In 2018, Erdal Karapinar introduced the concept of interpolative Kannan operators, a novel adaptation of the Kannan mapping originally defined in 1969 by Kannan. This new mapping condition is more lenient than the basic contraction condition. In this paper, we study the concept by introducing the IKC-iterated function/multi-function system using interpolative Kannan operators, including a broader area of mappings. Moreover, we establish the Collage Theorem endowed with the iterated function system (IFS) based on the IKC, and show the well-posedness of the IKC-IFS. Interpolative Kannan contractions are meaningful due to their applications in fractals, offering a more versatile framework for creating intricate geometric structures with potentially fewer constraints compared to classical approaches.

Financial fraud detection (FFD) is crucial for ensuring the safety and efficiency of financial transactions. This article presents the Regularised Memory Graph Attention Capsule Network (RMGACNet), an original architecture aiming at improving fraud detection using Bidirectional Long Short-Term Memory (BiLSTM) networks combined with advanced feature extraction and classification algorithms. The model is tested on two reliable datasets: the European Cardholder (ECH) transactions dataset, which contains 284,807 transactions and 492 fraud instances, and the IEEE-CIS dataset, which has more than 1 million transactions. Our approach enhances comparison to existing methods of feature selection and classification accuracy. On the ECH dataset, RMGACNet achieves an accuracy of 0.9772, a precision of 0.9768, and an F1 score of 0.9770 measures; on the IEEE-CIS dataset, it achieves an accuracy of 0.9882, a precision of 0.9876 and an F1 score of 0.9879. The findings indicate that RMGACNet routinely surpasses existing models’ efficiency and accuracy while ensuring strong execution time performance, especially when handling large-scale datasets. The suggested model demonstrates scalability and stability, making it suitable for real-time financial systems.

It is widely recognized that fuzzy number theory relies on the characteristic function. However, within the fuzzy realm, the characteristic function transforms into a membership function contingent upon the interval [0,1]. This implies that real numbers and intervals represent exceptional cases of fuzzy numbers. By considering this approach, this paper introduces a new ${\mathbb{L}}_P$ space and novel refinements for integral variations of Hölder's inequality which is known as Hölder's-like inequality over fuzzy domain. Numerous prevailing inequalities associated with Hölder's-like inequality can be enhanced through the newly acquired inequalities, as demonstrated through an application. By using newly defined special means, some new versions of integral inequalities have obtained where differentiable mappings are real-valued convex-like (or convex fuzzy) mappings Lastly, nontrivial numerical examples are also included to validate the accuracy of the presented inequalities as they vary with the parameter $ꙍ$.

Market crises pose significant challenges for businesses, emphasizing the importance of effective crisis management strategies. Central to these strategies is the ability to identify and retain loyal customers, who often serve as the bedrock of stability during tumultuous times. This paper investigates the application of deep meta-learning analysis to predict loyal customers as a cornerstone of market crisis management. Drawing upon an extensive literature review, the study explores previous research on market crises, customer loyalty, and the evolution of deep learning and meta-learning in crisis management contexts. Methodologically, the paper outlines data collection, model development, and evaluation procedures tailored for deep meta-learning-based customer prediction. In results, the overall accuracy of model 85%, a precision 0.88, recall 0.82, and F1-score 0.85 were obtained. The analysis demonstrates the effectiveness of deep meta-learning models in accurately identifying loyal customers during market crises, offering insights into their performance and applicability compared to traditional methods. Practical implications include potential applications in crisis management for businesses and considerations for real-world implementation.