Research On Value Distribution Of Difference Polynomials And Solutions Of Differential And Difference Equations

Value distribution theory of meromorphic functions was due to R. Nevanlinna in 1920’s. This theory is one of the most important achievements in mathematics in the preceding century. It is composed of two main theorems, which are called Nevanlinna’s first and second main theorems. Since then, Nevanlinna theory has been well developed in itself and widely applied to other fields of complex analysis such as the uniqueness of meromorphic functions, normal families, complex differential and difference equations, and several complex variables etc.The complex oscillation theory of differential equations studies differential equations by using the theory and method of complex analysis. It is the borderline field and intersected subject research. It became very popular after S. Bank and I. Laine made some original work in 1980s. Since then, many mathematicians have studied deeply and paid close attention to it.Difference counterparts of Nevanlinna theory have been established recently. The key result is the difference analogue of the lemma on the logarithmic derivative, which was obtained by Halburd-Korhonen [22,23] and Chiang-Feng [17], independently. Based on this theory, many scholars studied value distrubution on difference operators as well as the properties of solutions of complex difference equations and differential-difference equations.The present thesis involves some results of the author under the guidance of super-visor professor Lianzhong Yang. It consists of four parts and the matters are explained as below.In Chapter 1, we introduce some classic results of Nevanlinna theory as well as its difference analogue, which are the powerful tools in the field of value distribution of meromorphic functions, and complex differential and difference equations.In Chapter 2, we investigate the value distribution problems of difference polyno-mials, and obtain some related results on difference polynomials P(f)f(z+c) - α(z) and f(z)nL(f), which can be seen as the difference analogues of classical results given by Hayman [25], i.e. the value distributions of differential polynomials of the type fnf’.In Chapter 3, we investigate the complex oscillation problems of meromorphic so-lutions to some linear difference equations with meromorphic coefficients by using dif-ference analogue of Nevanlinna Theory, and obtain some results about the relationships between the exponent of convergence of zeros, poles and the order of growth of mero-morphic solutions to complex linear difference equations. We also study the existence of solution of certain types of nonlinear differential-difference equations, and partially answer a conjecture concerning the above problem posed by Yang and Laine [44] in 2010.In Chapter 4, we investigate the complex oscillation problems of meromorphic solu-tions to some linear differential equations. We obtain a precise estimation of the hyper order of solutions and the exponent of convergence of their fixed points for a class of higher order linear differential equation. Particularly, we investigate the exponents of convergence of the fixed points of solutions and their first derivatives for the correspond-ing second order case. In addition, we also study the existence of non-trivial subnormal solutions for second-order linear differential equations of another type, and estimate their hyper-order further. These results generalize the results of Gundersen-Steinbart [21], Wittich [42] and Chen-Shon [9,10,11], etc.