Non-orthogonal unmanned aerial vehicle (UAV)-aided secure downlink transmissions are investigated. A single-antenna UAV serves multiple ground users in the face of multiple colluding eavesdroppers (EVs) and only imperfect location information of the EVs is available. Specially, a power splitting based secure non-orthogonal downlink transmission scheme is considered, where the transmit power is divided into two parts for transmitting confidential information and artificial noise. Explicitly, we maximize the minimum average secrecy rate among all the users by optimizing the UAV trajectory, the transmit power and the power splitting ratio. In order to tackle this non-convex optimization problem, we propose an iterative algorithm based on the block coordinate descent and successive convex approximation techniques. Numerical results demonstrate the superiority of our proposed algorithm.