Based on the coupled higher-order nonlinear Schrdinger equation with varying coefficient, evolution characteristics of optical soliton and Airy pulse with initial chirp transmitting in the anomalous dispersion region of the inhomogeneous optical fiber are studied by the split-step Fourier method. The results indicate that the Airy pulse can produce trapped solitons in positive and negative chirp conditions. When the initial chirp is positive, a new Airy pulse whose propagation direction is opposite to the original acceleration direction is generated. When the initial chirp is negative, the Airy pulse is dispersed in the direction opposite to the accelerating direction. The transmission characteristics of the Airy pulse with initial positive chirp are investigated under different truncation coefficients and nonlinear conditions.