On an open problem of value sharing and differential equations

Abstract Let f be a non-constant meromorphic function. We define its linear differential polynomial $ L_k[f] $ by \begin{equation*} L_k[f]=\displaystyle b_{-1}+\sum_{j=0}^{k}b_jf^{(j)}, \text{where}\; b_j (j=0, 1, 2, \ldots, k) \; \text{are constants with}\; b_k\neq 0. \end{equation*} In this paper, we solve an open problem posed by Li [ J. Math. Anal. Appl. 310 (2005) 412-423] in connection with the problem of sharing a set by entire functions f and their linear differential polynomials $ L_k[f] $ . Furthermore, we study the Fermat-type functional equations of the form $ f^n+g^n=1 $ to find the meromorphic solutions ( f , g ) which enable us to answer the question of Li completely. This settles the long-standing open problem of Li.