|
|
| |
| On Meromorphic Solutions of Non-linear Differential Equations andTheir Applications |
| |
Citation： |
Linke MA · Liangwen LIAO.On Meromorphic Solutions of Non-linear Differential Equations andTheir Applications[J].Chinese Annals of Mathematics B,2025,46(6):859~936 |
| Page view： 107
Net amount： 49 |
Authors： |
Linke MA · Liangwen LIAO; |
Foundation： |
This work was supported by the National Natural Science Foundation of China (No. 12171232). |
|
|
| Abstract： |
In this paper, the authors consider meromorphic solutions of nonhomogeneous
differential equation
f n(f ′ + af) + Pd(z, f) = u(z)ev(z),
where n is a positive integer, a is a nonzero constant, Pd(z, f) is a differential polynomial in
f(z) of degree d with rational functions as its coefficients and d ≤ n ? 1, u(z) is a nonzero
rational function, v(z) is a nonconstant polynomial with v′(z) 6= (n+1)a, v′(z) 6= ?na and
v′(z) 6= ? (n+1) n 2 a. They prove that if it admits a meromorphic solution f(z) with finitely
many poles, then
f(z) = s(z)e
v(z)
n+1 and Pd(z, f) ≡ 0,
where s(z) is a rational function and sn[(n + 1)s′ + sv′] + (n + 1)asn+1 = (n + 1)u.
Using this result, they also prove that if f(z) is a transcendental entire function, then
f n(f ′ + af) + qm(f) assumes every complex number α infinitely many times, except for
a possible value qm(0), where n, m are positive integers with n ≥ m + 1 and qm(f) is a
polynomial in f(z) with degree m. |
Keywords： |
Non-linear differential equations, Differential polynomial, Meromorphic
functions, Entire functions, Nevanlinna theory, Normal family |
Classification： |
30D35 |
|
|
Download PDF Full-Text
|
|
|
|
