On Meromorphic Solutions of Nonlinear Complex Difference Equations

Abstract

In this paper, we study meromorphic solutions of the following nonlinear complex difference equation

$$\begin{aligned} f^n(z)+P(z)\Delta _{\eta }^{l}f(z)=H_{0}(z)+H_{1}(z)e^{p_1(z)}+\cdots +H_{m}(z)e^{p_m(z)}, \end{aligned}$$

where n, l, m, q are positive integers with $n\ge 2$, $\eta$ is a nonzero complex number satisfying $\Delta _{\eta }^{l}f(z)\not \equiv 0$, $p_1,\ldots ,p_m$ are polynomials of degree q, whose leading coefficients $\alpha _1,\ldots ,\alpha _m$ are distinct nonzero complex numbers, and $P,~H_{0},~H_{1},\ldots ,H_{m}$ are meromorphic functions of order less than q such that $PH_{1}\ldots H_{m}\not \equiv 0.$ In addition, we analyze meromorphic solutions of the above equation for $n=1$. Our proofs depend on Cartan’s theorem and the variation of Nevanlinna’s theorem regarding a set of meromorphic functions. Several examples are provided to demonstrate our results. These results generalize some very recent known results.