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In the era of continuous development in Internet of Things (IoT) technology, smart services are penetrating various facets of societal life, leading to a growing demand for interconnected devices. Many contemporary devices are no longer mere data producers but also consumers of data. As a result, massive amounts of data are transmitted to the cloud, but the latency generated in edge-to-cloud communication is unacceptable for many tasks. In response to this, this paper introduces a novel contribution—a layered computing network built on the principles of fog computing, accompanied by a newly devised algorithm designed to optimize user tasks and allocate computing resources within rechargeable networks. The proposed algorithm, a synergy of Lyapunov-based, dynamic Long Short-Term Memory (LSTM) networks, and Particle Swarm Optimization (PSO), allows for predictive task allocation. The fog servers dynamically train LSTM networks to effectively forecast the data features of user tasks, facilitating proper unload decisions based on task priorities. In response to the challenge of slower hardware upgrades in edge devices compared to user demands, the algorithm optimizes the utilization of low-power devices and addresses performance limitations. Additionally, this paper considers the unique characteristics of rechargeable networks, where computing nodes acquire energy through charging. Utilizing Lyapunov functions for dynamic resource control enables nodes with abundant resources to maximize their potential, significantly reducing energy consumption and enhancing overall performance. The simulation results demonstrate that our algorithm surpasses traditional methods in terms of energy efficiency and resource allocation optimization. Despite the limitations of prediction accuracy in Fog Servers (FS), the proposed results significantly promote overall performance. The proposed approach improves the efficiency and the user experience of Internet of Things systems in terms of latency and energy consumption. [ABSTRACT FROM AUTHOR]