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

We obtain necessary and sufficient conditions so that every solution of neutral delay difference equation Δ(yn − ∑kj=1 pj nyn−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.


Introduction
Consider the neutral delay difference equation of first order where Δ is the forward difference operator given by Δ  =  +1 −   ,   and   are members of infinite real sequences, and   are positive integers.Further, assume {   } are real sequences for each  ∈ 1, 2, . . . and that  ∈ (R, R) and () ≤  are monotonic increasing sequences which are unbounded.
We study the oscillatory behavior of solutions of neutral difference equation (1) under the following assumptions.
In addition to the above we assume some new conditions on    (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 under the assumption instead of (H4).The following neutral difference equations/delay difference equations are obtained as particular case of (2).
The neutral difference equations (5) are seen as the discrete analogue of the neutral differential equations 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][3][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 which has a solution   =  − tending to zero.It is because the primary assumption, lim inf 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][11][12] assume the condition that  is nondecreasing unlike this paper.Let  0 be a fixed nonnegative integer.Let max{ 1 ,  2 , . . .,   } =  and  = min{( 0 ),  0 − }.By a solution of (1) we mean a real sequence {  } which is defined for all positive integers  ≥  and satisfies (1) for  ≥  0 .Clearly if the initial condition is given then (1) has a unique solution satisfying the given initial condition (11).A solution {  } of ( 1) is said to be oscillatory if, for every positive integer  0 > 0, there exists  ≥  0 such that    +1 ≤ 0; otherwise {  } 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  sufficiently large.Here we assume the existence of solution of (1) and study its oscillatory and asymptotic behavior.

Sufficient Condition
In this section we present some results which prove that (H4) is sufficient for any solution of ( Then every solution of (1) oscillates or tends to zero as  → ∞.
From the continuity of  and assumption (H1) it follows that there exists a positive lower bound  for  on [, ].Hence there exists  5 such that ( () ) >  > 0 for  >  5 .Then summing (15) from  =  5 to  − 1 we obtain 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 inf →∞   = 0. From (H3), monotonic nature of   and ( 14), it follows that lim →∞   exists finitely.Let lim →∞   = .If  > 0, then a contradiction.If  ≤ 0 then Hence lim sup →∞   ≤ 0, by (12), which implies the desired result lim →∞   = 0.If   < 0 for  >  1 then proceeding as above we can arrive at lim →∞   = 0. Thus the theorem is proved.( Then every solution of (1) oscillates or tends to zero as  → ∞.
Proof.Proceeding as in the proof of Theorem 1 and setting   ,   as in ( 13) and ( 14), respectively, we obtain (15) and further prove   is bounded with lim inf →∞   = 0. From (H3) and the fact that   is monotonic it follows that lim →∞   = International Journal of Mathematics and Mathematical Sciences lim →∞   =  ∈ R. As   ≥ 0, so  ≥ 0. We claim  = 0; if not then  > 0, and this implies Hence we get Again a contradiction, due to inequality (24).Hence we conclude  = 0 and from   >   , it follows that lim →∞   ≤ 0. Hence lim →∞   = 0.
Example 5. Consider the first-order neutral delay difference equation with several delays and variable coefficients Note that    satisfy (30) for the above neutral delay difference equation ( 31).This neutral delay difference equation has an unbounded solution   = 2  tending to ∞ as  → ∞ 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 Theorem 4, it is omitted.Theorem 6. Suppose that (H1)-(H4) hold.Assume that there exists a positive constant  such that the sequences {   } for  = 1, 2, . . .,  satisfy the condition (30).Then every bounded solution of (1) oscillates or tends to zero as  → ∞.
Remark 7. The above Theorems 4 and 6 hold for  = 1 but not for  = 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 where and The neutral delay difference equation ( 32) satisfies all the conditions of Theorem 1.As such, it has an oscillatory solution   = (−1)  .
Example 9. Consider the first-order inhomogeneous neutral delay difference equation where  1  = 2 − +1/16 and  2  = 2 − +1/32.This neutral delay difference equation satisfies all the conditions of Theorem 2. As such, it has a bounded positive solution   = 2 − tending to zero as  → ∞.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  − tending to zero as  → ∞.Thus our work complements the work in [3,9].Further, since we do not assume  is nondecreasing, our Theorems 1, 2, 4, and 6 improve and generalize the results in [7].

Necessary Conditions
In this section we show that (H4) is necessary for every solution of (1) to be oscillatory or tending to zero as  → ∞.For this, we need the following lemma.
Lemma 11 (Krasnoselskii's fixed point theorem [13]).Let  be a Banach space and  be a bounded closed convex subset of 6 International Journal of Mathematics and Mathematical Sciences .Let ,  be operators from  to  such that  +  ∈  for every pair of ,  ∈ .If  is a contraction and  is completely continuous then the equation has a solution in .
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  → ∞.
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 Thus, all we need to show is the existence of a bounded solution   of ( Choose  1 >  2 such that Let  = ℓ  0 ∞ , Banach space of real bounded sequences  = {  } with  1 =  2 = ⋅ ⋅ ⋅ =   0 and supremum norm Define Clearly S is a bounded closed and convex subset of X.Now we define two operators  and  :  →  as follows.For  ∈ , define Thus  is uniformly cauchy.Hence it is relatively compact. Then by Lemma 11, we can find  0 in  such that  0 +  0 =  0 .Clearly, ( 0 )  is a bounded, positive solution of (1) with limit infimum greater than or equal to  > 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  → ∞.

International Journal of Mathematics and Mathematical Sciences
These prior studies (and datasets) are cited at relevant places within the text as references [#-#].