2000 Mathematics Subject Classification: 39A10.The oscillatory and nonoscillatory behaviour of solutions of the second order quasi linear neutral delay difference equation
Δ(an | Δ(xn+pnxn-τ)|α-1 Δ(xn+pnxn-τ) + qnf(xn-σ)g(Δxn) = 0
where n ∈ N(n0), α > 0, τ, σ are fixed non negative integers, {an}, {pn}, {qn} 
are real sequences and f and g real valued continuous functions are studied. 
Our results generalize and improve some known results of neutral delay difference equations

Pandian, S.

Revathi, T.

Thandapani, E.

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 36 (2010), 255–264OSCILLATION CRITERIA OF SECOND-ORDERQUASI-LINEAR NEUTRAL DELAY DIFFERENCEEQUATIONSE. Thandapani, S. Pandian, T. RevathiCommunicated by I. D. IlievAbstract. The oscillatory and nonoscillatory behaviour of solutions of thesecond order quasi linear neutral delay difference equation∆(an|∆(xn + pnxn−τ )|α−1∆(xn + pnxn−τ ) + qnf(xn−σ)g(∆xn) = 0where n ∈ N(n0), α > 0, τ, σ are fixed non negative integers, {an}, {pn},{qn} are real sequences and f and g real valued continuous functions arestudied. Our results generalize and improve some known results of neutraldelay difference equations.1. Introduction. In this paper, we consider the second order quasilinear neutral delay difference equation of the form(1) ∆(an|∆(xn + pnxn−τ )|α−1∆(xn + pnxn−τ ) + qnf(xn−σ)g(∆xn) = 02010 Mathematics Subject Classification: 39A10.Key words: Oscillation, quasi-linear, neutral type, delay difference equations.256 E. Thandapani, S. Pandian, T. Revathiwhere n ∈ N(n0) = {n0, n0 + 1, . . . } n0 a non negative integer, ∆ is the forwarddifference operator defined by ∆xn = xn+1 − xn, α > 0, τ , σ are fixed nonnegative integers.Throughout this paper we assume that the following conditions hold:(C1) {an} is a positive real sequence and {qn} is a non negative real sequencewith qn is not identically zero for large n,(C2) {pn} is a real sequence,(C3) g : R→ R such that g(u) ≥ c > 0 for u 6= 0,(C4) f : R → R is continuous and uf(u) > 0 for u 6= 0 and f(u) − f(v) =h(u, v)(u − v) for all u 6= 0 and h is a non negative function.Let m = max{τ, σ}. By a solution of equation (1) we mean a real sequence {xn}which is defined for all n ≥ n0 −m and satisfies (1) for large n ≥ n0. A solution{xn} of (1) is said to be nonoscillatory if all the terms xn are eventually of fixedsign, otherwise the solution {xn} is called oscillatory. A nonoscillatory solution{xn} of (1) is said to be weakly oscillatory if {∆xn} changes sign for arbitrarilylarge values of n.In this paper, we investigate oscillatory and asymptotic behaviour of nonoscillatory solution of equation (1), when qn is either non negative or changingsign for large n.Let S denote the set of all nontrivial solutions of (1). With respect totheir asymptotic nature, the nonoscillatory solutions of equation (1) may be apriori divided into the following classes:M+ = {{xn} ∈ S : there exists an integer N such thatxn∆xn ≥ 0, ∀n ≥ N}M− = {{xn} ∈ S : there exists an integer N such thatxn∆xn ≤ 0, ∀n ≥ N}OS = {{xn} ∈ S : there exists an integer N such thatxnxn+1 ≤ 0, ∀n ≥ N}WOS = {{xn} ∈ S : {xn} is nonoscillatory for every N ∃ n ≥ Nsuch that ∆xn∆xn+1 ≤ 0}In [1] and [3] the authors studied the oscillatory and asymptotic behaviourof nonoscillatory solution of equation (1) when g(u) ≡ 1, α = 1 and pn eitherOscillation Criteria of Second-order Quasi-linear Difference Equations 257identically zero or pn = p via the above said classification. Hence the resultsobtained in this paper generalize that in [3].2. Main results. Define(2) zn = xn + pnxn−τFirst we examine the non-existence of solutions of equation (1) in theclass M+.Theorem 2.1. With respect to difference equation (1), assume that(3) −1 < −h ≤ pn(4) qn is non negative and limn→∞supn−1∑s=n0qs =∞(5) and∞∑s=n01a1/αn=∞hold. Then for equation (1) we have M+ = φ.P r o o f. Suppose that equation (1) has a solution {xn} ∈ M+. Withoutloss of generality we can assume that there exists an integer n1 ≥ n0 such thatxn > 0, ∆xn ≥ 0, xn−m > 0, ∆xn−m ≥ 0 for all n ≥ n1 = n0 +m (the proof issimilar if xn < 0, ∆xn ≤ 0 for all large n). If pn ≥ 0, we have zn ≥ xn > 0. If−1 < −h ≤ pn < 0 we claim that zn > 0, for all n ≥ n1. Otherwise, there is an2 ≥ n1, such that zn2 ≤ 0, thenxn2 ≤ hxn2−τand thereforexn2+τ ≤ hxn2by inductionxn2+2τ ≤ hxn2+τ ≤ h2xn2258 E. Thandapani, S. Pandian, T. Revathiwe obtainxn2+jτ ≤ hjxn2implying that xn2+jτ ≤ 0 for large n, which contradicts the fact that xn > 0,∆xn ≥ 0 for n ≥ n1.Hence zn > 0 for all n ≥ n1.Now from the equation (1), it follows that(6) ∆(an|∆zn|α−1∆zn) = −qnf(xn−σ)g(∆xn) ≤ 0 n ≥ n1we claim that ∆zn ≥ 0 for n ≥ n1.Otherwise, there exists an integer n3 ≥ n1 such that ∆zn3 < 0.It follows from (6) thatzn ≤ zn3 − (−an3|∆zn3 |α−1∆zn3)1/αn−1∑s=n31/a1/αn n ≥ n3By using (5), we have limn→∞zn = −∞ which contradicts the fact that zn > 0 forn ≥ n1. So(7) ∆zn ≥ 0 for n ≥ n1Summing equation (6) and using (C1) - (C4)an(∆zn)αf(xn−σ)≤an1(∆zn1)αf(xn1−σ−n−1∑s=n1an(∆zn)αh(xs+1−σ, xs−σ)∆xs−σf(xs+1−σf(xs−σ)− cn−1∑s=n1qs.≤an1(∆zn1)αf(xn1−σ)− cn−1∑s=n1qs n ≥ n1From (4) we obtainlimn→∞infan(∆zn)αf(xn−σ)= −∞which contradicts (7). The proof is complete. Theorem 2.2. With respect to the difference equation (1), assume that(8) {pn} is non negative and nondecreasing for all n ∈ N(n0)Oscillation Criteria of Second-order Quasi-linear Difference Equations 259(9) limn→∞supn−1∑s=n0qs =∞hold. Then for equation (1) we have M+ = φ.P r o o f. Suppose that equation (1) has a solution {xn} ∈ M+. There isno loss of generality in assuming that there exists n1 ≥ n0 such that xn > 0,∆xn ≥ 0, xn−m > 0, ∆xn−m ≥ 0 for all n ≥ n1 = n0 +m. The proof is similar ifxn < 0, ∆xn ≤ 0 for all large n.By condition (8) we see that(10) zn > 0,∆zn ≥ 0, n ≥ n1.Similar to the proof of Theorem 2.1, we obtainlimn→∞infan(∆zn)αf(xn−σ)= −∞which contradicts (10). The proof is complete. Now we examine existence of solutions of equation (1) in the class M−.Theorem 2.3. Assume that τ ≤ σ. If the function1(f(u))1/αis locallyintegrable on (0, α) and (−α, 0) for all α > 0 and(11)∫ α0du(f(u))1/α<∞,∫0−αdu(f(u))1/α> −∞(12) f is sub multiplicative;(13) {pn} is non negative and nonincreasing for all n ∈ N(n0)(14) limn→∞supn∑s=N1(asf(1 + ps))1/α(s−1∑t=N(qτ )1/α)=∞ N ∈ N(n0)hold, then for equation (1) we have M− = φ.260 E. Thandapani, S. Pandian, T. RevathiP r o o f. Suppose that equation (1) has a solution {xn} ∈ M−. Thenthere is no loss of generality in assuming that there exists n1 ≥ n0 such thatxn > 0, ∆xn ≤ 0 xn−m > 0, ∆xn−m ≤ 0 for all n ≥ n1. The proof is similar ifxn < 0 ∆xn ≥ 0 for all large n. Then from (2) by using (13) we see thatzn > 0, ∆zn ≤ 0 n ≥ n1.Summing (6), using summation by parts from n1 to n− 1 and by (C3) and (C4)n−1∑s=n1∆[an(∆zn)α]f(xs−σ)≤ −cn−1∑s=n1qs n ≥ n1an(∆zn)αf(xn−σ)−an1(∆zn1)αf(xn1−σ)+n−1∑s=n1as(∆zs)αh(xs−σ, xs+1−σ)∆xs−σf(xs+1−σ)f(xs−σ)≤ −cn−1∑s=n1qs.an(∆zn)αf(xn−σ)≤an1(∆zn1)αf(xn1−σ)−n−1∑s=n1as(∆zs)αh(xs−σ, xs+1−σ∆xs−σf(xs+1−σ)f(xs−σ)− cn−1∑s=n1qs.≤ −cn−1∑s=n1qs n ≥ n1.By (C1)(15) −(∆zn)αf(xn−σ)≥ c/ann−1∑s=n1qs for n ≥ n1Since {xn} is non increasing and τ ≤ σ we have zn ≤ (1 + pn)xn−σ and hence byusing (12)(16) f(zn) ≤ f(1 + pn)f(xn−σ)Combining (15) and (16)−(∆zn)αf(zn)≥canf(1 + pn)n−1∑s=n1qs n ≥ n1.Oscillation Criteria of Second-order Quasi-linear Difference Equations 261Then we have−(∆zn)(f(zn))1/α≥ c1/α( ∑n−1s=n1qsanf(1 + pn))1/α, n ≥ n1.Using (by parts) summing the last inequality from n1 to n− 1(17)n−1∑s=n1−(∆zs)α(f(zs))1/α≥ c1/αn−1∑s=n11(asf(1 + ps))1/α(s−1∑t=n1qt)1/αn ≥ n1For t+ 1 ≤ zn ≤ t ∫ tt+1dtf(zt)1/α≥ −∆zsf(zs)1/αhence(18)∫ zn10dtf(zt)1/α≥n−1∑s=n1−∆zsf(zs)1/αCombining (17) and (18) and taking limit sup we get a contradiction to (11) and(14).The proof is complete. Next we establish sufficient conditions under which equation (1) has noweakly oscillatory solution.Theorem 2.4. Let qn ≥ 0 for all n ≥ n0. If(19) pn ≡ p ≥ 0 for n ∈ N(n0).Then for equation (1), WOS = φ.P r o o f. Let {xn} be a weakly oscillatory solution of (1). Without lossof generality we assume that there exists an integer n1 ≥ n0 such that xn > 0,xn−m > 0 for n ≥ n1.(The proof is similar if xn < 0 for all large n)Using (2) and (19), zn > 0∆zn = ∆xn + p∆xn−τ∆zn+1 = ∆xn+1 + p∆xn−τ+1∆zn∆zn+1 = ∆xn∆xn+1 + p(∆xn∆xn−τ+1 +∆xn+1∆xn−τ )+p2∆xn−τ∆xn−τ+1≤ 0.262 E. Thandapani, S. Pandian, T. RevathiHence zn > 0 and weakly oscillatory. In equation (1) putting Fn = an|∆zn|α−1∆znfor n ≥ n0 we get ∆Fn = −qnf(xn−σ)g(∆xn) ≤ 0 which implies {Fn} is non-increasing hence Fn is eventually of one sign which gives a contradiction, since{Fn} an oscillatory sequence. Theorem 2.5. Assume conditions (5), (9), (19) hold. Then every solu-tion of equation (1) is either oscillatory or weakly oscillatory.P r o o f. From Theorem 2.2 it follows that for equation (1) M+ = φ. Inorder to complete the proof it suffices to show that for (1) M− = φ.Suppose that {xn} ∈ M−. Then as earlier we can assume that xn > 0,∆xn ≤ 0, xn−m > 0, ∆xn−m ≤ 0 for all n ≥ n1 the proof is similar if xn < 0,∆xn ≥ 0 for large n.Then by using (2) and (19) we see thatzn > 0 ∆zn ≤ 0 n ≥ n1Let wn = an(∆zn)α, so that wn ≤ 0 for n ≥ n1. From (1)∆wn ≤ −cqnf(xn−σ)wn ≤ wn1 − cn−1∑s=n1qsf(xs−σ)using Abel’s transformation. (1, p. 35)wn ≤ wn1 − cf(xn−σ)n−1∑s=n1qs −n−1∑s=n1∆f(xs−σ)(s∑t=n1qt)From the above relationwn ≤ wn1(∆zn)α ≤wn1an< 0 for n ≥ n1zn − zn1 ≤ w1/αn1n−1∑s=n11a1/αs→ −∞ as n→∞which contradicts zn > 0. The proof is complete. From Theorems 2.4 and 2.5 we can easily get the following theorem.Oscillation Criteria of Second-order Quasi-linear Difference Equations 263Theorem 2.6. Let qn ≥ 0 for all n ≥ n0 and conditions (5), (9), (19)hold. Then every solution of equation (1) is oscillatory.Now we study the asymptotic behaviour of the eventually monotone so-lution of equation (1).Theorem 2.7. Assume conditions (12), (13), (14) are satisfied. Thenfor every solution xn ∈M− we have limn→∞xn = 0.P r o o f. The assertion follows from the same argument as given in theproof of Theorem 2.3. Taking into account (18) which implies limn→∞zn = 0,together with zn ≥ xn for all n ≥M we have limn→∞xn = 0.This completes the proof. Example 2.1. Consider the quasi linear neutral delay difference equation∆[1n2|∆xn + 2xn−1|α−1∆(xn + 2xn−1)]+nn− 2xn−2(1 + (∆xn)2) = 0 n ≥ 3.(E1)τ = 1 σ = 2 f(y) = y g(y) = 1 + y2 ≥ 1pn = 2 > 0 an = 1/n2 > 0 qn = n/n− 2All conditions of Theorem 2.6 are satisfied and hence (E1) is oscillatory by The-orem 2.6.REFERENCES[1] E. Thandapani, S. Pandian. Oscillatory and asymptotic behaviour ofa second order functional difference equation. Indian J. Math. 37 (1995),221–233.[2] E. Thandapani, S. Pandian, R. K. Balasubramaniam. Asymptotic be-haviour of solutions of a class of second order quasilinear difference equations.Kyungpook Math. J. 44 (2004), 173–185.264 E. Thandapani, S. Pandian, T. Revathi[3] E. Thandapani, M. Maria Susai Manuel. Summable criteria for a clas-sification of solutions of linear difference equations. Indian J. Pure Appl.Math. 28 (1997), 53–62.E. ThandapaniRamanujan Institute for Advanced Study in MathematicsUniversity of Madras, ChepaukChennai – 600 005, IndiaS. PandianInstitute of Advanced Study in EducationSaidapetChennai – 600 015, IndiaT. RevathiDepartment of MathematicsQueen Mary’s CollegeChennai – 600 004, Indiae-mail: kalyanrevathi@yahoo.com Received August 28, 2010

Oscillation Criteria of Second-Order Quasi-Linear Neutral Delay Difference Equations

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Oscillation Criteria of Second-Order Quasi-Linear Neutral Delay Difference Equations

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