Fermatean fuzzy sets (FFSs) constitute a formidable tool for tackling uncertainty in information, thereby enjoying widespread adoption across multiple domains. Similarity measures play a crucial role in determining the similarity between different FFSs. However, existing similarity measures for FFSs sometimes encounter issues that result in unreasonable outcomes when differentiating between FFSs. Therefore, efficiently quantifying the similarity between FFSs has emerged as a pressing issue demanding immediate resolution. In this paper, we introduce twenty-four novel similarity measures based on elementary function. Some properties of these similarity measures and weighted similarity measures are verified. Furthermore, we implement the proposed similarity measures in medical pattern recognition and multi-criteria decision-making to validate their feasibility in practical applications. Compared with some existing measures for FFSs, the outcomes conclusively demonstrate that the introduced similarity measures lead to significantly efficient outcomes.

Fractional order differential equations often possess inherent symmetries that play a crucial role in governing their dynamics in a variety of scientific fields. In this work, we consider numerical solutions for fractional-order linear delay differential equations. The numerical solution is obtained via the Laplace transform technique. The quadrature approximation of the Bromwich integral provides the foundation for several commonly employed strategies for inverting the Laplace transform. The key factor for quadrature approximation is the contour deformation, and numerous contours have been proposed. However, the highly convergent trapezoidal rule has always been the most common quadrature rule. In this work, the Gauss–Hermite quadrature rule is used as a substitute for the trapezoidal rule. Plotting figures of absolute error and comparing results to other methods from the literature illustrate how effectively the suggested approach works. Functional analysis was used to examine the existence of the solution and the Ulam–Hyers (UH) stability of the considered equation.

Thepaper represents the concept of a new type of cone metric space known as double composed- cone-metric-like space. It is a generalization of the notion of a double composed-cone-metric space, where in the new space the self-distance is not necessarily zero; meanwhile if the distance metric equals zero, it must be for identical points of distance. Moreover, this context introduces many results related to this novel concept in the literature, with proving some theorems showing that the existence of common fixed points for two maps holds generalized non-linear rational contractions in the proposed new space. Some examples establish the main results and relationship with other cone metric spaces. Finally, this paper demonstrates applications to the nonlinear integral equation and boundary value problem to support the work results.

This study addresses the challenge of effectively modeling uncertainty and hesitation in complex decision-making environments, where traditional fuzzy and vague set models often fall short. To overcome these limitations, we propose the Fermatean neutrosophic vague soft set (FNVSS), an advanced extension that integrates the concepts of neutrosophic sets with Fermatean membership functions into the framework of vague sets. The FNVSS model enhances the representation of truth, indeterminacy, and falsity degrees, providing greater flexibility and resilience in capturing ambiguous and imprecise information. We systematically develop new operations for the FNVSS, including union, intersection, complementation, the Fermatean neutrosophic vague normalized weighted average (FNVNWA) operator, the generalized Fermatean neutrosophic vague normalized weighted average (GFNVNWA) operator, and an adapted Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method. To demonstrate the practicality of the proposed methodology, we apply it to a solar panel selection problem, where managing uncertainty is crucial. Comparative results indicate that the FNVSS significantly outperforms traditional fuzzy and vague set approaches, leading to more reliable and accurate decision outcomes. This work contributes to the advancement of predictive decision-making systems, particularly in fields requiring high precision, adaptability, and robust uncertainty modeling.