On Meromorphic Solutions Of A Certain Type Of Nonlinear Differential Equations

In 1920s,the Finland mathematician Rolf Nevanlinna[1]gave the Nevanlinna characteristic function,and used it to prove the First and the Second Main The-orem.The Second Main Theorem extends the conclusion of Picard Theorem,and then he established the Value Distribution Theory,which is one of the most im-portant theories of modern mathematics.For remembering the great work of Rolf Nevanlinna,this theory is also called Nevanlinna Theory.This theory has been con-stantly improved and now it is a mature theory.At the same time,it also plays a key role in many other fields,such as the Uniqueness Theory of Meromorphic Func-tions,Complex Differential Equations,Complex Difference Equations,Diophantine Approximations,Non-Archimedean Analysis,Normal Families and Complex Dy-namics.For a long time,differential equation is an attractive branch of mathematics.many great mathematicians do a large amount of work in this field,they solve many difficult problems and get a number of important theorems.They also try to use axioms and notions from other fields of mathematics,which leads to the combination of these branches.Non-linear differential equation is an essential component,of differential equation,there are still many unsolved questions in it.During the process of research,mathematicians found that the Value Distribution Theory can be used to deal with some problems of differential equation,and they got many good results.In this way,more and more mathematicians began to apply the Value Distribution Theory to the research of differential equations.It is always an interesting and difficult problem to prove the existence of the entire or meromorphic solutions of a given differential equation and find out the solutions if the solutions exist.In this paper,I deal with a special kind of nonlinear differential equations fn+Pd?f?=h where h is a given entire or meromorphic function,Pd?f?is a differential polynomial in f and its derivatives of a total degree at most d with small functions of f as the coefficients.This problem was introduced by C.C.Yang in 2006 and there were some great results.I prove a lemma and part of a theorem.The specific arrangement is as follows:The first chapter introduces some basic notions of Nevanlinna thoery,like char-acteristic function and small function,and the First and the Second Main Theorem,Milloux theorem.The second chapter extends the results of Professor Xiaoqing Lu,I add an unknown item in this equation,my lemma and theorem axe:Lemma 2.6.Suppose that f?z?is meromorphic and not constant in the plane,that 9?z?=fn?m?z?fn-,?z?+Pn-2?f?where Pn-2?f?is a differential polynomial in of degree at most n-2,m?z?is a small function of f,and that N?r,f?+N?r,1/g?=S?r,f?.Then g?z?=?f+m/n?n,P,n-2?f?=?f+m/n?n-fn-mfn-1.Theorem.Let n?>3 be an integer,Pn-2?f?be a differential polynomial in f of degree at most n-2,pi?z?,p2?z?and h?z?be nonzero small functions of f?z?,and a1?z?,a2?z?be nonconstant entire functions.If f?z?is a transcendental meromorphic solution of the following nonlinear differential equation:fn+h?z?fn-1?z?+Pn?f?=p1?z?e?1?z?+p2?z?e?2?z?satisfying N?r,f?= S?r,f?.then there exist two cases:???T?r,e?1?= S?r,e?2?.In this case,?f+h/n?n= P2?z?e?2?z?.Similarly,if T?r,ea2?= S?r,e?1?,then?f+h/n?n = p1?z?e?2?z????T?r,e?1?= O?T?r,e?2??and T?r,e?2?= O?T?r,e?1??,In this case,T?r,f?=O?T?r,ea1??= O?T?r,e?2??,S?r,f?= S?r,e?1?)= S?r,e?2?.We use T?r?and S?r?to denote these two quantities.I get some results as following???If P1?p2+p2?2?-p2?p1+p1?1??0,then?f+h/n?n=??P1+p2?e?2.where?=e?1-?2;???When?p2+p2?2?Pn-2?f?-p2Pn-2?f??0,if Pn-2?f?? 0,then T?r,e?1/fn-1?=O?T?r,f??,T?r,e?2/fn-1?= O?T?r,f??;if Pn-2?f????0,when Pn-2?f?=p2?z?eaz?z?,then h?z??0,fn = p1?z?e?1?z?;if Pn-2?f??P2?z?ea2?z?,we have?f+h/n?n =P1?z?e?1?z??iii?When?P2+P2?2?Pn₂?f?-P2Pn-2?f????0,if?p2+p2?2?Pn-2?f?-P2Pn-2?f?=[P1?p2+p2?2?-P2?p1+p1?1?]e?1,then f satisfies?p2+ P2?2?f2+?hp2+ hp2?2-p2h?f-np2ff-?n-1?p2hf=0 I also give some examples in chapter 2.