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This paper investigates a new spectrum-sharing strategy using non-orthogonal multiple access (NOMA), a technique that is expected to lead toward the fifth generation of wireless networks. The proposed scheme, namely, successive bandwidth division (SBD), NOMA, is a hybrid approach exploiting the characteristics of both the conventional NOMA system and the orthogonal multiple access (OMA) system. Power allocation is being performed in successive bandwidth division NOMA to maximize the sum rate using a divide-and-allocate approach such that all users are allocated with optimal transmission power. Under Rayleigh fading, the probability density function (PDF) of the received signal-to-interference-plus-noise ratio (SINR) is derived, and closed-form expressions for the outage probability are presented when channel state information (CSI) is available at the base station. The performance evaluation is carried out in terms of receiver complexity, average sum rate, and outage probability. Simulations results are provided to access and compare the performance of the proposed scheme with other contemporary approaches.