Algebroid Solutions of Second Order Complex Differential Equations

Lingyun Gao,Yue Wang

doi:10.1155/2014/123049

Abstract

Using value distribution theory and maximum modulus principle, the problem of the algebroid solutions of second order algebraic differential equation is investigated. Examples show that our results are sharp.

Highlights

Introduction and Main ResultsWe use the standard notations and results of the Nevanlinna theory of meromorphic or algebroid functions; see, for example, [1, 2].In this paper we suppose that second order algebraic differential equation (3) admit at least one nonconstant ]-valued algebroid solution w(z) in the complex plane
Some authors had investigated the problem of the existence of algebroid solutions of complex differential equations, and they obtained many results ([2,3,4,5,6,7,8,9,10], etc.)
The purpose of this paper is to investigate algebroid solutions of the following second order differential equation in the complex plane with the aid of the Nevanlinna theory and maximum modulus principle of meromorphic or algebroid functions: n

Summary

Introduction and Main Results

We use the standard notations and results of the Nevanlinna theory of meromorphic or algebroid functions; see, for example, [1, 2]. Some authors had investigated the problem of the existence of algebroid solutions of complex differential equations, and they obtained many results ([2,3,4,5,6,7,8,9,10], etc.). The purpose of this paper is to investigate algebroid solutions of the following second order differential equation in the complex plane with the aid of the Nevanlinna theory and maximum modulus principle of meromorphic or algebroid functions: n (z, w). Let w(z) be a nonconstant ]-valued algebroid solution of differential equation (3) and all ajk are polynomials. Let w(z) be a nonconstant ]-valued algebroid solution of differential equation (3) and the orders of all ajk are finite.

Some Lemmas

Proof of Theorem 1

Proof of Theorem 2

Some Examples