Edge caching is a promising technology to alleviate the burden on backhaul and improve the quality of experience (QoE) of users in unmanned aerial vehicle (UAV)-enabled mobile edge networks, which has played an important role in wireless communication systems. However, owing to the selfish nature and limited battery life of the user equipments (UEs), only a limited part of the caching resources will be shared by the UEs. To drive the UEs to share more caching resources to improve the social welfare of the UAV-enabled mobile edge network, in this paper, we jointly design an edge caching strategy and reward contract optimization scheme. Aiming to maximize the utility of the UAV, a joint edge caching and contract optimization problem is formulated. Firstly, a novel edge caching strategy is proposed based on the content popularity. Then, an optimal reward contract is designed by reducing the restrictions. Finally, the system performance of the designed caching strategy and reward contract is evaluated compared with two benchmark caching schemes with no incentives. Simulation results show that the proposed scheme performs better than the other two schemes in terms of the utilities of the UAV and UEs, which proves the efficiency of our proposed scheme.