Mathematics

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Accurate detection of concealed items in X-ray baggage images is critical for public safety in high-security environments such as airports and railway stations. However, small objects with low material contrast, such as plastic lighters, remain challenging to identify due to background clutter, overlapping contents, and weak edge features. In this paper, we propose a novel architecture called the Contrast-Enhanced Feature Pyramid Network (CE-FPN), designed to be integrated into the YOLO detection framework. CE-FPN introduces a contrast-guided multi-branch fusion module that enhances small-object representations by emphasizing texture boundaries and improving semantic consistency across feature levels. When incorporated into YOLO, the proposed CE-FPN significantly boosts detection accuracy on the HiXray dataset, achieving up to a +10.1% improvement in mAP@50 for the nonmetallic lighter class and an overall +1.6% gain, while maintaining low computational overhead. In addition, the model attains a mAP@50 of 84.0% under low-resolution settings and 87.1% under high-resolution settings, further demonstrating its robustness across different input qualities. These results demonstrate that CE-FPN effectively enhances YOLO’s capability in detecting small and concealed objects, making it a promising solution for real-world security inspection applications.

In this paper, we introduce and investigate two generalized forms of classical contraction mappings, namely the p-Hardy–Rogers and p-Zamfirescu contractions. By incorporating the integer parameter $p\ge 1$, these new definitions extend the traditional Hardy–Rogers and Zamfirescu conditions to iterated mappings ${\hslash }^p$. We establish fixed-point theorems, ensuring both existence and uniqueness of fixed points for continuous self-maps on complete metric spaces that satisfy these p-contractive conditions. The proofs are constructed via geometric estimates on the iterates and by transferring the fixed point from the p-th iterate ${\hslash }^p$ to the original mapping ℏ. Our results unify and broaden several well-known fixed-point theorems reported in previous studies, including those of Banach, Hardy–Rogers, and Zamfirescu as special cases.