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This paper investigates an optimization problem corresponding to energy efficiency maximization of an unmanned aerial vehicle (UAV)-enabled relaying system, where a fixed-wing UAV acts as an amplify-and-forward mobile relay to assist data transmission between a source node and a destination node. On the premise of satisfying the speed and acceleration constraints of the UAV, the energy efficiency (EE) of the relaying system is maximized by jointly optimizing the UAV's trajectory and the individual transmit power levels of the source and the UAV relay. The initial joint optimization problem is non-convex and cannot be solved directly. Therefore, the joint optimization problem is decomposed into two sub-problems which are solved by applying the successive convex approximation technique and the Dinkelbach's algorithm. On this basis, an efficient iterative algorithm is proposed to tackle the joint optimization problem through the block coordinate descent technique. Simulation results demonstrated that by conducting the proposed algorithm, the flight trajectory of the UAV and the individual transmit power levels of the nodes can be flexibly adjusted according to the system conditions, and the proposed algorithm contributes to the higher EE compared with the benchmark schemes. [ABSTRACT FROM AUTHOR]