In this paper, we studied the optomechanically induced transparency (OMIT) in a cavity optomechanical system containing a cubic nonlinear oscillator. In our system, a partially transparent, dielectric membrane was placed in the middle of the F-P cavity. Due to the partial transmission and reflective property of the membrane, the membrane was combined with both the mirrors on the left and right sides to form two cavities. When the system was driven by two coupling fields, we calculated the quantum fluctuation of the optomechanical system operators and showed the response of the cavity optomechanical system to the probe field. We found that the cubic nonlinearity led to a shift of the OMIT window, which moved towards a frequency less than the resonance frequency, and the absorption peak became significantly asymmetrical when OMIT appeared. The shift of the OMIT dip provided a method to detect the nonlinear effects of the system due to the existence of cubic anharmonic potential.