On the Growth of Meromorphic Solutions of Certain Nonlinear Difference Equations

By Cartan's version of Nevanlinna's theory, we prove the following result: let m and n be two positive integers satisfying n >= 2+ m, let p not equivalent to 0 be a polynomial, let. eta not equal 0 be a finite complex number, let omega(1), omega(2), ... ,omega(m) be m distinct finite nonzero complex numbers, and let H-j be either exponential polynomials of degree less than q, or an ordinary polynomial in z for 0 <= j <= m, such that H-j not equivalent to 0 for 1 <= j <= m. Suppose that f not equivalent to infinity is a meromorphic solution of the difference equation: f(n)(z) + p(z)f(z + eta) = H-0(z) + H-1(z)e(omega 1zq) + H-2(z)e(omega 2zq) + center dot center dot center dot + (H)m(z)e(omega mzq), such that the hyper-order of f satisfies rho(2)(f) < 1. Then, f reduces to a transcendental entire function, such that either n = m+ 2 with H-0 not equivalent to 0 and lambda(f) = rho(f) = q, or m = 2, H-0 = 0 and: f(z) = H1(z - eta)e(omega 1(z-eta)q) /p(z -eta) with H-1(n)(z) = p(n)(z)H-2(z +eta)e(omega 2Pq-1(z)) and Pq-1(z) = Sigma(q)(k=1) ((k)(q))eta(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.