On the Growth of Meromorphic Solutions of Certain Nonlinear Difference Equations

Abstract

AbstractBy Cartan’s version of Nevanlinna’s theory, we prove the following result: let m and n be two positive integers satisfying $$n\ge 2+m,$$ n ≥ 2 + m , let $$p\not \equiv 0$$ p ≢ 0 be a polynomial, let $$\eta \ne 0$$ η ≠ 0 be a finite complex number, let $$\omega _{1}, \omega _{2}, \ldots , \omega _{m}$$ ω 1 , ω 2 , … , ω m be m distinct finite nonzero complex numbers, and let $$H_{j}$$ H j be either exponential polynomials of degree less than q,  or an ordinary polynomial in z for $$0\le j\le m$$ 0 ≤ j ≤ m , such that $$H_{j}\not \equiv 0$$ H j ≢ 0 for $$1\le j\le m.$$ 1 ≤ j ≤ m . Suppose that $$f\not \equiv \infty $$ f ≢ ∞ is a meromorphic solution of the difference equation: $$\begin{aligned} f^n(z)+p(z)f(z+\eta )&=H_0(z)+H_1(z)e^{\omega _{1}z^{q}}+H_2(z)e^{\omega _{2}z^{q}}\\&\quad +\cdots +H_m(z)e^{\omega _{m}z^{q}}, \end{aligned}$$ f n ( z ) + p ( z ) f ( z + η ) = H 0 ( z ) + H 1 ( z ) e ω 1 z q + H 2 ( z ) e ω 2 z q + ⋯ + H m ( z ) e ω m z q , such that the hyper-order of f satisfies $$\rho _2(f)<1.$$ ρ 2 ( f ) < 1 . Then, f reduces to a transcendental entire function, such that either $$n=m+2$$ n = m + 2 with $$H_0\not \equiv 0$$ H 0 ≢ 0 and $$\lambda (f)=\rho (f)=q,$$ λ ( f ) = ρ ( f ) = q , or $$m=2,$$ m = 2 , $$H_0=0$$ H 0 = 0 and: $$\begin{aligned} f(z)=\frac{H_1(z-\eta )e^{\omega _{1}(z-\eta )^{q}}}{p(z-\eta )} \end{aligned}$$ f ( z ) = H 1 ( z - η ) e ω 1 ( z - η ) q p ( z - η ) with $$\begin{aligned} H^n_1(z)=p^n(z)H_2(z+\eta )e^{\omega _2P_{q-1}(z)}\quad \text {and}\quad P_{q-1}(z)=\sum \limits _{k=1}^q\left( {\begin{array}{c}q\\ k\end{array}}\right) \eta ^kz^{q-k}. \end{aligned}$$ H 1 n ( z ) = p n ( z ) H 2 ( z + η ) e ω 2 P q - 1 ( z ) and P q - 1 ( z ) = ∑ k = 1 q q k η k z q - k . This result improves Theorems 1.1 and 1.3 from [19] by removing some assumptions of theirs. An example is provided to show that some results obtained in this paper, in a sense, are the best possible.