Oscillation and Property B for Third Order Difference Equations with Advanced Arguments

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1 Iter atioal Joural of Pure ad Applied Mathematics Volume 3 No , ISSN: (prited versio); ISSN: (o-lie versio) url: ijpam.eu Oscillatio ad Property B for Third Order Differece Equatios with Advaced Argumets R.Sriivasa ad C.Dharuma 2,2 Departmet of Mathematics, SRM Uiversity, Ramapuram Campus, Cheai , Idia. sriimaths986@gmail.com cdharuma55@gmail.com. February 9, 207 Abstract I this paper we preset sufficiet coditios for the third order oliear advaced differece equatio have property B or to be oscillatory. The results obtaied are improve kow results for delay differece equatios ad for ordiary differece equatios. Examples are provided to illustrate the mai results. AMS Subject Classificatio: 39A0 Key Words ad Phrases:Third order differece equatio, property B, advaced argumet, oscillatio. Itroductio Cosider the oliear third order differece equatio (a ( (b x )) α ) p f(x σ() ) = 0, 0, () where 0 is a oegative iteger, where α is quotiet of odd positive itegers, a }, b } ad p } are positive real sequeces, ad ijpam.eu

2 σ()} is a odecreasig sequece of itegers with σ() for all 0, f is a real valued cotiuous odecreasig fuctio, uf(u) > 0 for u 0, ad f(uv) f(u)f(v) for uv > 0, ad = 0 a /α = = 0 b =. By a solutio of equatio (), we mea a real sequece x } which is defied for all 0 ad satisfies equatio (). A otrivial solutio of equatio () is called ooscillatory if it is either evetually positive or evetually egative, ad otherwise it is called oscillatory. The study of oscillatory properties of third order ad higher order liear differece equatios received less attetio, see for example, 2] ad the refereces cited therei. The ivestigatio of property B for the liear third order differece equatio 3 x p y σ() = 0 (2) with σ() = + discussed i 7]. By property B of equatio (2) it is meat that every positive solutio x } of equatio (2) is strogly icreasig, that is x > 0, 2 x 0, 3 x 0. I 3 6], the authors preseted sharp criteria for the qualitative properties of solutios of third order differece equatios with delay argumet. I this paper, we exted the techique that yields property B of equatio (2) to equatio (). Thus the results essetially improve ad complemet may kow results for both delay ad ordiary differece equatios. 2 Prelimiary Results I this sectio, first we classify the possible ooscillatory solutios of equatio () which follows from the discrete Keser s theorem ]. We itroduce the followig classes of ooscillatory (let us say positive) solutios of equatio (): x S x > 0, (b x ) < 0, (a (b x )) > 0, ad x S 3 x > 0, (b x ) > 0, (a (b x )) > 0, for all N 0. ijpam.eu

3 Lemma. Assume that x } is a evetually positive solutio of equatio (), the x } S or x } S 3. I the followig we derive some importat mootoic properties ad estimates of ooscillatory solutios, that will be used i the mai results. Deote A = ad P = Lemma 2. (), ad a /α a /α s (, B =, C = b s s b s t=n p s ) α, for N is large eough. a /α t Let x } S 3 be a positive solutio of equatio p f(c σ() ) =. (3) The x C } is evetually icreasig for all N. Lemma 3. If x } S be a positive solutio of equatio (), the x B } is evetually decreasig. The proof of last two lemmas are similar to that of i 6]. 3 Criteria for Property B I this sectio, we provide several criteria for the class S of equatio () to be empty. If u Theorem 4. Assume that lim u ± = k f /α (u) <, ad ] /α p t =, (4) b a /α s f /α ( σ() + t=s ) f /α (B σ(s) )B s+ P s B s+ P s + =σ() P > k ijpam.eu

4 the equatio () has property B. Proof. Assume equatio () has a positive solutio x } S for all N. Summatio of equatio () from to twice yields b x a /α /α p s f(x σ(s) )] = f /α (x σ() )P, where we have used the mootoicity of f. Summig agai from N to, ad the chagig the order of summatio we have x f /α (x σ(s) )P s B s+ + B f /α (x σ(s) )P s. Usig the mootoicities of x } ad x B }, we obtai x σ() f /α (x σ() ) ( ) f /α f /α (B σ(s) )P s B s+ σ() + P s B s+ + s=σ() P s. (5) From the assumptio (4), we see that x as. Takig as o both sides of the iequality (5), we get a cotradictio ad the proof is ow complete. Theorem 5. Assume that lim u 0 u f /α (u) = k 2 <, ad a /α /α p s f(b σ(s) )] =, (6) If + f /α ( ) f /α (B σ(s) )B s+ P s σ() B s+ P s + f /α ( ) =σ() P > k 2, the equatio () has property B. ijpam.eu

5 Proof. Assume that x } is a evetually positive solutio of equatio () such that x } S for N. By Lemma 3, the sequece x B } is decreasig ad we shall show that (6) implies If ot, the lim x B f(x σ() ) = f x lim = 0. (7) B = d > 0. The x B d ad therefore ( ) xσ() f(d)f( ). Further, summig equatio () twice yields b N x N a /α /α p s f(x σ(s) )] f(d) a /α /α p s f(b σ(s) )], which cotradicts (6), ad we coclude that (7) holds. Set z = x σ(), coditio (5) implies z f /α (z ) f /α (B σ(s) )P s B s+ + f /α ( ) +f /α ( ) s=σ() σ() P s B s+ P s. (8) Takig as i (8), we have a cotradictio with the assumptios of the theorem. The proof is ow complete. Corollary 6. B β/α σ() Let coditio (4) holds ad σ() B β/α σ(s) B s+p s + the the superliear differece equatio has property B. B s+ P s + =σ() (a ( (b x )) α ) p x β σ() = 0, β > α (9) P > 0 ijpam.eu

6 Corollary 7. B σ() Let coditio (4) holds ad σ() B σ(s) B s+ P s + the the halfliear differece equatio has property B. Corollary 8. B s+ P s + s=σ() (a ( (b x )) α ) p x α σ() = 0 (0) Let coditio (4) holds ad B β/α σ(s) B s+p s the the subliear differece equatio has property B. + Bβ/α σ() σ() B s+ P s + B β/α σ(s) P s > =σ() (a ( (b x )) α ) p x β σ() = 0, α > β () P > 0 4 Oscillatio Results I this sectio, we preset oscillatio criteria for equatio (). Theorem 9. Let coditio (3) holds. Assume that lim u ± u f /α (u) = k 3 <. (2) If σ() C σ() b s t=s a /α t the the class S 3 = ϕ for equatio (). /α p j f(c σ(j) )] > k 3, j=t ijpam.eu

7 Proof. Assume that equatio () has a evetually positive solutio x } S 3 for N. Summig () from to s yields ( (b s x s )) α s p t f a s t= ( ) ( ) xσ(t) xσ(t) C σ(t) f C σ(t) C σ(t) a s s p t Summig the last iequality twice i s, ad the settig s = σ() ad z = x σ() C σ(), we obtai z f /α (z ) C σ() σ() t= b t j=t a /α j ( i=j ) /α p i. Takig as o both sides of the last iequality, we get a cotradictio ad this completes the proof. By usig S ad S 3 are empty we have the followig theorem. Theorem 0. Let all coditios of Theorem 4 (Theorem 5) ad Theorem 9 are hold. The every solutio of equatio () is oscillatory. 5 Examples I this sectio we preset three examples to illustrate the mai results. Example 5.. Cosider the differece equatio t= ( ( 2 x ) /3 ) 4/3 (2 + ) x /3 +4 = 0,. (3) ( + ) Simple computatio show that A 4, B 4, C The it is easy to see that all coditios of Corollary 7 ad Theorem 0 are satisfied ad hece every solutio of equatio (3) is oscillatory. I fact x } = ( ) } is oe such oscillatory solutio of (3). Example 5.2. ( 3 ( ( x )) 3 ) Cosider the differece equatio ( + ) x +4 = 0,. (4) ijpam.eu

8 Simple computatio show that A 2, B 2 2, C ad p =. It is easy to see that all coditios of Corollary 8 (+) are hold ad hece the equatio (4) has property B. Example 5.3. ( ( x )) Cosider the differece equatio ( + ) x3 +5 = 0,. (5) Simple computatio show that A = ( ), B 2 = ( ), C 2 = ( )(3 2 5+), ρ 6 =. Oe ca easily verify that all coditios of Corollary 6 are hold ad hece equatio (5) has property B. Coclusio: The results established i this paper are of high geerality ad improve ad complemet earlier results kow for special cases of equatio (). Refereces ] R.P.Agarwal, Differece equatios ad Iequalities, Theory, Methods ad Applicatios, Secod Editio, Marcel Dekker, New York, ] R.P.Agarwal, M.Boher, S.R.Grace, D.O Rega, Discrete Oscillatio Theory, Hidawi, New York, ] R.Grace, R.P.Agarwal ad J.R.Graef, Oscillatio criteria for certai third order oliear differece equatios, Appl.Aal.Discrete Math., 3(2009), ] J.Graef ad E.Thadapai, Oscillatory ad asymptotic behavior of solutios of third order delay differece equatios, Fuk.Ekvac., 42(999), ] S.H.Saker, Oscillatio of third order differece equatios, Port.Math., 6(2004). 6] S.H.Saker, J.O.Alzabut ad A.Mukheimer, O the oscillatory behavior ofor a certai class of third order oliear delay differece equatios, Elec.J.Qual.Theo. Diff.Eq., 67(200), -6. 7] B. Smith, Oscillatory ad asymptotic behavior i certai third order differece equatios, Rocky Moutai J. Math. 7 (987), ijpam.eu

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