OSCILLATION OF THREE DIMENSIONAL NEUTRAL DELAYDIFFERENCE SYSTEMS

6751 | P a g e N o v e m b e r 2 0 1 6 w w w . c i r w o r l d . c o m OSCILLATION OF THREE DIMENSIONAL NEUTRAL DELAYDIFFERENCE SYSTEMS K.THANGAVELU ASSOCIATE PROFESSOR ,DEPARTMENT OF MATHEMATICS, PACHIYAPPA’S COLLEGE ,CHENNAI600 030 . kthangavelu14@gmail.com G.SARASWATHI RESEARCH SCHOLAR,DEPARTMENT OF MATHEMATICS,PACHIYAPPA’S COLLEGE ,CHENNAI-600 030. ganesan_saraswathi@yahoo.co.in Abstract. This paper deals with the some oscillation criteria for the three dimensional neutral delay difference system of the form


1.Introduction
Consider a three dimensional neutral delay difference system of the form ∆ x n +p n x n-k =b n y n α ∆ y n =c n z n β (1.1) ∆ z n =-a n x n−l+1 γ n=1,2,…, subject to the following conditions  b n ∆ x n + p n x n−k + a n x n−l+1 γ = 0. whose oscillatory behaviour has been studied in, for example, [1][2][3][4] and the refrences cited therein. Also the oscillatory theory is considered for two-dimensional and three-dimensional difference systems (see, for example, [5][6][7][8][9][10] and the references cited therein).This observation motivated us to consider the three-dimensional neutral delay difference systems and to investigate its oscillatory behaviour. In section 2, we present some basic lemmas which will be used to prove the main theorems, and in Section 3, we obtain the sufficient conditions for the oscillation of system (1.2). Examples are provided in Section 4 to illustrate the main results.

SOME BASIC LEMMAS
In this section, we state and prove some basic lemmas, which will be used in establishing our main results.
Summing the last inequality from N1 to n-1 and then taking n→ ∞, we find that yn→ −∞ as n→ ∞. Then there is an integer N2≥ N1 and a constant η such that yn<η < 0 for n≥ N 2. ∆w n = η α bn , n≥ N 2.
Where wn =xn +pn x-k for n≥ N 2. Now taking summation from N2 to n-1 and then making n→ ∞, we see that wn→ −∞, as n→ ∞. This contradicts the fact that wn >0 for all n≥ N. Hence zn>0 for all n≥ N. The proof for the case wn <0 eventually is similar. This completes the proof of the lemma.
Proof. Proceeding as in Lemma 2.2, we have > 0 and > 0 for n≥ ≥ 1. From the first equation of the system Therefore > 0 and nondecreasing for all n≥ . From the definition of , we obtain This completes the proof of the lemma.

3.OSCILLATION RESULTS
In this section, we establish sufficient conditions for the oscillatory and asymptotic behaviour of the solutions of system (1.2).
and < 1. We shall prove that →∞ = 0. Let →∞ = 1 > 0. Then there exists an integer 1 ≥ , such that +1 > 1 > 0 for n≥ 1 . Now summing the third equation of (1.2) from n to ∞ and then using ≥ − +1 and ≥ 1 for n≥ 1 . We obtain Since is a ratio of odd positive integer, we have from the last inequality Summing the second equation of (1.2) from 1 to n-1 and then using (3.10) we obtain In view of (3.1) the last inequality implies for → ∞ that →∞ = ∞, which is a contradiction. Therefore where 0< < 1, ℎ ℎ conclusion of the Theorem 3.1 holds.
Proof. Let {( , , )} be a nonoscillatory solution of system (1.2). We see that Theorem 3.1 satisfies one of the two cases of Lemma 2.2 for n≥ . First we consider case(I). In this case, we have inequality (3.7). Using (3.11) Raising (3.13) to (1-) th power we obtain Since {xn} is monotonically nondecreasing, there exists an integer 2 ≥ 1 and a constant 1 > 0 such that − +1 ≥ 1 , ≥ 2. Multiplying (3.17) by −1 , using the third equation of (1.2), summing from 2 to n-1 and then using the fact that {zn} is positive and nondecreasing we have which contradicts (3.13). Therefore, case(I) cannot occur and for case(II), we proceed in the same way as in the proof of Theorem 3.1. This completes the proof.  → ∞ as n→ ∞. which is a contradiction to the fact that > 0 for m≥ 1 . Therefore, case(I) cannot occur and hence the solution of (1.2) satisfies case(II). The proof for case(II) is similar to that of Theorem 3.1 and this completes the proof. } is one such solution of the system (4.1). } is one such solution of the system (4.2).