二阶半线性中立型差分方程的振动性准则

262Vol.26,No.2 20065JOURNAL OF MATHEMATICAL RESEARCH AND EXPOSITION May,2006 Article ID:1000-341X(2006)02-0247-06Document code:AOscillation Criteria for Second-Order Semi-Linear NeutralDifference EquationsKANG Guo-lian(Academy of Mathematics and Systems Science,C.A.S.,Beijing100080,China)(E-mail:glkang@)Abstract:Consider the second-order semi-linear neutral difference equation∆[a n|∆(x n+p n x n−τ)|α−1∆(x n+p n x n−τ)]+ki=1q i(n)|x n−σi|α−1x n−σi=0.(1)The sufficient conditions are established for oscillation of the solutions of(1).These results generalize and improve some known results about both neutral and delay difference equation.Key words:Semi-linear;neutral difference equation;oscillation.MSC(2000):39A11CLC number:O1751.IntroductionIn the paper,we consider the semi-linear second-order neutral difference equation∆[a n|∆(x n+p n x n−τ)|α−1∆(x n+p n x n−τ)]+ki=1q i(n)|x n−σi|α−1x n−σi=0,(1)where n=1,2,3,···,αis a positive constant,andτand{σi}k i=1are nonnegative integers.∆is the usual forward difference operator.Throughout this paper,we assume that (h1)α≥1,0≤p n<1for n=0,1,2,···.(h2){q n}is a nonnegative sequence with infinitely many positive terms.(h3)a n>0,n=0,1,2,···,and ∞1/a1/αn=∞.A solution{x n}of(1)is defined for n≥−max{τ,σi,i=1,2,···,k}and satisfies(1)for n=1,2,3,···.A solution{x n}of(1)is said to be oscillatory if for every N>0,there exists an n≥N such that x n x n+1≤0.Otherwise,it is nonoscillatory.Most of the previous studies on the oscillation theory of(1)have been restricted to the case in whichα=1,p n=0and a n=1[1−4].We note that the following equation is related to the continuous version of(1)[a(t)|(x(t)+p(t)x(t−τ)) |α−1(x(t)+p(t)x(t−τ)) ] +q(t)|x(t−σ)|α−1x(t−σ)=0.248Journal of Mathematical Research and Exposition Vol.26(4)|z n−σ|α−1z n−σsatisfies the following Riccati inequality:α∆w n+No.2KANG Guo-lian:Oscillation criteria for second-order semi-linear neutral Diff.Eq.249zαn−σ−αa n+1(∆z n+1)αξα−1∆z n−σa1/αn−σ(∆z n+1 (1+α)1+1/α.Thenk(1+x)1+1/α≥x for x≥−1,where the equality holds if and only if k=α(1+α)1+1/α, such that∞s=n1a1/αs−σw1+1/αs+1+ls=n+1k i=1q i(s)(1−p s−σi)α≤w n+1−w l.(11)If(i)holds,then w l→−∞as l→+∞.This contradicts the fact that w n>0.If(ii)holds,by (11),we have∞s=n+1α250Journal of Mathematical Research and Exposition Vol.26a1/αs−σC1+1/αs+1,u(2)(n)=∞s=n+1αa1/αs−σ[C s+1+u(m)(s)]1+1/α,m=1,2,·(13)It is obvious that0<u(1)(n)≤u(2)(n)≤···u(m)(n)≤u(m+1)(n)≤···.By(12),we haveu(1)(n)+C n+1≤w n+1.(14)Suppose thatu(m)(n)+C n+1≤w n+1.(15) From(12),(14)and(15),we obtainu(m+1)(n)+C n+1≤∞s=n+1αa1/αs−σC1+1/αs+1≥kC n+1,where k=αρandu(2)(n)=∞ s=n+1αNo.2KANG Guo-lian:Oscillation criteria for second-order semi-linear neutral Diff.Eq.251(α+1)1+1/α.Thus the assumption thatlim m →∞d m =c is impossible.By (17)and (19)the sequence {u (m )(n )}can not be convergent.This contradicts (16)and so the proof is completed.Corollary 1Assume that conditions (h 1)–(h 3)and i =1hold and that(i) ∞q n (1−p n −σ)α=+∞,or(ii) ∞q n (1−p n −σ)α<+∞and there exists a positive constant ρ>1a 1/αs −σ∞m =s +1q m (1−p m −σ)α1+1/α≥ρ∞m =n +1q m (1−p m −σ)α.Then every solution of (1)is oscillatory.By taking α=1,a n =1in Corollary 1we obtainCorollary 2Assume that (h 1)–(h 3)hold and that(i) ∞q s (1−p s −σ)=∞,or(ii) ∞q s (1−p s −σ)<∞and there exists a positive constant ρ>1/4such that∞ s =n [∞ m =s +1q m (1−p m −σ)]2≥ρ∞m =n +1q m (1−p m −σ).Then,every solution of Equation (1)is oscillatory.Remark 1Corollary 1can be considered as discrete analogues of Theorem 1given in [5]forthe neutral delay equation[a (t )|(x (t )+p (t )x (t −τ)) |α−1(x (t )+p (t )x (t −τ)) ] +q (t )|x (t −σ)|α−1x (t −σ)=0.Remark 2When k =1,α=1and a n =1,Equation (1)reduces to∆2(x n +p n x n −τ)+q n x n −σ=0.(21)Hence Corollary 2is an extension of Theorem 2.5in [1].But we weakened the conditions0≤p n ≤p <1and ∞q n =+∞.To author’s knowledge,the results are even new for Equation (21).4.Some applicationsIn this section,we indicate some applications of our results.These applications are given as examples.Example 1Consider the neutral difference equation∆2(x n +(1−1252Journal of Mathematical Research and Exposition Vol.26n2x n−σ=0,(23) whereτ,σ>0,0≤p<1andδ>1s2(1−p)=δ(1−p)∞s=n+11s(s+1)=δ(1−p)1(s+1)2>C n+1/3.Choose a constantρ=(1/4,1/3).Then,the conditions of Corollary2are satisfied and therefore every solution of Equation(23)is oscillatory.But in Equation(23),the condition ∞q n=∞does not justify the oscillation of Equation(23).References:[1]LUO J W,BAINOV D D.Oscillatory and asymptotic behavior of second order neutral difference equationswith maxima[J]put.Appl.Math.,2001,131:333–341.[2]Jiang Jian-chu,Oscillatory criteria for second-order quasilinear neutral delay difference equations[J].Appl.put.,2002,125:287–293.[3]KANG Guo-lian,ZHANG Hui.Oscillation criteria of solutions of nonlinear difference equations of secondorder[J].Ann.Differential Equations,2004,20(1):41–48.[4]KANG Guo-lian.Oscillation criteria for second-order nonlinear difference equations with“Summation Small”coefficient[J].Bulletin of K.M.S.,2005,42(2):245–256.[5]LI Xiao-ping,JIANG Jian-chu.Oscillation of second order nonlinear neutral delay differential equations[J].Chinese Quart.J.Math.,2001,16(4):43–48.(100080):∆[a n|∆(x n+p n x n−τ)|α−1∆(x n+p n x n−τ)]+ki=1q i(n)|x n−σi|α−1x n−σi=0.(1)(1) :。