In this study, the $ (h $-$ \varphi)_R $ and $ (h $-$ \varphi)_M $-contractions with two metrics endowed with a directed graph are examined using auxiliary functions. We propose a set of criteria that guarantees the existence of common fixed points for our contractions. This leads to a generalization of previous results in the literature. Towards our accomplishments, we establish affirmative results that demonstrate solutions to a class of nonlinear two-term fractional differential equations with nonlocal boundary conditions. To further corroborate our major findings, we also provide instances.

This paper introduces a new concept of a connected-image set for a mapping, which extends the notion of edge-preserving properties with respect to mapping. We also present novel definitions of connected-image contractions, with a focus on fixed-point theorems involving auxiliary functions in b-metric spaces. The relationships between these mathematical concepts are explored, along with their applications to solving differential and integral equations. In particular, we discuss existence results for solving integral equations and second-order ordinary differential equations with inhomogeneous Dirichlet boundary conditions, as well as theorems related to contractions of the integral type.

This study presents an iterative method for approximating common fixed points of a finite set of G-nonexpansive mappings within a real Hilbert space with a directed graph. We establish definitions for left and right coordinate convexity and demonstrate both weak and strong convergence results based on reasonable assumptions. Furthermore, our algorithm’s effectiveness in solving the heat equation is highlighted, contributing to energy optimization and sustainable development.

Northern Thailand frequently suffers from severe PM2.5 air pollution, especially during the dry season, due to agricultural burning, local emissions, and transboundary haze. Understanding how pollution moves across regions and identifying source–receptor relationships are critical for effective air quality management. This study investigated the spatial and temporal dynamics of PM2.5 in northern Thailand. Specifically, it explored how pollution at one monitoring station influenced concentrations at others and revealed the seasonal structure of PM2.5 transmission using network-based analysis. We developed a Python-based framework to analyze daily PM2.5 data from 2022 to 2023, selecting nine representative stations across eight provinces based on spatial clustering and shape-based criteria. Delaunay triangulation was used to define spatial connections among stations, capturing the region’s irregular geography. Cross-correlation and Granger causality were applied to identify time-lagged relationships between stations for each season. Trophic coherence analysis was used to evaluate the hierarchical structure and seasonal stability of the resulting networks. The analysis revealed seasonal patterns of PM2.5 transmission, with certain stations, particularly in Chiang Mai and Lampang, consistently acting as source nodes. Provinces such as Phayao and Phrae were frequently identified as receptors, especially during the winter and rainy seasons. Trophic coherence varied by season, with the winter network showing the highest coherence, indicating a more hierarchical but less stable structure. The rainy season exhibited the lowest coherence, reflecting greater structural stability. PM2.5 spreads through structured, seasonal pathways in northern Thailand. Network patterns vary significantly across seasons, highlighting the need for adaptive air quality strategies. This framework can help identify influential monitoring stations for early warning and support more targeted, season-specific air quality management strategies in northern Thailand.