔵}^n)\in \mathcal{𝒵}(𝔄)$ for an integer $n=n(𝔵,𝔶)>1$ and sufficiently many $𝔵,𝔶$.

A typical quandary in geometric functions theory is to study a functional composed of amalgamations of the coefficients of the pristine function. Conventionally, there is a parameter over which the extremal value of the functional is needed. The present paper deals with consequential functional of this type. By making use of Hohlov operator, a new subclass ${{ℛ}}_{a,b}^c$ of analytic functions defined ins the open unit disk is introduced. For both real and complex parameter, the sharp bounds for the Fekete–Szegö problems are found. An attempt has also been taken to found the sharp upper bound to the second and third Hankel determinant for functions belonging to this class. All the extremal functions are express in term of Gauss hypergeometric function and convolution. Finally, the sufficient condition for functions to be in ${{ℛ}}_{a,b}^c$ is derived. Relevant connections of the new results with well-known ones are pointed out.

The author [S. M. Yusuf, Inverse semimodules, J. Natur. Sci. Math. 6 (1966) 111–117.] introduced inverse semimodules. In this paper, we introduce Clifford semimodules. The Clifford semimodules have been characterized here as a strong $R$-semilattice of $R$-modules. The congruences on inverse semimodules, specially, the $R$-module congruences, have been characterized here completely.

In [Kar, Purkait and Sarkar, Birkhoff center of $c$-semiring, Asian-Europ. J. Math. 12(1) (2019) 1950003, 11 pp.], Kar et al. introduced the notion of Birkhoff center of a $c$-semiring and obtained some characterizations of this center. In this paper, we study different types of lattice structures of Birkhoff center of a $c$-semiring. We also prove that if two $c$-semirings are isomorphic, then their Birkhoff centers are isomorphic but the converse is not necessarily true.

In this paper, we present a fixed point theorem for multivalued mappings satisfying new inequalities without using the Pompeiu–Hausdorff distance. The results generalize the famous Feng and Liu’s result [Feng and Liu, J. Math. Anal. Appl. 317 (2006) 103–112] and some known results in the literature. Some examples are given to illustrate our results.

In this paper, we introduce a notion of approximate Connes-biprojectivity for dual Banach algebras. We study the relation between approximate Connes-biprojectivity, approximate Connes amenability and $\phi$-Connes amenability. We propose a criterion to show that certain dual triangular Banach algebras are not approximately Connes-biprojective. Next, we show that for a locally compact group $G$, the Banach algebra $M(G)$ is approximately Connes-biprojective if and only if $G$ is amenable. Finally, for an infinite commutative compact group $G$, we show that the Banach algebra $L^2(G)$ with convolution product is approximately Connes-biprojective, but it is not Connes-biprojective.

In the work of Beidleman and Smith [On Frattini-like subgroups, Glasgow Math. J. 35 (1993) 95–98], the following question was raised: “If $H$ is a subnormal subgroup of a finite group $G$ containing $\mathrm{\Phi }(G)$, then whether the supersolvability of $H$ follows the supersolvability of $H/\mathrm{\Phi }(G)?$”. This problem was considered in works of Selkin [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)], Skiba [On the intersection of all maximal $𝔉$-subgroups of a finite group, Prob. Phys. Math. Tech. 3(4) (2010) 56–62], Ballester-Bolinches [On $𝔉$-subnormal subgroups and Frattini-like subgroups of a finite group, Glasgow Math. J. 36 (1994) 241–247] and many other authors (see monograph [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)]).

In this paper, we give the answer to the more general question: “Let $𝔉$ be a local formation. If $H$ is a subnormal subgroup of a group $G$, then in what case $H\in 𝔉$ will follow from $H/{\mathrm{\Phi }}_{𝜃}(G,A)\in 𝔉?$”.

If $f_n:X\to X$, $n=1,2,\dots ,$ is a sequence of mappings on a metric space $(X,d)$, a point $x$ in $X$ is called an inverse fixed point of the sequence $(f_n)$, if $inf\{d(x,y):y\in f_n^{-1}(x)\}$ tends to zero as $n$ tends to infinity. This definition is proposed and studied in this paper to obtain a few fixed point results.