On meromorphic solutions of Fermat type delay-differential equations with two exponential terms

The existence of the meromorphic solutions to Fermat type delay-differential equation {equation} f^n(z)+a(f⁽ˡ⁾(z+c))^m=p_1(z)ea_1z^k+p_2(z)ea_2z^k, {equation} is derived by using Nevanlinna theory under certain conditions, where k≥1, $m,$ n and l are integers, pᵢ are nonzero entire functions of order less than k, c, a and aᵢ are constants, $i=1,2$. These results not only improve the previous results from Zhu et al. [J. Contemp. Math. Anal. 59(2024), 209-219], Qi et al. [Mediterr. J. Math. 21(2024), article no. 122], but also completely solve two conjectures posed by Gao et al. [Mediterr. J. Math. 20(2023), article no. 167]. Some examples are given to illustrate these results.