The basic idea of this paper is to consider fixed points of solutions of the differential equation $f^{\left( k\right) }+A\left( z\right) f=0$, $k\geq 2$, where $A\left( z\right) $ is a transcendental meromorphic function with $\rho \left( A\right) =\rho >0$. Instead of looking at the zeros of $f\left( z\right) -z$, we proceed to a slight generalization by considering zeros of $f\left( z\right) -\varphi \left( z\right) $, where $\varphi $ is a meromorphic function of finite order, while the solution of respective differential equation is of infinite order.