Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 87, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu OSCILLATORY BEHAVIOR FOR NONLINEAR HOMOGENEOUS NEUTRAL DIFFERENCE EQUATIONS OF SECOND ORDER WITH COEFFICIENT CHANGING SIGN AJIT KUMAR BHUYAN, LAXMI NARAYAN PADHY, RADHANATH RATH Abstract. In this article, we obtain sufficient conditions so that all solutions of the neutral difference equation ∆2 ( yn − pnL(yn−s) ) + qnG(yn−k) = 0, and all unbounded solutions of the neutral difference equation ∆2 ( yn − pnL(yn−s) ) + qnG(yn−k)− unH(yα(n)) = 0 are oscillatory, where ∆yn = yn+1 − yn, ∆2yn = ∆(∆yn). Different types of super linear and sub linear conditions are imposed on G to prevent the solution approaching zero or ±∞. 1. Introduction In this article, we obtain sufficient conditions so that all solutions of the neutral difference equation ∆2 ( yn − pnL(yn−s) ) + qnG(yn−k) = 0, n ≥ n0, (1.1) and all unbounded solutions of the neutral difference equation ∆2 ( yn − pnL(yn−s) ) + qnG(yn−k)− unH(yα(n)) = 0, n ≥ n0 (1.2) are oscillatory, where ∆ is the forward difference operator ∆yn = yn+1−yn, ∆2yn = ∆(∆yn), {qn} and {un} are sequences of real numbers with qn > 0, un ≥ 0, and G,H,L ∈ C(R,R). We assume that α(n) < n− 1 and it approaches ∞ as n→∞, and s, k are positive integers. Further, we assume that G(−x) = −G(x), H(−x) = −H(x), L(−x) = −L(x), ∀x ∈ R xG(x) > 0, xH(x) > 0, xL(x) > 0 ∀x > 0. (1.3) Some of the following assumptions are used later in this article. (A1) There exists δ > 0 such that for each x > 0, L(x) ≤ δx; (A2) qn > 0 and ∑∞ n=n0 qn =∞; (A3) ∑∞ n=n1 q∗n =∞, where q∗ = min{qn, qn−s}; (A4) lim infn→∞ qn > 0; (A5) G is non decreasing; 2010 Mathematics Subject Classification. 39A10, 39A12. Key words and phrases. Oscillatory solution; nonoscillatory solution; asymptotic behavior; difference equation. c©2020 Texas State University. Submitted June 3, 2020. Published August 12, 2020. 1 2 A. K. BHUYAN, L. N. PADHY, R. N. RATH EJDE-2020/87 (A6) ∑∞ n=n0 nun <∞; (A7) H is bounded. For the sequence {pn} we state the following conditions: 0 ≤ pn ≤ p, (1.4) 0 ≤ pn ≤ 1, (1.5) −p ≤ pn < 0, (1.6) pn changes sign and − p ≤ pn ≤ p, (1.7) 1 ≤ pn ≤ p, (1.8) −1 < −b ≤ pn ≤ 0, (1.9) where p and b are positive constants. As of now, many researchers all over the world are engaged to find necessary or sufficient conditions for oscillation or non oscillation for neutral difference equations, because of its important applications in different fields of science and technology. For the fundamentals and some recent results on the subject, one may go through the monograph [1, 5] and the research articles [2, 4, 12, 14] and the references cited there in. Sufficient conditions are found, in [3, 4, 7, 12, 13, 14, 15, 16], and more recently in [2, 3], so that every solutions of the non linear neutral difference equation ∆2 ( yn − pnyn−s ) + qnG(yn−k)− unH(yn−r) = fn, n ≥ n0, (1.10) (or of its particular case un ≡ 0, fn ≡ 0) oscillates or tends to zero or to ±∞ at ∞. The asymptotic behavior of the solution is probably due to the presence of the forcing term fn in (1.10). The objective of this work is to find sufficient conditions so that all solutions of (1.2) are oscillatory under different cases of pn > 0, pn < 0 or pn changing sign. For that, we had to prevent the bounded solutions of (1.2) from approaching zero by imposing a sub linear condition (4.4) or (4.1) on G as well as stop the unbounded solution of (1.2) from approaching ±∞ by imposing a super linear condition (3.5) or (3.2) on G. Then the results for (1.2) are applied to study the oscillatory behavior of the unbounded solutions of neutral difference equation ∆2 ( yn − pnL(yn−s) ) + vnG(yn−k) = 0, n ≥ n0, (1.11) where vn changes sign. Our results generalize and extend some results in [2, 11]. Let n0 be a fixed nonnegative integer. Let ρ = min { n0−s, n0−k, infn≥n0 {α(n)} } . By a solution of (1.2) we mean a real sequence {yn} which is defined for all integers n ≥ ρ and satisfies (1.2) for n ≥ n0. Clearly if the initial condition yn = an for ρ ≤ n ≤ n0 + 1, (1.12) is given then equation (1.2) has a unique solution satisfying (1.12). A non trivial solution {yn} of (1.2) is said to be oscillatory if for every positive integer n0 > 0, there exists n ≥ n0 such that ynyn+1 ≤ 0, otherwise {yn} is said to be non- oscillatory. 2. Some lemmas In this section, we present some lemmas to be applied in next section. EJDE-2020/87 OSCILLATION FOR SECOND ORDER NEUTRAL EQUATIONS 3 Lemma 2.1. [5, Theorem 7.6.1, page 184] Let {rn} be a non negative sequence of real numbers, k a positive integer and lim inf n→∞ n−1∑ i=n−k ri > ( k k + 1 )k+1 . (2.1) Then the following statements are true. (a) ∆xn+rnxn−k ≤ 0 has no eventually positive solutions, which implies ∆xn+ rnxn−k ≥ 0 has no eventually negative solutions. (b) ∆xn−rnxn+k ≥ 0 has no eventually positive solutions, which implies ∆xn− rnxn+k ≤ 0 has no eventually negative solutions. Lemma 2.2. Suppose that (A6) and (A7) hold, and yn is an eventually positive solution of (1.2). Then the sequence cn = − ∞∑ i=n (i− n+ 1)uiH(yα(i)) (2.2) satisfies lim n→∞ cn = 0, cn ≤ 0, ∆cn ≥ 0, (2.3) for n large enough, and ∆2cn = −unH(yα(n)). (2.4) Proof. Clearly, applying ∆2 to (2.2), we obtain ∆2cn = −unH(yα(n)). By (A6) and (A7), ∑∞ i=n iuiH(yα(i)) < ∞. Comparing this infinite series with (2.2), we show that {cn} converges absolutely to zero. The other statements follow easily. � Note that if yn is eventually negative, then cn ≥ 0 and ∆cn ≤ 0. Next, we prove an important lemma to be used later. Lemma 2.3. Let (A1), (A6), (A7) hold, yn be an eventually positive solution of (1.2), and cn be defined by (2.2). Then for the sequences zn = yn − pnL(yn−s), (2.5) wn = zn + cn (2.6) we have the following statements: (a) If (A2) and (A5) hold and pn satisfy (1.4), then either ∆wn < 0 for large n which implies lim n→∞ wn = −∞, (2.7) or ∆wn > 0 for large n which implies lim n→∞ wn = 0, (2.8) wn < 0, lim n→∞ ∆wn = 0. (2.9) (b) If in addition pδ ≤ 1, then only (2.8) and (2.9) hold. Proof. Suppose that yn is an eventually positive solution of (1.2). Then there exits an integer n1 ≥ n0 such that yn > 0, yn−s > 0, yn−k and yα(n) > 0 for n ≥ n1. Then setting cn, zn and wn as in (2.2), (2.5), (2.6), and using (1.2), (2.5), (2.6), and Lemma 2.2, we obtain ∆2wn = −qnG(yn−k) ≤ 0 for n > n1 . (2.10) 4 A. K. BHUYAN, L. N. PADHY, R. N. RATH EJDE-2020/87 Then ∆wn is decreasing. Hence ∆wn is monotonic and of single sign for n large enough. It follows that either ∆wn < 0 or ∆wn > 0. If ∆wn < 0, then wn is decreasing, and using that ∆wn is decreasing, we have lim n→∞ ∆wn = −∞. (2.11) If ∆wn > 0, then wn is increasing, and using that ∆wn is decreasing, we have lim n→∞ ∆wn = ζ (a finite number). (2.12) Let us prove part (a). If (2.11) holds then clearly (2.7) follows. If (2.12) holds then, summing (2.10) from n2 > n1 to ∞ we obtain ∞∑ n=n2 qnG(yn−k) <∞, (2.13) which by using (A2) yields lim inf n→∞ yn = 0. (2.14) Then we find a subsequence {ynk } such that ynk → 0 as k →∞. Now using (1.4), (A1) and Lemma 2.2 we obtain wnk < ynk + cnk → 0 as k →∞ (2.15) and wnk+s > −pδynk + cnk+s → 0 as k →∞. (2.16) Since wn is monotonic, it follows that limn→∞ wn = 0, which is (2.8). Then (2.9) follows from (2.8). The proof of part (a) is complete. To prove part (b) of the lemma, we show that (2.7) cannot happen; therefore (2.8) and (2.9) must occur. To obtain a contradiction, let us assume that limn→∞ wn = −∞. Note that from (2.6) and Lemma 2.2 we have lim n→∞ wn = lim n→∞ zn; (2.17) thus limn→∞ zn = −∞. This implies that for large n, there exists η > 0, however large, such that for n ≥ n3 implies zn < −η which implies by (A1) that yn < −η + pδyn−s < yn−s. Then yn is bounded. Consequently zn and wn are bounded, which contradicts (2.7). As a result, (2.7) cannot hold and so, (2.8) holds, which implies (2.9). The proof is complete. � Remark 2.4. If yn is an eventually negative solution of (1.2), then using (1.3), we observe that xn = −yn is a positive solution of (1.2). So that all the oscillation results for the positive solutions also apply to negative solutions. Lemma 2.5. Let yn be an eventually positive solution of (1.2), with wn as in (2.6). Then the following statements hold. (a) If (2.7) holds, then (2.10) implies ∆wn+1 + qnG(yn−k) ≤ 0, (2.18) which further implies ∆zn+1 + qnG(yn−k) ≤ 0 . (2.19) (b) If (2.8) holds, then (2.10) implies ∆wn − qnG(yn−k) ≥ 0. (2.20) EJDE-2020/87 OSCILLATION FOR SECOND ORDER NEUTRAL EQUATIONS 5 Proof. If (2.7) holds then ∆wn < 0 and ∆w2 n < 0. We write (2.10) as ∆wn+1 + qnG(yn−k) = ∆wn ≤ 0. Thus, (2.18) holds. From (2.6), it follows that ∆wn+1 = ∆zn+1+∆cn+1. Therefore (2.18) implies ∆zn+1 + qnG(yn−k) = −∆cn+1 ≤ 0 by Lemma 2.2. Hence (a) is proved. Let us prove (b). If (2.8) holds then (2.9) follows as a consequence, which implies wn < 0 and ∆wn > 0. Using (2.9), we write (2.10), as −∆wn + qnG(yn−k) = −∆wn+1 ≤ 0, which implies ∆wn − qnG(yn−k) = ∆wn+1 ≥ 0. This proves of (b), and completes the proof. � Lemma 2.6. Let (A1), (A3), (A6), (A7) hold. Assume that there exists λ > 0 such that for all x, y ∈ R with x+ y > 0, we have G(x) +G(y) ≥ λG(x+ y). (2.21) Further, we assume that G(x)G(y) ≥ G(xy) for all x, y > 0. (2.22) Let yn be an eventually positive solution of (1.2). Define cn, zn and wn as in (2.2), (2.5) and (2.6) respectively. If pn satisfies (1.6) or (1.7), then limn→∞ wn = 0. Consequently, (2.9) holds. Proof. Suppose yn is an eventually positive or eventually negative solution of (1.2) and pn satisfies (1.6). From (1.2), using (2.6), (2.5), (2.2) and Lemma 2.2, we obtain (2.10). This implies wn and ∆wn are monotonic and single sign. Hence, it follows that (2.17) holds and let limn→∞ zn = β. Clearly, zn > 0 by (1.6). This implies, β in (2.17), cannot be in negative. If β > 0, then then there exists a positive scalar χ such that zn > χ > 0 for large n. Clearly, ∆wn > 0, otherwise, β = −∞, a contradiction. Since ∆wn is decreasing, limn→∞∆wn exists. If x > y then using (1.3) and (2.21), we note that 0 < λG(x−y) ≤ G(x)+G(−y) = G(x)−G(y). Thus, (A5) holds, i.e; G is non decreasing. Then using (A5), (A1) and (1.6) in (2.5), we have zn ≤ yn + pδyn−s. (2.23) From (2.10), by using (A3), (2.21), (2.22) and (2.23) it follows that 0 ≥ ∆2wn + qnG(yn−k) +G(pδ)[∆2wn−s + qnG(yn−s−k)] ≥ ∆2wn +G(pδ)∆2wn−s + q∗n ( G(yn−k) +G(pδ)G(yn−s−k) ) ≥ ∆2wn +G(pδ)∆2wn−s + λq∗n ( G(zn−k) ) ≥ ∆2wn +G(pδ)∆2wn−s + λG(χ)q∗n (2.24) for n ≥ n2 > n1. Then taking summation in (2.24) from n2 to l − 1 and then letting l → ∞, we obtain a contradiction to (A3). Thus β = limn→∞ wn = 0, which implies (2.9). Suppose pn satisfies (1.7). If β > 0 then proceeding as above, we obtain a similar contradiction. If β < 0 then using (1.7), we have wn ≥ −pδyn−s + cn. This implies yn ≥ cn+s pδ − wn+s pδ . Then taking limit inferior on both sides of this inequality, we obtain lim inf n→∞ yn ≥ lim inf n→∞ cn+s pδ + lim inf n→∞ −wn+s pδ ≥ −β/pδ > 0. 6 A. K. BHUYAN, L. N. PADHY, R. N. RATH EJDE-2020/87 In the above we used limn→∞ cn = 0 and limn→∞ wn = β < 0. For −β/(3pδ) = ε > 0, we find n3 ≥ n2 such that n > n3 implies yn > 2ε. As cn → 0, from (2.6) it follows that pnL(yn−s) > yn + cn > ε > 0. This further implies pn+s > ε L(yn) ≥ ε δyn > 0, for n ≥ n3, which contradicts that pn changes sign. Thus β cannot be in negative, hence limn→∞ wn = β = 0. Consequently (2.9) holds. Similarly, if yn be an eventually negative solution of (1.2) then proceeding with substitution xn = −yn and taking note of Remark 2.4, it could be shown β = limn→∞ wn = 0 and the proof is complete. � Next we have the following remark, which would be helpful in proving results concerned with neutral equation (1.1). Remark 2.7. Lemmas 2.3, 2.5 and 2.6 hold for un ≡ 0. In that case cn = 0 and wn = zn. The following Lemmas follow from Lemmas 2.3, 2.5, and 2.6 as a consequence of the above remark. Lemma 2.8. Assume (A1) holds. Let yn be an eventually positive solution of (1.1), and zn be defined as in (2.5). Then ∆2zn = −qnG(yn−k) ≤ 0, (2.25) and the following statements hold. (a) If (A2), (A5) hold and pn satisfies (1.4), then either ∆wn < 0 for large n which implies lim n→∞ zn = −∞, (2.26) or ∆wn > 0 for large n which implies lim n→∞ zn = 0, (2.27) zn < 0, ∆zn > 0, lim n→∞ ∆zn = 0. (2.28) (b) If in addition δ ≤ 1 and if pn satisfy (1.5), then only (2.27) and (2.28) hold. Lemma 2.9. If yn is any eventually positive solution of (1.1), with zn as in (2.5), then the following statements hold. (a) If (2.26) holds then, (2.25) implies (2.19), i.e; ∆zn+1 + qnG(yn−k) ≤ 0. (b) If (2.27) holds, then (2.25) implies ∆zn − qnG(yn−k) ≥ 0. (2.29) Lemma 2.10. Let (A1), (A3), (2.21), and (2.22) hold, let yn be an eventually positive or eventually negative solution of (1.1), and let zn be as in (2.5). If pn satisfies (1.6) or (1.7) then limn→∞ zn = 0. Consequently, zn < 0, ∆zn > 0 for yn > 0 and zn > 0, ∆zn < 0 for yn < 0. EJDE-2020/87 OSCILLATION FOR SECOND ORDER NEUTRAL EQUATIONS 7 3. Main results part I In this section, we find sufficient conditions, so that, all unbounded solutions of (1.2) oscillate. Remark 3.1 ([6, Remark 4.8]). Assumption (A4) and the condition ∞∑ j=1 qnj =∞, where qnj is any subsequence of qn (3.1) are equivalent. Theorem 3.2. Let (A1), (A4)–(A7) hold, and s > k + 1, (1.4) be satisfied. If∣∣ ∫ ∞ a du G(u) ∣∣ <∞, ∀a ∈ R, (3.2) then every unbounded solution of (1.2) oscillates. Proof. To obtain a contradiction, let yn be an eventually positive solution of (1.2). Setting zn, wn and cn as in (2.5), (2.6) and (2.2) respectively, we obtain (2.10). Note that (A4) implies (A2). Hence, by Lemma 2.3(a), we observe that either (2.7) or (2.8) holds. First we consider the case when (2.7) holds. Using Lemma 2.5(a), we show that (2.10) implies (2.19). From (2.7), (2.17) and Lemma 2.2, it follows that limn→∞ zn = −∞, which implies ∆zn < 0 and zn < 0 for large n. If pn = 0 then zn = yn < 0, a contradiction. Hence pn > 0. From (2.5), we find yn−k ≥ −zn+s−k/(pδ). Using this in (2.19), we obtain ∆zn+1 + qnG( −zn+s−k pδ ) ≤ 0. (3.3) Note that −zn/(pδ) = vn implies ∆zn = −pδ∆vn. Then, substituting this expres- sion in the above, we obtain pδ∆vn+1 − qnG(vn+s−k) ≥ 0. Note that vn > 0, limn→∞ vn = ∞ and vn is increasing. Dividing both sides by G(vn+s−k), we obtain pδ ∆vn+1 G(vn+s−k) ≥ qn. (3.4) Then writing ∆vn+1 = ∫ vn+2 vn+1 dx, where vn+1 ≤ x ≤ vn+2, and using s− k ≥ 2, we obtain qn ≤ pδ ∫ vn+2 vn+1 dx G(x) . Summing n2 to l − 1, and then taking limit l→∞, we obtain ∞∑ n=n2 qn ≤ pδ ∫ ∞ vn2+1 dx G(x) <∞ , by (3.2), which contradicts (A2). Now we consider the case when (2.8) holds. Consequently, we obtain (2.9). Then taking summation in (2.10) from n2 to∞ we find (2.13). As yn is unbounded, we can find a subsequence {ynj } of {yn} which approaches∞ as j →∞. Then there exists η > 0 such that ynj > η for large j. Then ∑∞ j=n3 qnj G(ynj ) > G(η) ∑∞ j=n3 qnj → 8 A. K. BHUYAN, L. N. PADHY, R. N. RATH EJDE-2020/87 +∞ by (A4). This contradicts (2.13) which follows from (2.8). The proof for the case yn < 0, and unbounded is similar. Thus, the proof is complete. � Theorem 3.3. Let (A1), (A4)–(A7), and (3.2), s > k+ 1, (1.8) be satisfied. Then every unbounded solution of (1.2) oscillates. The proof of the above theorem is similar to that of theorem 3.2; we omit it. Theorem 3.4. Let (A1), (A4)–(A7) hold. Suppose pn satisfies (1.8), s > k + 1, and lim inf |x|→∞ G(x) x > γ > 0. (3.5) Suppose that lim inf n→∞ n−1∑ i=n−s+k+1 qi > pδ γ (s− k − 1 s− k )s−k (3.6) Then every unbounded solution of (1.2) oscillates. Proof. To obtain a contradiction, let yn be an eventually positive solution of (1.2). Proceeding as in the proof of theorem 3.2, we show that if (2.7) holds then ∆zn+1 + qnG(−zn+s−k pδ ) ≤ 0. Applying (3.5) to the above inequality, we obtain ∆zn+1 − γqn( zn+s−k pδ ) ≤ 0. Note that zn < 0 for large n. Substituting (zn+1/(pδ)) = vn and ∆zn+1 = pδ∆vn, in the above we obtain ∆vn − γ pδ qnvn+s−k−1 ≤ 0. Since s−k−1 > 0 this is an advanced difference inequality with a negative solution vn, which contradicts Lemma 2.1(b). Next consider the case that (2.8) holds. Proceeding as in the proof of theorem 3.2 we obtain a contradiction. The proof for the case yn < 0, and unbounded is similar. Thus, the proof is complete. � Remark 3.5. Condition (3.6) implies (A2). If (3.6) holds and (A2) fails, we have∑∞ n=n1 qn <∞ which implies pδ γ (s− k − 1 s− k )s−k < lim inf n→∞ n−1∑ i=n−s+k qi ≤ lim sup n→∞ ( n−1∑ i=n1 qi − n−s+k−1∑ i=n1 qi ) = 0, a contradiction. Theorem 3.6. Suppose (A1), (A4)–(A7) hold, and (1.5) and δ ≤ 1 are satisfied. Then every unbounded solution of (1.2) oscillates. Proof. Let yn be an unbounded and eventually positive solution of (1.2). Setting cn, zn and wn as in (2.2), (2.5) and (2.6) respectively, we obtain (2.10). By Lemma 2.3(b), we have limn→∞ wn = 0. Using this, unboundedness of yn and (A4), and proceeding as in the last part of the proof of theorem 3.2 we obtain a contradic- tion. A similar contradiction could be obtained if yn be an eventually negative and unbounded solution of (1.2). This completes the proof. � EJDE-2020/87 OSCILLATION FOR SECOND ORDER NEUTRAL EQUATIONS 9 Theorem 3.7. Suppose (A1), (A3), (A4), (A6), (A7) hold, (1.6) or (1.7), and (2.21) and (2.22) be satisfied. Then every unbounded solution of (1.2) oscillates. Proof. On the contrary suppose yn be an eventually positive and unbounded solu- tion of (1.2). Setting zn and wn as in (2.5) and (2.6), we obtain (2.10). Application of Lemma 2.6 yields β = limn→∞ wn = 0, wn < 0 and ∆wn > 0. Then using this, unboundedness of yn, (A4) and proceeding as in the last part of the proof of theorem 3.2 we obtain a contradiction. A similar contradiction could be obtain if yn be an eventually negative and unbounded solution of (1.2). This completes the proof. � Note that the condition lim inf n→∞ |xn| > 0 implies lim inf n→∞ |G(xn)| > 0. (3.7) is equivalent to lim inf u→±∞ G(u) 6= 0 (3.8) and note that (A5) implies (3.8). Consequently, we quote a particular case of [13, theorem 2.5, p.236 ] for fn ≡ 0 as our next result. Theorem 3.8. Suppose (A2), (A6), (A7) hold, and (1.9) and (3.8) are satisfied. If L(x) = x, then every non-oscillatory solution of (1.2) is bounded. Or equivalently every unbounded solution of (1.2) oscillates. 4. Main results part II In this section, we find sufficient conditions so that all solutions of (1.1) oscillate under condition (A2), which is less restrictive than (A4). Theorem 4.1. Suppose (A1), (A2), (A5) hold, and (1.4) and s < k are satisfied. If ∣∣ ∫ ±c 0 du G(u) ∣∣ <∞, for any finite positive c ∈ R, (4.1) Then every bounded solution of (1.1) oscillates. Proof. On the contrary let yn be a bounded eventually positive solution of (1.1). Setting zn as in (2.5) we obtain (2.25). Then zn is bounded and by Lemma 2.8(a), we find that (2.26) cannot hold because boundedness of zn, as a result, (2.27) holds. Then (2.28) follows as a consequence which implies that zn < 0 and increasing. If pn = 0, then zn = yn < 0, is a contradiction. Hence pn > 0. From (A1) and (1.4) it follows that yn−k ≥ zn+s−k −pδ . Hence, (2.25) with Lemma 2.9 (b) yields ∆zn − qnG(zn+s−k/(−pδ)) ≥ 0. Substituting vn = zn/(−pδ), which implies −pδ∆vn = ∆zn, we find that pδ∆vn + qnG(vn+s−k) ≤ 0, (4.2) which together with s < k and vn is positive and decreasing, implies pδ∆vn + qnG(vn) ≤ 0, Then dividing both sides of the above by G(vn) we obtain pδ∆vn G(vn) + qn ≤ 0. 10 A. K. BHUYAN, L. N. PADHY, R. N. RATH EJDE-2020/87 Then using ∆vn = ∫ vn+1 vn dx and taking vn+1 ≤ x ≤ vn we have pδ ∫ vn+1 vn dx G(vn) + qn ≤ 0, which implies, because of the nondecreasing character of G that pδ ∫ vn+1 vn dx G(x) + qn ≤ 0. Summing from n = n1 to l − 1 we obtain pδ ∫ vl vn1 dx G(x) + l−1∑ n1 qn ≤ 0. As l→∞, vl → 0, and so in the limiting case, we obtain ∞∑ n1 qn ≤ pδ ∫ vn1 0 dx G(x) <∞ by (4.1), which contradicts (A2). The proof for the case yn being eventually negative is similar and this completes the proof. � Theorem 4.2. Suppose (A1), (A2), (A5) hold, and (1.5), (4.1) , δ ≤ 1 and s < k are satisfied. Then every solution of (1.1) oscillates. Proof. On the contrary, let yn be an eventually positive solution of (1.1). Setting zn as in (2.5), we obtain (2.25). Then by Lemma 2.8(b), we find that (2.27) holds. Then (2.28), follows as a consequence and zn < 0. If pn = 0 then zn = yn < 0 which is a contradiction. Hence from (2.25), and Lemma 2.9(b) we obtain ∆zn − qnG(yn−k) ≥ 0. Using δ ≤ 1 and (1.5), we find yn−k ≥ zn+s−k −δ ≥ −zn+s−k. Therefore, ∆zn − qnG(−zn+s−k) ≥ 0. Substituting −zn = vn, which implies ∆zn = −∆vn, in the above, we obtain ∆vn + qnG(vn+s−k) ≤ 0. (4.3) Then further using s < k and vn > 0 and decreasing, we obtain ∆vn + qnG(vn) ≤ 0. Dividing both sides of the above inequality, by G(vn), we obtain ∆vn G(vn) + qn ≤ 0. Taking vn+1 ≤ v ≤ vn and using ∆vn = ∫ vn+1 vn dv, we proceed as in the proof of theorem 4.1 to obtain ∞∑ n=n1 qn ≤ ∫ vn1 0 dv G(v) <∞ by (4.1), which contradicts (A2). The proof for the case when yn is eventually negative is similar. � EJDE-2020/87 OSCILLATION FOR SECOND ORDER NEUTRAL EQUATIONS 11 Theorem 4.3. Suppose (A1), (A2), (A5) hold, and (1.5), δ ≤ 1 and s < k are satisfied. If lim inf |x|→0 G(x) x > γ > 0. (4.4) and lim inf n→∞ n−1∑ i=n−k+s qi > 1 γ ( k − s k − s+ 1 )k−s+1 , (4.5) then every solution of (1.1) oscillates. Proof. As s < k and 0 < δ ≤ 1 proceeding as in the proof of the theorem 4.2, we obtain the first order delay difference inequality (4.3), which by (4.4), yields ∆vn + γqnvn+s−k ≤ 0 which has a positive solution. This, contradicts Lemma 2.1(a). The proof for the case when yn is eventually negative is similar. This completes the proof. � Theorem 4.4. Suppose (A1), (A3) hold, and (1.6), (2.21) and (2.22) are satisfied. Then every solution of (1.1) oscillates. Proof. On the contrary, suppose yn be an eventually positive solution of (1.1). Setting zn as in (2.5), we obtain (2.25). Then applying Lemma 2.10, we obtain β = limn→∞ zn = 0, which implies zn < 0, a contradiction because zn ≥ 0 by (1.6). The proof for the case when yn is eventually negative, is similar and thus, the proof is complete. � Theorem 4.5. Suppose (A1), (A3) hold, and (1.7), (2.21) and (2.22) are satisfied. Then every solution of (1.1) oscillates. Proof. On the contrary, assume yn be an eventually positive solution of (1.1). Set- ting zn as in (2.5), we obtain (2.25). Then application of Lemma 2.10 yields β = limn→∞ zn = 0. Consequently ∆zn > 0 and zn < 0. Again, this would lead to pn > yn L(yn−s) > 0 for large n, which is a contradiction, because pn changes sign. For the proof of the case, when yn is eventually negative, we may proceed with xn = −yn and complete the proof. � Theorem 4.6. Suppose (A2), (A5) hold, L(x) = x, and pn satisfies (1.9). Then every solution of (1.1) oscillates. Proof. On the contrary assume yn be an eventually positive solution of (1.1). Set- ting zn as in (2.5), we obtain (2.25). Note that (A5) implies (3.8). Then applying Theorem 3.8 for un = 0, we show that yn is bounded, which implies zn is bounded. As zn is monotonic, limn→∞ zn = β ∈ R. Summing (2.25) from n1 to ∞, we obtain (2.13), which implies lim infn→∞ yn = 0. By [9, Lemma 2.1], we have limn→∞ zn = 0. As a consequence (2.28) holds, which implies zn < 0. However, by (1.9) we have zn > 0, a contradiction. The proof for the case yn < 0 is similar. Thus proof is complete. � Next, we give some examples to illustrate the results. Example 4.7. Consider the neutral difference equation ∆2 ( yn − pyn−4 ) + 18 ( 1 22n + 1− p 16 ) yn−1 − ( 72 22n + 9 2n ) H(yn−3) = 0 (4.6) 12 A. K. BHUYAN, L. N. PADHY, R. N. RATH EJDE-2020/87 where |p| < 16. Suppose p = ±2 or p = ±1/2. Here, s = 4, k = 1, qn = 18 ( 1 22n + 1 − p 16 ) , un = ( 72 22n + 9 2n ) and H(u) = u/(1 + |u|). Clearly, the neutral difference equation (4.6) satisfies all the conditions of Theorems 3.4, 3.6, 3.7 and 3.8. As a result, it has an unbounded solution yn = 2n(−1)n, which is oscillatory. Example 4.8. Consider the neutral difference equation ∆2 ( yn − byn−2 ) + ( 9(1− b/4)(2n + 128) + 2−2n ) G(yn−7)− unH(yn−7) = 0 (4.7) where |b| < 4 is suitably selected constant. Suppose b = ±1/2. Here, s = 2, k = 7, un = 2+2n−7 22n(1+2n−7) , qn = ( 9(1 − b/4)(2n + 128) + 2−2n ) , G(u) = u/(1 + |u|) and H(u) = u/(2 + |u|). Clearly, the neutral difference equation (4.7) satisfies all the conditions of Theorems 3.6 and 3.8. Consequently, it has an unbounded solution yn = 2n(−1)n, which is oscillatory. Example 4.9. Consider the neutral difference equation ∆2 ( yn − pL(yn−3) ) + [ 4(1 + a+ p) (1 + a)(1 + γ) ] G(yn−5) = 0 (4.8) where p > 0 is any scalar. Here L(x) = x/(a + |x|) and G(u) = u(γ + |u|) where a and γ are positive constants. This neutral equation satisfies all the conditions of Theorems 4.3. As such, it has a solution yn = (−1)n, which is oscillatory. Example 4.10. Consider the neutral difference equation ∆2 ( yn + p(−1)nL(yn−5) ) + 4y 1/3 n−1 = 0 (4.9) where p > 0 is any scalar. Here pn changes sign and satisfies (1.7). Further, L(x) = x/(a + |x|). This neutral equation satisfies all the conditions of Theorem 4.5. Hence, it has a solution yn = (−1)3n, which is oscillatory. It seems, no result in the literature, could be applied to the neutral equations (4.8)–(4.9) given in the examples above, because of the non linear term inside ∆2, 5. Application to neutral difference equations with oscillating coefficients In this section, we find sufficient conditions so that every unbounded solution of the second order neutral difference equation (1.11) oscillates, where vn is allowed to change sign. Let v+n = max{vn, 0} and v−n = max{−vn, 0}. Then vn = v+n − v−n and the equation (1.11) can be written as ∆2 [ yn − pnL(yn−s) ] + v+nG(yn−k)− v−nG(yn−k) = 0. (5.1) Now we proceed as in the previous section by setting qn = v+n , un = v−n and H(x) = G(x). Assumptions (A4), (A3) and (A6) become ∞∑ n=n0 v+n =∞. (5.2) lim inf n→∞ v+n > 0. (5.3) ∞∑ n=n0 V + n =∞ where V + n = min{v+n , v+n−s}. (5.4) EJDE-2020/87 OSCILLATION FOR SECOND ORDER NEUTRAL EQUATIONS 13 ∞∑ n=n0 nv−n <∞. (5.5) respectively, which are feasible conditions. Therefore, the study of (1.11) reduces to the study of (5.1), which could be achieved, by following the study of (1.2) for different results in section 3. The following results for (1.11) (with vn changing sign) follow from Theorems 3.6, 3.7 and 3.8, by replacing qn by v+n , un by v−n and H by G. Theorem 5.1. Suppose that (A1) holds with δ ≤ 1, (A5) holds, pn satisfies (1.5), G is bounded, and (5.3) and (5.5) are satisfied. Then every unbounded solution of (1.11) (with vn changing sign) oscillates. Theorem 5.2. Suppose pn satisfies (1.6) or (1.7), G is bounded, (A1) holds, and (2.21), (2.22), (5.3), (5.4), and (5.5) are satisfied. Then every unbounded solution of (1.11) (with vn changing sign) oscillates. Theorem 5.3. Suppose pn satisfy (1.9), G is bounded, (3.8), (5.2) and (5.5) are satisfied. If L(x) = x, then every unbounded solution of (1.11) oscillates. 6. Final comments Before we close this article, we would like to give our concluding remarks, which may be helpful for further research. In this paper, some oscillatory results are obtained for the neutral difference equation (1.2) and (1.1) by imposing different super linear conditions like (3.5) or (3.2), and sublinear conditions like (4.4) or (4.1) on G. Note that the super linear condition (3.5) and the sub linear condition (4.4) on G include their corresponding linear case G(x) = x. Authors while studying the oscillatory and asymptotic behavior of (1.2) or (1.1), very often find difficulty in tackling, the case of pn ≥ 1, i.e; when (1.8) or (1.4) are satisfied. That is why, the results [10, Theorems 2.6 and 2.7] appear to be wrong, as the neutral equation ∆2(yn − 4yn−1) + 4(n+1)/3y 1/3 n−2 = 0 satisfies all the conditions of the theorems, but, it admits a non oscillatory solution yn = 2n, which tends to ∞, as n → ∞, contradicting the theorems. With the super linear G with (3.2) or (3.5), we proved in Theorems 3.2 and 3.4 that (A4) is sufficient for all unbounded solutions of (3.2) to be oscillatory which is more restrictive than (A2). Hence, one may extend this study to improve the results (Theorems 3.2 and 3.4)by attempting to answer the following problem. Problem 6.1. Suppose that L(x) = x, or (A1) holds, and 1 ≤ pn ≤ p. Assuming (A2), (A5) and (3.2) can we prove that every unbounded solution of (1.1) oscillates? Acknowledgment. The authors are obliged and thankful to the referee and the editor for their various suggestions to improve the presentation of this article. References [1] R. P. Agarwal; Difference Equations and Inequalities, Marcel Dekker, NewYork, 2000. [2] Chittaranjan Behera, Radhanath Rath, Prayag Prasad Mishra; Oscillation for second order neutral difference equations with variable delays, International J. of Mathematical, Engineer- ing and Management Sciences, 5, (2020), no. 4, 663–681. 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[14] Radhanath Rath, Chittaranjan Behera; Oscillatory and asymptotic behaviour of a first order neutral equation of discrete type with variable several delay under ∆ sign. International J. of Math. and Mathematical Sciences, Vol 2018, Article ID 4586176, https//doi.org10.1155/20184586176 [15] A. K. Tripathy, S. Panigrahi; Oscillation in non linear neutral difference equations with positive and negative coefficients,Int. J. of Difference equations 5(2010), no. 2, 251–265. [16] M. K. Yildiz; Oscillation of a class of nonlinear difference equations of second order with oscillating coefficients, Konuralp J. Math., 3(2015), no. 2, 211-218. Ajit Kumar Bhuyan Dept. Of Mathematics, Sai international School, Bhubaneswar, Odisha, India Email address: ajitbhuyan13@gmail.com Laxmi Narayan Padhy Dept. of Math and Computer Science, Konark Institute of Science and Technology, Bhubaneswar, Odisha, India Email address: padhyln@gmail.com Radhanath Rath (corresponding author) VSSUT Burla, 768018. Retired Principalhallikote Autonomous College, Berhampur, 760001. Center Point Apartment, Flat A-203, Shailashree Vihar ph-7, 751024, Bhubaneswar, Odisha, India Email address: radhanathmath@yahoo.co.in 1. Introduction 2. Some lemmas 3. Main results part I 4. Main results part II 5. Application to neutral difference equations with oscillating coefficients 6. Final comments Acknowledgment References