Given a positive definite matrix \(A\), the authors study three types of generalized Lyapunov equations, the first one is \[ A^3X+XA^3+t(A^2 XA+AX A^2)=B. \] the problem in question is whether this equation has a positive semidefinite solution \(X\) whenever \(B\) is positive semidefinite. This problem can be reduced to the following one: For which real \(t\) the matrix \(Y=\| y_{ij}\| \) with \[ y_{ij}^{-1}=\lambda_i^3+\lambda_j^3+t(\lambda_i^2\lambda_j+ \lambda_i\lambda_j^2) \] is positive semidefinite for all \(n\) and all positive numbers \(\lambda_1,\ldots,\lambda_n\)? It turns out that the latter holds as long as \(t>-1\).