About this Journal Submit a Manuscript Table of Contents
Journal of Complex Analysis
Volume 2013 (2013), Article ID 253168, 5 pages
http://dx.doi.org/10.1155/2013/253168
Research Article

Some Properties of Solutions of Second-Order Linear Differential Equations

Laboratory of Pure and Applied Mathematics, Department of Mathematics, University of Mostaganem (UMAB), BP 227, 27000 Mostaganem, Algeria

Received 13 August 2012; Accepted 25 September 2012

Academic Editor: Rabha W. Ibrahim

Copyright © 2013 Zinelaâbidine Latreuch and Benharrat Belaïdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the growth and oscillation of , where and are entire functions of finite order not all vanishing identically and and are two linearly independent solutions of the linear differential equation .

1. Introduction and Main Results

Throughout this paper, we assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory (see [14]). In addition, we will use and to denote, respectively, the exponents of convergence of the zero sequence and distinct zeros of a meromorphic function , to denote the order of growth of .

Definition 1 (see [4, 5]). Let be a meromorphic function. Then the hyperorder of is defined by

Definition 2 (see [4, 5]). Let be a meromorphic function. Then the hyper-exponent of convergence of zeros sequence of is defined by where is the counting function of zeros of in . Similarly, the hyperexponent of convergence of the sequence of distinct zeros of is defined by where is the counting function of distinct zeros of in .

Suppose that and are two linearly independent solutions of the complex linear differential equation and the polynomial of solutions where and are entire functions of finite order in the complex plane. It is clear that if are complex numbers or where is a complex number, then is a solution of (4) or has the same properties of the solutions.

It is natural to ask what can be said about the properties of in the case when where is a complex number and under what conditions keeps the same properties of the solutions of (4).

In [6], Chen studied the fixed points and hyper-order of solutions of second-order linear differential equations with entire coefficients and obtained the following results.

Theorem A (see [6]). For all nontrivial solutions of (4) the following hold.(i)If is a polynomial with , then one has (ii)If is transcendental and , then one has

Before we state our results we define and by where is entire function of finite order and The subject of this paper is to study the controllability of solutions of the differential equation (4). In fact, we study the growth and oscillation of where  and are two linearly independent solutions of (4) and   and are entire functions of finite order not all vanishing identically and satisfying where is a complex number, and we obtain the following results.

Theorem 3. Let be a transcendental entire function of finite order. Let be finite-order entire functions that are not all vanishing identically such that . If and   are two linearly independent solutions of (4), then the polynomial of solutions (5) satisfies

Theorem 4. Under the hypotheses of Theorem 3, let be an entire function with finite order such that . If and are two linearly independent solutions of (4), then the polynomial of solutions (5) satisfies

Theorem 5. Let be a polynomial of . Let    be finite-order entire functions that are not all vanishing identically such that and . If are two linearly independent solutions of (4), then the polynomial of solutions (5) satisfies

Theorem 6. Under the hypotheses of Theorem 5, let be an entire function with such that . If and are two linearly independent solutions of (4), then the polynomial of solutions (5) satisfies

2. Auxiliary Lemmas

Lemma 7 (see [7, 8]). Let be finite-order meromorphic functions. If is a meromorphic solution of the equation with and , then satisfies

Here, we give a special case of the result due to Cao et al. in [9].

Lemma 8. Let be finite-order meromorphic functions. If is a meromorphic solution of (14) with then

3. Proofs of the Theorems

Proof of Theorem 3. Suppose that and are two linearly independent solutions of (4). Then by Theorem A, we have Suppose that , where is a complex number. Then, by (5) we obtain Since is a solution of (4) and , then we have Suppose now that where is a complex number. Differentiating both sides of (5), we obtain Differentiating both sides of (21), we obtain Substituting into (22), we obtain Differentiating both sides of (23) and by substituting , we obtain By (5), (21), (23), and (24) we have To solve this system of equations, we need first to prove that . By simple calculations we obtain To show that , we suppose that Dividing both sides of (27) by , we obtain equivalent to which implies that where is a complex number and this is a contradiction. Since and , we can deduce from (26) that Hence . By Cramer's method we have where are meromorphic functions of finite order defined in (9). Suppose now that , then by (31) we obtain , which is a contradiction, hence . By (5) we have . Suppose that , then by (31) we obtain , which is a contradiction. Hence .

Proof of Theorem 4. By Theorem 3 we have and . Set . Since , then we have and . In order to prove that ,   we need to prove only that and . By we get from (31) where Substituting (32) into (4), we obtain where are meromorphic functions of finite order. Since and , it follows that is not a solution of (4), which implies that . Then by applying Lemma 7 we obtain (11).

Proof of Theorem 5. Suppose that and are two linearly independent solutions of (4). Then by Theorem A By the same reasoning as in Theorem 3, we have Since and by Cramer's method we have where are meromorphic functions with defined in (9). By (5) we have . Suppose that , then by (37) we obtain , which is a contradiction. Hence, .

Proof of Theorem 6. By Theorem 5 we have . Set . Since , then we have . In order to prove that , we need to prove only that . By we get from (37) where Substituting (38) into (4), we obtain where are meromorphic functions with . Since and , it follows that is not a solution of (4), which implies that . Then by applying Lemma 8 we obtain (13).

Acknowledgments

The authors would like to thank the referees for their helpful remarks and suggestions to improve the paper. This research is supported by Agence Nationale pour le Développement de la Recherche Universitaire (ANDRU) and University of Mostaganem (UMAB), (PNR Project Code 8/u27/3144).

References

  1. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964. View at MathSciNet
  2. I. Laine, Nevanlinna Theory and Complex Differential Equations, vol. 15 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  3. R. Nevanlinna, Eindeutige analytische Funktionen, vol. 46 of Die Grundlehren der mathematischen Wissenschaften, Springer, Berlin, Germany, 2nd edition, 1974. View at MathSciNet
  4. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. View at MathSciNet
  5. Z. X. Chen and C. C. Yang, “Some further results on the zeros and growths of entire solutions of second order linear differential equations,” Kodai Mathematical Journal, vol. 22, no. 2, pp. 273–285, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Z. X. Chen, “The fixed points and hyper-order of solutions of second order complex differential equations,” Acta Mathematica Scientia A, vol. 20, no. 3, pp. 425–432, 2000 (Chinese). View at Zentralblatt MATH · View at MathSciNet
  7. B. Belaïdi, “Growth and oscillation theory of solutions of some linear differential equations,” Matematicki Vesnik, vol. 60, no. 4, pp. 233–246, 2008. View at MathSciNet · View at Scopus
  8. Z. X. Chen, “Zeros of meromorphic solutions of higher order linear differential equations,” Analysis, vol. 14, no. 4, pp. 425–438, 1994. View at Zentralblatt MATH · View at MathSciNet
  9. T. B. Cao, Z. X. Chen, X. M. Zheng, and J. Tu, “On the iterated order of meromorphic solutions of higher order linear differential equations,” Annals of Differential Equations, vol. 21, no. 2, pp. 111–122, 2005. View at Zentralblatt MATH · View at MathSciNet