Fermat Type Functional Matrix Equations

In this paper,the systems of non-linear functional equations will be simplified to matrix forms.Using the Nevanlinna theory and the properties of matrix,we deeply studies the meromorphic function matrix solutions of many Fermat type functional matrix equations and the entire solutions of some non-linear functional equations.In chapter 1,firstly,we introduce the research background,including the basic knowledge of Nevanlinna theory and the long development of Fermat type function equations.Secondly,we give the main research issues and results of Fermat type function equations.In chapter 2,we mainly introduce some basic concepts,theorems and lemmas of Nevanlinna theory,which will be widely used in this paper.In chapter 3,we mainly introduce some properties on the meromorphic function matrix solutions of Fermat type functional matrix differential equation A(z)n+A’(z)n=E for n=2 and n=3.We consider the entire function solutions of two types non-linear differential equations,one of them is called Bi-Fermat differential equation f(z)2+f’(z)2+g(z)2+g’(z)2=1.We also obtain the transcendental entire matrix solutions of Malmquist type matrix differential equation A’(z)=αA(z)2+βA(z)+γE.In chapter 4,we mainly introduce the meromorphic function matrix solutions of Fermat type functional matrix difference equation A(z)n+A(z+c)n=E and Fermat type functional matrix q-difference equation A(z)n+A(qz)n=E.In addition,the entire function solutions of Bi-Fermat type functional difference equation f(z)2+f(z+c)2+g(z)2+g(z+c)2=1 and Bi-Fermat type functional q-difference equation f(z)2+f(qz)2+g(z)2+g(qz)2=1 are also considered.In chapter 5,conclusion and prospect.Firstly,the solutions of Fermat functional matrix equations and nonlinear equations obtained in this paper are summarized.Secondly,the general solutions of some equations in this paper are forecasted.