Complex delay-differential equations
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 exponent \(\lambda(f)\), \(\lambda(1/f)\)) for meromorphic or entire solutions to linear delay-differential equations, beginning with an equation of the form \(f(z+1)=\alpha(z)f(z).\) For the equation, say, \[ \alpha_n(z) f^{(n)}(z+c_n) +\cdots + \alpha_1(z) f'(z+c_1)+\alpha_0 f(z+c_0) =0 \] with polynomials \(\alpha_j(z)\) such that \(\mathrm{deg} (\alpha_0) >\mathrm{deg} (\alpha_j)\) (\(j\ge 1\)) it is shown that a transcendental meromorphic solution satisfies \(\rho(f) \ge 1.\) \par In Chapter 6, delay-differential versions of Fermat-type functional equations are considered. In place of a classical Fermat-type equation \(f(z)^n+g(z)^n =1\), for various equations including \( f(z)^2+f(z+c)^2=1\), \( f'(z)^2+f(z+c)^2=1\), meromorphic or entire solutions are studied. \par Chapter 7 is devoted to delay-differential variants of the Riccati equation. After the difference Riccati equation including \[ f(z+1)=\frac{A(z)+f(z)}{1-f(z)} \] are surveyed, for delay-differential equations including \[ f'(z+c)^n= \frac{a_0(z)+ a_1(z)f(z) +\cdots + a_p(z)f(z)^p} {b_0(z)+ b_1(z)f(z) +\cdots + b_q(z)f(z)^q}, \] Malmquist-type theorems are described. \par Discrete Painlevé equations \(\mathrm{dP}_{\mathrm{I}}\), \(\mathrm{dP}_{\mathrm{II}}\), \(\mathrm{dP}_{\mathrm{III}}\) are written in the form \(f(z+1)+f(z-1)= R(z,f(z))\), \(f(z+1)f(z-1)= R(z,f(z))\). In Chapter 8, for the delay-differential equation \[ f(z+1)*f(z-1)+a(z)\frac{f'(z)}{f(z)}= \frac{P(z,f(z))}{Q(z,f(z))}, \] where \(f*g=f-g\) or \(fg\), and equations of more general form, various Malmquist-type theorems are discussed. \par Chapter 9 examines entire solutions to the equation \(f(z)^3+q(z)f(z+1)=c\sin bz\) and other equations related to \(e^{\alpha z}\) or trigonometric functions. \par Chapter 10 is concerned with problems related to the \(q\)-difference \(f(qz)\). The beginning part of this chapter explains basic facts on \(q\)-difference version of value distribution theory. Value distribution of \(q\)-difference polynomials and \(q\)-difference uniqueness problems are considered. Furthermore various types of equations containing \(f(qz)\) are discussed. \par Chapter 11 includes results on meromorphic or entire solutions to systems of delay-differential equations of Malmquist-type, of Fermat-type and of \(q\)-difference type. \par The final chapter examines the periodicity of delay-differential polynomials of an entire function. \par This book is recommended to people who are interested in complex analytic functional equations, especially, value distribution properties of them.
scientific article; zbMATH DE number 7358891