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

Journal of Inequalities and Applications

Yunfei Du¹  Minfeng Chen¹  Zongsheng Gao¹ 

[1] LMIB and School of Mathematics and Systems Science, Beihang University;

关键词: differential-difference equation;    admissible entire solution;    finite order;   

DOI  :  10.1186/s13660-017-1368-1

来源: DOAJ

【 摘 要 】

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

【 授权许可】

Unknown