Based on the inner-outer iteration sequence for solving the PageRank vector, a relaxed inner-outer iteration method is obtained by introducing a relaxed factor. Combining the multi-splitting iteration framework with two different relaxed factors, a relaxed multi-splitting iteration method for solving the PageRank vector is proposed, and its convergence property is analyzed. Furthermore, by using the relaxed inner-outer iteration format, a preconditioned matrix for accelerating the projection subspace methods is constructed, the spectral distribution is theoretically investigated, and choice criteria of the parameters in the relaxed multi-splitting iteration method and preconditioner are provided. Several numerical examples validate the effectiveness of the relaxed multi-splitting iteration method and preconditioner, the relaxed multi-splitting iteration method is more efficient compared to the multi-splitting iteration method with appropriate relaxed factors.