Meromorphic Solutions Of Complex Difference And Differential Equations And Their Properties

In this thesis, we mainly investigate the meromorphic solutions of complex differ-ential equations and systems of complex difference equations and their properties, and have some results. This thesis is divided into four parts.In chapter1, we give some definitions, basic results in Nevanlinna’s value distribu-tion theory and Wiman-Valiron Theory, some important results of complex difference equations and complex differential equations.In chapter2, we research a system of complex difference equations of the form where c1, c2,…, Cn are distinct, nonzero complex numbers, coefficients αλi,μi(z)(λi(?) Iil,μi(?)Ji1),βφi,Ti<(z)(φi(?)Ii2,Ti(?)Ji2)(i=1,2),α(z)(i=0,1,…,p), bj(z)(j=0,1,…,q), dk(z)(k=0,1,…, s) and el(z)(l=0,1,…,t) are small functions relative to f(z) and g(z), Ii1={λi=(lλi,1,lλi,2,…,lλi,n)|lλi,v(?)N∪{0}, v= 1,2,…,n}{i=1,2),Jj1={μj=(mμj,1,mμj,2,…,mμj,n)|mμj,v∈N∪{0}, v=1,2,…,n}(j=1,2),Ii2={φi=(lφi,1,lφi,2,…,lφi,n)|lφi,v.∈N∪{0},v=1,2,…,n}(i=1,2),Jj2={Tj=(mTj,1,mTJ,2,…,mTj,n)|mTj,v∈N∪{0},v=1,2,…,n}(j=1,2)are finite index sets.In this section we get the Malmaquist type result and the growth property of the meromorphic solutions of the system of complex difference equations as above.In chapter3,we focus on the nonlinear differential equations of the following form w2+R(z)(w(k))2=Q(z),where R(z),Q(z)are nonzero rational functions.We proved(1)if the differential equation w2+R(z)(w’)2=Q(z),where R(z),Q(z)are nonze-ro rational functions,admits a transcendental meromorphic solution f,then Q≡C(constant),the multiplicines of the zeros of R(z) are no greater than2and,(z)=√Ccos α(z),where α(z)is a primitive of such that√Ccos α(z)is a tran. scendental meromorphic function.(2)if the differential equation w2+R(z)(w(k))2=Q(z),where k≥2is an integer and R,Q are nonzero rational functions,admits a transcendental meromorphic solution f, then f satisfies the following linear second order differential equationwhere a is a nonzero rational function.Furthermore,if Q is a constant C,then k is an odd integer,R(z)三A (constant)and f(z)=√C cos(az+b),where a2k=1/A.In chapter4,we investigate the higher order algebraic differential equation differ-ential equationswhere m≥ΓP+1,Pm≠0,I is a finite set of multi-indices(λ0,μ1,…,λn)=λ and the coefficients αλ(z)(λ∈I)and Pk(k=0,1,…,m)are polynomials in z.We give an upper bound of the number of linearly independent meromorphic solutions of the higher order algebraic differential equation.