Abstract
We obtain necessary and sufficient conditions so that every solution of neutral delay difference equation oscillates or tends to zero as , where and are real sequences and , , and are positive integers. Here is the forward difference operator given by and is an increasing unbounded sequences with . This paper complements, improves, and generalizes some past and recent results.
1. Introduction
Consider the neutral delay difference equation of first orderwhere is the forward difference operator given by , and are members of infinite real sequences, and are positive integers. Further, assume are real sequences for each and that and are monotonic increasing sequences which are unbounded.
We study the oscillatory behavior of solutions of neutral difference equation (1) under the following assumptions.(H1) for .(H2)There exists a bounded sequence such that .(H3)The sequence in (H2) satisfies .(H4), .
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 equationunder the assumptioninstead of (H4). The following neutral difference equations/delay difference equations are obtained as particular case of (2).and
The neutral difference equations (5) are seen as the discrete analogue of the neutral differential equationsThe 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 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 equationwhich has a solution tending to zero. It is because the primary assumption,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 is nondecreasing unlike this paper.
Let be a fixed nonnegative integer. Let and . By a solution of (1) we mean a real sequence which is defined for all positive integers and satisfies (1) for . Clearly if the initial conditionis 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 , there exists such that ; 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.
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 . 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 constant such that the sequences for satisfy the conditionThen every solution of (1) oscillates or tends to zero as .
Proof. Let be any solution of (1) for , where is a fixed positive integer. If it oscillates then there is nothing to prove; otherwise, it leads to two distinct possibilities, either or for . Consider the first one, i.e., eventually. There exits positive integer such that for each , and for . For , letandFrom (1), (13), and (14), it follows due to (H1) thatThen there exists such that is monotonic and is of constant sign for . For the sake of a contradiction assume that is not bounded. Then there exists a subsequence such that andSince as , we may choose large enough so that . For , because of (H3), we can find a positive integer such that implies . As (12) holds, then using (13), (14), and (17) we obtain Taking , we find , a contradiction as is monotonic decreasing. Hence is bounded which implies and are bounded and exists. Further it follows that and exist. We claim . Otherwise, let . Next boundedness of yields . Hence we have , which will be used for bounding the term in (1) from below.
From the continuity of and assumption (H1) it follows that there exists a positive lower bound for on . Hence there exists such that for . Then summing (15) from to 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 . From (H3), monotonic nature of and (14), it follows that exists finitely. Let . If , thena contradiction. If then Hence , by (12), which implies the desired result . If for then proceeding as above we can arrive at . Thus the theorem is proved.
Theorem 2. Suppose that (H1)–(H4) hold. Assume that there exists a positive constant such that the sequences for satisfy the conditionThen 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 . From (H3) and the fact that is monotonic it follows that . As , so . We claim ; if not then , and this impliesHence we getAgain a contradiction, due to inequality (24). Hence we conclude and from , it follows that . Hence .
The proof for the case for large is similar. Hence the theorem is proved.
Remark 3. Theorems 1 and 2 hold good for and . 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 constant such that the sequences for satisfy the conditionThen 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 . From (H3) and that is monotonic it follows that . As , so . We claim . If not, then , and this impliesAgain we haveFrom (27) and (28), it follows that Using (26), we obtain . Thus the theorem is proved.
Next, we intend to present a result where , , satisfy the following condition: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 coefficientsNote that satisfy (30) for the above neutral delay difference equation (31). This neutral delay difference equation has an unbounded solution 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 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 but not for . 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 equationwhere andThe neutral delay difference equation (32) satisfies all the conditions of Theorem 1. As such, it has an oscillatory solution .
Example 9. Consider the first-order inhomogeneous neutral delay difference equationwhere and . This neutral delay difference equation satisfies all the conditions of Theorem 2. As such, it has a bounded positive solution 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].
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 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 . 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. HenceThus, all we need to show is the existence of a bounded solution of (1) with From (H2), we find a positive constant and a positive integer such that Choose a positive constant such that . Since , let Let Then using (37) one can fix such that for it follows that Choose such that Let , Banach space of real bounded sequences with and supremum norm Define Clearly S is a bounded closed and convex subset of X. Now we define two operators and as follows. For , defineFirst we show that if then Hence, for and and for we obtain On the other handHence Thus, we proved that for any Next we show that is a contraction on In fact for and we have This implies A is a contraction because . Next we show that is completely continuous. For this as a first step we show that is continuous. Suppose the sequence in as (with taken from the index set). Since is closed then For we have Since is continuous, therefore as . Hence is continuous. Next what remained to show is is relatively compact. Using the result [14, Theorem 3.3], we need only show that is uniformly cauchy. Let be a sequence in . From (H2) and (37), it follows that, for , there exists such that, for , Then for we have Thus is uniformly cauchy. Hence it is relatively compact. Then by Lemma 11, we can find in such that . Clearly, is a bounded, positive solution of (1) with limit infimum greater than or equal to . 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 .
Proof. Suppose that 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 of (1) with From (H2), we find a positive constant and a positive integer such that By (30), we can find a small positive real , a lower bound for , and upper bounds for ( and ) such that . Let . Hence . Next choose an upper bound for such that . The nonexistence of such an upper bound for would lead to the fact that, for all and . Taking , we have , a contradiction. Choose a real as follows:LetFrom (54) and (55) it follows thatSince , let Let Then using (37), one can fix such that for it follows that Choose such that Let , Banach space of real bounded sequences with and supremum norm Define Clearly S is a bounded closed and convex subset of X. Now we define two operators and as follows. For , defineProceeding as in the proof of above theorem we show that (i) if then by (56) and by (55), so that , (ii) , hence is a contraction on , and (iii) is completely continuous. This completes the proof of the theorem.
Remark 14. For the results in this section, we assume none of conditions (H3), is nondecreasing, and , 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., Theorems 1, 2, 4, 6, 12, and 13, we obtain the following theorem.
Theorem 15. Suppose that (H1)-(H3) hold. Assume 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 .
Remark 16. The results of this work hold for and , i.e., for the linear homogeneous equation associated with (1).
Data Availability
Previously reported data were used to support this study and are available at [DOI or OTHER PERSISTENT IDENTIFIER]. These prior studies (and datasets) are cited at relevant places within the text as references [#-#].
Disclosure
This work is done for the Ph.D. thesis work of the second author.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are thankful to Professor Prayag Prasad Mishra for his valuable guidance during the completion of this paper.