On existence of meromorphic solutions for nonlinear q-difference equation

In this paper, we mainly consider the existence of meromorphic solutions of nonlinear q-difference equation of type f(qz) + f(z/q) = P(z, f(z))/Q(z, f(z)), where the right-hand side is irreducible, P(z, f(z)) and Q(z, f(z)) are polynomials in f with rational coefficients, and q is a nonzero complex constant. We obtain that such equation has no transcendental meromorphic solution when vertical bar q vertical bar =1 and m = deg(f) (P)-deg(f)(Q) > 1. And we investigate the growth of transcendental meromorphic solutions of nonlinear q-difference equation and find lower bounds for their characteristic functions for transcendental meromorphic solutions of such equation for the case vertical bar q vertical bar not equal 1.