On the solutions of the linear matrix equations $AX+f(X)B=C$

Many applications in applied mathematics give rise to the unique solutions of Sylvester-like matrix equations associate with an underlying structured matrix operator $f$. In this paper, we shall discuss the solvability of the Sylvester-like matrix equations through an auxiliary standard or generalized Sylvester equations. We also show that when this Sylvester-like matrix equation is uniquely solvable, the closed-form solutions can be obtained by utilizing the previously result. In addition, with the aid of Kronecker map some useful results about the solvability of this matrix equation are provided.