Complex delay-differential equations
type: Item
instance of: scholarly article
title: Complex delay-differential equations
zbMATH Open document ID: 1486.34001
DOI: 10.1515/9783110560565
author: Kai Liu
author: Ilpo Laine
author: Lianzhong Yang
published in: De Gruyter Studies in Mathematics
publication date: 2021-06-14T00:00:00Z
review text: Nevanlinna theory has been applied to the study of meromorphic solutions to complex differential equations, and a difference version of Nevanlinna theory to complex difference equations. This book presents recent developments and typical results in various topics on value distribution for delay-differential functional equations and related fields, where a delay-differential equation means one expressed in terms of both difference operators \(f(z+c)\) (or \(q\)-difference operators \(f(qz)\)) and the derivatives of \(f(z).\) \par Chapter 1 reviews basic facts on Nevanlinna theory. In addition to basic classical versions, some results of difference Nevanlinna theory are exhibited, e.g., for a nonconstant meromorphic function \(f(z)\), \( m\bigl(r, {f(z+c)}/{f(z)} \bigr) = o\bigl({T(r+|c|, f)^{1+\varepsilon}} {r^{-\delta}} \bigr)\), where \(c\in \mathbb{C}\), \(0< \delta <1\), \(\varepsilon >0\), outside a possible exceptional set of finite logarithmic measure. This corresponds to the well-known estimate \(m(r, f'/f) =S(r,f)\). Difference variants are explained also for the second main theorem and Clunie's lemma. \par Chapters 2 and 3 are devoted to delay-differential variants of classical problems, respectively, on value distribution of differential polynomials of a meromorphic or entire function, and on the uniqueness for two meromorphic functions sharing some values. In place of the differential polynomial, say, \(f(z)^nf'(z)\) by Hayman, value distribution of \(f(z)^n f(z+c)\) is examined. \par The Wiman-Valiron theory is useful in treating entire solutions to differential equations. The difference and \(q\)-difference versions of it are explained in Chapter 4. \par Linear and nonlinear delay-differential equations are discussed in the remaining chapters. Many of the results on the value distribution of meromorphic or entire solutions are asserted under the supposition of the existence of such solutions. \par Chapter 5 examines the growth order \(\rho(f)\) (and the…
reviewed by: Shun Shimomura
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 34K41
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 39A13
zbMATH DE Number: 7358891
zbMATH Keywords: delay-differential equations
zbMATH Keywords: value distribution
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110560565
OpenAlex ID: W3161404584
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description: scientific article; zbMATH DE number 7358891