The Properties Of Solutions Of Differential Equations And Uniqueness Theorem On Sharing Value With Weight

In this paper, we mainly investigate the complex oscillation properties of solutions of differential equations and the uniqueness problems of meromorphic functions sharing one value with weight. It consists of four chapters.In chapter1, we briefly introduce the basic results of the Nevanlinna theory and the Wiman-Valiron theory, which are the powerful tools in the research of the complex oscillation theory of differential equations and the uniqueness theory of meromorphic functions.In chapter2, the complex oscillation properties concerning the second non-homogeneous linear differential equation f"+A1(z)eaznf’+A0(z)ebzn f=F(z) have been investigated, where Aj(z)((?)0)(j=0,1) are polynomials satisfying deg(A0)<deg(A1)<n-1, F(z))((?)0) is an entire function of order less than n. It is proved that every solution/of the above equation satisfies A(f)=A(f)=a(f)=∞,λ2(f)=λ2(f)=σ2(f)=n.In chapter3, we investigate the growth of meromorphic solutions of the d-ifferential equation f(k)+Ak-1(z)eak-1z f(k1)+…+A1(z)ea1zf’+(B1(z)ebz+B2(z)edz) f=0, and also investigate the relation between its solutions and mero-morphic functions of small growth, where Aj(z)((?)0)(j=1,…, k-1), Bi(z)((?)0)(i=1,2) are meromorphic functions of order less than1. It is proved that λ(f-φ)=A(f’-φ)=σ(f)=∞hold for every nonzero meromorphic solution f of the above equation with5(∞, f)>0.In chapter4, by using the weighted sharing method, we study the unique-ness problem of meromorphic functions concerning their differential polynomials sharing one value, and obtain one result which extends those results obtained by Bhoosnurmath, Dyavanal and Fang.