This article presents a two-dimensional extension of divisibility networks, constructed on generalized integer lattice, and developed to explore their applications in inequality structures. Edges join nodes that share either the same multiple index \(n\) or the same divisor index \(k\), which form a rook–divisibility network that unites arithmetic structure and graph topology within a deterministic grid. The resulting finite graph \(G_N=\{(k,n)\in\mathbb{N}^2:1\le n\le N,\,k\mid n\}\) admits exact analysis of its main invariants. Closed forms are derived for the local degree \(\deg(k,n)\) and clustering coefficient \(C(k,n)\); they reveal how small \(k\) columns act as hubs and highly composite rows yield strong local cohesion. A constructive proof via projection maps establishes global connectivity for all \(N\), and asymptotic evaluation shows that the average degree grows as \(\langle k\rangle_N\!\sim\!(\pi^2/6)\,N/\log N\), much faster than in the one-dimensional divisor network. The results provide a heavy–tailed degree distribution governed by a logarithmic factor, while empirical simulations and log-binned spectra confirm close agreement between measured and analytic clustering across degree ranges. Further visual analyses illustrate the emergence of hubs, stretching similarity, and stable scaling of local clustering. In addition, the rook-divisibility framework is shown to generate new forms of discrete and fractional inequalities. By interpreting row- and column-averaging operations as convex and fractional mean processes, the model yields Hermite-Hadamard-Mercer-type bounds and degree-clustering inequalities.
This article presents a generalization of the Hardy–Littlewood–Pólya majorization theorem by employing a weighted Montgomery identity derived from Taylor’s formula. We establish new identities and inequalities for n-convex functions, provide Čebyšev-type bounds for the remainders, and derive associated Ostrowski and Grüss-type inequalities. Our results significantly extend the classical theory of majorization and provide a comprehensive framework for analyzing n-convex functions in the context of weighted integral inequalities.
This work is dedicated to some generalized upper bounds obtained for the Jensen gap by using different generalized convex functions including strongly convex functions, s-convex functions, η-convex functions, strongly η-convex functions, m-convex functions, and (α, m)-convex functions. The results are then extended to the integral form of Jensen’s inequality. The main results enable us to establish such bounds for Hölder and Hermite-Hadamard inequalities as well. Finally, estimates for the Csiszár divergence are presented as direct applications of the main outcomes.
This paper deals with the finite element approximation of the elliptic impulse control quasi-variational inequality (QVI), when the impulse control cost goes to zero. By means of the concepts of subsolutions for QVIs and a Lipschitz dependence property with respect to the impulse cost, an \(L^{\infty}\) error estimate is derived for both the impulse control QVI and the correponding asymptotic problem.
In this paper, we give extensions of Jensen-Mercer inequality for functions whose derivatives in the absolute values are uniformly convex considering the class of \(k-\)fractional integral operators.
This paper investigates the application of the Hardy-Littlewood-Sobolev inequality to analyze the global existence of solutions for a wave equation incorporating distributed delay effects and Hartree-type nonlinearities under suitable analytical conditions.
We study a coercive quasi-variational inequality (QVI) system and propose a generalized Schwarz method using finite element approximations. The discrete solution is iteratively constructed through monotone upper and lower sequences, and its convergence is rigorously established in the \(\mathfrak{L}^\infty\) norm. This framework ensures stability, geometric convergence, and efficient computation on overlapping subdomains.
Let \(\left( H;\left\langle \cdot ,\cdot \right\rangle \right)\) be a complex Hilbert space. Denote by \(\mathcal{B}\left( H\right)\) the Banach \(C^{\ast }\) -algebra of bounded linear operators on \(H\). For \(A\in \mathcal{B}\left( H\right)\) we define the modulus of \(A\) by \(\left\vert A\right\vert :=\left( A^{\ast }A\right) ^{1/2}.\) We say that the continuous function \(B:\left[ a,b \right] \rightarrow \mathcal{B}\left( H\right)\) is square modulus convex (concave) on \(\left[ a,b\right]\) if \[\begin{equation*} \left\vert B\left( \left( 1-t\right) u+tv\right) \right\vert ^{2}\leq \left( \geq \right) \left( 1-t\right) \left\vert B\left( u\right) \right\vert ^{2}+t\left\vert B\left( v\right) \right\vert ^{2}, \end{equation*}\] in the operator order of \(\mathcal{B}\left( H\right) ,\) for all \(u,\) \(v\in \left[ a,b\right]\) and \(t\in \left[ 0,1\right] .\) In this paper, we show among others that, if \(B:\left[ m,M\right] \subset \mathbb{R\rightarrow } \mathcal{B}\left( H\right)\) is square modulus convex on \(\left[ m,M\right]\) and \(f:\Omega \rightarrow \left[ m,M\right]\) so that \(f,\) \(\left\vert B\circ f\right\vert ^{2},\) \(Re\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) ,\) \(fRe\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) \in L_{w}\left( \Omega ,\mu ,\mathcal{B}\left( H\right) \right) ,\) where \(w\geq 0\) \(\mu\) -a.e. on \(\Omega\) with \(\int_{\Omega }wd\mu =1,\) then \[0 \leq \int_{\Omega }w\left( s\right) \left\vert B\circ f\right\vert ^{2}d\mu \left( s\right) -\left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2}\] \[\qquad\qquad\qquad~\qquad\qquad \leq \frac{1}{2}\left( M-m\right) \left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) – Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert .\] The discrete versions are also provided.
This work advances the stability theory of fractional differential equations by establishing superstability criteria for a significant class of problems involving the Caputo-Katugampola derivative. Utilizing a generalized Taylor series as a foundational tool, we prove that these equations exhibit superstable behavior under specific conditions. Our results generalize a wide range of existing stability theorems, creating a unified framework that encompasses systems governed by the Caputo fractional derivative as special cases of the more general Katugampola operator.
This paper examine new versions of the Hermite-Hadamard (H-H) inequality in the context of \((p,q)-h\) integrals on finite intervals. Using the properties of convex and differentiable functions, we derive generalized inequalities that consolidate and generalize a number of existing results in quantum calculus. Specifically, the presented approach offers new implicit inequalities whose special cases result in well-known findings for \((p,q)-, (q,h)\)- and \(q\)-integrals, previously found in recent research. The newly established results not just recover and extend known inequalities but also bring further insights into convexity structure in the context of post-quantum calculus. Such contributions yet again enrich the current advancement of integral inequalities within fractional and quantum analysis, with possible uses in optimization, theory of approximation, and related topics.
