Affiliation:
1. School of Mathematical Sciences, Beijing Normal University , Beijing , 100875 , P. R. China
Abstract
Abstract
Let
n
n
and
m
m
be two positive integers, and the second-order Fermat-type functional equation
f
n
+
(
f
″
)
m
≡
1
{f}^{n}+{({f}^{^{\prime\prime} })}^{m}\equiv 1
does not have a nonconstant meromorphic solution in the complex plane, except
(
n
,
m
)
∈
{
(
1
,
1
)
,
(
1
,
2
)
,
(
1
,
3
)
,
(
2
,
1
)
,
(
3
,
1
)
}
\left(n,m)\in \left\{\left(1,1),\left(1,2),\left(1,3),\left(2,1),\left(3,1)\right\}
. The research gives a ready-to-use scheme to study certain Fermat-type functional differential equations in the complex plane by using the Nevanlinna theory, the complex method, and the Weierstrass factorization theorem.