Existence of entire solutions of some non-linear differential-difference equations

摘要

In this paper, we investigate the admissible entire solutions of finite order of the differential-difference equations (f^′(z))² + P²(z)f²(z + c) = Q(z)e^α(z) and (f^′(z))² + [f(z+c)−f(z)]² = Q(z)e^α(z), where P(z), Q(z) are two non-zero polynomials, α(z) is a polynomial and c ∈ ℂ∖{0}. In addition, we investigate the non-existence of entire solutions of finite order of the differential-difference equation (f^′(z))ⁿ + P(z)f^m(z + c) = Q(z), where P(z), Q(z) are two non-constant polynomials, c ∈ ℂ∖{0}, m, n are positive integers and satisfy 1m+1n<2 except for m = 1, n = 2.