Cooperative Non-Orthogonal Multiple Access (NOMA) with Simultaneous Wireless Information and Power Transfer (SWIPT) communication can effectively improve the spectrum efficiency and energy efficiency of the wireless networks with extend coverage. An important design issue is to incentivize a relaying center user to participate in the cooperative process and achieve a win-win situation to both the BS and the center user. Some private information of the center users are hidden from the BS in the networks. We apply a contract theory-based incentive mechanism under such asymmetric information scenario to incentives center user to join the cooperative communication to maximize the BS profit utility and to guarantee the center user’s expect payoff. A match theory-based Gale-Shapley algorithm is proposed to obtain the optimal strategy with low computation complexity. Simulation results indicated the network performance of our proposed cooperative transmission is much better than the conventional NOMA transmission and the benefit utility of the BS with the stable match strategy is nearly close to the complete channel state information multi-users scenario while the center users get the satisfied expect payoffs.