‎We present a Picone's identity for the‎ ‎$mathcal{A}_{p(x)}$-Laplacian‎, ‎which is an extension of the classic‎ ‎identity for the ordinary Laplace‎. ‎Also‎, ‎some applications of our‎ ‎results in Sobolev spaces with variable exponent are suggested.

Abstract Palu defined the index with respect to a cluster tilting object in suitable triangulated category, order better understand Caldero–Chapoton map that exhibits connection between algebras and representation theory. We push this further by proposing an contravariantly finite, rigid subcategory, we show behaves similarly classical index. Let ${\mathcal{C}}$ be skeletally small category spl...

‎We present some inequalities for operator space numerical radius of $2times 2$ block matrices on the matrix space $mathcal{M}_n(X)$‎, ‎when $X$ is a numerical radius operator space‎. ‎These inequalities contain some upper and lower bounds for operator space numerical radius.

The main purpose of this manuscript is to prove a common fixed point theorem for two weakly compatible maps satisfying the following integral type contraction in \(G\)-metric space:\[\int^{G(\mathcal{F}x,\mathcal{F}y,\mathcal{F}z)}_0\varphi (t)dt\le \alpha \int^{L(x,y,z)}_0\varphi (t)dt,\] all \(x,y, z\in X\), where\begin{align*}L(x,y,z)&=\max\{G(gx, gy, gz), G(gx, \mathcal{F}x, \mathcal{F}...

n this paper, we consider continuity properties(especially, regularity, also viewed as an approximation property) for $%mathcal{p}_{0}(x)$-valued set multifunctions ($x$ being a linear,topological space), in order to obtain egoroff and lusin type theorems forset multifunctions in the vietoris hypertopology. some mathematicalapplications are established and several physical implications of thema...

In this paper, we introduce the concept of ternary antiderivation on Banach algebras and investigate stability in algebras, associated to $(\alpha,\beta)$-functional inequality: \begin{align*} &\Vert \mathcal{F}(x+y+z)-\mathcal{F}(x+z)-\mathcal{F}(y-x+z)-\mathcal{F}(x-z)\Vert \nonumber\\ &\leq \Vert \alpha (\mathcal{F}(x+y-z)+\mathcal{F}(x-z)-\mathcal{F}(y))\Vert + \beta (\mathcal{F}(x-...

We investigate the quotient algebra $\mathfrak{A}_X^{\mathcal I}:=\mathcal I(X)/\overline{\mathcal F(X)}^{||\cdot||_{\mathcal I}}$ for Banach operator ideals $\mathcal I$ contained in ideal of compact operators, where $X$ is a space that fails I$-approximation property. The main results concern nilpotent algebras $\mathfrak A_X^{\mathcal{QN}_p}$ and A_X^{\mathcal{SK}_p}$ quasi $p$-nuclear opera...

We determine the relative rank of semigroup $$ \mathcal{T}\left(X,Y\right) all transformations on a finite chain X with restricted range Y ⊆ modulo set \mathcal{OP}\left(X,Y\right) orientation-preserving in \mathcal{T}\left(X,Y\right). Moreover, we \mathcal{O}\left(X,Y\right) order-preserving \mathcal{OP}\left(X,Y\right). In both cases, characterize minimal generating sets.

A graph class $\mathcal{F}$ has the induced Erd\H{o}s-P\'osa property if there exists a function $f$ such that for every graph $G$ and every positive integer $k$, $G$ contains either $k$ pairwise vertex-disjoint induced subgraphs that belong to $\mathcal{F}$, or a vertex set of size at most $f(k)$ hitting all induced copies of graphs in $\mathcal{F}$. Kim and Kwon (SODA'18) showed that for a cy...

We obtain some inequalities related to the powers of numerical radius inequalities of Hilbert space operators. Some results that employ the Hermite-Hadamard inequality for vectors in normed linear spaces are also obtained. We improve and generalize some inequalities with respect to Specht's ratio. Among them, we show that, if $A, Bin mathcal{B(mathcal{H})}$ satisfy in some conditions, it follow...