fasciculi mathemat ici

FASCICULI

MATHEMAT ICI

Nr 56

2016 DOI:10.1515/fascmath-2016-0011

E. Thandapani, S. Selvarangam, R. Rama and M. Madhan IMPROVED OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR DELAY DIFFERENCE EQUATIONS WITH NON-POSITIVE NEUTRAL TERM Abstract. In this paper, we present some oscillation criteria for second order nonlinear delay difference equation with non-positive neutral term of the form ∆(an (∆zn )α ) + qn f (xn−σ ) = 0,

n ≥ n0 > 0,

where zn = xn − pn xn−τ , and α is a ratio of odd positive integers. Examples are provided to illustrate the results. The results obtained in this paper improve and complement to some of the existing results. Key words: 39A11. AMS Mathematics Subject Classification: Ooscillation, second order delay difference equation, non-positive neutral term.

1. Introduction In this paper, we study the oscillatory behavior of the following second order difference equation with non-positive neutral term of the form (1)

∆(an (∆zn )α ) + qn f (xn−σ ) = 0,

n ≥ n0 > 0,

where zn = xn − pn xn−τ , subject to the following hypotheses: (H1 ) {an }, {pn }, and {qn } are sequences of real numbers with 0 ≤ pn ≤ p < 1, qn ≥ 0 and qn is not identically zero for P infinitely many values of n, and {an } is a positive real sequence with ns=n0 11 → ∞ as n → ∞; asα

(H2 ) f ∈ C(R, R) such that uf (u) > 0 for all u 6= 0, and there exists a positive constant M such that fu(u) α ≥ M for all u 6= 0;

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E. Thandapani, S. Selvarangam, R. Rama and M. Madhan

(H3 ) α is a ratio of odd positive integers, σ and τ are positive integers. Let θ = max{τ, σ}. By a solution of equation (1) we mean a real sequence {xn } defined for all n ≥ n0 −θ, and satisfying the equation (1) for all n ≥ n0 . As usual a solution {xn } of equation (1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is said to be non-oscillatory. We consider only nontrivial solutions of equation (1), and we assume that equation (1) possesses such solutions. From the review of literature, it is known that there are many results available on the oscillatory and asymptotic behavior of solutions of equation (1) when the neutral term is nonnegative, i.e., pn ≤ 0; see for example [1, 2, 3, 8, 14] and the references cited therein. However, there are few results available on the oscillatory behavior of solutions of equation (1) when the neutral term is non-positive; see, for example [5, 6, 7, 10, 11, 12, 13, 15, 16] and the references cited therein. In [1], we see that the oscillatory behavior of the equation (2) ∆2 xn − pxn−τ + qn xn−σ = 0, n ∈ N(n0 ), is discussed and in [13], the authors studied the oscillatory and asymptotic behavior of the equation (3) ∆ an (∆(xn − pn xn−τ )) + qn xαn−σ = 0, n ∈ N(n0 ), ∞ P 1 = ∞. Also Thandapani et al. [11] studied the oscillation of n=n0 an the equation (4) ∆ an (∆(xn − pn xn−τ ))α + qn f (xn−σ ) = 0,

with

under the conditions f (u) ≥ M, uα

and

∞ X 1 1

n=n0

= ∞.

anα

In these papers the authors obtained that every solution of equations (2), (3) and (4) is either oscillatory or tends to zero as n → ∞. This observation motivated us to study when all solutions of equation (1) are oscillatory. In Section 2, we present some lemmas which will be useful to prove our main results. In Section 3, we obtain some new sufficient conditions for the oscillation of all solutions of equation (1), and in Section 4, we provide some examples to illustrate the main results. Thus the results presented in this paper improve and complement to those established in [5, 6, 7, 10, 11, 12, 13, 15, 16].

Improved oscillation criteria for second order . . .

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2. Some preliminary lemmas In this section, we provide two lemmas which are useful in proving the main results. Since α is a ratio of odd positive integers it is enough to prove all the results for positive solutions of equation (1) since the proof for the negative case in similar. Lemma 1. Assume that hypotheses (H1 ) − (H3 ) hold. If {xn } is a positive solution of equation (1), then {zn } satisfies only one of the following two cases for all n ≥ N ≥ n0 : (C1 ) zn > 0, ∆zn > 0 and ∆(an (∆zn )α ) ≤ 0; (C2 ) zn < 0, ∆zn > 0 and ∆(an (∆zn )α ) ≤ 0. Proof. Let {xn } be a positive solution of equation (1). Then there exists an integer n1 ≥ n0 such that xn > 0, xn−τ > 0 and xn−σ > 0 for all n ≥ n1 . Now from equation (1), we have ∆(an (∆zn )α ) = −qn f (xn−σ ) ≤ 0

(5)

for all n ≥ n1 . Therefore an (∆zn )α is non-increasing and hence (∆zn )α is non-increasing and of one sign eventually. That is, there exists an integer n2 ≥ n1 such that either ∆zn > 0 or ∆zn < 0 for all n ≥ n2 . We shall show that ∆zn > 0 for all n ≥ n2 . If not, then ∆zn < 0 for all n ≥ n2 , and an (∆zn )α ≤ an2 (∆zn2 )α < 0 for all n ≥ n2 . Then

1

anα ∆zn2 ∆zn ≤ 2 an for all n ≥ n2 . Summing the last inequality from n2 to n − 1, we obtain 1 α n2

zn − zn2 ≤ a ∆zn2

n−1 X

1

s=n2

asα

1

.

Letting n → ∞, and using (H1 ), we see that zn → −∞. Now we consider the two cases for {xn }. Case (i): Suppose {xn } is an unbounded sequence. Then there exists a sequence {nj } such that nj → ∞ and xnj → ∞, where xnj = max{xs : n0 ≤ s ≤ nj }. Then xnj −τ = max{xs : n0 ≤ s ≤ nj − τ } ≤ max{xs : n0 ≤ s ≤ nj } = xnj . Therefore for large values of n znj = xnj − pnj xnj −τ ≥ (1 − pnj )xnj > 0,

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which is a contradiction for zn → −∞ as n → ∞. Case (ii): Suppose {xn } is a bounded sequence then {zn } is also a bounded sequence. This contradicts zn → −∞ as n → ∞. Hence ∆zn > 0 for all n ≥ n2 and zn satisfies only one of the two possible cases (C1 ) and (C2 ). Lemma 2. If {xn } is an eventually positive solution of equation (1) such that (C1 ) of Lemma 1 holds, then xn ≥ zn ≥ Rn an ∆zn where Rn =

Pn−1

1

s=n0

1

.

asα

Proof. The proof can be found in [11].

3. Oscillation results In this section, we obtain some sufficient conditions which ensure that all solutions of equation (1) are oscillatory. In what follows, we denote 1

ξn = anα

n−1 X

−1

asα ,

s=n1

where n1 ≥ n0 is sufficiently large. Theorem 1. Assume that hypotheses (H1 )-(H3 ) hold, and σ > τ . If there exists a positive and nondecreasing sequence {ρn } of real numbers, such that ∞ X ∆ρn an−σ (6) M ρn+1 qn − = ∞, α ξn−σ n=n 1

and (7)

lim sup

n→∞

n−1 X s=n−σ+τ

1 as

n−1 X t=s

! α1 qt

>

p 1

Mα

for sufficiently large n1 ≥ n0 , then every solution of equation (1) is oscillatory. Proof. Suppose {xn } is a positive solution of equation (1). Then there exists an integer n1 ≥ n0 such that xn > 0, xn−τ > 0 and xn−σ > 0 for all n ≥ n1 . Choose an integer N ≥ n1 such that the sequence {zn } satisfies


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only one of the two Cases (C1 ) and (C2 ) of Lemma 1. First we assume that {zn } satisfies Case (C1 ) of Lemma 1. From the definition of zn , we have (8)

xn = zn + pn xn−τ ≥ zn

for all n ≥ N . Since an (∆zn )α is non-increasing, we have (9)

an (∆zn )α ≤ an−σ (∆zn−σ )α .

Now (10)

zn = zN +

n−1 X

∆zs

s=N 1

≥ (an (∆zn )α ) α

n−1 X

1

s=N

asα

1

= ξn ∆zn . Define (11)

wn = ρn

an (∆zn )α α zn−σ

for all n ≥ N . Then wn > 0 for all n ≥ N , and (12)

an (∆zn )α ∆(an (∆zn )α ) + ρ n+1 α α zn−σ zn−σ ρn+1 an+1 (∆zn+1 )α α − ∆(zn−σ ) α zα zn−σ n−σ+1 ∆(an (∆zn )α ) an (∆zn )α + ρ . ≤ ∆ρn n+1 α α zn−σ zn−σ

∆wn = ∆ρn

Using (5) and (H1 ) in (12) we have (13)

∆wn ≤ ∆ρn

ρn+1 qn xαn−σ an (∆zn )α − M . α α zn−σ zn−σ

Applying (8) and (10) in (13), we have an−σ (∆zn−σ )α − M ρn+1 qn α zn−σ ∆ρn an−σ ≤ − M ρn+1 qn , n ≥ N. α ξn−σ

∆wn ≤ ∆ρn

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Summing the last inequality from N to n − 1, we obtain n−1 X

wn − wN ≤

s=N

∆ρs as−σ − M ρs+1 qs . α ξs−σ

That is, n−1 X s=N

∆ρs as−σ ≤ wN . M ρs+1 qs − α ξs−σ

Letting n → ∞ in the last inequality, we obtain a contradiction to (6). Next consider the Case (C2 ) of Lemma 1. From the definition of zn , we have (14)

xn−k > (−

zn ), p

n ≥ N.

Using (H2 ) and (14) in equation (1), we obtain 1

∆(an (∆zn ) α ) −

M qn α z ≤ 0, pα n+τ −σ

n ≥ N.

Summing the last inequality from s to n − 1 for n > s + 1, we have n−1 MX α qt zt−σ+τ ≤ 0, an (∆zn ) − as (∆zs ) − α p t=s α

α

or 1

Mα −∆zs ≤ p

n−1 1 X α qt zt−σ+τ as t=s

! α1 n ≥ N.

,

Again summing the last inequality from n − σ + τ to n − 1 for s and then using monotonicity of {zn } we have zn−σ+τ

1 n−1 X Mα − zn ≤ zn−σ+τ p s=n−σ+τ

or p M

1 α

≥

n−1 X s=n−σ+τ

1 as

n−1 X

n−1 1 X qt as t=s

! α1

! α1 qt

t=s

which is a contradiction to (7). Now the proof is complete.

Next assume that ρn ≡ 1. then by Theorem 1, we have the following corollary. P Corollary 1. Assume that condition (7) is satisfied. If ∞ n=n0 qn = ∞, then every solution of equation (1) is oscillatory.


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Theorem 2. Assume that hypotheses (H1 ), (H2 ), and condition (7) hold. If α > 1, and there exists a positive and non-decreasing sequence {ρn } of real numbers such that ! ∞ X an−σ(∆ρn )α+1 (15) =∞ M ρn+1 qn − α+1 α ρn ρn+1 (α + 1)α+1 n=n1 for sufficiently large n1 ≥ n0 , then every solution of equation (1) is oscillatory. Proof. Suppose {xn } is a positive solution of equation (1). Then as in the proof of Theorem 1, {zn } satisfies (C1 ) or (C2 ) of Lemma 1. First, assume that {zn } satisfies (C1 ) of Lemma 1. Define a sequence {wn } as in Theorem 1. Then wn > 0 for all n ≥ N ≥ n1 ≥ n0 , and ∆wn = ∆ρn

an (∆zn )α ∆(an (∆zn )α ) ρn+1 an+1 ∆(zn+1 )α α + ρ − ∆zn−σ . n+1 α α α zα zn−σ zn−σ zn−σ n−σ+1

Using the definition of wn and the monotonicity of an (∆zn )α , we get an (∆zn )α ∆ρn α wn − M ρn+1 qn − ρn+1 α ∆zn−σ α ρn zn−σ zn−σ+1 wn ∆z α ∆ρn ≤ wn − M ρn+1 qn − ρn+1 α n−σ ρn zn−σ+1

∆wn ≤ (16)

for all n ≥ N . By Mean value theorem, we have α = αtα−1 ∆zn−σ ∆zn−σ

(17)

where zn−σ ≤ t ≤ zn−σ+1 . Therefore (18)

z α−1 ∆zn−σ ∆ρn wn − M ρn+1 qn − αρn+1 wn n−σ α ρn zn−σ+1 ∆ρn ∆zn−σ ≤ wn − M ρn+1 qn − αρn+1 wn , n ≥ N. ρn zn−σ

∆wn ≤

Now using (38) in the last inequality, we have 1

∆ρn (an ) σ ∆zn ∆wn ≤ wn − M ρn+1 qn − αρn+1 wn 1 ρn aα z ≤

∆ρn wn − M ρn+1 qn − ρn

n−σ n−σ 1 α+ α αρn+1 wn 1 α

an−σ

,

n ≥ N.

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E. Thandapani, S. Selvarangam, R. Rama and M. Madhan 1

α+1

α

Using the inequality Au − Buα+ α ≤ (α1α)α+1 AB α with u = wn , A = α+1 B = αρ in the last inequality, we obtain aα

∆ρn ρn ,

n−σ

∆wn ≤ −M ρn+1 qn +

an−σ (∆ρn )α+1 , n ≥ N. α α+1 ρα+1 n ρn+1 (α + 1)

Summing the last inequality from N to n − 1, we have " # n−1 X as−σ (∆ρs )α+1 M ρs+1 qs − α+1 α ≤ wN − wn ≤ wN . ρs ρs+1 (α + 1)α+1 s=N Letting n → ∞ in the last inequality, we obtain a contradiction to (15). If Case (C2 ) of Lemma 1 is satisfied by {zn }, then as in the proof of Theorem 1 we obtain a contradiction to (7). This completes the proof. Next we consider the case α = 1. Theorem 3. Assume hypotheses (H1 ), (H2 ) hold, and α = 1. If condition (3.2) and (19)

lim inf

n→∞

n−1 X s=n−σ

qs Rs−σ >

1 σ σ+1 ( ) M σ+1

are satisfied, then every solution of equation (1) is oscillatory. Proof. Suppose {xn } is a positive solution of equation (1). Then as in the proof of Theorem 1, {zn } satisfies Case (C1 ) or (C2 ) of Lemma 1. Let {zn } satisfies Case (C1 ). By the definition of zn , we have (20)

xn = zn + pn xn−τ ≥ zn for all n ≥ N.

Applying (19) and (H2 ) in equation (1), we have ∆(an ∆zn ) + M qn zn−σ ≤ 0,

n ≥ N.

Using Lemma 2 in the last inequality, we obtain ∆(an ∆zn ) + M qn Rn−σ an−σ ∆zn−σ ≤ 0,

n ≥ N.

Let wn = an ∆zn for n ≥ N . Then {wn } is a positive solution of (21)

∆wn + M qn Rn−σ wn−σ ≤ 0,

n ≥ N.

But by Theorem 7.6.1 of [4] and (19), the inequality (21) has no positive solution, which is a contradiction. Next assume Case (C2 ) of Lemma 1 holds. Then as in the proof of Theorem 1 we again obtain a contradiction with (7). The proof is now completed.

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3. Examples In this section, we provide three examples in support of our results. Example 1. Consider the following second order neutral difference equation (22)

1 1 1 3 ∆((∆(xn − xn−2 )) 3 ) + 2xn−3 = 0, 2

n ≥ 3.

It is easy to see that all conditions of Theorem 3.1 are satisfied and hence every solution of equation (4.1) is oscillatory. In fact {xn } = {(−1)3n } is one such oscillatory solution of equation (4.1). Example 2. Consider the following second order neutral difference equation (23)

1 ∆(n3 (∆(xn − xx−2 ))3 ) + (2n2 + 2n + 1)x3n−3 = 0, 2

n ≥ 3.

It is easy to see that all conditions of Theorem 2 are satisfied and hence every solution of equation (23) is oscillatory. In fact {xn } = {(−1)n } is one such oscillatory solution of equation (23). Example 3. Consider the following second order neutral difference equation (24)

1 ∆2 (xn − xn−2 ) + xn−3 (1 + x2n−3 ) = 0, 2

n ≥ 3.

It is easy to see that all the conditions of Theorem 3 are satisfied and hence every solution of equation (24) is oscillatory. In fact {xn } = {(−1)n } is one such oscillatory solution of equation (24). We conclude the paper with the following remark. Remark 1. If results presented in [5-7, 10-13, 15, 16] are applied to examples 1-3, we obtain that the solutions of the equations (6)-(8) are either oscillatory or tend to zero as n → ∞. But our Theorems 1 to 3 give stronger results in the sense that all solutions of equations (22) to (24) are oscillatory. Thus our results improve that of in [5-7, 10-13, 15, 16]. Acknowledgment. The author E. Thandapani thanks the University Grants Commission of India for awarding EMERITUS FELLOWSHIP (No.F.6-6/2013-14/ EMERITUS-2013-14-GEN-2747) to carry out this research.

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References [1] Agarwal R.P., Difference Equations and Inequalities, Second Edition, MarcelDekker, New York, 2000. [2] Agarwal R.P., Bohner M., Grace S.R., O’Regan D., Discrete Oscillation Theory, Hindawi Publ. Corp., New York, 2005. [3] Agarwal R.P., Grace S.R., Thandapani E., Oscillatory and nonoscillatory behavior of second order neutral delay difference equations, Math. Comp. Modelling, 24(1996), 5-11. [4] Gyori I., Ladas G., Oscillation Theory of Delay Differential Equations, Clarendon press, Oxford, 1991. [5] Saker S.H., Oscillation Theory of Delay Differential and Difference Equations: Second and Third Order, Verlag dr Muler, Germany, 2010. [6] Saker S.H., O’Regan D., New oscillation criteria for second order neutral functional dynamic equations via the generalized Riccati substitution, Commu. Nonlin. Sci. Numer. Simu., 16(2011), 423-434. [7] Saker S.H., O’Regan D., New oscillation criteria for second order neutral functional dynamic equations on time scales via Riccati substitution, Hiroshima Math. J., 42(2012), 77-98. [8] Sun Y.G., Saker S.H., Oscillation for second order nonlinear neutral delay difference equations, Appl. Math. Comput., 163(2005), 909-918. [9] Tang X.H., Liu Y., Oscillation for nonlinear delay difference equations, Tamkang J. Math., 32(2001), 275-280. [10] Thandapani E., Sundaram P., Graef J.R., Spikes P.W., Asymptotic properties of solutions of nonlinear second order neutral delay difference equations, Dynamic Sys. Appl., 4(1995), 125-136. [11] Thandapani E., Balasubramanian V., Graef J.R., Oscillation criteria for second order neutral difference equations with negative neutral term, Inter. J. Pure. Appl. Math., 87(2013), 283-292. [12] Thandapani E., Liu Z., Arul R., Raja P.S., Oscillation and asymptotic behavior of second order difference equations with nonlinear neutral terms, Appl.Math. E-Notes, 4(2004), 59-67. [13] Thandapani E., Mahalingam K., Necessary and sufficient conditions for oscillation of second order neutral delay difference equations, Tamkang J. Math., 34(2)(2003), 137-145. [14] Wang D.M., Xu Z.T., Oscillation of second order quasi-linear neutral delay difference equations, Acta Math. Appl. Sinica., 27(2011), 93-104. [15] Zafer A., Dahiya R.S., Oscillation of a neutral difference equations, Appl. Math. Lett., 6(1993), 71-74. [16] Zhou Z., Yu J., G. Lei, Oscillation for even order neutral difference equations, Korean J. Comput. Appl. Math., 7(2000), 601-610. Ethiraju Thandapani Ramanujan Institute for Advanced Study in Mathematics University of Madras Chennai - 600 005, India e-mail: [email protected]

Improved oscillation criteria for second order . . . Srinivasan Selvarangam Department of Mathematics Presidency College Chennai - 600 005, India e-mail: [email protected] Renu Rama Department of Mathematics Quaid-e-Millath Government College for Women Chennai - 600 002, India e-mail: [email protected] Mayakrishnan Madhan Department of Mathematics Presidency College Chennai - 600 005, India e-mail: [email protected] Received on 14.08.2015 and, in revised form, on 04.04.2016.

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