This study aims to present novel fixed-point results within the structure of a newly introduced abstract structure known as perturbed metric spaces. As expected, these spaces naturally extend and generalize the classical metric spaces. Consequently, the key results of this study broaden, refine, and broaden the existing fixed-point results in the published outcomes.

The present article presents a system of symmetric equations defining multi-cubic mappings (M-CMs). Next, we describe how these mappings are structured and obtain an equation for describing them. Moreover, we Address the Hyers-Ulam stability (H-UStab) in the sense of Găvruţa for a symmetric multi-cubic equation through the application of the so-called Hyers (direct) method in the setting of 2-Banach spaces. For a typical case, by means of a norm, induced from a 2-norm of ${\mathbb{R}}^d$, we examine the stability and hyperstability of a mapping $f:{\mathbb{R}}^{dn}⟶{\mathbb{R}}^d$ by using a fixed point (FP) result.

In this study, we present generalized multi-quadratic mappings (GM-QMs) which differ from earlier ones that were previously available in the literature. We then express these mappings (specified by a system of generalized quadratic functional equations (GQFEs)) in a single equation. The fixed-point (FP) methodology and the direct approach (Hyers) method are also used to generate a number of stability findings for a system of generalized FEs in the setting of fuzzy norm spaces (FNSs). In terms of the results obtained by the aforementioned methods, we find that in comparison to the direct method, the FP tool provides a more accurate estimate of GM-QMs while requiring fewer conditions for the proofs.

The most generalized form of the soft neutrosophic set used in this article is the single valued heptapartitioned neutrosophic soft set (HPNSS). A set theoretic research is completed and some operations on them are defined. We went on to do a thorough investigation on HPNSS and discovered a number of odd characteristics. We built heptapartitioned neutrosophic soft topological spaces (HP-NSTS). Characterization of triangle theory of love is addressed with the help of HPNSTS. Heat map is developed for young boys and young girls separately. The normalized correlation matrix heat map (NCMHM) for young boys and girls is discussed. The analyzation and visualization of the correlation matrix of a data set containing data for Sternberg’s Eight Types of Love for young boys and girls is addressed. K-mean-HPNS-clustering is applied to group the data into K clusters. The elbow approach is used to determine the optimal number of clusters (K) for the K-mean clustering algorithm.

The paper proposes a new adaptive algorithm for solving variational inequality problems (VIPs) with pseudo-monotone operators. Such problems are central to modeling equilibrium in supply chain networks, where asymmetric cost structures lead to pseudo-monotonicity. The proposed method integrates convex combination techniques with an adaptive step-size criterion, requiring only a single metric projection per iteration. We prove linear convergence under strong pseudo-monotonicity and weak convergence under pseudo-monotonicity. Numerical experiments on a blood supply chain network with 24 routes and three demand scenarios show that the algorithm performs competitively with recent methods in terms of residual reduction and achieves the lowest execution time. The results demonstrate the algorithm's efficiency and scalability for large-scale network optimization, with potential impact on healthcare logistics and related applications.