Research Article Nonexistence of Homoclinic Solutions for a Class of Discrete Hamiltonian Systems

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1 Abstract ad Applied Aalysis Volume 203, Article ID 39868, 6 pages Research Article Noexistece of Homocliic Solutios for a Class of Discrete Hamiltoia Systems Xiaopig Wag Departmet of Mathematics, Xiaga College, Chezhou, Hua , Chia Correspodece should be addressed to Xiaopig Wag; wxp345@sia.com Received 3 September 202; Accepted 27 November 202 Academic Editor: Jide Cao Copyright 203 Xiaopig Wag. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We give several sufficiet coditios uder which the first-order oliear discrete Hamiltoia system Δx) = α)x + ) + β) y) μ 2 y), Δy) = γ) x + ) ν 2 x + ) α)y) has o solutio x), y)) satisfyig coditio 0 < = [ x) ν + + β)) y) μ ]<,whereμ, ν > ad + = ad α), β), ad γ) are real-valued fuctios defied o Z.. Itroductio I 907, Lyapuov []established the first so-called Lyapuov iequality: if Hill s equatio has a real solutio xt) such that b b a) q t) dt > 4, ) a x t) +qt) x t) =0 2) x a) =xb) =0, xt) 0, t [a, b], 3) ad the costat 4 i )caotbereplacedbyalargerumber, where qt) is a piecewise cotiuous ad oegative fuctio defied o R. Sice this result has foud applicatios i the study of various properties of solutios such as oscillatio theory, discojugacy, ad eigevalue problems of 2), a large umber of Lyapuov-type iequalities were established i the literature which geeralized or improved ); see [ 20]. I 983, Cheg [3] first obtaied the discrete aalogy of Lyapuov iequality ) for the secod-order differece equatio: Δ 2 x ) +q) x +) =0, 4) where, ad i the sequel, Δ deotes the forward differece operator defied by Δx) = x + ) x). Whe a = ad b =, that is, system 4) hasa solutio x) satisfyig lim x) = 0, whichiscalled homocliic solutio, whether oe ca obtai Lyapuov-type iequalities for 4)? To the best of our kowledge, there are o results. I 2003, Sh. Guseiov ad Kaymakçala [7] partly geeralized the Cheg s result to the discrete liear Hamiltoia system: Δx ) =α) x +) +β) y ), 5) Δy ) = γ) x +) α) y ), where α), β), adγ) are real-valued fuctios defied o Z ad a ad b are ot ecessarily usual zeros, but rather, geeralized zeros. Later, some better Lyapuov-type iequalities for system 5)wereobtaiedi[9, 20]. Very recetly, He ad Zhag [0]furthergeeralizedthe result i [9] to the followig first-order oliear differece system: Δx ) =α) x +) +β) y ) μ 2 y ), Δy ) = γ) x +) ν 2 x +) α) y ), where μ, ν > ad + = ad α), β),adγ) are real-valued fuctios defied o Z. 6)

2 2 Abstract ad Applied Aalysis Whe μ=ν=2,system6) reducesto5). I additio, the special forms of system 6) cotai may well-kow differece equatios which have bee studied extesively ad have much applicatios i the literature [2 23], such as the secod-order liear differece equatio: Δ[p) Δx )]+q) x +) =0, 7) ad the secod-order half-liear differece equatio: Δ[p) Δx ) r 2 Δx )]+q) x +) r 2 x +) =0, 8) where r>, p) ad q) are real-valued fuctios defied o Z ad p) > 0.Let y ) =p) Δx ) r 2 Δx ), 9) the 8)cabewritteastheformof6): Δx ) =[p)] / r) y ) 2 r)/r ) y ), Δy ) = q) x +) r 2 x +), 0) where μ = r/r ), ν=rad α) = 0, β) = [p)] / r) ad γ) = q). I this paper, we will establish several Lyapuov-type iequalities for systems 5) ad6) iftheyhaveasolutio x), y)) satisfyig coditios [ x ) 2 ++β)) y ) 2 ]<, ) [ x ) ν + +β)) y ) μ ] <, 2) respectively. Takig advatage of these Lyapuov-type iequalities, we are able to establish some criteria for oexistece of homocliic solutios of systems 5) ad 6). As we kow, there are o results o o-existece of homocliic solutios for Hamiltoia systems i previous literature. 2. Lyapuov-Type Iequalities for System 6) I this sectio, we shall establish some Lyapuov-type iequalities for system 6). For the sake of coveiece, we list some assumptios o α) ad β) as follows: A0) α) <,forall Z, [ αs)] < ; A) α) <,forall Z, αs) < ; B0) β) ) 0,forall Z; B) 0 βτ)0 [ αs)] μ + τ= βτ) s=0 [ αs)]μ <. Deote ζ ) := [ η ) := [ τ=+ s=+ ν/μ [ αs)] μ ], [ αs)] μ ] ν/μ. 3) Theorem. Suppose that hypotheses A0), B0), ad B) are system 6) has a solutio x), y)) satisfyig [ x ) ν + +β)) y ) μ ] <, 4) = the oe has the followig iequality: = where γ + ) = max{γ), 0}. ζ ) η ) ζ ) +η) γ+ ), 5) Proof. Hypothesis B) implies that fuctios ζ) ad η) are well defied o Z. Without loss of geerality, we ca assume that = From 4) ad B0), oe has ζ ) η ) ζ ) +η) γ+ ) <. 6) lim x ) = lim y ) =0, 7) y τ) μ <. 8) It follows from 3), 8), ad the Hölder iequality that y τ) μ [ αs)] =[ζ)] [ <, Z, [ αs)] μ ] [ y τ) μ ] y τ) μ [ αs)] τ=+ s=+ τ=+ =[η)] [ <, s=+ τ=+ Z. [ αs)] μ ] [ y τ) μ ] τ=+ y τ) μ ] 9) y τ) μ ] 20)

3 Abstract ad Applied Aalysis 3 From A0), 7), 9), 20), ad the first equatio of system 6), we have x +) = τ=+ y τ) μ 2 y τ) [ αs)], x +) = y τ) μ 2 y τ) [ αs)], Combiig 9)with2), oe has x +) ν = ζ) s=+ Z, 2) Z. y τ) μ 2 y τ) [ αs)] y τ) μ, Z. Similarly, it follows from 20) ad22) that x +) ν = τ=+ η) y τ) μ 2 y τ) τ=+ Combiig 23)with24), oe has x +) ν ζ ) η ) ζ ) +η) s=+ y τ) μ, Z. Now, it follows from 6), 8), ad 25)that By 6), we obtai γ + ) x +) ν = = = [ αs)] y τ) μ, Z. ζ ) η ) ζ ) +η) γ+ )] β ) y ) μ <. 22) ν 23) ν 24) 25) 26) Δx) y )) =β) y ) μ γ) x +) ν. 27) Summig the above from to ad usig 7)ad8), we obtai γ ) x +) ν = β ) y ) μ, 28) = = which, together with 26), implies that γ + ) x +) ν = =[ = = = ζ ) η ) ζ ) +η) γ+ )] ζ ) η ) ζ ) +η) γ+ )] ζ ) η ) ζ ) +η) γ+ )] = = = β ) y ) μ γ ) x +) ν γ + ) x +) ν. 29) We claim that γ + ) x +) ν >0. 30) = If 30)isottrue,the γ + ) x +) ν =0. 3) = From 28)ad3), we have 0 β ) y ) μ = γ ) x +) ν = = It follows that γ + ) x +) ν =0. = 32) β ) y ) μ 2 y ) 0, Z. 33) Combiig 2)with33), we obtai that x ) 0, Z, 34) which, together with the secod equatio of system 6), implies that Δy ) = α) y ), Z. 35) Combiig the above with 7), oe has y ) 0, Z. 36) Both 34) ad36) cotradictwith4). Therefore, 30) holds. Hece, it follows from 29)ad30)that5)holds. Corollary 2. Suppose that hypotheses A), B0), ad B) are system 6) has a solutio x), y)) satisfyig 4), the oe has the followig iequality: = γ + ) [ τ=+ ] 2 = {Θ [α )]} ν/2, 37)

4 4 Abstract ad Applied Aalysis where ad i the sequel, Θ [α )] = mi { α + ),[+α )] }, α + ) = max {α ),0}, α ) = max { α ),0}. 38) Proof. Obviously, A) implies that [ αs)] <, 39) ad so A0) holds, ad which, together with B), implies that βτ) <.Sice it follows that ζ ) η ) ζ ) +η) γ+ ) = 2 = = = = = = ζ ) +η) 2[ζ) η )] /2, 40) [ζ ) η )] /2 γ + ) γ + ) { γ + ) { γ + ) [ τ=+ τ=+ [ αs)] μ s=+ [ αs)] μ } [ α + s)] μ s=+ τ=+ [ + α s)] μ } ] [ α + s)] ν/2 [ + α s)] ν/2 γ + ) [ {Θ [α s)]} ν/2, which implies that 37)holds. Sice [ τ=+ s=+ b τ=+ /2 ] ] = 4) β ), 42) the it follows from 37)thatthefollowigcorollaryistrue. Corollary 3. Suppose that hypotheses A), B0), ad B) are system 6) has a solutio x), y)) satisfyig 4),the = β )) = γ + )) 2 = {Θ [α )]} /2. 43) Applyig Theorem ad Corollary 2 to system 8) i.e., 0)), we have immediately the followig two corollaries. Corollary 4. Suppose that r>ad p) > 0 for Z,ad that If 8) has a solutio x) satisfyig the = <. /r ) 44) [p τ)] [ x ) r +p) +[p )] /r ) ) Δx ) r ]<, = 45) { { r [p τ)] /r ) } +{ { τ=+ r [p τ)] /r ) } τ=+ r [p τ)] /r ) } r [p τ)] /r ) } ) )q + ). 46) Corollary 5. Suppose that r>ad p) > 0 for Z,ad that 44) holds. If 8) has a solutio x) satisfyig 45),the = q + ) { [p τ)] /r ) τ=+ r )/2 [p τ)] /r ) } Lyapuov-Type Iequalities for System 5) 47) Whe μ = ν = 2, assumptio B) reduces the followig form: B2) 0 βτ)0 [ αs)] 2 + τ= βτ) s=0 [ αs)]2 <. Applyig the results obtaied i last sectio to the firstorder liear Hamiltoia system 5), we have immediately the followig corollaries.

5 Abstract ad Applied Aalysis 5 Corollary 6. Suppose that hypotheses A0), B0), ad B2) are system 5) has a solutio x), y)) satisfyig the = [ x ) 2 ++β)) y ) 2 ]<, 48) = { { τ=+ [ αs)] 2 } + τ=+ s=+ [ αs)] 2 } [ αs)] 2 s=+ [ αs)] 2 ) )γ + ). 49) Corollary 7. Suppose that hypotheses A), B0), ad B2) are system 5) has a soldutio x), y)) satisfyig 48),the = γ + ) [ τ=+ /2 ] 2 = Θ [α )]. Corollary 8. Suppose that p) > 0 for Z,adthat If 7) has a solutio x) satisfyig the = 50) <. 5) p τ) [ x ) 2 +p) +p)) Δx ) 2 ] <, = 52) q + ) [ p τ) τ=+ p τ) ] = 4. Noexistece of Homocliic Solutios p ). 53) Applyig the results obtaied i Sectios 2 ad 3, we ca drive the followig criteria for o-existece of homocliic solutios of systems 5)ad6)immediately. Corollary 9. Suppose that hypotheses A0), B0), ad B) are = ζ ) η ) ζ ) +η) γ+ ) <, 54) the system 6) has o solutio x), y)) satisfyig [ x ) ν ++β)) y ) μ ]<. 55) = Corollary 0. SupposethathypothesesA),B0),adB)are = γ + ) [ τ=+ ] <2 = {Θ [α )]} ν/2, thesystem 6) has o solutio x), y)) satisfyig 55). 56) Corollary. Suppose that hypotheses A), B0), ad B) are = β )) = γ + )) <2 = {Θ [α )]} /2, thesystem 6) has o solutio x), y)) satisfyig 55). 57) Corollary 2. Suppose that hypotheses A0), B0), ad B2) are = { { τ=+ + [ αs)] 2 } τ=+ s=+ [ αs)] 2 } [ αs)] 2 s=+ [ αs)] 2 ) )γ + ) <, the system 5) has o solutio x), y)) satisfyig 58) [ x ) 2 ++β)) y ) 2 ]<. 59) = Corollary 3. Suppose that hypotheses A), B0), ad B2) are = γ + ) [ τ=+ /2 ] <2 = Θ [α )], thesystem 5) has o solutio x), y)) satisfyig 59). 60)

6 6 Abstract ad Applied Aalysis Corollary 4. Suppose that p) > 0 for Z,adthat5) holds. If q + ) = p τ) τ=+p τ) ) < the 7) has o solutio x) satisfyig 52). = p ), 6) Example 5. Cosider the secod-order differece equatio: Δ[+ 2 )Δx)]+q) x +) =0, 62) where q) is real-valued fuctio defied o Z. I view of Corollary 4,if = [ +τ 2 τ=+ +τ 2 )] q+ ) < = + 2, 63) the 62)hasosolutiox) satisfyig = [ x ) ) 2 Δx ) 2 ] <. 64) Ackowledgmet ThisworkissupportedbytheScietificResearchFudof Hua Provicial Educatio Departmet 07A066). Refereces [] A. M. Lyapuov, Probléme gééral de la stabilité du mouvemet, Ade La Faculté,vol.2,o.9, pp ,907. [2] M. Boher, S. Clark, ad J. Ridehour, Lyapuov iequalities for time scales, Joural of Iequalities ad Applicatios, vol.7, o., pp. 6 77, [3] S.S.Cheg, AdiscreteaalogueoftheiequalityofLyapuov, Hokkaido Mathematical Joural,vol.2,o.,pp.05 2,983. [4] S.-S. Cheg, Lyapuov iequalities for differetial ad differece equatios, Polytechica Posaiesis, o. 23, pp. 25 4, 99. [5] S. Clark ad D. Hito, A Liapuov iequality for liear Hamiltoia systems, Mathematical Iequalities & Applicatios,vol.,o.2,pp ,998. [6] S. Clark ad D. Hito, Discrete Lyapuov iequalities, Dyamic Systems ad Applicatios,vol.8,o.3-4,pp , 999. [7] G. Sh. Guseiov ad B. Kaymakçala, Lyapuov iequalities for discrete liear Hamiltoia systems, Computers & Mathematics with Applicatios,vol.45,o.6 9,pp ,2003. [8] G. Sh. Guseiov ad A. Zafer, Stability criteria for liear periodic impulsive Hamiltoia systems, JouralofMathematical Aalysis ad Applicatios,vol.335,o.2,pp ,2007. [9] P. Hartma, Differece equatios: discojugacy, pricipal solutios, Gree s fuctios, complete mootoicity, Trasactios of the America Mathematical Society, vol.246,pp. 30, 978. [0] X. He ad Q.-M. Zhag, A discrete aalogue of Lyapuovtype iequalities for oliear differece systems, Computers & Mathematics with Applicatios, vol.62,o.2,pp , 20. [] L. Jiag ad Z. Zhou, Lyapuov iequality for liear Hamiltoia systems o time scales, Joural of Mathematical Aalysis ad Applicatios,vol.30,o.2,pp ,2005. [2] S. H. Li ad G. S. Yag, O discrete aalogue of Lyapuov iequality, Tamkag Joural of Mathematics,vol.20,o.2,pp , 989. [3] X. Wag, Stability criteria for liear periodic Hamiltoia systems, Joural of Mathematical Aalysis ad Applicatios,vol. 367,o.,pp ,200. [4] X.-H. Tag ad M. Zhag, Lyapuov iequalities ad stability for liear Hamiltoia systems, Joural of Differetial Equatios,vol.252,o.,pp ,202. [5] X. H. Tag, Q.-M. Zhag, ad M. Zhag, Lyapuov-type iequalities for the first-order oliear Hamiltoia systems, Computers & Mathematics with Applicatios, vol.62,o.9,pp , 20. [6] A. Tiryaki, M. Üal, ad D. Çakmak, Lyapuov-type iequalities for oliear systems, Joural of Mathematical Aalysis ad Applicatios, vol. 332, o., pp , [7] M. Üal, D. Çakmak, ad A. Tiryaki, A discrete aalogue of Lyapuov-type iequalities for oliear systems, Computers & Mathematics with Applicatios,vol.55,o.,pp , [8] M. Üal ad D. Çakmak, Lyapuov-type iequalities for certai oliear systems o time scales, Turkish Joural of Mathematics, vol. 32, o. 3, pp , [9] Q.-M. Zhag ad X. H. Tag, Lyapuov iequalities ad stability for discrete liear Hamiltoia systems, Applied Mathematics ad Computatio,vol.28,o.2,pp ,20. [20] Q.-M. Zhag ad X. H. Tag, Lyapuov iequalities ad stability for discrete liear Hamiltoia systems, Joural of Differece Equatios ad Applicatios, vol.8,o.9,pp , 202. [2] R. Agarwal, C. Ahlbradt, M. Boher, ad A. Peterso, Discrete liear Hamiltoia systems: a survey, Dyamic Systems ad Applicatios,vol.8,o.3-4,pp ,999. [22] C. D. Ahlbradt ad A. C. Peterso, Discrete Hamiltoia Systems, vol.6ofkluwer Texts i the Mathematical Scieces, Kluwer Academic, Dordrecht, The Netherlads, 996. [23] S. N. Elaydi, A Itroductio to Differece Equatios, Spriger, New York, NY, USA, 3rd editio, 2004.

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