Uniqueness and value distribution of differences of entire functions

We consider the existence of transcendental entire solutions of certain type of non-linear difference equations. As an application, we investigate the value distribution of difference polynomials of entire functions. In particular, we are interested in the existence of zeros of f n ( z ) ( λ f m ( z + c ) + μ f m ( z ) ) − a , where f is an entire function, n, m are two integers such that n ⩾ m > 0 , and λ, μ are non-zero complex numbers. We also obtain a uniqueness result in the case where shifts of two entire functions share a small function.