On The Solutions Of Linear Differential Equations (Equation Group) And Its Applications

Nevanlinna Theory([21, 10, 18, 15]) is one of the most perfect achievements in the 20th century. As a new branch of function theory, it inherits the great value in application from function analysis. Besides its value in this field, it contributes a lot to numbertheory and differential equation theory. Nevanlinna Theory paves the path of research on value distribution of meromorphic functions, constructs the characteristic function, which overcomes the restriction of meaningless of maximum module function on meromorphic functions. Due to the broad correlation with other fields and mutual influence, both Nevanlinna Theory and others take big strides. The value distribution of module of function directly based on Nevanlinna Theory provides a new approach in research on differential equations. It gives a decent example for the angular value distribution theory and value distribution theoryon p-adic field, complex Euclidian space, and even on general manifold.In Chapter 1, we describe the basic Nevanlinna theory and basic Wiman-Valiron theory briefly. In Chapter 2, we consider the existence of solutions to a certaintype of linear differential equation group, which is related to Br(?)ck's conjecture[2], and prove the Lemma^*, which provides a tool in improving the result of Yang[20]. Then we obtain Theorem A and use it as an application to give a shot proof of TheoremB[16].Lemma^* Let f be a solution ofwhereαandβ(?)αare nonconstant entire function, Q is a polynomial and deg(Q)=q,n＞q. ThenTheorem A Letαandβbe nonconstant entire functions such that e^α-β(?)1.Then(where Q is a polynomial, deg(Q) = q,n＞q) has no solutions.In Chapter 3, we discuss on the subnormal solutions to lineardifferential equations of order 2. Through considering more generalized equation(where A(z) is polynomial in z of order k), we obtain Theorem C and Corollary D, which improve the result of Chen[6]. Theorem C Let P_j(z),Q_j(z),j=1,2 be the polynomialsin z, A(z)=a_kz^k+a_k-1z^k-1+…+a₀,(a_k≠0 is a real number) be a nonconstant polynomial. Ifthen the differential equationhas no nontrivial subnormal solution, and every solution of it satisfiesσ₂(f)=k.Corollary D Assume that the assumptions of Theorem C hold, then the following equationat most has one subnormal solution, where R₁(z) and R₂(z) are the polynomials in z that satisfy R₁+R₂(?)0.