On the Growth of Meromorphic Solutions of Certain Nonlinear Difference Equations

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