Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 82 https://internationalpubls.com Qualitative Behavior Of Second Order Difference Equation with Non Positive Neutral Term S. Kaleeswari 1, J. Gowri 2 1,2 Department of Mathematics, Nallamuthu Gounder Mahalingam College, Pollachi, Tamilnadu, India. e-mail: kaleesdesika@gmail.com, e-mail: pranithagowri86@gmail.com Article History: Received: 15-09-2023 Revised: 24-10-2023 Accepted: 18-11-2023 Abstract: The oscillation of second order difference equations with a nonlinear nonpositive neutral component is the subject of this study. We come up with a sufficient condition that guarantees that all solutions to the examined equation are either oscillatory or going towards zero. Through examples, the improvement of our primary findings is demonstrated. Keywords: Oscillatory, neutral term, non positive, second order. MSC: 39A10. 1. Introduction In this article, we study some oscillatory manners of a second order non linear non positive neutral delay difference equation of the form βˆ†(π‘Ÿ(β„˜)βˆ†(z(β„˜) βˆ’ 𝑝(β„˜)𝑧𝛾1(𝜏1(β„˜)))) + π‘ž(β„˜)𝑧𝛾2(𝜎1(β„˜)), β„˜ β‰₯ β„˜0 > 0. (1) subject to the restrictions outlined below : (R1) 𝛾2 and 0 < 𝛾1 ≀ 1 are ratio of odd positive integers; (R2) {π‘Ÿ(β„˜)}, {π‘ž(β„˜)} and {𝑝(β„˜)} are positive real sequences such that 0 < 𝑝(β„˜) ≀ 𝑝 < 1, βˆ€ β„˜ β‰₯ β„˜0 and (R3) 𝜎1 and 𝜏1 are positive integers with 𝜏1β„˜) ≀ β„˜, Ξ”πœ1(β„˜) > 0, 𝜎1(β„˜) ≀ β„˜, Ξ”πœŽ1(β„˜) > 0, limβ„˜β†’βˆž β€Šπœ1(β„˜) = limβ„˜β†’βˆž β€ŠπœŽ1(β„˜) = ∞. A real sequence {z(β„˜)} is said to be a solution of (1) if it is defined for all β„˜ β‰₯ β„˜_0. A nontrivial solution of (1) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, the solution is said to be non oscillatory. An equation is oscilltory if all its solutions oscillate. Since neutral type equations are prevalent in the study of economics, mathematical biology, and many other fields of mathematics, determining oscillation conditions for these equations has garnered a lot of attention in recent years. (see for example [1] βˆ’ [9] ) . To the best of our knowledge, there are no results in the literature that guarantee that all solutions for the second order difference equation are just oscillatory. This conclusion is drawn from a review of the literature. All results ( [10] βˆ’ [14]) established for neutral type difference equations are guaranteed that every solution is either oscillatory or tends to zero monotonically. In order to define conditions for the oscillation of all solutions under the following condition, the authors considered (1) with 𝑝(β„˜) < 0, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 83 https://internationalpubls.com βˆ‘ β€Šβˆž 𝑖=β„˜0 1 π‘Ÿ(𝑖) = ∞ (2) In this article we arrive at some new oscillation results. 2. Oscillatory Results We begin with the following Lemmas, which are critical in establishing our key results. We represent 𝑠(β„˜) = 𝑧(β„˜) βˆ’ 𝑝(β„˜)𝑧𝛾1(𝜏1(β„˜)), π‘Œ(β„˜) = βˆ‘ β€Š β„˜βˆ’1 𝑖=β„˜1 β€Š 1 π‘Ÿ(𝑖) , for every β„˜ β‰₯ β„˜1 β‰₯ β„˜0. Lemma 2.1. Let (2) hold and if z is a positive solution of (1), then the corresponding function s meets one of the following two requirements : (I) 𝑠(β„˜) > 0, Δ𝑠(β„˜) > 0 and Ξ”(π‘Ÿ(β„˜)Δ𝑠(β„˜)) < 0; (II) 𝑠(β„˜) < 0, Δ𝑠(β„˜) > 0 and Ξ”(π‘Ÿ(β„˜)Δ𝑠(β„˜)) < 0, for all β„˜ β‰₯ β„˜1, where β„˜1 β‰₯ β„˜0 is sufficiently large. Proof. It is sufficient to state and prove the results for positive solutions. Because the proof of the other case is same. Suppose that 𝑧(β„˜) > 0, 𝑧(𝜏1(β„˜)) > 0 and 𝑧(𝜎1(β„˜)) > 0 for every β„˜ β‰₯ β„˜1 for some β„˜1 β‰₯ β„˜0. By the representation of 𝑠(β„˜) and (1), we get Ξ”(π‘Ÿ(β„˜)Δ𝑠(β„˜)) = βˆ’π‘ž(β„˜)𝑧𝛾2(𝜎1(β„˜)) < 0. (3) Hence π‘Ÿ(β„˜)(Δ𝑠(β„˜)) is decreasing and of one sign for large β„˜, that means, βˆƒ β„˜2 β‰₯ β„˜1 and Δ𝑠(β„˜) > 0 (or) Δ𝑠(β„˜) < 0 for all β„˜ β‰₯ β„˜2. If Δ𝑠(β„˜) < 0 for β„˜ β‰₯ β„˜2, then π‘Ÿ(β„˜)(Δ𝑠(β„˜) ≀ βˆ’π‘‘1 for β„˜ β‰₯ β„˜2 where 𝑑1 = βˆ’π‘Ÿ(β„˜2)Δ𝑠(β„˜2) > 0. Then, we obtain 𝑠(β„˜) ≀ 𝑠(β„˜2) βˆ’ 𝑑1 βˆ‘ β€Š β„˜βˆ’1 𝑖=β„˜2 1 π‘Ÿ(𝑖) . By the condition (2), the above inequality implies limβ„˜β†’βˆž β€Šπ‘ (β„˜) = βˆ’βˆž. We will now examine each of the next two situations separately. Case (I): If z is unbounded, then βˆƒ a π‘ π‘’π‘žπ‘’π‘’π‘›π‘π‘’ {β„˜π‘›} such that limπ‘›β†’βˆž β€Šβ„˜π‘› = ∞ and limπ‘›β†’βˆž β€Šπ‘§(β„˜π‘›) = ∞, where 𝑧(β„˜π‘›) = max{𝑧(𝑖), β„˜0 ≀ 𝑖 ≀ β„˜π‘›}. Since limβ„˜β†’βˆž β€Šπœ1(β„˜) = ∞, 𝜏1(β„˜π‘›) > β„˜0 for large β„˜ and 𝜏1(β„˜) ≀ β„˜, then we obtain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 84 https://internationalpubls.com 𝑧(𝜏1(β„˜π‘›)) = max{𝑧(𝑖): β„˜0 ≀ 𝑖 ≀ 𝜏1(β„˜π‘›)} ≀ max{𝑧(𝑖): β„˜0 ≀ 𝑖 ≀ β„˜π‘›} = 𝑧(β„˜π‘›). That is, 𝑧(𝜏1(β„˜π‘›)) ≀ 𝑧(β„˜π‘›). Consequently, 𝑠(β„˜π‘›) = 𝑧(β„˜π‘›) βˆ’ 𝑝(β„˜π‘›)𝑧𝛾1(𝜏1(β„˜π‘›)) β‰₯ 𝑧(β„˜π‘›)[1 βˆ’ 𝑝(β„˜π‘›)𝑧𝛾1βˆ’1(β„˜π‘›)] β†’ ∞ as 𝑛 β†’ ∞, since 𝛾1 ∈ (0,1] and 𝑝(β„˜) is bounded, which contradicts limβ„˜β†’βˆž β€Šπ‘ (β„˜) = βˆ’βˆž. Case (II): If z is bounded, then s is also bounded, because 𝑝(β„˜) is bounded, which contradicts that limβ„˜β†’βˆž β€Šπ‘ (β„˜) = βˆ’βˆž. So 𝑠(β„˜) fulfills one of the cases (I) and (II). Lemma 2.2. Let the condition (2) be true. Assume 𝑧 be a positive solution of (1) there exists case (I) of Lemma 2.1. Then 𝑧(β„˜) > 𝑠(β„˜) > π‘Œ(β„˜)π‘Ÿ(β„˜)Δ𝑠(β„˜) (4) for β„˜ β‰₯ β„˜1 and 𝑠(β„˜)/π‘Œ(β„˜) is eventually decreasing. Proof. By the representation of 𝑠(β„˜) and above (𝑅2), we can write 𝑧(β„˜) > 𝑠(β„˜) for β„˜ β‰₯ β„˜1 β‰₯ β„˜0. From the case (I), we get 𝑠(β„˜) = 𝑠(β„˜1) + βˆ‘ β€Šβ„˜βˆ’1 𝑖=β„˜1 β€Š π‘Ÿ(𝑖)Δ𝑠(𝑖) π‘Ÿ(𝑖) , > π‘Œ(β„˜)π‘Ÿ(β„˜)Δ𝑠(β„˜), β„˜ β‰₯ β„˜1. (5) Also, Ξ” ( 𝑠(β„˜) π‘Œ(β„˜) ) = π‘Œ(β„˜)π‘Ÿ(β„˜)Δ𝑠(β„˜) βˆ’ 𝑠(β„˜) π‘Ÿ(β„˜)π‘Œ(β„˜)π‘Œ(𝑙 + 1) < 0, β„˜ β‰₯ β„˜1. Thus { 𝑠(β„˜) π‘Œ(β„˜) } is strictly decreasing for all β„˜ β‰₯ β„˜1. Theorem 2.3. Let 𝛾2 < 𝛾1, 𝜎1(β„˜) < 𝜏1(β„˜) and condition (2) hold. If βˆ‘ β€Šβˆž β„˜1 π‘ž(β„˜)π‘Œπ›Ύ2(𝜎1(β„˜)) = ∞ (6) and lim β„˜β†’βˆž β€Šsup βˆ‘ β€Š β„˜βˆ’1 𝑗=𝜏1 βˆ’1(𝜎1(β„˜)) 1 π‘Ÿ(𝑗) βˆ‘ β€Š π‘—βˆ’1 𝑒=𝑗3 π‘ž(𝑒) 𝑝 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(𝑒))) > 0 , (7) then every solution of equation (1) is oscillatory. Proof. Let z be a non oscillatory solution of (1). Then 𝑧(β„˜) > 0, 𝑧(𝜎1(β„˜)) > 0, 𝑧(𝜏1(β„˜)) > 0, β„˜ β‰₯ β„˜1 β‰₯ β„˜0. By Lemma 2.1, the corresponding function 𝑠(β„˜) fullfills either case (I) or case (II). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 85 https://internationalpubls.com First, we assume that 𝑠(β„˜) satisfie s case (I). From the representation of 𝑠(β„˜), we get 𝑧(β„˜) β‰₯ 𝑠(β„˜), 𝑧𝛾2(πœŽπ‘™(β„˜)) β‰₯ 𝑠𝛾2(πœŽπ‘™(β„˜)). Applying above inequality in (1), we get Ξ”(π‘Ÿ(β„˜)Δ𝑠(β„˜)) + π‘ž(β„˜)𝑠𝛾2(𝜎1(β„˜)) ≀ 0. (8) Substituting (4) in (8) and taking 𝑦(β„˜) = π‘Ÿ(β„˜)Δ𝑠(β„˜), we clear that 𝑦(β„˜) is a positive solution of the inequality Δ𝑦(β„˜) + π‘ž(β„˜)π‘Œπ›Ύ2(𝜎1(β„˜))(π‘Ÿ(β„˜)Δ𝑠(β„˜))𝛾2 ≀ 0, Δ𝑦(β„˜) + π‘ž(β„˜)π‘Œπ›Ύ2(𝜎1(β„˜))𝑦𝛾2(𝜎1(β„˜)) ≀ 0, β„˜ β‰₯ β„˜1 (9) On the other hand, from [6], we can see that condition (6) assures that (9) has no eventually positive solution, which is contradiction . Next , assume that 𝑠(β„˜) satisfies case (II) of Lemma 2.1. Then, by the representation of 𝑠(β„˜), we get 𝑧(𝜏1(β„˜)) > ( βˆ’π‘ (β„˜) 𝑝(β„˜) ) 1 𝛾1 . (10) Applying (10) in (1), we get Ξ”(π‘Ÿ(β„˜)Δ𝑠(β„˜)) βˆ’ 1 𝑝 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(β„˜))) π‘ž(β„˜)𝑠 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(β„˜))) ≀ 0. (11) Since 𝑠(β„˜) is negative and increasing, we obtain limβ„˜β†’βˆž β€Šπ‘ (β„˜) = 𝑐1 ≀ 0. We prove that 𝑐1 = 0. If not, then 𝑐1 < 0 and 𝑠(β„˜) ≀ 𝑐1 and 𝑠(𝜏1 βˆ’1(𝜎1(β„˜))) ≀ 𝑐1 for large β„˜. Therefore, 𝑠 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(β„˜))) ≀ 𝑐1 𝛾2 𝛾1 (12) Summing (11) from β„˜ to ∞ and using (12), we get π‘Ÿ(β„˜)Δ𝑠(β„˜) βˆ’ π‘Ÿ(β„˜1)Δ𝑠(β„˜1) ≀ βˆ‘ β€Š ∞ 𝑖=𝑙 β€Š π‘ž(𝑖) 𝑝 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(𝑖))) 𝑠 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(𝑖))), βˆ’π‘Ÿ(β„˜)Δ𝑠(β„˜) ≀ 𝑐1 𝛾2 𝛾1 βˆ‘ β€Š ∞ 𝑖=𝑙 β€Š π‘ž(𝑖) 𝑝 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(𝑖))) . Again summing from β„˜1 to ∞, we have 𝑠(β„˜1) ≀ 𝑐1 𝛾2 𝛾1 βˆ‘ β€Šβˆž 𝑗=β„˜1 1 π‘Ÿ(𝑗) βˆ‘ β€Šβˆž π‘š=𝑗1 π‘ž(π‘š) 𝑝 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(π‘š))) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 86 https://internationalpubls.com which is contradiction with (7) and from (7), we claim lim β„˜β†’βˆž β€Šsup βˆ‘ β€Š ∞ 𝑗=β„˜1 1 π‘Ÿ(𝑗) βˆ‘ β€Š ∞ π‘š=𝑗1 π‘ž(π‘š) 𝑝 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(π‘š))) = ∞. Thus, limβ„˜β†’βˆž β€Šπ‘ (β„˜) = 0 and s(β„˜) is negative and increasing. Summing (11) from β„˜2 to β„˜ βˆ’ 1 for β„˜ > 𝑖, we get βˆ’π‘Ÿ(β„˜2)(Δ𝑠(β„˜2)) ≀ βˆ‘ β€Š β„˜βˆ’1 𝑠=β„˜2 β€Š π‘ž(𝑖) 𝑝 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(𝑖))) 𝑠 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(𝑖))) Again summing from 𝜏1 βˆ’1(𝜎1(β„˜)) to β„˜ βˆ’ 1 and using 𝑠(β„˜) is increasing and we have 𝑠(𝜏1 βˆ’1(𝜎1(β„˜))) βˆ’ 𝑠(β„˜) ≀ 𝑠 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(β„˜))) βˆ‘ β€Š β„˜βˆ’1 𝑗=𝜏1 βˆ’1(𝜎1(β„˜)) 1 π‘Ÿ(𝑗) βˆ‘ β€Š π‘—βˆ’1 π‘š=𝑗3 π‘ž(π‘š) 𝑝 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(π‘š))) or 𝑠(𝜏1 βˆ’1(𝜎1(β„˜))) 𝑠 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(β„˜))) β‰₯ βˆ‘ β€Š β„˜βˆ’1 𝑗=𝜏1 βˆ’1(𝜎1(β„˜)) 1 π‘Ÿ(𝑗) βˆ‘ β€Š π‘—βˆ’1 π‘š=𝑗3 π‘ž(π‘š) 𝑝 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(π‘š))) . (13) Since 𝑠(𝜏1 βˆ’1(𝜎1(β„˜))) 𝑠 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(β„˜))) = |𝑠(𝜏1 βˆ’1(𝜎1(β„˜)))| 1βˆ’ 𝛾2 𝛾1 and 1 βˆ’ 𝛾2 𝛾1 > 0, we get lim β„˜β†’βˆž β€Šsup βˆ‘ β€Š β„˜βˆ’1 𝑗=𝜏1 βˆ’1(𝜎1(β„˜)) 1 π‘Ÿ(𝑗) βˆ‘ β€Š π‘—βˆ’1 π‘š=𝑗3 π‘ž(π‘š) 𝑝 𝛾2 𝛾1(𝜏1 βˆ’1(𝜎1(π‘š))) ≀ 0 which contradicts (7). Theorem 2.4. Assume 𝛾2 = 1 and condition (2) holds. If lim π‘‘β†’βˆž β€Šinf βˆ‘ β€Š β„˜βˆ’1 𝜎1(β„˜) π‘ž(𝑠)π‘Œ(𝜎1(𝑠)) > 1 𝑒 , (14) then every solution of (1) is either oscillatory or tends to zero as β„˜ β†’ ∞. Proof. We assume that a non-oscillatory solution z of (1), 𝑧(β„˜) > 0, 𝑧 (𝜎1(β„˜)) > 0, 𝑧(𝜏1(β„˜)) > 0, β„˜ β‰₯ β„˜1 β‰₯ β„˜0 and that for s one of the case (I) and case (II) holds. Assume that 𝑠(β„˜) meets case (I) of Lemma 2.1 and from the proof of case (I) of theorem 2.1, we have for 𝛾2 = 1 that 𝑦(β„˜) = π‘Ÿ(β„˜)Δ𝑠(β„˜) is a positive solution of the inequality Δ𝑦(β„˜) + π‘ž(β„˜)π‘Œ(𝜎1(β„˜))𝑦(𝜎1(β„˜)) ≀ 0. (15) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 87 https://internationalpubls.com On the other hand, from [6], we can notice that equation (14) guarantees that (15) has no positive solution, which implies contradiction. Let 𝑠(β„˜) meets case (II) of Lemma 2.1. From this 𝑠(β„˜) < 0 and Δ𝑠(β„˜) > 0 and also limβ„˜β†’βˆž β€Šπ‘ (β„˜) = 𝑐1 ≀ 0, where 𝑐1 is a constant. i.e. s is bounded and as in the proof of Lemma 2.1, we can say that z is also bounded. Therefore, limβ„˜β†’βˆž β€Šπ‘§(β„˜) = π‘š1, 0 ≀ π‘š1 < ∞. We claim that π‘š1 = 0. Suppose π‘š1 > 0, there is a sequence {β„˜π‘›} such that limπ‘›β†’βˆž β€Šβ„˜π‘› = ∞ and limπ‘›β†’βˆž β€Šπ‘§(β„˜π‘›) = π‘š1. Thus 𝑠(β„˜π‘›) = 𝑧(β„˜π‘›) βˆ’ 𝑝(β„˜π‘›)𝑧𝛾1(𝜏1(β„˜π‘›)), 𝑧(𝜏1(β„˜π‘›)) = (𝑧(β„˜π‘›) βˆ’ 𝑠(β„˜π‘›)) 1 𝛾1 𝑝 1 𝛾1(β„˜π‘›) . Taking 𝑛 β†’ ∞, we get π‘š1 β‰₯ lim π‘›β†’βˆž β€Šπ‘§(𝜏1(β„˜π‘›)) β‰₯ ( π‘š1 𝑝 ) 1 𝛾1 We conclude that π‘š1 = 0, because of 𝑝 ∈ (0,1), that is limπ‘›β†’βˆž β€Šπ‘§(β„˜) = 0. 3. Examples Example 3.1. Examine second order neutral delay difference equation Ξ” (β„˜Ξ” (𝑧(β„˜) βˆ’ 𝑝𝑧 1 3 ( β„˜ 2 ))) + 8β„˜ ( β„˜ 3 ) = 0, β„˜ β‰₯ 1, (16) where 𝑝 ∈ (0,1) which is a constant. Here π‘Ÿ(β„˜) = β„˜, 𝑝(β„˜) = 𝑝, π‘ž(β„˜) = 8β„˜, 𝜏1(β„˜) = β„˜ 2 , 𝜎1(β„˜) = β„˜ 3 for β„˜ β‰₯ β„˜1 = 1, 𝛾1 = 1/3, 𝛾2 = 1/5 and π‘Œ(β„˜) = 1βˆ’β„˜ β„˜ . Clearly, these calculations shows that above conditions (6) and (7) are fullfilled. So that by Theorem 2.3, every solution of (16) is oscillatory. Example 3.2. Examine second order neutral delay difference equation Ξ” ( 1 β„˜ Ξ” (𝑧(β„˜) βˆ’ 𝑝𝑧 1 3 ( β„˜ 2 ))) + β„˜π‘§ ( β„˜ 3 ) = 0, β„˜ β‰₯ 1, (17) where 𝑝 ∈ (0,1) which is a constant. Here π‘Ÿ(β„˜) = 1 β„˜ , 𝑝(β„˜) = 𝑝, π‘ž(β„˜) = β„˜, 𝜏1(β„˜) = β„˜ 2 , 𝜎1(β„˜) = β„˜ 3 for β„˜ β‰₯ β„˜1 = 1, 𝛾1 = 1/3, 𝛾2 = 1 and π‘Œ(β„˜) = 1 βˆ’ β„˜. Each and every conditons of Theorem 2.4 with 𝛾2 = 1 are satisfied, so the equation (17) is oscillatory. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 88 https://internationalpubls.com 4. Conclusion The solutions of nonlinear equations behave in peculiar ways and these ways can be developed by means of different strategies included in the method. An attempt was made here to establish the sufficient conditions with the fact that the solution space of nonlinear non positive neutral term of difference equation is reducing to the solution of its limiting equation and we assumed with 𝛾2 = 1. By these discussions, (1) is oscillatory or asymptotically zero as β„˜ β†’ ∞. References [1] Agarwal, R.P., Grace, S.R., O'Regan.,D Oscillation of higher order difference equations via comparision, Glasnik.Mat., 39 (2004), 289-301. [2] Agarwal, R.P., Bohner, M., Grace, S.R., O'Regan, D., Discrete oscillation theory, Hindawi, New York, 2005. [3] Agarwal, R.P., Difference equations and inqualities, theory, methods and applications, Second Edition, Revised and Expanded, New York, Marcel Dekker, 2000. 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