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| Entire Solutions of Certain Types of Delay Differential Equations |
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Citation： |
Shuangting LAN · Zhibo HUANG · Ranran ZHANG.Entire Solutions of Certain Types of Delay Differential Equations[J].Chinese Annals of Mathematics B,2025,46(6):911~936 |
| Page view： 106
Net amount： 48 |
Authors： |
Shuangting LAN · Zhibo HUANG · Ranran ZHANG; |
Foundation： |
This work was supported by the National Natural Science Foundation of China (Nos. 11801110,
12471072, 11801093) and the National Science Foundation of Guangdong (No. 2018A030313508). |
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| Abstract： |
In this paper, the authors investigate a delay differential equation of the form
w(z + 1) ? w(z ? 1) + a(z)w′(z)
w(z) =
P (z, w)
Q(z, w),
where a(z) is a nonzero rational function, P (z, w) and Q(z, w) are prime polynomials in
w with rational coefficients. They remove the restriction that the order of meromorphic
solutions of the above difference equation is σ2(w) < 1, and obtain the growth of transcendental meromorphic solutions. The exact forms of all transcendental entire solutions
are obtained when degw P = degw Q = 0, or degw P = 1 and degw Q = 0, respectively.
If degw P ≥ 2 and degw Q = 0, or degw Q ≥ 1 and Q(z, 0) 6≡ 0, they prove that the
above equation has no transcendental entire solution. They show that the existence of
transcendental entire solutions of the above equation depends on the degrees of P (z, w)
and Q(z, w). |
Keywords： |
Delay differential equation, Entire solution, Growth, Existence |
Classification： |
30D35, 34K40, 34M55 |
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