Research Article Convergence of Variational Iteration Method for Second-Order Delay Differential Equations

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1 Applied Mahemaics Volume 23, Aricle ID 63467, 9 pages hp://dx.doi.org/.55/23/63467 Research Aricle Convergence of Variaional Ieraion Mehod for Second-Order Delay Differenial Equaions Hongliang Liu, Aiguo Xiao, and Lihong Su Hunan Key Laboraory for Compuaion and Simulaion in Science and Engineering and Key Laboraory of Inelligen Compuing and Informaion Processing of Minisry of Educaion, Xiangan Universiy, Xiangan, Hunan 45, China Correspondence should be addressed o Hongliang Liu; lhl@xu.edu.cn Received 24 Ocober 22; Revised 7 December 22; Acceped 7 December 22 Academic Edior: Changbum Chun Copyrigh 23 Hongliang Liu e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. This paper employs he variaional ieraion mehod o obain analyical soluions of second-order delay differenial equaions. The corresponding convergence resuls are obained, and an effecive echnique for choosing a reasonable Lagrange muliplier is designed in he solving process. Moreover, some illusraive examples are given o show he efficiency of his mehod.. Inroducion The second-order delay differenial equaions ofen appear in he dynamical sysem, celesial mechanics, kinemaics, and so forh. Some numerical mehods for solving secondorder delay differenial equaions have been discussed, which include θ-mehod [], rapezoidal mehod [2], and Runge- Kua-Nysröm mehod [3]. The variaional ieraion mehod (VIM) was firs proposed by He [4, 5] and has been exensively applied due o is flexibiliy, convenience, and efficiency. So far, he VIM is applied o auonomous ordinary differenial sysems [6], panograph equaions [7], inegral equaions [8], delay differenial equaions [9], fracional differenial equaions [], he singular perurbaion problems [], and delay differenial-algebraic equaions [2].Rafei e al.[3]and Marinca e al. [4] applied he VIM o oscillaions. Taari and Dehghan [5] consider he VIM for solving second-order iniial value problems. For a more comprehensive survey on his mehod and is applicaions, he readers refer o he review aricles [6 9] and he references herein. Bu he VIM for second-order delay differenial equaions has no been considered. The aricle apply he VIM o second-order delay differenial equaions o obain he analyical or approximae analyical soluions. The corresponding convergence resuls are obained. Some illusraive examples confirm he heoreical resuls. 2. Convergence 2.. The Firs Kind of Second-Order Delay Differenial Equaions. Consider he iniial value problems of second-order delay differenial equaions y () =f(,y(),y(α ())), [, T], y () =φ (), [ τ, ], y () =φ(), [ τ, ], where φ() is a differeniable funcion, α() C [, T] is a sricly monoone increasing funcion and saisfies ha τ α() and α() = τ, here exiss [,T] such ha α( )=,andf:d=[,t] R R Ris a given coninuous mapping and saisfies he Lipschiz condiion f(,u,v) f(,u 2,v) β u u 2, (2) f(,u,v ) f(,u,v 2 ) β v v 2, where β,β are Lipschiz consans; denoes he sandard Euclidean norm. ()

2 2 Applied Mahemaics NowheVIMfor()canread λ (, ξ) [y m (ξ) f(ξ,y m (ξ),φ(α (ξ)))] dξ, << ; (3) Moreover, he general Lagrange muliplier λ (, ξ) =ξ (8) can be readily idenified by (7). Thus, he variaional ieraion formula can be wrien as (ξ )[y m (ξ) f(ξ,y m (ξ),φ(α (ξ)))] dξ, << ; λ (, ξ) [y m (ξ) f(ξ,y m (ξ),φ(α (ξ)))] dξ λ (, ξ) [y m (ξ) f(ξ,y m (ξ),y m (α (ξ)))] dξ, (9) >, (4) where y m () = φ() for [ τ,]; f denoes he resricive variaion, ha is, δ f =.Thus,wehave δ =δy m () δλ (, ξ) [y m (ξ) f(ξ,y m (ξ),φ(α (ξ)))] dξ =δy m () δλ (, ξ) y m (ξ) dξ, <<. Using inegraion by pars o (4), we have δ =δy m () δλ (, ξ) [y m (ξ) f(ξ,y m (ξ),φ(α (ξ)))] dξ δλ (, ξ) [y m (ξ) f(ξ,y m (ξ),y m (α (ξ)))] dξ =δy m () +δλ(, ξ)y m (ξ) ξ= λ (, ξ) δy ξ m (ξ) ξ= 2 λ (, ξ) ξ 2 δy m (ξ) dξ. From he above formula, he saionary condiions are obained as 2 λ (, ξ) ξ 2 =, λ (, ξ) =, ξ ξ= λ (, ξ) =. ξ= (5) (6) (7) (ξ )[y m (ξ) f(ξ,y m (ξ),φ(α (ξ)))] dξ (ξ )[y m (ξ) f(ξ,y m (ξ),y m (α (ξ)))] dξ, >. () Theorem. Suppose ha he iniial value problems () saisfy he condiion (2), andy(), y i () C 2 [, T], i =, 2,.... Then he sequence {y m ()} m= defined by (9) and () wih y () converges o he soluion of (). Proof. From (), we have y () =y() (ξ )[y (ξ) f(ξ,y(ξ),φ(α (ξ)))] dξ, << ; () y () =y() (ξ )[y (ξ) f(ξ,y(ξ),φ(α (ξ)))] dξ (ξ )[y (ξ) f(ξ,y(ξ),y(α (ξ)))] dξ, >. (2) Le E i () = y i () y(), i =,,.... If,henE i () =, i =,,.... From (9)and(), we have E m+ () =E m () (ξ )[E m (ξ) (f(ξ,y m (ξ),φ(α (ξ))) f (ξ, y (ξ),φ(α (ξ))))] dξ, <<. (3)

3 Applied Mahemaics 3 From ()and(2), we have E m+ () =E m () (ξ )[E m (ξ) (f(ξ,y m (ξ),φ(α (ξ))) f(ξ, y (ξ),φ(α (ξ)))) ]dξ (ξ )[E m (ξ) (f(ξ,y m (ξ),y m (α (ξ))) Using inegraion by pars, we have E m+ () =E m () (ξ ) E m (ξ) dξ f(ξ,y(ξ),y(α (ξ)))) ]dξ, (ξ )[f (ξ, y m (ξ),φ(α (ξ))) f(ξ,y(ξ),φ(α (ξ)))] dξ = (ξ )[f (ξ, y m (ξ),φ(α (ξ))) f(ξ,y(ξ),φ(α (ξ)))] dξ, E m+ () =E m () (ξ ) E m (ξ) dξ << ; >. (4) Since [α ()] is bounded, M = max τ ξ α(t) (α (ξ)) is bounded. Moreover, i follows from (2) and he inequaliy ξ T ha E m+ () ξ f(ξ,y m (ξ),φ(α (ξ))) f(ξ,y(ξ),φ(α (ξ))) dξ Tβ y m (ξ) y(ξ) dξ = Tβ E m (ξ) dξ, < < ; E m+ () ξ f(ξ,y m (ξ),φ(α (ξ))) f(ξ,y(ξ),φ(α (ξ))) dξ ξ f(ξ,y m (ξ),y m (α (ξ))) Tβ y m (ξ) y(ξ) dξ f(ξ,y(ξ),y(α (ξ))) dξ T(β E m (ξ) +β E m (α (ξ)) )dξ (6) = Tβ E m (ξ) dξ Tβ E m (α (ξ)) dξ = Tβ E m (ξ) dξ α() Tβ E m (ξ) α( (α (ξ)) dξ ) (ξ )[f (ξ, y m (ξ),φ(α (ξ))) f(ξ,y(ξ),φ(α (ξ)))] dξ (ξ )[f (ξ, y m (ξ),y m (α (ξ))) f(ξ,y(ξ),y(α (ξ)))] dξ = (ξ )[f (ξ, y m (ξ),φ(α (ξ))) f(ξ,y(ξ),φ(α (ξ)))] dξ (ξ )[f (ξ, y m (ξ),y m (α (ξ))) f(ξ,y(ξ),y(α (ξ)))] dξ, >. (5) TMβ E m (ξ) dξ, >, where β=max β i, i=,2.moreover, E m+ () (TMβ)2 (TMβ) 3 (TMβ) 4 s E m (s 2 ) ds 2ds s s (TMβ) m+ s 2 E m 2 (s 3 ) ds 3ds 2 ds s 2 s (7) s 3 E m 3 (s 4 ) ds 4ds 3 ds 2 ds s 2 s m E (s m+ ) ds m+ ds 3 ds 2 ds, (8)

4 4 Applied Mahemaics where E () is consan. Therefore, we have E m+ () E (TMβ) m+ (), (m ). (m+)! (9) 2.2. The Second Kind of Second-Order Delay Differenial Equaions. Consider he iniial value problems of secondorder delay oscillaion differenial equaions y () = ω 2 y () f(,y(),y(α ())), [, T], y () =φ (), [ τ, ], y () =φ(), [ τ, ], (2) where φ() is a differeniable funcion, α() C [, T] is a sricly monoone increasing funcion and saisfies ha τ α() and α() = τ, here exiss [,T] such ha α( )=, ω is a consan, and f:d=[,t] R R R is a given coninuous mapping and saisfies he Lipschiz condiion f(,u,v) f(,u 2,v) κ u u 2, (2) f(,u,v ) f(,u,v 2 ) κ v v 2, where κ,κ are Lipschiz consans. NowheVIMfor(2)canread λ (, ξ) [y m (ξ) +ω2 y m (ξ) + f(ξ,y m (ξ),φ m (α (ξ)))]dξ, << ; λ (, ξ) [y m (ξ) +ω2 y m (ξ) + f(ξ,y m (ξ),φ m (α (ξ)))]dξ λ (, ξ) [y m (ξ) +ω2 y m (ξ) + f(ξ,y m (ξ),y m (α (ξ)))]dξ, (22) >, (23) where y m () = φ() for [ τ,]; f denoes he resricive variaion, ha is, δ f =.Thus,wehave δ =δy m () δλ (, ξ) [y m (ξ) +ω2 y m (ξ) + f(ξ,y m (ξ),φ m (α (ξ)))] dξ =δy m () δλ (, ξ) [y m (ξ) +ω2 y m (ξ)]dξ, <<. (24) Using inegraion by pars o (23), we have δ =δy m () δλ (, ξ) [y m (ξ) +ω2 y m (ξ) + f(ξ,y m (ξ),φ m (α (ξ)))] dξ δλ (, ξ) [y m (ξ) +ω2 y m (ξ) + f(ξ,y m (ξ),y m (α (ξ)))] dξ =δy m () δλ (, ξ) [y m (ξ) +ω2 y m (ξ)]dξ =δy m () λ (, ξ) δy ξ m (ξ) ξ= +λ(, ξ) δy m (ξ) ξ= [ω 2 λ (, ξ) + 2 λ (, ξ) ξ 2 ]δy m (ξ) dξ. (25) From he above formula, he saionary condiions are obained as ω 2 λ (, ξ) + 2 λ (, ξ) ξ 2 =, λ (, ξ) =, ξ ξ= λ(, ξ) =. ξ= Moreover, he general Lagrange muliplier (26) λ (, ξ) = sin ω (ξ ) (27) ω

5 Applied Mahemaics 5 can be readily idenified by (26). Thus, he variaional ieraion formula can be wrien as ω sin ω (ξ )[y m (ξ) +ω2 y m (ξ) +f(ξ,y m (ξ),φ(α (ξ)))]dξ, << ; ω sin ω (ξ )[y m (ξ) +ω2 y m (ξ) +f(ξ,y m (ξ),φ(α (ξ)))]dξ ω sin ω (ξ )[y m (ξ) +ω2 y m (ξ) +f(ξ,y m (ξ),y m (α (ξ)))]dξ, >. (28) Theorem 2. Suppose ha he iniial value problems (2) saisfy he condiion (2), andy(), y i () C 2 [, T], i =,2,... Then he sequence {y m ()} m= defined by (28) wih y () converges o he soluion of (2). Proof. The proof process is similar ha in Theorem The Third Kind of Second-Order Delay Differenial Equaions. In order o improve he ieraion speed, we modify he above ieraive formulas and reconsruc he Lagrange muliplier. Consider he iniial value problems of secondorder delay differenial equaions y () +a() y () +b() y () +N(,y(),y(α ()))=, [, T], y () =φ (), [ τ, ], NowheVIMfor(29)canread λ (, ξ) [y m (ξ) +a(ξ) y m (ξ) +b(ξ) y m (ξ) + N(ξ,y m (ξ),φ(α (ξ)))] dξ, << ; (3) + λ (, ξ) [y m (ξ)+a(ξ) y m (ξ) +b(ξ) y m (ξ) + N(ξ,y m (ξ),φ(α (ξ)))] dξ + λ (, ξ) [y m (ξ) +a(ξ) y m (ξ) +b(ξ) y m (ξ) + N(ξ,y m,y m (α (ξ)))] dξ, >, (32) where y m () = φ() for [ τ,]; N denoes he resricive variaion, ha is, δ N =.Thus,wehave δ =δy m () δλ (, ξ) [y m (ξ) +a(ξ) y m (ξ) +b(ξ) y m (ξ) + N(ξ,y m (ξ),φ(α (ξ)))]dξ =δy m () δλ (, ξ) [y m (ξ)+a(ξ) y m (ξ)+b(ξ) y m (ξ)]dξ, <<. (33) y () =φ(), [ τ, ], (29) Using inegraion by pars o (32), we have where φ() is a differeniable funcion, α() C [, T] is a sricly monoone increasing funcion and saisfies ha τ α() and α() = τ, here exiss [,T] such ha α( ) =, a(), b() are bounded funcions, and N : D = [, T] R R R is a given coninuous mapping and saisfies he Lipschiz condiion N(,u,v) N(,u 2,v) γ u u 2, N(,u,v ) N(,u,v 2 ) γ v v 2, (3) where γ, γ are Lipschiz consans. δ =δy m () δλ (, ξ) [y m (ξ) +a(ξ) y m (ξ) +b(ξ) y m (ξ) + N(ξ,y m (ξ),φ(α (ξ)))] dξ δλ (, ξ) [y m (ξ) +a(ξ) y m (ξ) +b(ξ) y m (ξ) + N(ξ,y m (ξ),y m (α (ξ)))] dξ

6 6 Applied Mahemaics =δy m () δλ (, ξ) [y m (ξ) +a(ξ) y m (ξ) +b(ξ) y m (ξ)]dξ =δy m () λ (, ξ) δdy m (ξ) λ (, ξ) a (ξ) δdy m (ξ) δλ (, ξ) b (ξ) y m (ξ) dξ =( λ (, ξ) ξ +λ(, ξ) δy m (ξ) ξ= +a(ξ) λ (ξ))δy m (ξ) ξ= [ 2 λ (, ξ) (a (ξ) λ (, ξ)) 2 +b(ξ) λ (, ξ)] (ξ) ξ δy m (ξ) dξ. (34) From he above formula, he saionary condiions are obained as 2 λ (, ξ) 2 ξ (a (ξ) λ (, ξ)) ξ λ (, ξ) ξ +b(ξ) λ (, ξ) =, +a(ξ) λ (, ξ) =, ξ= λ(, ξ) ξ= =. (35) We suppose ha λ (, ξ), λ 2 (, ξ) are he fundamenal soluions of (35); he corresponding general soluion of (35) is λ (, ξ) =c λ (, ξ) +c 2 λ 2 (, ξ). (36) Using he iniial condiions of (35), we have λ Noe ha W(, ) = (,) λ 2 (,) of λ (, ξ), λ 2 (, ξ).wehave c λ (, ) +c 2 λ 2 (, ) =, (37) c λ (, ) +c 2λ 2 (, ) =. λ (,) λ 2 (,) is he Wronski deerminan λ (, ξ) = λ 2 (, ) λ (, ξ) +λ (, ) λ 2 (, ξ). (38) W (, ) Using he Liouville formula, we have So λ(, ξ) canbeexpressedas W (, ) =W(, ) e a(ξ)dξ. (39) λ (, ξ) = λ 2 (, ) λ (, ξ) +λ (, ) λ 2 (, ξ) λ (, ) λ 2 (, ) λ (, ) λ 2 (, ) e a(ξ)dξ. (4) Noe ha u (ξ) = λ (, ξ) W (, ), u 2 (ξ) = λ 2 (, ξ) W (, ). (4) Equaion (4) can be expressed as λ (, ξ) = e a(ξ)dξ [u (ξ) u 2 () u 2 (ξ) u ()]. (42) Subsiuing (42) ino(3) and(32), we obain e a(ξ)dξ [u (ξ) u 2 () u 2 (ξ) u ()] [y m (ξ) +a(ξ) y m (ξ) +b(ξ) y m (ξ) +N(ξ,y m (ξ),φ(α (ξ)))]dξ, << ; e a(ξ)dξ [u (ξ) u 2 () u 2 (ξ) u ()] [y m (ξ) +a(ξ) y m (ξ) +b(ξ) y m (ξ) +N (ξ, y m (ξ),φ(α (ξ)))]dξ e a(ξ)dξ [u (ξ) u 2 () u 2 (ξ) u ()] [y m (ξ) +a(ξ) y m (ξ) +b(ξ) y m (ξ) (43) +N(ξ,y m,y m (α (ξ)))]dξ, >. (44) Theorem 3. Suppose ha he iniial value problems (29) saisfy he condiion (3), andy(), y i () C 2 [, T], i =, 2,.... Then he sequence {y m ()} m= defined by (43) and (44) wih y () converges o he soluion of (29). Proof. The proof process is similar o ha in Theorem. 3. Illusraive Examples In his secion, some illusraive examples are given o show he efficiency of he VIM for solving second-order delay differenial equaions. Example 4. Consider he iniial value problem of secondorder differenial equaion wih panograph delay y () = y( 2 ) y2 () + sin 4 () + sin 2 ( ) + 8, >, 2 φ () =, φ () =2, (45)

7 Applied Mahemaics 7 wih he exac soluion y() = (5 cos 2)/2.UsingheVIM given in formulas(9) and(), we consruc he correcion funcional (ξ )(y m (ξ) +y m ( ξ 2 )+y2 m (ξ) sin4 (ξ) sin 2 (ξ) 8) dξ, m =, 2,.... (46) We ake y () = 2 as he iniial approximaion and obain ha y () = cos cos2 6 cos4, y 2 () = sin ( 2 ) + sin sin cos +3cos ( ) + cos cos2 6 cos4, y 3 () = sin ( 5 ) sin ( 2 ) + sin sin cos + 2 cos ( 4 ) cos ( 2 ) cos cos2 6 cos cos 32 2 cos ( 2 ) 2 2 cos ( 4 ),. (47) The exac and approximae soluions are ploed in Figure, which shows ha he mehod gives a very good approximaion o he exac soluion. Example 5. Consider he second-order delay differenial equaion y () = 6y () +y 2 ( 4 ) sin2, >, () Figure : Resuls for Example 4. Using he VIM given in formulas (28), we consruc he correcion funcional 4 sin (4ξ 4) (y m (ξ) + 6y m (ξ) y 2 m ( ξ 4 ) + sin 2 ξ)dξ, m=,2,... (49) We ake y () = 4 as he iniial approximaion, and obain ha y () = sin (4) cos (2) cos (4), y 2 () = o( 4 ) + ( o ( 2 )) sin + ( o ( 2 )) cos sin (2) 2cos 4, φ () =4, φ () =. (48). (5)

8 8 Applied Mahemaics Table : The errors of he ieraion soluions. =. =.5 =. =.5 Ieraive formula (43).948E 9.95E 6.948E E 5 Ieraive formula (9).89278E 5.972E E E 2 Example 6. Consider he second-order delay differenial equaion y () = 2 y () + 6y 2 ( 2 )+6 4, >, φ () =, φ () =, (5) wih he exac soluion y() = 2.From(35), we can solve ha λ(, ξ) = ξ + ξ 2 /. Using he VIM given in formulas (43)and (44), we consruc he correcion funcional ( ξ + ξ2 )(y m (ξ) + 2 ξ y m (ξ) 6y2 m ( ξ 2 ) 6+ξ 4 )dξ, m=,2,... (52) We ake y () = 2 as he iniial approximaion and obain ha y () = , y 2 () = ,. (53) We use he ieraive formulas (9) and(43) for Example 6, respecively. When he ieraion number n = 2, he corresponding relaive errors are showed in Table. Table shows ha he ieraion speed of he ieraive formula (43) for Example 6 is much faser han ha of ieraive formula (9).This demonsraes ha i is imporan o choose a reasonable Lagrange muliplier. 4. Conclusion In his paper, we apply he VIM o obain he analyical or approximae analyical soluions of second-order delay differenial equaions. Some illusraive examples show ha his mehod gives a very good approximaion o he exac soluion. The VIM is a promising mehod for second-order delay differenial equaions. Acknowledgmens The auhors would like o hank he projecs from NSF of China (226322, 273), he Program for Changjiang Scholars and he Innovaive Research Team in he Universiy of China (IRT79), he Aid Program for Science and Technology Innovaive Research Team in Higher Educaional Insiuions of Hunan Province of China, and he Fund Projec of Hunan Province Educaion Office (C2). References []Y.Xu,J.J.Zhao,andM.Z.Liu, TH-sabiliyofθ-mehod for a second-order delay differenial equaion, Mahemaica Numerica Sinica,vol.26,no.2,pp.89 92,24. [2] C.M.HuangandW.H.Li, Delay-dependensabiliyanalysis of he rapezium rule for a class of second-order delay differenial equaions, Mahemaica Numerica Sinica,vol.29,no.2,pp , 27. [3] X.H.Ding,J.Y.Zou,andM.Z.Liu, P-sabiliy of coninuous Runge-Kua-Nysröm mehod for a class of delayed second order differenial equaions, Numerical Mahemaics, vol.27, no. 2, pp , 25. [4] J.-H. He, Variaional ieraion mehod: a kind of non-linear analyical echnique: some examples, Inernaional Non-Linear Mechanics, vol. 34, no. 4, pp , 999. [5] J.-H. He, Some asympoic mehods for srongly nonlinear equaions, Inernaional Modern Physics B, vol.2, no.,pp.4 99,26. [6] J.-H. He, Variaional ieraion mehod for auonomous ordinary differenial sysems, Applied Mahemaics and Compuaion,vol.4,no.2-3,pp.5 23,2. [7] Z.-H. Yu, Variaional ieraion mehod for solving he mulipanograph delay equaion, Physics Leers A, vol.372,no.43, pp , 28. [8] L. Xu, Variaional ieraion mehod for solving inegral equaions, Compuers & Mahemaics wih Applicaions,vol.54,no. 7-8, pp. 7 78, 27. [9] S.-P. Yang and A.-G. Xiao, Convergence of he variaional ieraion mehod for solving muli-delay differenial equaions, Compuers & Mahemaics wih Applicaions, vol.6,no.8,pp , 2. [] S. Yang, A. Xiao, and H. Su, Convergence of he variaional ieraion mehod for solving muli-order fracional differenial equaions, Compuers & Mahemaics wih Applicaions,vol.6, no.,pp ,2. [] Y. Zhao and A. Xiao, Variaional ieraion mehod for singular perurbaion iniial value problems, Compuer Physics Communicaions,vol.8,no.5,pp ,2. [2] H. Liu, A. Xiao, and Y. Zhao, Variaional ieraion mehod for delay differenial-algebraic equaions, Mahemaical & Compuaional Applicaions,vol.5,no.5,pp ,2. [3]M.Rafei,D.D.Ganji,H.Daniali,andH.Pashaei, Thevariaional ieraion mehod for nonlinear oscillaors wih disconinuiies, JournalofSoundandVibraion,vol.35,no.4-5,pp , 27.

9 Applied Mahemaics 9 [4] V. Marinca, N. Herişanu, and C. Boa, Applicaion of he variaional ieraion mehod o some nonlinear one-dimensional oscillaions, Meccanica,vol.43,no.,pp.75 79,28. [5] M. Taari and M. Dehghan, On he convergence of He s variaional ieraion mehod, Compuaional and Applied Mahemaics,vol.27,no.,pp.2 28,27. [6] J.-H. He and X.-H. Wu, Variaional ieraion mehod: new developmen and applicaions, Compuers & Mahemaics wih Applicaions, vol. 54, no. 7-8, pp , 27. [7] J.-H. He, Variaional ieraion mehod some recen resuls and new inerpreaions, Compuaional and Applied Mahemaics,vol.27,no.,pp.3 7,27. [8] J.-H. He, A shor remark on fracional variaional ieraion mehod, Physics Leers A, vol. 375, no. 38, pp , 2. [9] J.-H. He, Asympoic mehods for soliary soluions and compacons, Absrac and Applied Analysis, vol.22,aricleid 96793, 3 pages, 22.

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