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.

This paper investigates the existence of solutions for initial value problems (IVPs) involving implicit fractional differential inclusions defined via the Hilfer-Katugampola fractional derivative. The Hilfer-Katugampola operator, recently introduced as a generalization of Katugampola and Caputo-Katugampola derivatives, encompasses a wide class of fractional operators. We establish existence results for the multivalued fractional differential problem under convexity and compactness assumptions on the multivalued right-hand side, leveraging Bohnenblust-Karlin fixed point theorem and contraction principles for multivalued maps. An illustrative example is provided to demonstrate the applicability of the main theoretical results. Our work contributes to the emerging theory of fractional differential inclusions governed by fractional derivatives of generalized type.

By examining the properties of a certain linear transformation of functionals, we present applications of Cauchy, Aczel, Callebaut, and Beckenbach type inequalities. Additionally, we provide results for complex functionals.