# Oscillatory and Asymptotic Behavior of a First-Order Neutral Equation of Discrete Type with Variable Several Delay under Δ Sign

Authors: Radhanath Rath; Chittaranjan Behera
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

We obtain necessary and sufficient conditions so that every solution of neutral delay difference equationΔyn-∑j=1kpnjyn-mj+qnG(yσ(n))=fn oscillates or tends to zero as n→∞, where {qn} and {fn} are real sequences and G∈C(R,R), xG(x)>0, and m1,m2,…,mk are positive integers. Here Δ is the forward difference operator given by Δxn=xn+1-xn, and {σn} is an increasing unbounded sequences with σn≤n. This paper complements, improves, and generalizes some past and recent results.


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## Body

## 1. Introduction

Consider the neutral delay difference equation of first order(1)Δyn-∑j=1kpnjyn-mj+qnGyσn=fnwhere Δ is the forward difference operator given by Δxn=xn+1-xn, qn and fn are members of infinite real sequences, and mj are positive integers. Further, assume {pnj} are real sequences for each j∈1,2,…k and that G∈C(R,R) and σ(n)≤n are monotonic increasing sequences which are unbounded.We study the oscillatory behavior of solutions of neutral difference equation (1) under the following assumptions.(H1)
xG(x)>0 for x≠0.(H2)
There exists a bounded sequence{Fn} such that ΔFn=fn.(H3)
The sequence{Fn} in (H2) satisfies limn→∞Fn=0.(H4)
qn>0, ∑n=n0∞‍qn=∞.In addition to the above we assume some new conditions onpnj (see (12), (22), (26), and (30) in next section). It is important to note that our results hold good for the solutions of the neutral equation(2)Δyn-∑j=1kpnjyn-mj+∑j=1lqnjGyσjn=fnunder the assumption(3)∑n=n0∞∑j=1lqnj=∞,qnj>0instead of (H4). The following neutral difference equations/delay difference equations are obtained as particular case of (2).(4)Δyn-pnyn-m+∑j=1lqnjGyσjn=fn,(5)Δyn-pnyn-m+qnGyσn=fn,(6)Δyn+∑j=1lqnjGyσjn=0,and(7)Δyn+qnGyσn=0.The neutral difference equations (5) are seen as the discrete analogue of the neutral differential equations(8)yt-ptyt-τ′+qtGyt-σ=ft.The oscillatory and asymptotic behavior of delay difference equations and neutral difference equations have been intensively studied in recent years due to its various application in different field of science and technology [1]. It is observed that several articles (see [2–4]) exist in literature for the study of neutral difference equations/delay difference equations with several delay, i.e., for (4) or (6), respectively. However study of neutral equations with several delay term under Δ symbol, i.e., (1) or (2), seems to be relatively scarce in literature. Use of lemmas from [1, Lemma 1.5.1 and 1.5.2] or its discrete analogue (see [5]) plays an important role in studying (4) [6], (5) [7], and (8) [8]. In this context, one may note these lemmas cannot be applied to the study of (1) or (2). Hence study of (1) and (2) needs a different approach.The work in this paper complements and generalizes the work in [3, 9]. This can be verified that the results in [3, 9] which are concerned with the study of (6) and (7) cannot be applied to the delay difference equation(9)Δyn+e-2-e-3yn-2=0.which has a solution yn=e-n tending to zero. It is because the primary assumption,(10)lim⁡infk→∞∑i=σkk-1qi>1e,is not satisfied. However, note that (10) implies (H4) and (H4) is satisfied in (9) and hence the results of this paper give an answer to the behavior of solutions of neutral equations like (9). While working on nonlinear neutral equations most of the authors [7, 8, 10–12] assume the condition that G is nondecreasing unlike this paper.Letn0 be a fixed nonnegative integer. Let max{m1,m2,…,mk}=m and ρ=min{σ(n0),n0-m}. By a solution of (1) we mean a real sequence {yn} which is defined for all positive integers n≥ρ and satisfies (1) for n≥n0. Clearly if the initial condition(11)yn=anforρ≤n≤n0,is given then (1) has a unique solution satisfying the given initial condition (11). A solution {yn} of (1) 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 nonoscillatory. In the sequel, unless otherwise specified, when we write a functional inequality, it will be assumed to hold for all n sufficiently large. Here we assume the existence of solution of (1) and study its oscillatory and asymptotic behavior.


## 2. Sufficient Condition

In this section we present some results which prove that (H4) is sufficient for any solution of (2) to be oscillatory or tending to zero as n→∞. Moreover we give some examples to illustrate and signify our results. Our first result and the subsequent ones are as follows.Theorem 1.
Suppose that (H1)–(H4) hold. Assume that there exists a positive constantp such that the sequences {pnj} for j=1,2,…,k satisfy the condition(12)pnj≥0,for  everyj=1,2,…,kand∑j=1klim⁡supn→∞⁡pnj<p<1.Then every solution of (1) oscillates or tends to zero as n→∞.Proof.
Lety={yn} be any solution of (1) for n≥n0, where n0 is a fixed positive integer. If it oscillates then there is nothing to prove; otherwise, it leads to two distinct possibilities, either yn>0 or yn<0. for n≥n1>n0. Consider the first one, i.e., yn>0 eventually. There exits positive integer n2≥n1 such that yn>0,yn-mj>0 for each j, and yσ(n)>0 for n≥n2. For n≥n2, let(13)zn=yn-∑j=1kpnjyn-mj,and(14)wn=zn-Fn.From (1), (13), and (14), it follows due to (H1) that(15)Δwn=-qnGyσn≤0.Then there exists n3≥n2 such that wn is monotonic and is of constant sign for n≥n3. For the sake of a contradiction assume that yn is not bounded. Then there exists a subsequence {ynr} such that (16)nr→∞,ynr→∞asr→∞,and(17)ynr=max⁡yn:n3≤n≤nr.Since σ(n)→∞ as n→∞, we may choose r large enough so that σ(nr)≥n3. For 0<ϵ, because of (H3), we can find a positive integer n4 such that n≥n4≥n3 implies |Fn|<ϵ. As (12) holds, then using (13), (14), and (17) we obtain (18)wnr=ynr-∑j=1kpnjynr-mj-Fnr≥1-∑j=1kpnjynr-ϵ>1-pynr-ϵ.
Takingr→∞, we find limn→∞wn=∞, a contradiction as wn is monotonic decreasing. Hence yn is bounded which implies wn and zn are bounded and limn→∞wn exists. Further it follows that lim⁡infn→∞yn and lim⁡supn→∞yn exist. We claim lim⁡infn→∞yn=0. Otherwise, let yn≥α>0. Next boundedness of yn yields yn≤β. Hence we have 0≤α<yn≤β, which will be used for bounding the G term in (1) from below.
From the continuity ofG and assumption (H1) it follows that there exists a positive lower bound m for G on [α,β]. Hence there exists n5 such that G(yσ(n))>m>0 for n>n5. Then summing (15) from n=n5 to s-1 we obtain (19)wn5-ws=∑j=n5s-1qjGyσj≥m∑j=n5∞qj.Since the left hand side is the member of a bounded sequence, while the right hand side approaches +∞, we have a contradiction. This yields lim⁡infn→∞yn=0. From (H3), monotonic nature of wn and (14), it follows that limn→∞zn exists finitely. Let limn→∞zn=δ. If δ>0, then(20)0<δ=lim⁡infn→∞⁡zn=lim⁡infn→∞⁡yn-∑j=1kpnjyn-mj≤lim⁡infn→∞⁡yn+lim⁡supn→∞⁡-∑j=1kpnjyn-mj=-lim⁡infn→∞⁡∑j=1kpnjyn-mj≤-∑j=1klim⁡infn→∞⁡pnjyn-mj≤-∑j=1klim⁡infn→∞⁡pnjlim⁡infn→∞⁡yn-mj≤0,a contradiction. If δ≤0 then (21)0≥δ=lim⁡supn→∞⁡zn=lim⁡supn→∞⁡yn-∑j=1kpnjyn-mj≥lim⁡supn→∞⁡yn+lim⁡infn→∞⁡-∑j=1kpnjyn-mj≥lim⁡supn→∞⁡yn-lim⁡supn→∞⁡∑j=1kpnjyn-mj≥lim⁡supn→∞⁡yn-∑j=1klim⁡supn→∞⁡pnjyn-mj≥lim⁡supn→∞⁡yn-∑j=1klim⁡supn→∞⁡pnjlim⁡supn→∞⁡yn-mj≥lim⁡supn→∞⁡yn1-∑j=1klim⁡supn→∞⁡pnj≥lim⁡supn→∞⁡yn1-p.Hence lim⁡supn→∞yn≤0, by (12), which implies the desired result limn→∞yn=0. If yn<0 for n>n1 then proceeding as above we can arrive at limn→∞yn=0. Thus the theorem is proved.Theorem 2.
Suppose that (H1)–(H4) hold. Assume that there exists a positive constantp such that the sequences {pnj} for j=1,2,…,k satisfy the condition(22)pnj≤0,for  everyj=1,2,…,kand∑j=1klim⁡infn→∞⁡pnj>-p>-1.Then every solution of (1) oscillates or tends to zero as n→∞.Proof.
Proceeding as in the proof of Theorem1 and setting zn, wn as in (13) and (14), respectively, we obtain (15) and further prove yn is bounded with lim⁡infn→∞yn=0. From (H3) and the fact that wn is monotonic it follows that limn→∞wn=limn→∞zn=δ∈R. As zn≥0, so δ≥0. We claim δ=0; if not then δ>0, and this implies(23)δ=lim⁡infn→∞⁡zn=lim⁡infn→∞⁡yn-∑j=1kpnjyn-mj≤lim⁡infn→∞⁡yn+lim⁡supn→∞⁡-∑j=1kpnjyn-mj≤∑j=1klim⁡supn→∞⁡-pnjlim⁡supn→∞⁡yn-mj=∑j=1k-lim⁡infn→∞⁡pnjlim⁡supn→∞⁡yn-mj≤plim⁡supn→∞⁡yn≤pα.Hence we get(24)α≥δp>δ.Again (25)δ=lim⁡supn→∞⁡zn=lim⁡supn→∞⁡yn-∑j=1kpnjyn-mj≥lim⁡supn→∞⁡yn+lim⁡infn→∞⁡-∑j=1kpnjyn-mj=lim⁡supn→∞⁡yn+lim⁡infn→∞⁡∑j=1k-pnjyn-mj≥lim⁡supn→∞⁡yn+∑j=1klim⁡infn→∞⁡-pnjyn-mj≥lim⁡supn→∞yn+∑j=1klim⁡infn→∞⁡-pnjlim⁡infn→∞⁡yn-mj=lim⁡supn→∞⁡yn=α,a contradiction, due to inequality (24). Hence we conclude δ=0 and from zn>yn, it follows that limn→∞yn≤0. Hence limn→∞yn=0.
The proof for the caseyn<0 for large n is similar. Hence the theorem is proved.Remark 3.
Theorems1 and 2 hold good for k=0 and k=1. Hence these results could be compared with results concerned with the difference equations (4), (5), (6), and (7).Theorem 4.
Suppose that (H1)–(H4) hold. Assume that there exists a positive constantp such that the sequences {pnj} for j=1,2,…,k satisfy the condition(26)pnj<0for  everyj=1,2,…,kand  there  exists,i∈1,2,3,…,ksuch  thatlim⁡supn→∞⁡pni-∑j≠ilim⁡infn→∞⁡pnj<-1.Then every solution of (1) oscillates or tends to zero as n→∞.Proof.
Proceeding as in the proof of Theorem1 and setting zn, wn as in (13) and (14), respectively, we obtain (15) and further prove yn is bounded with liminfn→∞yn=0. From (H3) and that wn is monotonic it follows that limn→∞wn=limn→∞zn=δ∈R. As zn≥0, so δ≥0. We claim δ=0. If not, then δ>0, and this implies(27)
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.Again we have(28)δ=lim⁡supn→∞⁡zn=lim⁡supn→∞⁡yn-∑j=1kpnjyn-mj≥lim⁡infn→∞⁡yn+lim⁡supn→∞⁡-∑j=1kpnjyn-mj=lim⁡supn→∞⁡-pniyn-mi+lim⁡infn→∞⁡∑j≠i-pnjyn-mj≥lim⁡supn→∞⁡yn-mjlim⁡infn→∞⁡-pni+∑j≠ilim⁡infn→∞⁡-pnjyn-mj≥lim⁡supn→∞⁡yn-lim⁡supn→∞⁡pni+∑j≠ilim⁡infn→∞⁡-pnjlim⁡infn→∞⁡yn-mj≥lim⁡supn→∞⁡yn-lim⁡supn→∞⁡pni.From (27) and (28), it follows that (29)lim⁡supn→∞⁡yn∑j≠ilim⁡infn→∞⁡pnj-1-lim⁡supn→∞⁡pni≤0Using (26), we obtain limn→∞yn=0. Thus the theorem is proved.Next, we intend to present a result wherepnj, j=1,2,3,…,k, satisfy the following condition:(30)pnj>0for  everyj=1,2,…,kand  there  exists,i∈1,2,3,…,ksuch  thatlim⁡infn→∞⁡pni-∑j≠ilim⁡supn→∞⁡pnj>1.For that purpose we give an example which would lead us to our next result.Example 5.
Consider the first-order neutral delay difference equation with several delays and variable coefficients(31)Δyn-1+2-nyn-1-4+2-nyn-2+22n+1/3yn-4=0.
Note thatpnj satisfy (30) for the above neutral delay difference equation (31). This neutral delay difference equation has an unbounded solution yn=2n tending to ∞ as n→∞ unlike other results presented so far.
The above example is the motivating point to the statement of our next result. Since the proof is almost similar to that of Theorem4, it is omitted.Theorem 6.
Suppose that (H1)–(H4) hold. Assume that there exists a positive constantp such that the sequences {pnj} for j=1,2,…,k satisfy the condition (30). Then every bounded solution of (1) oscillates or tends to zero as n→∞.Remark 7.
The above Theorems4 and 6 hold for k=1 but not for k=0. Hence these results can be compared with results concerned with neutral delay difference equations (4) and (5).
Few examples are noted below to illustrate our results and establish its significance.Example 8.
Consider the first-order neutral delay difference equation(32)Δyn-pn1yn-1-pn2yn-4+9760yn-23=0,n≥5,where (33)pn1=14,ifnisodd,15,ifniseven,and(34)pn2=13,ifnisodd,12,ifniseven.The neutral delay difference equation (32) satisfies all the conditions of Theorem 1. As such, it has an oscillatory solution yn=-1n.Example 9.
Consider the first-order inhomogeneous neutral delay difference equation(35)Δyn+pn1yn-4+pn2yn-5+342n-4yn-33=-32n+1-322n-2,n≥5,where pn1=2-n+1/16 and pn2=2-n+1/32. This neutral delay difference equation satisfies all the conditions of Theorem 2. As such, it has a bounded positive solution yn=2-n tending to zero as n→∞. Note that, no result in the papers cited under reference can be applied to the neutral delay difference equations (32) and (35).Remark 10.
Results of [3, 9] cannot be applied to the delay difference equation (9), because the condition (10) is not satisfied. However, due to Remark 3, Theorem 1 can be applied to the delay equation (9) as all the conditions are satisfied and as such the delay equation has a positive bounded solution e-n tending to zero as n→∞. Thus our work complements the work in [3, 9]. Further, since we do not assume G is nondecreasing, our Theorems 1, 2, 4, and 6 improve and generalize the results in [7].


## 3. Necessary Conditions

In this section we show that (H4) is necessary for every solution of (1) to be oscillatory or tending to zero as n→∞. For this, we need the following lemma.Lemma 11 (Krasnoselskii’s fixed point theorem [13]).
LetX be a Banach space and S be a bounded closed convex subset of X. Let A,B be operators from S to X such that Ax+By∈S for every pair of x,y∈S. If A is a contraction and B is completely continuous then the equation (36)Ax+Bx=xhas a solution in S.Theorem 12.
Assume that (H2) holds. Further, assume that one of the conditions of (12) and (22) hold. Then (H4) is a necessary condition for all solution of (1) to be oscillatory or tending to zero as n→∞.Proof.
Suppose the condition (12) holds. The proof for the case when (22) holds would follow on similar lines. Assume for the sake of contradiction that (H4) does not hold. Hence(37)∑n=n0∞qn<∞.Thus, all we need to show is the existence of a bounded solution yn of (1) with lim⁡infn→∞yn>0. From (H2), we find a positive constant c and a positive integer n1>n0>0 such that (38)Fn<cforn≥n1.Choose a positive constant L such that L≥5c/1-p. Since G∈C(R,R), let (39)μ=max⁡Gx:c≤x≤L.Let (40)η=max⁡m1,m2,…,mk.Then using (37) one can fix n2>n1 such that for n≥n2 it follows that (41)μ∑i=n∞qi<c.Choose N1>n2 such that (42)N0=min⁡σN1,N1-η.Let X=l∞N0, Banach space of real bounded sequences x={xn} with x1=x2=⋯=xN0 and supremum norm (43)x=sup⁡xn:n≥N0.Define (44)S=y∈X:c≤yn≤L,n≥N0.Clearly S is a bounded closed and convex subset of X. Now we define two operators A and B:S→X as follows. For y∈S, define(45)Ayn=AyN1,N0≤n≤N1∑j=1kpnjyn-mj+Fn+3c,n≥N1.Byn=ByN1,N0≤n≤N1∑i=n∞qiGyσi,n≥N1.First we show that if x,y∈S then Ax+By∈S. Hence, for x={xn} and y={yn}∈S and for n≥N1 we obtain (46)Axn+Byn≤∑j=1kpnjxn-mj+3c+∑i=n∞qiGyσi+Fn≤pL+5c≤L.On the other hand(47)Axn+Byn≥3c-∑i=n∞qiGyσi-Fn≥3c-c-c≥c.Hence (48)c≤Axn+Byn≤Lforn≥N1.Thus, we proved that Ax+By∈S for any x,y∈S. Next we show that A is a contraction on S. In fact for x,y∈S and n≥N1 we have (49)Axn-Ayn≤∑j=1kpnjxn-mj-yn-mj≤px-yThis implies A is a contraction because 0<p<1. Next we show that B is completely continuous. For this as a first step we show that B is continuous. Suppose the sequence xl≡{xnl}→x≡{xn} in S as l→∞ (with l taken from the index set). Since S is closed then x∈S. For n≥N1 we have (50)Bxln-Bxn≤∑i=n∞qiGxσil-GxσiSince G is continuous, therefore |G(xσ(i)l)-G(xσ(i))|→0 as l→∞. Hence B is continuous. Next what remained to show is BS is relatively compact. Using the result [14, Theorem 3.3], we need only show that BS is uniformly cauchy. Let x≡{xn} be a sequence in S. From (H2) and (37), it follows that, for ϵ>0, there exists N∗≥N1 such that, for n≥N∗, (51)∑i=n∞qiGxσi<ϵ2.Then for n3>n4≥N∗ we have (52)Bxn3-Bxn4<ϵ.Thus BS is uniformly cauchy. Hence it is relatively compact. Then by Lemma 11, we can find x0 in S such that Ax0+Bx0=x0. Clearly, x0n is a bounded, positive solution of (1) with limit infimum greater than or equal to c>0. Thus the theorem is proved.Theorem 13.
Assume that (H2) holds. Further assume that one of the conditions of (26) and (30) holds. Then (H4) is a necessary condition for all solution of (1) to be oscillatory or tending to zero as n→∞.Proof.
Suppose thatpn satisfies (30). The proof for the case when (26) holds is similar. Assume for the sake of contradiction that (H4) does not hold. Hence (37) holds. Thus, all we need to show is the existence of a bounded solution yn of (1) with lim⁡infn→∞yn>0. From (H2), we find a positive constant L and a positive integer n1>n0>0 such that (53)Fn<Lforn≥n1.By (30), we can find a small positive real ϵ, a lower bound c for pni, and upper bounds dj for pnj (j≠i and 1≤j≤k) such that c-∑‍dj-1=ϵ. Let ∑‍dj=d. Hence c=d+1+ϵ. Next choose an upper bound b for pni such that b<(c2-c)/d. The nonexistence of such an upper bound for pni would lead to the fact that, for all δ>0,b=c+δ and b≥(c2-c)/d. Taking δ=ϵ, we have ϵ2+ϵ≤0, a contradiction. Choose a real λ as follows:(54)0<λ=L+ϵd+c-1c+L+ϵc2-bd+c.Let(55)H=bλ+L+ϵc-1.From (54) and (55) it follows that(56)λ-Hd+L+ϵc=1.Since G∈C(R,R), let (57)μ=max⁡Gx:1≤x≤H.Let η=max{m1,m2,…,mk}. Then using (37), one can fix n2>n1 such that for n≥n2 it follows that (58)μ∑i=n∞qi<ϵ.Choose N1>n2 such that (59)N0=min⁡σN1,N1-η.Let X=l∞N0, Banach space of real bounded sequences x={xn} with x1=x2=⋯=xN0 and supremum norm (60)x=sup⁡xn:n≥N0.Define (61)S=y∈X:1≤yn≤H,n≥N0.Clearly S is a bounded closed and convex subset of X. Now we define two operators A and B:S→X as follows. For y∈S, define(62)Ayn=AyN1,N0≤n≤N1yn+mipn+mii-∑j≠ipn+mijyn-mj+mipn+mii+bλpn+mii-Fn+mipn+mii,n≥N1.Byn=ByN1,N0≤n≤N1-∑j=n+mi∞qjGyσjpn+mii,n≥N1.Proceeding as in the proof of above theorem we show that (i) if x,y∈S then Ax+By>1 by (56) and Ax+By<H by (55), so that Ax+By∈S, (ii) Axn-Ayn<[(d+1)/c]xn-yn, hence A is a contraction on S, and (iii) B is completely continuous. This completes the proof of the theorem.Remark 14.
For the results in this section, we assume none of conditions (H3),G is nondecreasing, and xG(x)>0, whereas the authors [7, 8] assumed these three conditions in their corresponding results. Hence the results of this article generalize and improve the corresponding results of these papers.
Combining all the above results, i.e., Theorems1, 2, 4, 6, 12, and 13, we obtain the following theorem.Theorem 15.
Suppose that (H1)-(H3) hold. Assumepnj in (1) to satisfy one of the four conditions (12), (22), (26), and (30). Then (H4) is both necessary and sufficient condition for every solution of (1) to be oscillatory or tending to zero as n→∞.Remark 16.
The results of this work hold forG(x)=x and f(x)=0, i.e., for the linear homogeneous equation associated with (1).



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