Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON DELAY DIFFERENTIAL EQUATIONS WITH MEROMORPHIC SOLUTIONS OF HYPER-ORDER LESS THAN ONE

  • Risto Korhonen (Department of Physics and Mathematics University of Eastern Finland) ;
  • Yan Liu (Department of Physics and Mathematics University of Eastern Finland)
  • Received : 2023.02.16
  • Accepted : 2023.06.03
  • Published : 2024.01.31
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Abstract

We consider the delay differential equations $$b(z)w(z+1)+c(z)w(z-1)+a(z)\frac{w'(z)}{w^k(z)}=\frac{P(z, w(z))}{Q(z, w(z))}$$, where k ∈ {1, 2}, a(z), b(z) ≢ 0, c(z) ≢ 0 are rational functions, and P(z, w(z)) and Q(z, w(z)) are polynomials in w(z) with rational coefficients satisfying certain natural conditions regarding their roots. It is shown that if this equation has a non-rational meromorphic solution w with hyper-order ρ2(w) < 1, then either degw(P) = degw(Q) + 1 ≤ 3 or max{degw(P), degw(Q)} ≤ 1. In addition, it is shown that in the case max{degw(P), degw(Q)} = 0 the equations above can have such a solution, with an additional zero density requirement, only if the coefficients of the equation satisfy certain strict conditions.

Keywords

Acknowledgement

The second author thanks the support of the China Scholarship Council (No. 202006330038).

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