Nonlinear matrix equations are encountered in many
applications of control and engineering problems. In this work, we establish a complete study
for a class of nonlinear matrix equations. With the aid of Sherman Morrison Woodbury formula,
we have shown that any equation in this class has the maximal positive definite solution under a certain condition. Furthermore, A thorough study of properties about this class of matrix equations is provided. An acceleration of iterative method with R-superlinear convergence with order $r>1$ is then designed to solve the maximal positive definite solution efficiently.