In this paper, an adaptive channel access mechanism is proposed to optimize the performance of licensed- assisted access (LAA) long-term evolution (LTE) small cell networks through joint allocation of licensed and unlicensed spectrum resources all the while ensuring a fair coexistence with Wi-Fi service sets on the unlicensed spectrum. A listen-before- talk (LBT) access mechanism featuring an adaptive distributed control function (DCF) protocol is adopted for the small cell base stations (SBSs), whereby the minimum backoff window size is adaptively adjusted according to the available licensed spectrum bandwidth and Wi-Fi traffic load to satisfy the quality-of-service (QoS) requirements of small cell users (SUs) and minimize the collision probability of Wi-Fi users. Meanwhile, both licensed and unlicensed spectrum bands are jointly allocated to optimize spectrum efficiency. An admission control mechanism is further developed for the SBSs to limit collision with Wi-Fi traffic. Extensive numerical results are presented to demonstrate the effectiveness of the proposed schemes.