Lyapunov-type inequalities and disconjugacy for some nonlinear difference system

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1 Zhang et al. Advances in Difference Equations 013, 013:16 R E S E A R C H Open Access Lyapunov-type inequalities and disconjugacy for some nonlinear difference system Qi-Ming Zhang 1*,XiaofeiHe and XH Tang * Correspondence: zhqm008008@sina.com 1 College of Science, Hunan University of Technology, Zhuzhou, Hunan 41007, P.R. China Full list of author information is available at the end of the article Abstract 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. MSC: 34D0; 39A99 Keywords: difference system; nonlinearity; Lyapunov-type inequality; generalized zero; disconjugacy 1 Introduction Consider the following nonlinear difference system: x(n)=α(n)x(n +1)+β(n) y(n) y(n), y(n)= γ (n) x(n +1) ν x(n +1) α(n)y(n), (1.1) where, ν >1and =1,α(n), β(n) andγ (n) are real-valued functions defined on Z, ν denotes the forward difference operator defined by x(n)= x(n +1) x(n). Throughout this paper, we always assume that β(n) 0, n Z. (1.) When = ν =,system(1.1)reducestoadiscretelinearhamiltoniansystem x(n)=α(n)x(n +1)+β(n)y(n), y(n)= γ (n)x(n +1) α(n)y(n). (1.3) Similar to 1], we first give the following two definitions. Definition 1.1 1] A function f : Z R is said to have a generalized zero at n 0 Z provided either f (n 0 )=0orf(n 0 )f (n 0 +1)<0. Definition 1. 1] Leta, b Z with a b. A function f : Za, b] R is said to be disconjugate if it has at most a generalized zero on Za, b]; otherwise, it is conjugate on Za, b]. 013 Zhang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Zhangetal. Advances in Difference Equations 013, 013:16 Page of 11 In addition, the definition that system (1.1) is relatively disconjugate here is the same as Definition 4 in 1]. For system (1.1), there are some generalizations and extensions related to Lyapunovtype 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 to1 6] and the references therein. Based on 3]and5], we further discuss system (1.1)inthispaperandestablishsomenew Lyapunov-type inequalities. In 1], 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. Now, we state several relative results in 1]and3]. Theorem 1.3 3] Suppose that (1.) holds and 1 α(n)>0, n Z. (1.4) Let n Za, b] with a b 1.Assume (1.1) has a real solution (x(n), y(n)) satisfying x(a)=0 or x(a)x(a +1)<0; x(b)=0 or x(b)x(b +1)<0; max a+1nb Then one has the following Lyapunov-type inequality: (1.5) x(n) >0. ( ) 1/ ( b 1 1/ν α(n) + β(n) γ (n)) +, (1.6) where, and in what follows, Za, b]=a, a +1,...,b +1,b} for any a, b Z with a < band γ + (n)=maxγ (n), 0}. Theorem 1.4 3] Suppose that (1.) and (1.4) hold and let n Za, b] with a b.assume (1.1) has a real solution (x(n),y(n)) such that x(a)=0or x(a)x(a +1)<0and x(b)=0 and x(n) is not identically zero on a, b]. Then one has the following Lyapunov-type inequality: ( b b 1 ) 1/ ( b 1/ν α(n) + β(n) γ (n)) +. (1.7) Theorem 1.5 1] Suppose that (1.4) holds and β(n)>0, n Za, b]. (1.8) If ( )( b 1 ) β(n)e α (n, τ) γ + (n) 1 (1.9) holds for all τ Za +1,b], then system (1.3) is relatively disconjugate on Za, b +1].

3 Zhangetal. Advances in Difference Equations 013, 013:16 Page 3 of 11 Regarding the discrete exponential function e α (n, τ)of(1.9), we refer the reader to 1]. Lyapunov-type inequalities In this section, we establish some new Lyapunov-type inequalities. Denote n ] ζ (n):= β(τ), (.1) ] η(n):= β(τ), (.) and for λ 0, 1), denote n ] n ] ζ λ (n):= (1 λ)β(a) + β(τ), (.3) b 1 η λ (n):= λβ(a) s=a ] + +1 ] β(τ). (.4) Theorem.1 Suppose that (1.) and (1.4) hold, and let n Za, b] with a b 1.Assume (1.1) has a real solution (x(n), y(n)) such that (1.5) holds. Then one has the following inequality: ζ (n)+η(n) γ + (n) 1. (.5) Proof It follows from (1.5) that there exist ξ 1, ξ 0, 1) such that (1 ξ 1 )x(a)+ξ 1 x(a +1)=0 (.6) and (1 ξ )x(b)+ξ x(b +1)=0. (.7) Multiplying the first equation of (1.1) byy(n) and the second one by x(n +1),andthen adding, we get x(n)y(n) ] = β(n) y(n) γ (n) x(n +1) ν. (.8) Summing equation (.8)froma to b 1, we can obtain x(b)y(b) x(a)y(a)= β(n) y(n) γ (n) x(n +1) ν. (.9) From the first equation of (1.1), we have 1 α(n) ] x(n +1)=x(n)+β(n) y(n) y(n). (.10)

4 Zhangetal. Advances in Difference Equations 013, 013:16 Page 4 of 11 Combining (.10) with(.6), we have ξ 1 β(a) x(a)= y(a) y(a). (.11) 1 (1 ξ 1 )α(a) Similarly, it follows from (.10)and(.7)that ξ β(b) x(b)= y(b) y(b). (.1) 1 (1 ξ )α(b) Substituting (.11)and(.1) into(.9), we have β(n) y(n) γ (n) x(n +1) ν ξ β(b) = y(b) ξ 1 β(a) + y(a), 1 (1 ξ )α(b) 1 (1 ξ 1 )α(a) which implies that (1 ξ 1 )1 α(a)] 1 (1 ξ 1 )α(a) β(a) y(a) + +1 β(n) y(n) ξ β(b) + y(b) 1 (1 ξ )α(b) = γ (n) x(n +1) ν. (.13) Denote that β(a)= (1 ξ 1)1 α(a)] 1 (1 ξ 1 )α(a) β(a), β(b)= ξ β(b) (.14) 1 (1 ξ )α(b) and β(n)=β(n), n Za +1,b 1]. (.15) Then we can rewrite (.13)as β(n) y(n) = γ (n) x(n +1) ν. (.16) From (.10), (.11), (.14)and(.15), we obtain n ] 1 x(n +1)=x(a) + β(τ) n y(τ) ] 1 y(τ) s=a ξ 1 β(a) n = y(a) ] 1 y(a) 1 (1 ξ 1 )α(a) s=a + β(τ) n y(τ) ] 1 y(τ) = β(τ) n y(τ) ] 1, y(τ) n Za, b 1]. (.17)

5 Zhangetal. Advances in Difference Equations 013, 013:16 Page 5 of 11 Similarly, from (.10), (.1), (.14)and(.15), we have Since b 1 x(n +1)=x(b) ] β(τ) y(τ) ] y(τ) ξ β(b) b 1 = y(b) ] y(b) 1 (1 ξ )α(b) = β(τ) y(τ) ] y(τ) β(τ) y(τ) ] y(τ), n Za, b 1]. (.18) 0 β(n) β(n), n Za, b], (.19) it follows from (.1)and(.17) and the Hölder inequality that x(n +1) ν β(τ) y(τ) 1 ( n β(τ) ( n β(τ) = ζ (n) n ] 1 ] ) ν β(τ) y(τ) β(τ) y(τ) ] ν ) β(τ) y(τ), n Za, b 1]. (.0) Similarly, it follows from (.), (.18), (.19) and the Hölder inequality that x(n +1) ν β(τ) y(τ) 1 τ 1 ] ( ) ν ] β(τ) β(τ) y(τ) ( ) ν ] β(τ) β(τ) y(τ) = η(n) From (.0)and(.1), we obtain x(n +1) ν ζ (n)+η(n) β(τ) y(τ), n Za, b 1]. (.1) β(τ) y(τ), n Za, b 1]. (.)

6 Zhangetal. Advances in Difference Equations 013, 013:16 Page 6 of 11 Combining (.) with(.16), we have γ + (n) x(n +1) ν = ζ (n)+η(n) γ + (n) β(n) y(τ) ζ (n)+η(n) γ + (n) γ (n) x(n +1) ν ζ (n)+η(n) γ + (n) γ + (n) x(n +1) ν. (.3) We claim that γ + (n) x(n +1) ν > 0. (.4) If (.4)isnottrue,then γ + (n) x(n +1) ν =0. (.5) From (.16)and(.5), we have 0 β(n) y(n) = γ (n) x(n +1) ν γ + (n) x(n +1) ν =0. (.6) It follows that β(n) y(n) =0, n Za, b]. (.7) Combining (.0) with(.7), we obtain that x(a + 1) = x(a + ) = = x(b) = 0, which together with (.6) impliesthatx(a) = 0. This contradicts (1.5). Therefore, (.4) holds. Hence, it follows from (.3)and(.4)that(.5)holds. In the case x(b)=0,i.e., ξ =0,andso β(b) = 0, we have the following equation: β(n) b y(n) = γ (n) x(n +1) ν (.8) and inequality x(n +1) ν ( b 1 β(τ) y(τ) 1 τ 1 ] β(τ) ) ν ] β(τ) y(τ)

7 Zhangetal. Advances in Difference Equations 013, 013:16 Page 7 of 11 ( b 1 = η 0 (n) β(τ) ) ν ] β(τ) y(τ) β(τ) y(τ), n Za, b 1] (.9) instead of (.16)and(.1), respectively. Similar to the proof of Theorem.1, wehavethe following theorem. Theorem. Suppose that (1.) and (1.4) hold, and let n Za, b] with a b.assume (1.1) has a real solution (x(n), y(n)) such that x(a)=0or x(a)x(a +1)<0and x(b)=0and x(n) is not identically zero on Za, b]. Then one has the following inequality: b ζ (n)η 0 (n) ζ (n)+η 0 (n) γ + (n) 1, (.30) where ζ (n) and η 0 (n) are defined by (.1) and (.4), respectively. Corollary.3 Suppose that (1.) and (1.4) hold, and let n Za, b] with a b 1.Assume (1.1) has a real solution (x(n), y(n)) such that (1.5) holds. Then one has the following inequality: γ + (n) β(τ) β(τ) ] ν b 1 α(n) ], (.31) where, and in the sequel, α(n) ] = min 1 α + (n), 1+α (n) ] 1} and α + (n)=max α(n), 0 }, α (n)=max α(n), 0 }. Proof Since ζ (n)+η(n) ] 1, it follows that 1 ζ (n)+η(n) γ + (n) 1 ] 1 γ + (n) = 1 n γ + ] (n) β(τ) β(τ) ]

8 Zhangetal. Advances in Difference Equations 013, 013:16 Page 8 of 11 1 n γ + (n) β(τ) 1 α + (s) ] b β(τ) 1+α (s) ] 1 γ + (n) β(τ) 1 γ + (n) β(τ) β(τ) ] ν s=a n 1 α + (s) ] ν b 1 1+α (s) ] ν s=a ] ν b 1 β(τ) α(s) ]} ν, (.3) which implies (.31)holds. Since ] 1 β(τ) β(τ) 1 we have the following result. β(n), Corollary.4 Suppose that (1.) and (1.4) hold, and let n Za, b] with a b 1.Assume (1.1) has a real solution (x(n), y(n)) such that (1.5) holds. Then one has the following Lyapunov-type inequality: ( ) 1 ( 1ν β(n) γ (n)) + b 1 α(n) ]} 1. (.33) In a fashion similar to the proofs of Corollaries.3 and.4, we can prove the following corollaries by using Theorem. instead of Theorem.1. Corollary.5 Suppose that (1.) and (1.4) hold, and let n Za, b] with a b.assume (1.1) has a real solution (x(n), y(n)) such that x(a)=0or x(a)x(a +1)<0and x(b)=0and x(n) is not identically zero on Za, b]. Then one has the following inequality: b γ + (n) β(τ) β(τ) ] ν b α(n) ]. (.34) Corollary.6 Suppose that (1.) and (1.4) hold, and let n Za, b] with a b.assume (1.1) has a real solution (x(n), y(n)) such that x(a)=0or x(a)x(a +1)<0and x(b)=0and x(n) is not identically zero on Za, b]. Then one has the following Lyapunov-type inequality: ( b 1 ) 1 ( b 1ν β(n) γ (n)) + b α(n) ]} 1. (.35) Remark.7 While the coefficient α(n) 0and = ν =insystem(1.1), inequality (1.6) of Theorem 1.3 can be derived from inequality (.33). Similarly, forinequality (1.7)ofThe- orem 1.4 and inequality (.35), this result also holds.

9 Zhangetal. Advances in Difference Equations 013, 013:16 Page 9 of 11 On the one hand, when α(n) 0, according to the definition of α(n)], we have α(n) ] = min 1 α + (n), 1+α (n) ] 1} = 1+ α(n) ] 1. (.36) On the other hand, for any u 0, we have ln(1 + u) u (.37) and e u 1 u. (.38) Using above inequalities (.36), (.37)and(.38), we obtain b 1 α(n) ]} 1 = b 1 1+ α(n) ] 1 } 1 b 1 ( =exp ln 1+ α(n) ) ]} 1 } exp 1 α(n) ] 1 1 α(n) = α(n). (.39) Then Remark.7 holds immediately. 3 Disconjugacy criteria Let a, b Z with a b.forsystem(1.1), we develop the following disconjugacy criterion on Za, b]inthissection. Theorem 3.1 Assume that (1.) and (1.4) hold. If ( ) 1 ( 1ν β(n) γ (n)) + < b 1 α(n) ]} 1 (3.1) holds, then system (1.1) is relatively disconjugate on Za, b +1]. Proof Suppose that system (1.1) is not relatively disconjugate on Za, b + 1]. Then there is a real solution (x, y)withx nontrivial and containing two generalized zeros at least. Without loss of generality, we assume that x(a)=0orx(a)x(a + 1) < 0 and the next generalized zero at c Za +,b], i.e., x(c)=0orx(c)x(c + 1) < 0. Hence, applying Corollary.4,wehave ( c ) 1 ( c 1 1ν β(n) γ (n)) + c 1 α(n) ]} 1, (3.) which clearly contradicts (3.1).

10 Zhangetal. Advances in Difference Equations 013, 013:16 Page 10 of 11 When = ν =insystem(1.1), then for system (1.3), the corresponding disconjugate condition (3.1)reducesto b 1 β(n) γ + (n)<4 α(n) ]. (3.3) While α(n)=0,system(1.3)reducesto x(n)=β(n)y(n), y(n)= γ (n)x(n + 1). (3.4) Then disconjugate conditions (1.9)and(3.3)reduceto β(n) γ + (n) 1 (3.5) and β(n) γ + (n)<4, (3.6) respectively. Remark 3. For system (3.4), it is obvious that disconjugate condition (3.5) impliesthat disconjugate condition (3.6) holds, but(3.5) cannot be derived from (3.6). Moreover, condition (1.) is weaker than condition (1.8). It is well-known that the second-order difference equation x(n)+q(n)x(n +1)=0 (3.7) is just a special case of system (3.4) whenβ(n)=1,γ (n)=q(n). And so, we have the following corollary. Corollary 3.3 Assume that (1.) and (1.4) hold. If inequality (3.6) holds, then equation (3.7) is relatively disconjugate on Za, b +1]. Competing interests The authors declare that they have no competing interests. Authors contributions All authors carried out the proof and conceived of the study. All authors read and approved the final manuscript. Author details 1 College of Science, Hunan University of Technology, Zhuzhou, Hunan 41007, P.R. China. School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan , P.R. China. Acknowledgements 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: ) and the Scientific Research Fund of Hunan Provincial Education Department (No: 1B034). Received: 18 July 01 Accepted: 4 January 013 Published: 18 January 013

11 Zhangetal. Advances in Difference Equations 013, 013:16 Page 11 of 11 References 1. Mert, R, Zafer, A: On stability and disconjugacy criteria for discrete Hamiltonian systems. Comput. Math. Appl. 6(8), (011). Tang, XH, Zhang, Q, Zhang, M: Lyapunov-type inequalities for the first-order nonlinear Hamiltonian systems. Comput. Math. Appl. 9, (011) 3. He, X, Zhang, Q: A discrete analogue of Lyapunov-type inequalities for nonlinear difference systems. Comput. Math. Appl., (011) 4. Zhang, Q, He, X, Jiang, J: On Lyapunov-type inequalities for nonlinear dynamic systems on time scales. Comput. Math. Appl. 11, (011) 5. Ünal, M, Cakmak, D, Tiryaki, A: A discrete analogue of Lyapunov-type inequalities for nonlinear systems. Comput. Math. Appl. 55, (008) 6. Zhang, Q, Tang, XH: Lyapunov inequalities and stability for discrete linear Hamiltonian systems. J. Differ. Equ. Appl. 18(9), (01) doi: / Cite this article as: Zhang et al.: Lyapunov-type inequalities and disconjugacy for some nonlinear difference system. Advances in Difference Equations :16.