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In uplink non-orthogonal multiple access (NOMA) channels, the existing cooperative successive interference cancellation (SIC) and power control (PC) schemes lack the capability of achieving the full capacity region, which restricts the outage performance of uplink NOMA users. For the uplink cognitive radio (CR) inspired NOMA system, we propose a new rate splitting (RS) strategy to maximize the achievable rate of the secondary user without deteriorating the primary user's outage performance. Based on the interference threshold and its own channel gain, the secondary user adaptively conducts RS, transmit power allocation and SIC, which utilizes the transmit power efficiently. Closed-form expression of the outage probability is derived for the secondary user. Numerical results show that the proposed RS scheme achieves the best outage performance for the secondary user among the existing cooperative SIC and PC schemes.