Abstract

In this paper, we consider the entire solutions of nonlinear difference equation \(f^{3}+q(z)\Delta f=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}\), where \(q\) is a polynomial, and \(p_{1},p_{2},\alpha_{1},\) and \(\alpha_{2}\) are nonzero constants with \(\alpha_{1}\neq\alpha_{2}\). It is showed that if \(f\) is a nonconstant entire solution of \(\rho_{2}(f)<1\) to the above equation, then \(f(z)=e_{1}e^{\frac{\alpha_{1}z}{3}}+e_{2}e^{\frac{\alpha_{2}z}{3}}\), where \(e_{1}\) and \(e_{2}\) are two constants. Meanwhile, we give an affirmative answer to the conjecture posed by Zhang et al in [18].

中文翻译：

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一类非线性差分方程的精确整体解

摘要

在本文中，我们考虑了非线性差分方程\（f ^ {3} + q（z）\ Delta f = p_ {1} e ^ {\ alpha_ {1} z} + p_ {2} e ^的整体解{\ alpha_ {2} z} \），其中\（q \）是多项式，而\（p_ {1}，p_ {2}，\ alpha_ {1}，\）和\（\ alpha_ {2} \）是带有\（\ alpha_ {1} \ neq \ alpha_ {2} \）的非零常量。结果表明，如果\（f \）是上述方程的\（\ rho_ {2}（f）<1 \）的非恒定整体解，则\（f（z）= e_ {1} e ^ { \ frac {\ alpha_ {1} z} {3}} + e_ {2} e ^ {\ frac {\ alpha_ {2} z} {3}} \），其中\（e_ {1} \）和\ （e_ {2} \）是两个常数。同时，我们对张等人在[18]中提出的猜想给出肯定的答案。