Research On The Value Distribution Of Complex Difference Polynomials And Solutions Of Complex Difference Equations

In this dissertation,we mainly use Nevanlinna theory and its difference analogues to survey the value distribution of difference polynomials,the existence and growth of mero-morphic solutions of difference equations.Moreover,we also consider the properties of entire solutions of a kind of the nonlinear differential-difference equation.The contents of this thesis are organized as follows:In the first chapter,we introduce some backgrounds and give the main results of present thesis.We also give some related definitions and lemmas.In the second chapter,we survey the value distribution of q-difiference polynomial f(qz)-a(f(z))n and f(q1z)f(q2z)...f(qmz)-a(f(z))n,here f(z)is a transcendental entire function with zero order.Furthermore,we also study the property of entire solutions of a certain q-difference equation.In the third chapter,we study the value distribution of q-difference product f(z)f(qz)and fn(z)(f(qz)-f(z)),here f(z)is a transcendental entire function with finite positive order,q ? 0,1.We also consider the property of entire solutions of a certain q-difference linear equation.In the fourth chapter,we investigate the existence and growth of solutions of the q-difference equation ?i=1n f=(qiz)=R(z,f(z)),where R(z,f(z))is an irreducible rational function in f(z).We also give an estimation of the growth of transcendental meromorphic solutions of the equation ?i=1n f(qiz)=f(z)m.In the fifth chapter,we investigate the deficiencies and growth of meromorphic solu-tions of difference equation An(z)f(z+n)+…+A1(z)f(z)+ A0(z)f(z)= F(z),where An(z),…,A0(z),F(z)are meromorphic functions with An(z)F(z)(?)0.In the sixth chapter,we study entire solutions of the nonlinear differential-difference equation of the form where p1(z),p2(z)are nonzero polynomials,q1(z),q2(z)are nonconstant polynomials,q(z),a(z)are nonzero entire functions of finite order,n ? 2 is an integer.For a special case,of n = 3,q1(z)=-q2(z),and p1(z),p2(z)are nonzero constants,we also obtain some results.