On meromorphic solutions of a certain type of nonlinear differential equations

We consider transcendental meromorphic solutions with N(r, f) = S(r, f) of the following type of nonlinear differential equations: fn+Pn−2(f)=p1(z)eα1(z)+p2(z)eα2(z),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f^n} + {P_{n - 2}}\left( f \right) = {p_1}\left( z \right){e^{{\alpha _1}\left( z \right)}} + {p_2}\left( z \right){e^{{\alpha _2}\left( z \right)}},$$\end{document} where n ≥ 2 is an integer, Pn−2(f) is a differential polynomial in f of degree not greater than n−2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z) are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.