Miskolc Mathematical Notes Vol. 16 (2015), No 1, pp. 237-248 HU e-ISSN 1787-2413 DOI: 10.18514/MMN.2015.704 On the zeros of solutions and their derivatives of second order non-homogenous linear dierential equations Zinelaâbidine Latreuch and Benharrat Belaïdi Miskolc Mathematical Notes Vol. 16 (2015), No. 1, pp. 237–248 HU e-ISSN 1787-2413 ON THE ZEROS OF SOLUTIONS AND THEIR DERIVATIVES OF SECOND ORDER NON-HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS ZINELÂABIDINE LATREUCH AND BENHARRAT BELAÏDI Received 24 September, 2013 Abstract. This paper is devoted to studying the growth and oscillation of solutions and their derivatives of equations of the type f 00 C A .´/ f 0 C B .´/ f D F .´/ ; where A .´/ ; B .´/ .6 0/ and F .´/ .6 0/ are meromorphic functions of finite order. 2010 Mathematics Subject Classification: 34M10, 30D35 Keywords: Linear differential equations, Meromorphic functions, Exponent of convergence of the sequence of zeros 1. I NTRODUCTION AND MAIN RESULTS We assume that the reader is familiar with the usual notations and basic results of the Nevanlinna theory [8, 17]. In addition, we will use  .f / and  .f / to denote respectively the exponents of convergence of the zero-sequence and distinct zeros of a meromorphic function f ,  .f / to denote the order of growth of f . A meromorphic function ' .´/ is called a small function with respect to f .´/ if T .r; '/ D o .T .r; f // as r ! C1 except possibly a set of r of finite linear measure; where T .r; f / is the Nevanlinna characteristic function of f: In the following, we give the necessary notations and basic definitions. Definition 1 ([17]). Let f be a meromorphic function. Then the hyper-order of f .´/ is defined by log log T .r; f / : 2 .f / D lim sup log r r!C1 Definition 2 ([8,11]). The type of a meromorphic function f of order  .0 <  < 1/ is defined by T .r; f /  .f / D lim sup : r r!C1 c 2015 Miskolc University Press 238 ZINELÂABIDINE LATREUCH AND BENHARRAT BELAÏDI If f is entire function of order  .0 <  < 1/, we can define the type by log M .r; f / : r r!C1 M .f / D lim sup 1  Remark 1. We have not always the equality M .f / D  .f / ; for example  .e ´ / D < 1 D M .e ´ / : Definition 3 ([7,17]). Let f be a meromorphic function. Then the hyper-exponent of convergence of zeros sequence of f .´/ is defined by   log log N r; f1 2 .f / D lim sup ; log r r!C1   where N r; f1 is the counting function of zeros of f .´/ in f´ W j´j 6 rg. Similarly, the hyper-exponent of convergence of the sequence of distinct zeros of f .´/ is defined by   log log N r; f1 2 .f / D lim sup ; log r r!C1   where N r; f1 is the counting function of distinct zeros of f .´/ in f´ W j´j 6 rg. The study of oscillation of solutions of linear differential equations has attracted many interests since the work of Bank and Laine [1, 2], for more details, see [9]. The main subject of this research is the zeros distribution of solutions and their derivatives of linear differential equations. In this paper, we first discuss the growth of solutions of second order linear differential equation f 00 C A .´/ f 0 C B .´/ f D F .´/ ; (1.1) where A .´/ ; B .´/ .6 0/ and F .´/ .6 0/ are meromorphic functions of finite order. Some results on the growth of entire solutions of (1.1) have been obtained by several researchers (see [5, 6, 12, 14]). Li and Wang (see [12]) investigated the nonhomogeneous linear differential equation f 00 C e ´ f 0 C h .´/ e b´ f D H .´/ ; (1.2) where h .´/ is a transcendental entire function of finite order  .h/ < 12 ; and b is a real constant. They proved that all nontrivial solutions of (1.2) are of infinite order, provided that  .H / < 1: After their, Wang and Laine (see [14]) studied the differential equation f 00 C A1 .´/ e a´ f 0 C A0 .´/ e b´ f D H .´/ ; (1.3) where A0 .´/ ; A1 .´/ ; H .´/ are entire functions of order less than one, and a; b 2 C; and obtained. ON THE ZEROS OF SOLUTIONS AND THEIR DERIVATIVES 239 Theorem 1 ([14]). Suppose that A0 6 0; A1 6 0; H are entire functions of order less than one, and the complex constants a; b satisfy ab ¤ 0 and a ¤ b: Then every nontrivial solution f of (1.3) is of infinite order. J. Tu and co-authors investigated the hyper-exponent of convergence of zeros of f .j / .´/ ' .´/ .j D 0; 1; 2; : : :/, where f is a solution of f 00 C A .´/ f 0 C B .´/ f D 0 (1.4) and ' .´/ is an entire function satisfying  .'/ <  .f / or 2 .'/ < 2 .f / ; and obtained the following result. Theorem 2 ([13]). Let A .´/ and B .´/ be entire functions with finite order. If  .A/ <  .B/ < 1 or 0 <  .A/ D  .B/ < 1 and M .A/ < M .B/ ; then for every solution f 6 0 of (1.4) and for any entire function ' .´/ 6 0 satisfying 2 .'/ < 2 .f / ; we have   2 f .j / ' D 2 .f / D  .B/ .j D 0; 1; 2; : : :/ : Recently in [15, 16], H. Y. Xu, J. Tu, X. M. Zheng and H. Y. Xu, J. Tu have investigated the relationship between small functions and the derivatives of solutions of higher order linear differential equations with entire and meromorphic functions. It is a natural to ask what about the exponent of convergence of zeros of f .j / .´/ .j D 0; 1; 2; : : :/ ; where f is a solution of (1.1). The main purpose of this paper is to give an answer to this question. The method used in the proofs of our theorems is quite different from the method used in the papers [13, 16]. Before we state our results we need to define the following notations Aj .´/ D Aj Bj .´/ D Aj0 1 .´/ Aj 1 .´/ 1 .´/ Bj0 1 .´/ Bj 1 .´/ Bj0 1 .´/ Bj 1 .´/ and Fj .´/ D Fj0 1 .´/ for j D 1; 2; 3; : : : ; C Bj Bj0 1 .´/ for j D 1; 2; 3; : : : (1.5) (1.6) 1 .´/ for j D 1; 2; 3; : : : ; (1.7) Bj 1 .´/ where A0 .´/ D A .´/ ; B0 .´/ D B .´/ and F0 .´/ D F .´/ : We obtain the following results. Fj 1 .´/ Theorem 3. Let A .´/ ; B .´/ 6 0 and F .´/ 6 0 be meromorphic functions with finite order such that Bj .´/ 6 0 and Fj .´/ 6 0 .j D 1; 2; 3; : : :/ : If f is a meromorphic solution of (1.1) with  .f / D 1 and 2 .f / D ; then f satisfies      f .j / D  f .j / D C1 .j D 0; 1; 2; : : :/ 240 ZINELÂABIDINE LATREUCH AND BENHARRAT BELAÏDI and     2 f .j / D 2 f .j / D  .j D 0; 1; 2; : : :/ : Theorem 4. Let A .´/ ; B .´/ 6 0 and F .´/ 6 0 be meromorphic functions with finite order such that Bj .´/ 6 0 and Fj .´/ 6 0 .j D 1; 2; 3; : : :/ : If f is a meromorphic solution of (1.1) with  .f / > max f .A/ ;  .B/ ;  .F /g ; then      f .j / D  f .j / D  .f / .j D 0; 1; 2; : : :/ : Remark 2. The conditions Bj .´/ 6 0 and Fj .´/ 6 0 .j D 1; 2; 3; : : :/ are neces´ 1 sary. For example f .´/ D e ´ C1 satisfies (1.1), where A .´/ D ´C1 ; B .´/ D ´C1 1 and F .´/ D ´C1 : On the other hand A1 D A B0 B0 C B  0; F1 D F 0 F 0 B B    .f / D 1 >  f .j / D 0 .j D 1; 2; 3; : : :/ : B1 D A0 and B0 D 1; B A Here, we will give some sufficient conditions on the coefficients which guarantee Bj .´/ 6 0 and Fj .´/ 6 0 .j D 1; 2; 3; : : :/: Theorem 5. Let A .´/ ; B .´/ 6 0 and F .´/ 6 0 be entire functions with finite order such that  .B/ > max f .A/ ;  .F /g : Then all nontrivial solutions of (1.1) satisfy      f .j / D  f .j / D C1 .j D 0; 1; 2; : : :/ with at most one possible exceptional solution f0 such that n o  .f0 / D max  .f0 / ;  .B/ : Remark 3. The condition  .B/ > max f .A/ ;  .F /g does not ensure that all solu2 tions of (1.1) are of infinite order. For example we can see that f0 .´/ D e ´ satisfies the differential equation 2 f 00 C 2´f 0 C .e ´ C 2/f D 1; where  .f0 / D 0 <  .f0 / D  .B/ D 2: ON THE ZEROS OF SOLUTIONS AND THEIR DERIVATIVES 241 In the next, we note log m .r; f / : log r r!C1  .f / D lim sup Theorem 6. Let A .´/ ; B .´/ 6 0 and F .´/ 6 0 be meromorphic functions with finite order such that  .B/ > max f .A/ ;  .F /g : If f is a meromorphic solution of (1.1) with  .f / D 1 and 2 .f / D ; then f satisfies      f .j / D  f .j / D C1 .j D 0; 1; 2; : : :/ and     2 f .j / D 2 f .j / D  .j D 0; 1; 2; : : :/ : Theorem 7. Let A .´/ ; B .´/ 6 0 and F .´/ 6 0 be entire functions with finite order such that  .B/ D  .A/ >  .F / and  .B/ > k .A/ ; k > 1 is an integer: If f is a nontrivial solution of (1.1) with  .f / D 1 and 2 .f / D ; then f satisfies      f .j / D  f .j / D C1 .j D 0; 1; : : : ; k/ and     2 f .j / D 2 f .j / D  .j D 0; 1; : : : ; k/ : Corollary 1. Suppose that A0 6 0; A1 6 0; H 6 0 are entire functions of order less than one, and the complex constants a; b satisfy ab ¤ 0 and jbj > k jaj ; k > 1 is an integer: Then every nontrivial solution f of (1.3) satisfies      f .j / D  f .j / D C1 .j D 0; 1; : : : ; k/ : 2. P RELIMINARY LEMMAS Lemma 1 ([8]). Let f be a meromorphic function and let k > 1 be an integer: Then ! f .k/ m r; D S .r; f / ; f where S .r; f / D O .log T .r; f / C log r/ ; possibly outside of an exceptional set E  .0; C1/ of r with finite linear measure. If f is of finite order of growth, then ! f .k/ m r; D O .log r/ : f Lemma 2 ([3,4]). Let A0 ; A1 ; : : : ; Ak 1 ; F 6 0 be finite order meromorphic functions. .i/ If f is a meromorphic solution of the equation f .k/ C Ak 1f .k 1/ C


C A1 f 0 C A0 f D F (2.1) 242 ZINELÂABIDINE LATREUCH AND BENHARRAT BELAÏDI with  .f / D C1 , then f satisfies  .f / D  .f / D  .f / D C1: .ii/ If f is a meromorphic solution of equation (2.1) with  .f / D C1 and 2 .f / D ; then  .f / D  .f / D  .f / D C1; 2 .f / D 2 .f / D 2 .f / D : Lemma 3 ([13]). Let A0 ; A1 ; : : : ; Ak 1 ; F 6 0 be finite order meromorphic functions. If f is a meromorphic solution of equation (2.1) with ˚
max  Aj .j D 0; 1; : : : ; k 1/ ;  .F / <  .f / < 1; then  .f / D  .f / D  .f / : Lemma 4 ([10]). Let f and g be meromorphic functions in the complex plane such that 0 <  .f / ;  .g/ < 1 and 0 <  .f / ;  .g/ < 1: Then we have .i/ If  .f / >  .g/ ; then we obtain  .f C g/ D  .fg/ D  .f / : .ii/ If  .f / D  .g/ and  .f / ¤  .g/ ; then we get  .f C g/ D  .fg/ D  .f / D  .g/ : Lemma 5 ([6]). Let A; B1 ; : : : ; Bk 1 ; F 6 0 be entire functions of finite order, where k >
2: Suppose that either .i/ or .ii/ below holds: .i/  Bj <  .A/ .j D 1; : : : ; k 1/ ; .ii/ B1 ; : : : ; Bk 1 are polynomials and A is transcendental. Then we have .a/ All solutions of the differential equation f .k/ C Bk 1f .k 1/ C


C B1 f 0 C Af D F satisfy  .f / D  .f / D  .f / D C1 with at most one possible solution f0 of finite order. .b/ If there exists an exceptional solution f0 in case .a/ ; then f0 satisfies n o  .f0 / 6 max  .A/ ;  .F / ;  .f0 / < 1: Furthermore, if  .A/ ¤  .F / and  .f0 / <  .f0 / ; then  .f0 / D max f .A/ ;  .F /g : (2.2) ON THE ZEROS OF SOLUTIONS AND THEIR DERIVATIVES 243 3. P ROOF OF THE THEOREMS AND COROLLARY Proof of Theorem 3. We prove this theorem by using mathematical induction. Since B 6 0; F 6 0; then by using Lemma 2, we have  .f / D  .f / D  .f / D C1 and 2 .f / D 2 .f / D 2 .f / D : Dividing both sides of (1.1) by B; we obtain 1 00 A 0 F f C f Cf D : B B B Differentiating both sides of equation (3.1), we have   0   0  0 1 .3/ A A F 1 00 0 f C f C C C1 f D : B B B B B (3.1) (3.2) Multiplying now (3.2) by B; we get f .3/ C A1 f 00 C B1 f 0 D F1 ; where (3.3) B0 ; B B0 A CB B A1 D A B1 D A0 and B0 : B Since B1 6 0; F1 6 0 are meromorphic functions with finite order, then by using Lemma 2, we obtain

 f 0 D  f 0 D  .f / D C1 and

2 f 0 D 2 f 0 D 2 .f / D : Dividing now both sides of (3.3) by B1 ; we obtain 1 .3/ A1 00 F1 f C f Cf 0 D : (3.4) B1 B1 B1 Differentiating both sides of equation (3.4) and multiplying by B1 ; we get F1 D F 0 F f .4/ C A2 f .3/ C B2 f 00 D F2 ; (3.5) where A2 ; B2 6 0 and F2 6 0 are meromorphic functions defined in (1.5)-(1.7). By using Lemma 2, we obtain

 f 00 D  f 00 D  .f / D C1 244 ZINELÂABIDINE LATREUCH AND BENHARRAT BELAÏDI and

2 f 00 D 2 f 00 D 2 .f / D : We suppose now that          f .k/ D  f .k/ D  .f / D C1; 2 f .k/ D 2 f .k/ D 2 .f / D  (3.6) for all k D 0; 1; 2; : : : ; j 1; and we prove that (3.6) is true for k D j: By the same method as before, we can obtain f .j C2/ C Aj f .j C1/ C Bj f .j / D Fj ; where Aj ; Bj 6 0 and Fj 6 0 are meromorphic functions defined in (1.5) -(1.7). By using Lemma 2, we obtain      f .j / D  f .j / D  .f / D C1 and     2 f .j / D 2 f .j / D 2 .f / D :  Thus, the proof of Theorem 3 is completed. Proof of Theorem 4. By a similar reasoning as in the proof of Theorem 3, and by using Lemma 3, we obtain      f .j / D  f .j / D  .f / .j D 0; 1; 2; : : :/ :  Proof of Theorem 5. By Lemma 5, all nontrivial solutions of (1.1) are of infinite order with at most one exceptional solution f0 of finite order. By using (1.5) and Lemma 1 we have

m r; Aj 6 m r; Aj 1 C O .log r/ for all j D 1; 2; 3; : : : ; which we can rewrite as
m r; Aj 6 m .r; A/ C O .log r/ .j D 1; 2; 3; : : :/ : (3.7) On the other hand, we have from (1.6) Bj D Aj D Aj 1 Aj0 Aj Aj0 1 1 Bj0 1 1 Bj 1 D jX 1 Aj ! kD0 Bj0 1 1 Bj 1 C Aj  Ak 1 2 Aj0 Aj A0k Bk0 Ak Bk ! C Bj 1 2 Bj0 2 2 Bj 2 ! C Bj 2  C B: (3.8) ON THE ZEROS OF SOLUTIONS AND THEIR DERIVATIVES 245 Now we prove that Bj 6 0 for all j D 1; 2; 3; : : : : Suppose there exists an integer j D 1; 2; 3; : : : such that Bj  0: By (3.7) and (3.8) T .r; B/ D m .r; B/ 6 jX 1 m .r; Ak / C O .log r/ kD0 6 j m .r; A/ C O .log r/ D j T .r; A/ C O .log r/ (3.9) which implies the contradiction  .B/ 6  .A/ : Hence Bj 6 0 for all j D 1; 2; 3; : : : : Suppose now there exists an integer j D 1; 2; 3; : : : that is the first index for which Fj  0: Then, by (1.7) and Fj 1 .´/ 6 0 we have Fj0 1 .´/ Fj 1 .´/ Bj0 1 .´/ Bj 1 .´/ D0 which implies Fj 1 .´/ D cBj where c 2 Cn f0g : By (3.8) we have  0 jX 2 Ak 1 Fj 1 D Ak c Ak kD0 1 .´/ ; Bk0 Bk  C B: (3.10) On the other hand, we have from (1.7)
m r; Fj 6 m .r; F / C O .log r/ .j D 1; 2; 3; : : :/ : (3.11) By (3.10), (3.11) and Lemma 1, we obtain T .r; B/ D m .r; B/ 6 jX 2 m .r; Ak / C m r; Fj 1
C O .log r/ kD0 6 .j 1/ m .r; A/ C m .r; F / C O .log r/ D .j 1/ T .r; A/ C T .r; F / C O .log r/ (3.12) which implies the contradiction  .B/ 6 max f .A/ ;  .F /g : Since Bj 6 0 and Fj 6 0 .j D 1; 2; 3; : : :/ ; then by applying Theorem 3 and Lemma 5 we have      f .j / D  f .j / D C1 .j D 0; 1; 2; : : :/ with at most one exceptional solution f0 of finite order. Since  .B/ > max f .A/ ;  .F /g ; then by (2.2) we obtain o n  .f0 / 6 max  .B/ ;  .f0 / : (3.13) On the other hand by (1.1), we can write  00  f0 f0 F BD CA 0 : f0 f0 f0 246 ZINELÂABIDINE LATREUCH AND BENHARRAT BELAÏDI It follows that   F T .r; B/ D m .r; B/ 6 m r; C m .r; A/ C O .log r/ f0 6 T .r; f0 / C T .r; F / C T .r; A/ C O.log r/ which implies  .B/ 6 max f .f0 / ;  .A/ ;  .F /g D  .f0 / : Since  .f0 / 6  .f0 / ; then by using (3.13) and (3.14) we obtain n o  .f0 / D max  .B/ ;  .f0 / : (3.14)  This completes the proof of Theorem 5. Proof of Theorem 6. By using the same reasoning as in the proof of Theorem 5, we can prove Theorem 6.  Proof of Theorem 7. First, we prove that Bj 6 0 for all j D 1; 2; : : : ; k: Suppose there exists an integer s; 1 6 s 6 k such that Bs  0: By (3.7) and (3.8), we have T .r; B/ D m .r; B/ 6 s 1 X m .r; Ak / C O .log r/ kD0 6 sm .r; A/ C O .log r/ D sT .r; A/ C O .log r/ (3.15) which implies the contradiction k .A/ <  .B/ 6 s .A/ : Hence Bj 6 0 for all j D 1; 2; : : : ; k: Now, we prove that Fj 6 0 for all j D 1; 2; : : : ; k: Suppose there exists an integer s; 1 6 s 6 k such that Fs  0: From (3.12), we have T .r; B/ 6 .s 1/ m .r; A/ C m .r; F / C O .log r/ D .s 1/ T .r; A/ C T .r; F / C O .log r/ (3.16) which implies the contradiction k .A/ <  .B/ 6 .s 1/  .A/ : Hence Fj 6 0 for all j D 1; 2; : : : ; k: Since Bj 6 0 and Fj 6 0 .j D 1; : : : ; k/ ; then by Theorem 3 we have      f .j / D  f .j / D C1 .j D 0; 1; : : : ; k/ and     2 f .j / D 2 f .j / D  .j D 0; 1; : : : ; k/ :  Proof of Corollary 1. Since ab ¤ 0; jbj > k jaj ; then by Theorem 1, every nontrivial solution f of (1.3) is of infinite order. By using Lemma 4 we have     jbj
jaj  A0 e b´ D  e b´ D >k D k A1 e a´ :   ON THE ZEROS OF SOLUTIONS AND THEIR DERIVATIVES 247 Then by Theorem 7, we obtain      f .j / D  f .j / D  .f / D C1 .j D 0; 1; : : : ; k/ :  4. O PEN PROBLEM It’s interesting to study whether the condition jbj > k jaj ; where k > 1 is an integer is necessary in Corollary 1. For that, we pose the following problem. Conjecture 1. Suppose that A0 6 0; A1 6 0; H 6 0 are entire functions of order less than one, and the complex constants a; b satisfy ab ¤ 0 and jbj > jaj : Then every nontrivial solution f of (1.3) satisfies      f .j / D  f .j / D C1 .j D 0; 1; : : :/ : ACKNOWLEDGEMENT The authors would like to thank the referees for their valuable comments and suggestions to improve the paper. R EFERENCES [1] S. Bank and I. Laine, “On the oscillation theory of f 00 C af D 0 where a is entire,” Trans. Amer. Math. Soc., vol. 273, no. 1, pp. 351–363, 1982. [2] S. Bank and I. Laine, “On the zeros of meromorphic solutions of second order linear differential equations,” Comment. Math. Helv., vol. 58, no. 4, pp. 656–677, 1983. [3] B. Belaı̈di, “Growth and oscillation theory of solutions of some linear differential equations,” Mat. Vesnik, vol. 60, no. 4, pp. 233–246, 2008. [4] Z. X. Chen, “Zeros of meromorphic solutions of higher order linear differential equations,” Analysis, vol. 14, no. 4, pp. 425–438, 1994. [5] Z. X. Chen, “The fixed points and hyper-order of solutions of second order complex differential equations,” Acta Math. Sci. Ser. A Chin. Ed., vol. 20, no. 3, pp. 425–432, 2000. [6] Z. X. Chen and S. A. Gao, “The complex oscillation theory of certain nonhomogeneous linear differential equations with transcendental entire coefficients,” J. Math. Anal. Appl., vol. 179, no. 2, pp. 403–416, 1993. [7] 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 Math. J., vol. 22, no. 2, pp. 273–285, 1999. [8] W. K. Hayman, Meromorphic functions. Oxford, 1964. [9] I. Laine, Nevanlinna theory and complex differential equations. [10] Z. Latreuch and B. Belaı̈di, “Estimations about the order of growth and the type of meromorphic functions in the complex plane,” An. Univ. Oradea, fasc. Mat., vol. XX, no. 1, pp. 179–186, 2013. [11] B. Y. Levin, Lectures on entire functions. Translations of Mathematical Monographs. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko. Providence, RI: American Mathematical Society, 1996, no. 150. [12] Y. Z. Li and J. Wang, “Oscillation of solutions of linear differential equations,” Acta Math. Sin. (Engl. Ser.), vol. 24, no. 1, pp. 167–176, 2008. 248 ZINELÂABIDINE LATREUCH AND BENHARRAT BELAÏDI [13] J. Tu, H. Y. Xu, and C. Y. Zhang, “On the zeros of solutions of any order of derivative of second order linear differential equations taking small functions,” Electron. J. Qual. Theory Differ. Equ., no. 23, pp. 1–17, 2011. [14] J. Wang and I. Laine, “Growth of solutions of second order linear differential equations,” J. Math. Anal. Appl., vol. 342, no. 1, pp. 39–51, 2008. [15] H. Y. Xu and J. Tu, “Oscillation of meromorphic solutions to linear differential equations with coefficients of Œp; q-order,” Electron. J. Diff. Equ., vol. 2014, no. 73, pp. 1–14, 2014. [16] H. Y. Xu, J. Tu, and X. M. Zheng, “On the hyper exponent of convergence of zeros of f .j / ' of higher order linear differential equations,” Adv. Difference Equ., vol. 114, p. 16 pp, 2012. [17] C. C. Yang and H. X. Yi, Uniqueness theory of meromorphic functions. Mathematics and its Applications. Dordrecht: Kluwer Academic Publishers Group, 2003, no. 557. Authors’ addresses Zinelâabidine Latreuch University of Mostaganem, Department of Mathematics, Laboratory of Pure and Applied Mathematics, B. P. 227 Mostaganem, Algeria E-mail address: z.latreuch@gmail.com Benharrat Belaı̈di University of Mostaganem, Department of Mathematics, Laboratory of Pure and Applied Mathematics, B. P. 227 Mostaganem, Algeria E-mail address: belaidi@univ-mosta.dz