We study energy-efficient power allocation among relays for lifetime maximization in a dual-hop relay network operated by amplify-and-forward relays with battery limitations. Power allocation algorithms are proposed for three different scenarios. First, we study the relay cooperation case where all the relays jointly support transmissions for a targeted data rate. By exploring the correlation of time-varying relay channels, we develop a prediction-based relay cooperation method for optimal power allocation strategy to improve the relay network lifetime over existing methods that do not predict the future channel state, or assume the current channel state remains static in the future. Next, we consider energy-efficient relay selection for the single source-destination case. Assuming finite transmission power levels, we propose a stochastic shortest path approach which gives the optimal relay selection decision to maximize the network lifetime. Due to the high computational complexity, a suboptimal prediction-based relay selection algorithm, directly coming from previous problem, is created. Finally, we extend our study to multiple source-destination case, where relay selection needs to be determined for each source-destination pair simultaneously. The network lifetime in the presence of multiple source-destination pairs is defined as the longest time when all source-destination pairs can maintain the target transmission rate. We design relay-to-destination mapping algorithms to prolong the network lifeii time. They all aim at maximizing the perceived network lifetime at the current time slot. The optimal max-min approach and suboptimal user-priority based approach are proposed with different levels of computational complexity. / UOIT