In this manuscript, we prove the existence and uniqueness of a common best proximity point for a pair of non-self mappings satisfying the iterative mappings in a complete fuzzy multiplicative metric space. We consider the pair of non-self mappings $X:\mathcal{P}\to \mathcal{G}$ and $Z:\mathcal{P}\to \mathcal{G}$ and the mappings do not necessarily have a common fixed-point. In a complete fuzzy multiplicative metric space, if $\phi$ satisfy the condition $\phi \left(b,Zb,\varsigma \right)=\phi \left(\mathcal{P},\mathcal{G},\varsigma \right)=\phi \left(b,Xb,\varsigma \right)$, then $b$ is a common best proximity point. Further, we obtain the common best proximity point for the real valued functions $\mathcal{L},\mathcal{M}:(0,1]\to \mathbb{R}$ by using a generalized fuzzy multiplicative metric space in the setting of $(\mathcal{L},\mathcal{M})$-iterative mappings. Furthermore, we utilize fuzzy multiplicative versions of the $(\mathcal{L},\mathcal{M})$-proximal contraction, $(\mathcal{L},\mathcal{M})$-interpolative Reich-Rus-Ciric type proximal contractions, $(\mathcal{L},\mathcal{M})$-Kannan type proximal contraction and $(\mathcal{L},\mathcal{M})$-interpolative Hardy-Rogers type proximal contraction to examine the common best proximity points in fuzzy multiplicative metric space. Moreover, we provide differential non-trivial examples to support our results.

This paper presents the dynamical aspects of a nonlinear multi-term pantograph-type system of fractional order. Pantograph equations are special differential equations with proportional delays that are employed in many scientific disciplines. The pantograph mechanism, for instance, has been applied in numerous scientific disciplines like electrodynamics, engineering, and control theory. Because of its key rule in diverse fields, the current study establishes some necessary criteria for its controllability. The main idea of the proof is based on converting the system into a fixed point problem and introducing a suitable controllability Gramian matrix ${\mathcal{G}}_c$. The Gramian matrix ${\mathcal{G}}_c$ is used to demonstrate the linear system's controllability. Controllability criteria for the associated nonlinear system have been established in the sections that follow using the Schaefer fixed-point theorem and the Arzela-Ascoli theorem, as well as the controllability of the linear system and a few key assumptions. Finally, a computational example is listed.

We aimed to analyze a new class of sequential fractional differential inclusions that involves a combination of $\varsigma$-Hilfer and $\varsigma$-Caputo fractional derivative operators, along with non-separated boundary conditions. Two cases of convex-valued and non-convex-valued set-valued maps are considered. Our outcomes are obtained from some famous theorems of fixed point method within the framework of the set-valued analysis. Additionally, some examples are provided to illustrate the applicability of our outcomes.

This paper investigates the boundedness and approximation properties of Hilbert-type singular integral operators within the framework of Triebel–Lizorkin spaces $F_{p,q}^s(\mathbb{R})$, a refined class of function spaces central to microlocal and harmonic analysis. We introduced a regularized Hilbert-type operator and established its strong convergence in Triebel–Lizorkin norms. Using Fourier analytic and interpolation techniques, we rigorously proved new boundedness results under optimal smoothness and integrability conditions. Furthermore, we demonstrated how this functional analytic framework enables robust modeling of non-smooth volatility structures in financial derivatives pricing, particularly under stochastic volatility regimes and market turbulence. This approach unifies singular operator theory and financial mathematics, offering a novel path for analyzing irregular price dynamics through spectral and geometric regularity tools.