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Lyapunov-type inequalities and disconjugacy for some nonlinear difference system
Advances in Difference Equations volume 2013, Article number: 16 (2013)
We establish several new Lyapunov-type inequalities for some nonlinear difference system when the end-points are not necessarily usual zeros, but rather generalized zeros. Our results generalize in some sense the known ones. As an application, we develop disconjugacy criteria by making use of the obtained inequalities.
Consider the following nonlinear difference system:
where and , , and are real-valued functions defined on ℤ, △ denotes the forward difference operator defined by . Throughout this paper, we always assume that
When , system (1.1) reduces to a discrete linear Hamiltonian system
Similar to , we first give the following two definitions.
Definition 1.1 
A function is said to have a generalized zero at provided either or .
Definition 1.2 
Let with . A function is said to be disconjugate if it has at most a generalized zero on ; otherwise, it is conjugate on .
In addition, the definition that system (1.1) is relatively disconjugate here is the same as Definition 4 in .
For system (1.1), there are some generalizations and extensions related to Lyapunov-type inequalities; and for recent work in the literature on discrete and continuous cases as well as its special case, i.e., system (1.3), we refer to [1–6] and the references therein. Based on  and , we further discuss system (1.1) in this paper and establish some new Lyapunov-type inequalities. In , the authors have gained many interesting results about Lyapunov-type inequalities and their applications for system (1.3) including developing several disconjugacy criteria by adopting some techniques. As an application, we also develop disconjugacy criteria in the last section. Moreover, we make a comparison with some existing ones.
Theorem 1.3 
Suppose that (1.2) holds and
Let with . Assume (1.1) has a real solution satisfying
Then one has the following Lyapunov-type inequality:
where, and in what follows, for any with and .
Theorem 1.4 
Suppose that (1.2) and (1.4) hold and let with . Assume (1.1) has a real solution such that or and and is not identically zero on . Then one has the following Lyapunov-type inequality:
Theorem 1.5 
Suppose that (1.4) holds and
holds for all , then system (1.3) is relatively disconjugate on .
Regarding the discrete exponential function of (1.9), we refer the reader to .
2 Lyapunov-type inequalities
In this section, we establish some new Lyapunov-type inequalities.
and for , denote
Theorem 2.1 Suppose that (1.2) and (1.4) hold, and let with . Assume (1.1) has a real solution such that (1.5) holds. Then one has the following inequality:
Proof It follows from (1.5) that there exist such that
Multiplying the first equation of (1.1) by and the second one by , and then adding, we get
Summing equation (2.8) from a to , we can obtain
From the first equation of (1.1), we have
Combining (2.10) with (2.6), we have
Similarly, it follows from (2.10) and (2.7) that
Substituting (2.11) and (2.12) into (2.9), we have
which implies that
Then we can rewrite (2.13) as
From (2.10), (2.11), (2.14) and (2.15), we obtain
Similarly, from (2.10), (2.12), (2.14) and (2.15), we have
it follows from (2.1) and (2.17) and the Hölder inequality that
Similarly, it follows from (2.2), (2.18), (2.19) and the Hölder inequality that
From (2.20) and (2.21), we obtain
Combining (2.22) with (2.16), we have
We claim that
If (2.24) is not true, then
From (2.16) and (2.25), we have
It follows that
Combining (2.20) with (2.27), we obtain that , which together with (2.6) implies that . This contradicts (1.5). Therefore, (2.24) holds. Hence, it follows from (2.23) and (2.24) that (2.5) holds. □
In the case , i.e., , and so , we have the following equation:
instead of (2.16) and (2.21), respectively. Similar to the proof of Theorem 2.1, we have the following theorem.
Theorem 2.2 Suppose that (1.2) and (1.4) hold, and let with . Assume (1.1) has a real solution such that or and and is not identically zero on . Then one has the following inequality:
where and are defined by (2.1) and (2.4), respectively.
Corollary 2.3 Suppose that (1.2) and (1.4) hold, and let with . Assume (1.1) has a real solution such that (1.5) holds. Then one has the following inequality:
where, and in the sequel,
it follows that
which implies (2.31) holds. □
we have the following result.
Corollary 2.4 Suppose that (1.2) and (1.4) hold, and let with . Assume (1.1) has a real solution such that (1.5) holds. Then one has the following Lyapunov-type inequality:
In a fashion similar to the proofs of Corollaries 2.3 and 2.4, we can prove the following corollaries by using Theorem 2.2 instead of Theorem 2.1.
Corollary 2.5 Suppose that (1.2) and (1.4) hold, and let with . Assume (1.1) has a real solution such that or and and is not identically zero on . Then one has the following inequality:
Corollary 2.6 Suppose that (1.2) and (1.4) hold, and let with . Assume (1.1) has a real solution such that or and and is not identically zero on . Then one has the following Lyapunov-type inequality:
Remark 2.7 While the coefficient and in system (1.1), inequality (1.6) of Theorem 1.3 can be derived from inequality (2.33). Similarly, for inequality (1.7) of Theorem 1.4 and inequality (2.35), this result also holds.
On the one hand, when , according to the definition of , we have
On the other hand, for any , we have
Using above inequalities (2.36), (2.37) and (2.38), we obtain
Then Remark 2.7 holds immediately.
3 Disconjugacy criteria
Let with . For system (1.1), we develop the following disconjugacy criterion on in this section.
Theorem 3.1 Assume that (1.2) and (1.4) hold. If
holds, then system (1.1) is relatively disconjugate on .
Proof Suppose that system (1.1) is not relatively disconjugate on . Then there is a real solution with x nontrivial and containing two generalized zeros at least. Without loss of generality, we assume that or and the next generalized zero at , i.e., or . Hence, applying Corollary 2.4, we have
which clearly contradicts (3.1). □
When in system (1.1), then for system (1.3), the corresponding disconjugate condition (3.1) reduces to
While , system (1.3) reduces to
Then disconjugate conditions (1.9) and (3.3) reduce to
Remark 3.2 For system (3.4), it is obvious that disconjugate condition (3.5) implies that disconjugate condition (3.6) holds, but (3.5) cannot be derived from (3.6). Moreover, condition (1.2) is weaker than condition (1.8).
It is well-known that the second-order difference equation
is just a special case of system (3.4) when , . And so, we have the following corollary.
Corollary 3.3 Assume that (1.2) and (1.4) hold. If inequality (3.6) holds, then equation (3.7) is relatively disconjugate on .
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The authors thank the referees for valuable comments and suggestions which improved the presentation of this manuscript. This study was partially supported by the National Natural Science Foundation of China (No: 11201138) and the Scientific Research Fund of Hunan Provincial Education Department (No: 12B034).
The authors declare that they have no competing interests.
All authors carried out the proof and conceived of the study. All authors read and approved the final manuscript.
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Zhang, QM., He, X. & Tang, X. Lyapunov-type inequalities and disconjugacy for some nonlinear difference system. Adv Differ Equ 2013, 16 (2013). https://doi.org/10.1186/1687-1847-2013-16
- difference system
- Lyapunov-type inequality
- generalized zero