LAUSR

Take complex numbers $$\alpha ,\beta ,c,a_j,b_j$$α,β,c,aj,bj$$(j=0,1,2)$$(j=0,1,2) such that $$c\ne 0$$c≠0 and $$\begin{aligned} \mathrm{rank} \left( \begin{array}{ccc} a_{0} &{} a_{1} &{} a_{2}\\ b_{0} &{} b_{1} &{} b_{2}\\ \end{array} \right) =2. \end{aligned}$$ranka0a1a2b0b1b2=2.We show that if the following functional equation of Fermat type $$\begin{aligned} \begin{aligned} \left\{ a_{0}f(z)+a_{1}f(z+c)+a_{2}f'(z)\right\} ^3+\left\{ b_{0}f(z)+b_{1}f(z+c)+b_{2}f'(z)\right\} ^3=e^{\alpha z+\beta } \end{aligned} \end{aligned}$$a0f(z)+a1f(z+c)+a2f′(z)3+b0f(z)+b1f(z+c)+b2f′(z)3=eαz+βhas meromorphic solutions of finite order, then it has only entire solutions of the form $$f(z)=Ae^{\frac{\alpha z+\beta }{3}}+Ce^{Dz},$$f(z)=Aeαz+β3+CeDz, where A, C, D are constants, which generalizes some results due to Han and Lü.