The promotion of electric power energy sharing transactions plays a crucial role in facilitating the consumption of clean energy and achieving a green, low-carbon transition. However, this novel transaction model may significantly impact traditional power markets and pose challenges to the economic and secure operation of distribution networks. To explore the effects of energy sharing transaction strategies on modern distribution networks, this study considers the participation of power load aggregators in peer-to-peer transactions and establishes a distribution network-load aggregator game framework based on dual fixed-point mapping. Within this framework, the study examines the equilibrium states of energy sharing transactions between the distribution network and load aggregators, as well as among the load aggregators themselves. By modeling the objectives and decision spaces of the load aggregators' stakeholders, an equivalent unified optimization decision model for the load aggregator group is derived. The Lagrange multiplier relaxation method is proposed to achieve distributed decoupling of the unified problem, thereby ensuring the privacy and fairness of the participants' behaviors in the energy transactions. Furthermore, the study introduces a modeling approach for the distribution network transaction strategy based on a dual fixed-point mapping model and demonstrates the existence and uniqueness of the equilibrium state of transaction strategies between the distribution network and load aggregators through game theory. To effectively solve this issue, the equivalence between the energy sharing mechanism of the load aggregator group and the distributed optimization problem is analyzed, and an algorithm for solving the distribution network energy sharing transaction strategy is proposed. Through simulation experiments based on a 33-node example, the reliability of the proposed dual fixed point model and algorithm in effectively describing the energy sharing trading strategy of the distribution network was verified, and at the same time, it was achieved on the premise of ensuring the economic operation of the distribution network. Taking into account the income of power aggregators.