Second Main Theorem Of Holomorphic Curves And Solutions Of Some Complex Equations Related With Differential Or Difference Operators

In 1920s,the famous mathematician R.Nevanlinna extended Picard's the-orem by establishing the value distribution of meromorphic functions.In honor of him,we call it Nevanlinna theory.Nevanlinna theory is composed of two main theorems,which are called the first main theorem and second main the-orem.Nevanlinna theory has not only its own theoretical value but also nu-merous applications in some other complex branches,for example,uniqueness theory of meromorphic functions,normal family,complex dynamical systems,complex differential equations and so on.In 1933,Cartan[3]extended Nevan-linna's result to holomorphic curves in projective spaces that intersect hyper-planes in general position and obtained Cartan theorem.In 1941,Ahlfors[1]gave a geometric approach to the theory of holomorphic curves in projective spaces by following Wetls' work[57].Cartan theorem has turned out to be very useful to many complex problems,for instance,Waring's problem of analytic functions,and Fermat-type equations etc.Recently,motivated by investigating the value distribution of complex difference polynomials and solutions of complex difference equations.Halburd and Korhonen[19-21]set up the difference analogues of Nevanlinna theory.Later,Haburd et al.[22]and Wong et al.[58]independently got the differ-ence analogue of the second main theorem of holomorphic curves in complex projective spaces.Moreover,Korhonen et al.[28]also obtained the difference analogue of Cartan theorem concerning slowly moving periodic targets.Re-cently,the research related with the difference analogue of Nevanlinna theory has drawn greater attention.Meanwhile,the topic about researching the com-plex difference equations by using the difference analogue of Nevanlinna theory is very important,and many results have been obtained.In this dissertation,we study the second main theorem of holomorphic curves concerning differential or difference operators and some complex equa-tions.It consists of four parts and the matters are the following.In Chapter 1,we introduce the basic knowledge and some main results about value distribution of Nevanlinna theory and the difference analogue of Nevanlinna theory.We also introduce the basic knowledge and some results about value distribution of holomorphic curves for fixed hyperplanes and mov-ing hyperplanes.In Chapter 2,we firstly define a new Wronskian which involves derivative and difference.It is a generalization of difference Wronskian in Wong[58].By-using this new Wronskian,we investigate the second main theorem for a kind of special holomorphic curves.In addition,we shows a difference analogue truncated second main theorem for general linearly degenerate holomorphic curves,with hyper-order strictly less that 1,following Fujimoto[16]and Ko-rnonen[28].This problem was originally proposed by Cartan[3]and solved by Nochka[37,38].Our results generalize the results in Wong[58]and[19-21,28].In Chapter 3,following the technique of Ru[46],we study the truncated second main theorem for a kind of special holomorphic curves with moving hyperplanes.By using the results and technique in Gundersen[18],Korhonen et al.[28]firstly obtained related results.They generalized the results due to Halburd et al.[22].It is noteworthy that the reduced representation function of holomorphic curves considered in their results must be linearly independent over a function field which contains the functions that are periodic with period c ? C,with hyper-order strictly less than one.However,for the linearly dependent case,they didn't consider it.By using the technique of Ru[46],we consider this case and supplement the results in[28].In Chapter 4,we firstly discuss the transcendental meromorphic solutions of a kind of differential equations.Our results generalize the results in[67].In addition,we investigate the finite order transcendental meromorphic solutions of two difference equations and get two important results which improve the results in[35].