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Cognitive radio (CR) technology has been developed to solve the spectrum-underutilization problem. In CR networks, the CR users have opportunities to access the licensed spectrum bands assigned to the primary users (PUs). Since the PUs have priorities to use the bands, the CR users are not allowed to generate unacceptable interference to them. In this paper, we investigate power allocation schemes for CR networks with both direct and relay-aided transmissions. We first formulate an overall rate optimization problem with interference constraints to the PU and peak power constraints at each node and obtain solutions by theoretical analysis. To take the fairness among CR users into consideration, we further investigate the overall rate optimization problem with an additional sum power constraint and achieve fairness between two CR users by adjusting the sum power threshold. Numerical results are provided to show the impact of the relay node and the PU locations on power allocation solutions.