Oscillations in Difference Equations with Deviating Arguments and Variable Coefficients

Abstract

New sufficient conditions for the oscillation of all solutions of difference equations with several deviating arguments and variable coefficients are presented. Examples illustrating the results are also given.

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G. E. Chatzarakis 1 and H. Péics 2 and I. P. Stavroulakis 3

Academic Editor:Patricia J. Y. Wong

1, Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), 14121 N. Heraklio, Athens, Greece
2, Faculty of Civil Engineering, University of Novi Sad, 24000 Subotica, Serbia
3, Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 19 May 2014; Accepted 21 July 2014; 18 August 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper we study the oscillation of all solutions of difference equation with several variable retarded arguments of the form [figure omitted; refer to PDF] and the (dual) difference equation with several variable advanced arguments of the form [figure omitted; refer to PDF] where m ∈ N , ( _{p i} ( n ) ) , 1 ...4; i ...4; m , are sequences of positive real numbers and ( _{τ i} ( n ) ) , 1 ...4; i ...4; m , are sequences of integers such that [figure omitted; refer to PDF] and ( _{σ i} ( n ) ) , 1 ...4; i ...4; m , are sequences of integers such that [figure omitted; refer to PDF] Here, Δ denotes the forward difference operator Δ x ( n ) = x ( n + 1 ) - x ( n ) and ∇ denotes the backward difference operator ∇ x ( n ) = x ( n ) - x ( n - 1 ) .

Strong interest in ( _{E R} ) is motivated by the fact that it represents a discrete analogue of the differential equation (see [1-3] and the references cited therein) [figure omitted; refer to PDF] where, for every i ∈ { 1 , ... , m } , _{p i} is a continuous real-valued function in the interval [ 0 , ∞ ) and _{τ i} is a continuous real-valued function on [ 0 , ∞ ) such that [figure omitted; refer to PDF] while ( _{E A} ) represents a discrete analogue of the advanced differential equation (see [1, 2] and the references cited therein) [figure omitted; refer to PDF] where, for every i ∈ { 1 , ... , m } , _{p i} is a continuous real-valued function in the interval [ 1 , ∞ ) and _{σ i} is a continuous real-valued function on [ 1 , ∞ ) such that [figure omitted; refer to PDF]

By a solution of ( _{E R} ) , we mean a sequence of real numbers ( x ( n ) _{) n ...5; - w} which satisfies ( _{E R} ) for all n ...5; 0 . Here [figure omitted; refer to PDF] It is clear that, for each choice of real numbers _{c - w} , _{c - w + 1} , ... , _{c - 1} , _{c 0} , there exists a unique solution ( x ( n ) _{) n ...5; - w} of ( _{E R} ) which satisfies the initial conditions x ( - w ) = _{c - w} , x ( - w + 1 ) = _{c - w + 1} , ... , x ( - 1 ) = _{c - 1} , x ( 0 ) = _{c 0} .

By a solution of ( _{E A} ) , we mean a sequence of real numbers _{( x ( n ) ) n ...5; 0} which satisfies ( _{E A} ) for all n ...5; 1 .

A solution ( x ( n ) _{) n ...5; - w} (or _{( x ( n ) ) n ...5; 0} ) of ( _{E R} ) (or ( _{E A} ) ) is called oscillatory, if the terms x ( n ) of the sequence are neither eventually positive nor eventually negative. Otherwise, the solution is said to be nonoscillatory .

In the last few decades, the oscillatory behavior of the solutions of difference and differential equations with several deviating arguments and variable coefficients has been studied. See, for example, [1-14] and the references cited therein.

In 2006, Berezansky and Braverman [5] proved that if [figure omitted; refer to PDF] where τ ( n ) = _{max ... 1 ...4; i ...4; m τ i} ( n ) , for all n ...5; 0 , then all solutions of ( _{E R} ) oscillate.

Recently, Chatzarakis et al. [7-9] established the following theorems.

Theorem 1 (see [9]).

Assume that the sequences ( _{τ i} ( n ) ) [ ( _{σ i} ( n ) ) ] , 1 ...4; i ...4; m , are increasing, (1) [ (2) ] holds, and [figure omitted; refer to PDF] where τ ( n ) = _{max ... 1 ...4; i ...4; m τ i} ( n ) , for all n ...5; 0 , [ σ ( n ) = _{min ... 1 ...4; i ...4; m σ i} ( n ) , for all n ...5; 1 ] , or [figure omitted; refer to PDF] then all solutions of ( _{E R} ) [ ( _{E A} ) ] oscillate.

Theorem 2 (see [7, 8]).

Assume that the sequences ( _{τ i} ( n ) ) [ ( _{σ i} ( n ) ) ] , 1 ...4; i ...4; m , are increasing and (1) [ (2) ] holds. Set [figure omitted; refer to PDF] If 0 < α ...4; 1 / e , and [figure omitted; refer to PDF] or [figure omitted; refer to PDF] then all solutions of ( _{E R} ) [ ( _{E A} ) ] oscillate.

The authors study further ( _{E R} ) and ( _{E A} ) and derive new sufficient oscillation conditions. These conditions are the improved and generalized discrete analogues of the oscillation conditions for the corresponding differential equations, which were studied in 1982 by Ladas and Stavroulakis [2]. Examples illustrating the results are also given.

2. Oscillation Criteria

2.1. Retarded Difference Equations

We present new sufficient conditions for the oscillation of all solutions of ( _{E R} ) .

Theorem 3.

Assume that ( _{τ i} ( n ) ) , 1 ...4; i ...4; m , are increasing sequences of integers such that (1) holds and ( _{p i} ( n ) ) , 1 ...4; i ...4; m , are sequences of positive real numbers and define _{α i} , 1 ...4; i ...4; m , by (11). If _{α i} > 0 , 1 ...4; i ...4; m , and [figure omitted; refer to PDF] then all solutions of ( _{E R} ) oscillate.

Proof.

Assume, for the sake of contradiction, that ( x ( n ) _{) n ...5; - w} is a nonoscillatory solution of ( _{E R} ) . Then it is either eventually positive or eventually negative. As ( - x ( n ) _{) n ...5; - w} is also a solution of ( _{E R} ) , we may restrict ourselves only to the case where x ( n ) > 0 for all large n . Let _{n 1} ...5; - w be an integer such that x ( n ) > 0 for all n ...5; _{n 1} . Then, there exists _{n 2} ...5; _{n 1} such that [figure omitted; refer to PDF] In view of this, ( _{E R} ) becomes [figure omitted; refer to PDF] which means that the sequence ( x ( n ) ) is eventually decreasing.

Next choose a natural number _{n 3} > _{n 2} such that [figure omitted; refer to PDF] Set [figure omitted; refer to PDF] It is obvious that [figure omitted; refer to PDF] Now we will show that _{[straight phi] i} < ∞ for i = 1,2 , ... , m . Indeed, assume that _{[straight phi] i} = ∞ for some i , i = 1,2 , ... , m . For this i , by ( _{E R} ) , we have [figure omitted; refer to PDF]

At this point, we will establish the following claim.

Claim 1 (cf. [8]). For each n ...5; _{n 3} , there exists an integer ^{n i *} ...5; n for each i = 1,2 , ... , m such that _{τ i} ( ^{n i *} ) ...4; n - 1 , and [figure omitted; refer to PDF] where [varepsilon] is an arbitrary real number with 0 < [varepsilon] < _{α i} .

To prove this claim, let us consider an arbitrary real number [varepsilon] with 0 < [varepsilon] < _{α i} . Then by (11) we can choose an integer _{n 3} ...5; _{n 2} such that [figure omitted; refer to PDF] Assume, first, that _{p i} ( n ) ...5; ( _{α i} - [varepsilon] ) / 2 and choose ^{n i *} = n . Then _{τ i} ( ^{n i *} ) = _{τ i} ( n ) ...4; n - 1 . Moreover, we have [figure omitted; refer to PDF] and, by (23), [figure omitted; refer to PDF] So, (21) and (22) are fulfilled. Next, we suppose that _{p i} ( n ) < ( _{α i} - [varepsilon] ) / 2 . It is not difficult to see that (23) guarantees that ^{∑ j = 0 ∞} _{p i} ( j ) = ∞ . In particular, it holds [figure omitted; refer to PDF] Thus, as _{p i} ( n ) < ( _{α i} - [varepsilon] ) / 2 , there always exists an integer ^{n i *} > n so that [figure omitted; refer to PDF] and (21) holds. We assert that _{τ i} ( ^{n i *} ) ...4; n - 1 . Otherwise, _{τ i} ( ^{n i *} ) ...5; n . We also have _{τ i} ( ^{n i *} ) ...4; ^{n i *} - 1 . Hence, in view of (27), we get [figure omitted; refer to PDF] On the other hand, (23) gives [figure omitted; refer to PDF] We have arrived at a contradiction, which shows our assertion that _{τ i} ( ^{n i *} ) ...4; n - 1 . Furthermore, by using (23) (for the integer ^{n i *} ) as well as (27), we obtain [figure omitted; refer to PDF] and consequently (22) holds true. Our claim has been proved.

Now, summing up (20) from n to ^{n i *} , we find [figure omitted; refer to PDF] or [figure omitted; refer to PDF] Summing up (20) from _{τ i} ( ^{n i *} ) to n - 1 , we find [figure omitted; refer to PDF] or [figure omitted; refer to PDF] Combining (32) and (34), we obtain [figure omitted; refer to PDF] or [figure omitted; refer to PDF] which means that ( _{z i} ( n ) ) is bounded. This contradicts our assumption that _{[straight phi] i} = ∞ . Therefore _{[straight phi] i} < ∞ for every i = 1,2 , ... , m .

Dividing both sides of ( _{E R} ) by x ( n ) , for n ...5; _{n 3} , we obtain [figure omitted; refer to PDF] or [figure omitted; refer to PDF] Summing up (38) from _{τ ρ} ( n ) to n - 1 for ρ = 1,2 , ... , m , we find [figure omitted; refer to PDF] But [figure omitted; refer to PDF] or [figure omitted; refer to PDF] Combining (39) and (41), we obtain [figure omitted; refer to PDF] or [figure omitted; refer to PDF] Taking limit inferiors on both sides of the above inequalities (43), we obtain [figure omitted; refer to PDF] and by adding we find [figure omitted; refer to PDF] Set [figure omitted; refer to PDF] Clearly [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] for [figure omitted; refer to PDF] the function f has a maximum at the critical point [figure omitted; refer to PDF] since the quadratic form [figure omitted; refer to PDF] Since f ( _{[straight phi] 1} , _{[straight phi] 2} , ... , _{[straight phi] m} ) ...5; 0 , the maximum of f at the critical point should be nonnegative. Thus, [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] or [figure omitted; refer to PDF] which contradicts (14).

The proof of the theorem is complete.

Theorem 4.

Assume that ( _{τ i} ( n ) ) , 1 ...4; i ...4; m , are increasing sequences of integers such that (1) holds and ( _{p i} ( n ) ) , 1 ...4; i ...4; m , are sequences of positive real numbers and define _{α i} , 1 ...4; i ...4; m , by (11). If _{α i} > 0 , 1 ...4; i ...4; m , and [figure omitted; refer to PDF] then all solutions of ( _{E R} ) oscillate.

Proof.

Assume, for the sake of contradiction, that ( x ( n ) _{) n ...5; - w} is a nonoscillatory solution of ( _{E R} ) . Then it is either eventually positive or eventually negative. As ( - x ( n ) _{) n ...5; - w} is also a solution of ( _{E R} ) , we may restrict ourselves only to the case where x ( n ) > 0 for all large n . Let _{n 1} ...5; - w be an integer such that x ( n ) > 0 for all n ...5; _{n 1} . Then, there exists _{n 2} ...5; _{n 1} such that [figure omitted; refer to PDF] In view of this, ( _{E R} ) becomes [figure omitted; refer to PDF] which means that the sequence ( x ( n ) ) is eventually decreasing.

Taking into account the fact that _{[straight phi] i} < ∞ for i = 1,2 , ... , m (see proof of Theorem 3), by using (44) and the fact that [figure omitted; refer to PDF] we obtain [figure omitted; refer to PDF] Adding these inequalities we have [figure omitted; refer to PDF] or [figure omitted; refer to PDF] which contradicts (56).

The proof of the theorem is complete.

2.2. Advanced Difference Equations

Similar oscillation theorems for the (dual) advanced difference equation ( _{E A} ) can be derived easily. The proofs of these theorems are omitted, since they follow a similar procedure as in Section 2.1.

Theorem 5.

Assume that ( _{σ i} ( n ) ) , 1 ...4; i ...4; m , are increasing sequences of integers such that (2) holds and ( _{p i} ( n ) ) , 1 ...4; i ...4; m , are sequences of positive real numbers and define _{α i} , 1 ...4; i ...4; m , by (11). If _{α i} > 0 , 1 ...4; i ...4; m , and [figure omitted; refer to PDF] then all solutions of ( _{E A} ) oscillate.

Theorem 6.

Assume that ( _{σ i} ( n ) ) , 1 ...4; i ...4; m , are increasing sequences of integers such that (2) holds and ( _{p i} ( n ) ) , 1 ...4; i ...4; m , are sequences of positive real numbers and define _{α i} , 1 ...4; i ...4; m , by (11). If _{α i} > 0 , 1 ...4; i ...4; m , and [figure omitted; refer to PDF] then all solutions of ( _{E A} ) oscillate.

2.3. Special Cases

In the case where _{p i} , i = 1,2 , ... , m , are positive real constants and _{τ i} are constant retarded arguments of the form _{τ i} ( n ) = n - _{k i} , [ _{σ i} are constant advanced arguments of the form _{σ i} ( n ) = n + _{k i} ], _{k i} ∈ N , i = 1,2 , ... , m , equation ( _{E R} ) [ ( _{E A} ) ] takes the form [figure omitted; refer to PDF] For this equation, as a consequence of Theorems 3 [5] and 4 [6], we have the following corollary.

Corollary 7.

Assume that [figure omitted; refer to PDF] or [figure omitted; refer to PDF] Then all solutions of ( E ) oscillate.

Remark 8.

A research question that arises is whether Theorems 3-6 are valid, even in the case where the coefficients p ( n ) oscillate (see [15, 16]). Then our results would be comparable to those in [15, 16]. This is a question that we currently study and expect to have some results soon.

3. Examples

The following two examples illustrate that the conditions for oscillations (65) and (66) are independent. They are chosen in such a way that only one of them is satisfied.

Example 1.

Consider the retarded difference equation [figure omitted; refer to PDF] Here m = 3 , _{τ 1} ( n ) = n - 1 , _{τ 2} ( n ) = n - 2 , _{τ 3} ( n ) = n - 3 , and [figure omitted; refer to PDF] It is easy to see that [figure omitted; refer to PDF] That is, condition (65) of Corollary 7 is satisfied and therefore all solutions of equation (67) oscillate.

However, [figure omitted; refer to PDF] That is, condition (66) of Corollary 7 is not satisfied.

Observe that [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Also, [figure omitted; refer to PDF] and therefore none of the conditions (9), (12), (13), (8), and (10) are satisfied.

Example 2.

Consider the advanced difference equation [figure omitted; refer to PDF] Here m = 2 , _{σ 1} ( n ) = n + 1 , _{σ 2} ( n ) = n + 2 , and [figure omitted; refer to PDF] It is easy to see that [figure omitted; refer to PDF] That is, condition (66) of Corollary 7 is satisfied and therefore all solutions of (74) oscillate.

However, [figure omitted; refer to PDF] That is, condition (65) of Corollary 7 is not satisfied.

Observe that [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Also, [figure omitted; refer to PDF] and therefore none of the conditions (9), (12), and (13) are satisfied.

At this point, we give an example with general retarded arguments illustrating the main result of Theorem 3. Similarly, one can construct examples to illustrate Theorems 4-6.

Example 3.

Consider the delay difference equation [figure omitted; refer to PDF] with c = 17 / 50 .

Here _{τ 1} ( n ) = [ 0.5 n ] and _{τ 2} ( n ) = [ 0.2 n ] denote the integer parts of 0.5 n and 0.2 n . Observe that the sequences _{τ 1} ( n ) and _{τ 2} ( n ) are increasing, _{lim ... n [arrow right] ∞ τ 1} ( n ) = + ∞ , _{lim ... n [arrow right] ∞ τ 2} ( n ) = + ∞ , and [figure omitted; refer to PDF] Observe that, for a positive decreasing function f ( x ) , the following inequality holds: [figure omitted; refer to PDF] Based on the above inequality, we will show that [figure omitted; refer to PDF] for any a , b ∈ N , a < b , and any real number Q . Indeed, [figure omitted; refer to PDF] It is easy to see that [figure omitted; refer to PDF] From the above, it follows that [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] That is, condition (14) of Theorem 3 is satisfied and therefore all solutions of (81) oscillate.

Observe, however, that [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Also, [figure omitted; refer to PDF] and therefore none of the conditions (8), (9), (12), and (13) are satisfied.

Acknowledgment

The second author was supported by the Serbian Ministry of Science, Technology and Development for Scientific Research Grant no. III44006.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] N. Fukagai, T. Kusano, "Oscillation theory of first order functional-differential equations with deviating arguments," Annali di Matematica Pura ed Applicata. Serie Quarta , vol. 136, pp. 95-117, 1984.

[2] G. Ladas, I. P. Stavroulakis, "Oscillations caused by several retarded and advanced arguments," Journal of Differential Equations , vol. 44, no. 1, pp. 134-152, 1982.

[3] B. Li, "Oscillation of first order delay differential equations," Proceedings of the American Mathematical Society , vol. 124, no. 12, pp. 3729-3737, 1996.

[4] O. Arino, I. Györi, A. Jawhari, "Oscillation criteria in delay equations," Journal of Differential Equations , vol. 53, no. 1, pp. 115-123, 1984.

[5] L. Berezansky, E. Braverman, "On existence of positive solutions for linear difference equations with several delays," Advances in Dynamical Systems and Applications , vol. 1, no. 1, pp. 29-47, 2006.

[6] L. Berezansky, E. Braverman, "Positive solutions for a scalar differential equation with several delays," Applied Mathematics Letters , vol. 21, no. 6, pp. 636-640, 2008.

[7] G. E. Chatzarakis, T. Kusano, I. P. Stavroulakis, "Oscillation conditions for difference equations with several variable," to appear in Mathematica Bohemica

[8] G. E. Chatzarakis, J. Manojlovic, S. Pinelas, I. P. Stavroulakis, "Oscillation criteria of difference equations with several deviating arguments," to appear in Yokohama Mathematical Journal

[9] G. E. Chatzarakis, S. Pinelas, I. P. Stavroulakis, "Oscillations of difference equations with several deviated arguments," to appear in Aequationes Mathematicae

[10] M. K. Grammatikopoulos, R. Koplatadze, I. P. Stavroulakis, "On the oscillation of solutions of first order differential equations with retarded arguments," Georgian Mathematical Journal , vol. 10, no. 1, pp. 63-76, 2003.

[11] B. R. Hunt, J. A. Yorke, "When all solutions of ^{x '} = ^{∑ j = 1 n} _{q j} t x t - _{T j} t oscillate," Journal of Differential Equations , vol. 53, no. 2, pp. 139-145, 1984.

[12] J. Jaros, I. P. Stavroulakis, "Necessary and sufficient conditions for oscillations of difference equations with several delays," Utilitas Mathematica , vol. 45, pp. 187-195, 1994.

[13] X. N. Luo, Y. Zhou, C. F. Li, "Oscillation of a nonlinear difference equation with several delays," Mathematica Bohemica , vol. 128, no. 3, pp. 309-317, 2003.

[14] X. H. Tang, R. Y. Zhang, "New oscillation criteria for delay difference equations," Computers & Mathematics with Applications , vol. 42, no. 10-11, pp. 1319-1330, 2001.

[15] L. Berezansky, G. E. Chatzarakis, A. Domoshnitsky, I. P. Stavroulakis, "Oscillations of difference equations with several oscillating coefficients," Abstract and Applied Analysis , vol. 2014, 2014.

[16] M. Bohner, G. E. Chatzarakis, I. P. Stavroulakis, "Qualitative behavior of solutions of difference equations with several oscillating coefficients," Arabian Journal of Mathematics , vol. 3, no. 1, pp. 1-13, 2014.