Oscillation criteria for third-order nonlinear neutral differential equations with distributed deviating arguments

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1 Avilble online t J. Nonliner Sci. Appl. 9 (206), Reserch Article Oscilltion criteri for third-order nonliner neutrl differentil equtions with distributed deviting rguments Cuimei Jing, Tongxing Li b,c, Qingdo Technologicl University, Feixin, Shndong , P. R. Chin. b LinD Institute of Shndong Provincil Key Lbortory of Network Bsed Intelligent Computing, Linyi University, Linyi, Shndong , P. R. Chin. c School of Informtics, Linyi University, Linyi, Shndong , P. R. Chin. Communicted by X. Z. Liu Abstrct The im of this pper is to investigte the oscilltion nd symptotic behvior of clss of thirdorder nonliner neutrl differentil equtions with distributed deviting rguments. By mens of Riccti trnsformtion technique nd some inequlities, we estblish severl sufficient conditions which ensure tht every solution of the studied eqution is either oscilltory or converges to zero. Two exmples re provided to illustrte the min results. c 206 All rights reserved. Keywords: Oscilltion, symptotic behvior, third-order neutrl differentil eqution, distributed deviting rgument. 200 MSC: 34K.. Introduction During the pst few decdes, n incresing interest in obtining sufficient conditions for oscilltory nd nonoscilltory behvior of different clsses of differentil equtions hs been stimulted due to their pplictions in nturl sciences nd engineering (see Hle 9 nd Wong 24). This resulted in publiction of severl monogrphs,, numerous reserch ppers 2 6, 8, 0, 2 23, 25, 26, nd the references cited Corresponding uthor Emil ddresses: jingcuimei2004@63.com (Cuimei Jing), litongx2007@63.com (Tongxing Li) Received

2 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), therein. Anlysis of qulittive properties of neutrl differentil equtions is importnt not only for the ske of further development of the oscilltion theory, but for prcticl resons too. In fct, neutrl differentil equtions re used in modeling of networks contining lossless trnsmission lines (see, for instnce, the pper by Driver 7). In wht follows, let us present some bckground detils which motivte our study. Assuming tht the oscilltion of second-order neutrl differentil eqution 0 p(t) p 0 <, (.) r(t)(x(t) + p(t)xτ(t)) + q(t)f(x(σ(t))) = 0, nd its prticulr cses were investigted by Bculíková nd Džurin 4, 5, Fišnrová nd Mřík 8, Li nd Rogovchenko 4, 5, Li et l. 6, nd Xing et l. 25. For the oscilltion of second-order neutrl differentil equtions with distributed deviting rguments, Li et l. 2 nd Li nd Thndpni 7 estblished severl oscilltion criteri for (r(t) z (t) α z (t)) + q(t, ξ) xg(t, ξ) α xg(t, ξ)dσ(ξ) = 0, where z(t) := x(t) + p(t)xτ(t). Compred with second-order neutrl differentil equtions, there re few oscilltion results for third-order neutrl differentil equtions. Bculíková nd Džurin 2, 3, Jing nd Li 0, nd Li et l. 8 exmined n eqution of the form under the ssumption tht (r(t)((x(t) + p(t)x(τ(t))) ) γ ) + q(t)x γ (σ(t)) = 0, (.2) 0 p(t) p 0 <, wheres Li nd Rogovchenko 4, Thndpni nd Li 20, nd Xing et l. 25 deduced oscilltion of (.2) ssuming tht condition (.) is stisfied. On the bsis of the ides exploited by Li et l. 2, 8, the objective of this pper is to estblish severl oscilltion criteri for (r(t) z (t) α z (t)) + q(t, ξ) xg(t, ξ) α xg(t, ξ)dσ(ξ) = 0, (.3) where t t 0 > 0, α > 0 is constnt, nd z(t) := x(t) + p(t)xτ(t). As usul, solution x of (.3) is clled oscilltory, if the set of its zeros is unbounded from bove, otherwise, it is sid to be nonoscilltory. In order to ccomplish these tsks, it is necessry to mke the following ssumptions hold throughout this pper: (A ) r C (t 0, ), (0, )) nd p C(t 0, ), 0, )); (A 2 ) q C(t 0, ), b, 0, )) nd q(t, ξ) is not eventully zero on ny t µ, ), b, t µ t 0, ); (A 3 ) g C(t 0, ), b, R) is nondecresing function for ξ stisfying lim inf t g(t, ξ) = for ξ, b; (A 4 ) τ C (t 0, ), R), τ (t) > 0, lim t τ(t) =, nd g(τ(t), ξ) = τg(t, ξ); (A 5 ) σ C(, b, R) is nondecresing nd the integrl of (.3) is tken in the sense of Riemnn Stieltijes. Min results in this pper re orgnized into two prts in ccordnce with different ssumptions on the coefficient r. In Section 2, oscilltion results for (.3) re estblished in the cse where t 0 r /α (t)dt =. (.4)

3 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), By ssuming tht t 0 r /α (t)dt <, (.5) oscilltion criteri for (.3) re obtined in Section 3. To illustrte the results reported in Sections 2 nd 3, we give two exmples in Section 4. In the sequel, we use the following nottions for compct presenttion of our results: Q(t, ξ) := min{q(t, ξ), q(τ(t), ξ)}, R(t) := mx{r(t), rτ(t)}, ρ +(t) := mx{0, ρ (t)}, ϑ(t) := ζ(t) r /α (s)ds, where ρ nd ζ will be explined lter, nd ll functionl inequlities re tcitly ssumed to hold for ll t lrge enough, unless mentioned otherwise. 2. Oscilltion criteri for the cse (.4) In this section, we consider two cses g(t, ) τ(t) nd g(t, ) τ(t). Let us strt with the first cse. Theorem 2.. Let conditions (A )-(A 5 ), (.), (.4), nd α be stisfied. Suppose tht g(t, ) C (t 0, ), R), g (t, ) > 0, g(t, ) t, nd g(t, ) τ(t) for t t 0. If there exists function ρ C (t 0, ), (0, )) such tht, for ll sufficiently lrge t 0 nd for some >, ( ) 2 α ρ(t)g (t) Q(t, ξ)dσ(ξ) (α + ) α+ + pα 0 rg(t, )(ρ + (t)) α+ (ρ(t)g (t, )) α dt = (2.) nd where u R(u) G (t) := u ( g(t,) s g(t,) Q(s, ξ)dσ(ξ)ds /α du =, (2.2) then every solution x of (.3) is either oscilltory or stisfies lim t x(t) = 0. ) α r /α (u)duds, (2.3) r /α (u)du Proof. Assume tht (.3) hs nonoscilltory solution x. Without loss of generlity, we my suppose tht there exists t 0 such tht x(t) > 0, xτ(t) > 0 for t, nd xg(t, ξ) > 0 for (t, ξ), ), b. Then we hve z > 0. It follows from (.3) tht nd (r(t) z (t) α z (t)) + (r(t) z (t) α z (t)) 0, (2.4) q(t, ξ)x α g(t, ξ)dσ(ξ) + pα 0 τ (t) (rτ(t) z τ(t) α z τ(t)) + By virtue of (2.4) nd τ (t) > 0, we obtin (r(t) z (t) α z (t)) + + q(t, ξ)x α g(t, ξ)dσ(ξ) p α 0 q(τ(t), ξ)x α g(τ(t), ξ)dσ(ξ) = 0. p α 0 q(τ(t), ξ)x α g(τ(t), ξ)dσ(ξ) + pα 0 (rτ(t) z τ(t) α z τ(t)) 0.

4 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), By using the ltter inequlity nd condition g(τ(t), ξ) = τg(t, ξ), we hve (r(t) z (t) α z (t)) + pα 0 (rτ(t) z τ(t) α z τ(t)) In view of 0 p(t) p 0 < nd the inequlity (see 5, Lemm ) (q(t, ξ)x α g(t, ξ) + p α 0 q(τ(t), ξ)x α g(τ(t), ξ)) dσ(ξ) Q(t, ξ) (x α g(t, ξ) + p α 0 x α τ(g(t, ξ))) dσ(ξ). (2.5) A α + B α 2 α (A + B)α for A 0, B 0, nd α, we rrive t x α g(t, ξ) + p α 0 x α τ(g(t, ξ)) (xg(t, ξ) + p 0xτ(g(t, ξ))) α 2 α zα g(t, ξ) 2 α. (2.6) By combining (2.5) nd (2.6), we conclude tht (r(t) z (t) α z (t)) + pα 0 (rτ(t) z τ(t) α z τ(t)) 2 α Bsed on condition (.4), z stisfies two possible cses: (I) z > 0, z > 0, z > 0, nd (r z α z ) 0; (II) z > 0, z < 0, z > 0, nd (r z α z ) 0. Q(t, ξ)z α g(t, ξ)dσ(ξ). (2.7) Assume first tht cse (I) holds. By using z > 0, z > 0, nd the fct tht g(t, ξ) is nondecresing function for ξ, b, we hve by (2.7) tht (r(t)(z (t)) α ) + pα 0 (rτ(t)(z τ(t)) α ) zα g(t, ) 2 α Define Riccti trnsformtion ω by Clerly, ω > 0 nd Q(t, ξ)dσ(ξ). (2.8) ω(t) := ρ(t) r(t)(z (t)) α (z g(t, )) α, t. (2.9) ω (t) = ρ (t) ρ(t) ω(t) + ρ(t)(r(t)(z (t)) α ) (z g(t, )) α αr(t)ρ(t)g (t, ) (z (t)) α z g(t, ) (z g(t, )) α+. Applying the monotonicity of r z α z nd g(t, ) t implies tht z g(t, ) ( ) r(t) /α z (t). rg(t, ) Then, combining the ltter inequlity nd (2.9), we conclude tht ω (t) ρ +(t) ρ(t) ω(t) + ρ(t)(r(t)(z (t)) α ) (z g(t, )) α αg (t, ) (ρ(t)rg(t, )) /α ω(α+)/α (t). (2.0) Furthermore, we define nother function ν by ν(t) := ρ(t) rτ(t)(z τ(t)) α (z g(t, )) α, t. (2.)

5 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), Thus, we hve ν > 0 nd ν (t) = ρ (t) ρ(t) ν(t) + ρ(t)(rτ(t)(z τ(t)) α ) (z g(t, )) α αρ(t)g (t, )rτ(t) (z τ(t)) α z g(t, ) (z g(t, )) α+. We derive from the monotonicity of r z α z nd g(t, ) τ(t) tht z g(t, ) By using the ltter inequlity nd (2.), we deduce tht ( ) rτ(t) /α z τ(t). rg(t, ) ν (t) ρ +(t) ρ(t) ν(t) + τ(t)) α ) αg (t, ) ρ(t)(rτ(t)(z (z g(t, )) α (ρ(t)rg(t, )) /α ν(α+)/α (t). (2.2) It follows from (2.0) nd (2.2) tht ω (t) + pα 0 (r(t)(z ν (t)) α ) (t) ρ(t) (z g(t, )) α + pα 0 (rτ(t)(z τ(t)) α ) (z g(t, )) α + ρ +(t) ρ(t) ω(t) αg (t, ) (ρ(t)rg(t, )) /α ω(α+)/α (t) + pα 0 ρ + (t) ρ(t) ν(t) αg (t, ) (ρ(t)rg(t, )) /α ν(α+)/α (t). Let By using the inequlity (see 3) C := ρ +(t) ρ(t) nd D := αg (t, ) (ρ(t)rg(t, )) /α. Cu Du (α+)/α α α C α+ (α + ) α+, D > 0, (2.3) Dα we conclude tht nd ρ +(t) ρ(t) ω(t) αg (t, ) (ρ(t)rg(t, )) /α ω(α+)/α (t) ρ +(t) ρ(t) ν(t) αg (t, ) (ρ(t)rg(t, )) /α ν(α+)/α (t) By combining the ltter inequlities nd (2.8), we obtin rg(t, )(ρ +(t)) α+ (α + ) α+ (ρ(t)g (t, )) α rg(t, )(ρ +(t)) α+ (α + ) α+ (ρ(t)g (t, )) α. ω (t) + pα 0 ν (t) ρ(t) ( zg(t, ) 2 α z g(t, ) + (α + ) α+ ( + pα 0 ) α Q(t, ξ)dσ(ξ) ) rg(t, )(ρ + (t)) α+ (ρ(t)g (t, )) α. (2.4) By virtue of (r(z ) α ) 0, z (t) = z ( ) + z (s)ds = z (r(s)(z (s)) α ) ( ) + r /α (s) /α ds r /α (t)z (t) r /α (s)ds.

6 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), Hence, we get which implies tht, for t >, ( ) z (t) 0, r /α (s)ds nd so Then, we hve z(t) = z( ) + z (t) r /α (u)du z (s)ds = z( ) + s z(t) z (t) r /α (u)duds, s z (s) s r /α (u)du s r /α (u)duds r /α (u)duds. (2.5) r /α (u)du ( ) zg(t, ) α z G (t), g(t, ) where G is defined by (2.3). Substitution of this inequlity into (2.4) yields t 3 ω (t) + pα 0 ν (t) 2 α ρ(t)g (t) + (α + ) α+ ( + pα 0 Q(t, ξ)dσ(ξ) ) rg(t, )(ρ + (t)) α+ (ρ(t)g (t, )) α. Integrting the ltter inequlity from t 3 (t 3 > ) to t, we conclude tht ρ(s)g (s) b 2 α Q(s, ξ)dσ(ξ) + pα 0 / rg(s, )(ρ +(s)) α+ (α + ) α+ (ρ(s)g (s, )) α ds ω(t 3 ) + pα 0 ν(t 3 ), which contrdicts (2.). Assume now tht cse (II) holds. On the bsis of the monotonicities of z nd g(t, ξ), we hve zg(t, ξ) zg(t, b). By tking into ccount tht z > 0, inequlity (2.7) becomes (r(t)(z (t)) α ) + pα 0 (rτ(t)(z τ(t)) α ) zα g(t, b) 2 α Q(t, ξ)dσ(ξ). By using similr proof of 4, Theorem 5, we cn obtin lim t x(t) = 0 when using (2.2). completes the proof. This Now, we turn our ttention to the cse when g(t, ) τ(t). Theorem 2.2. Let conditions (A )-(A 5 ), (.), (.4), (2.2), nd α be stisfied. Suppose tht τ(t) t nd g(t, ) τ(t) for t t 0. If there exists function ρ C (t 0, ), (0, )) such tht for ll sufficiently lrge t 0 nd for some >, where 2 α ρ(t)g 2 (t) Q(t, ξ)dσ(ξ) G 2 (t) := ( (α + ) α+ + pα 0 ( τ(t) s then the conclusion of Theorem 2. remins intct. ) rτ(t)(ρ + (t)) α+ ( ρ(t)) α dt =, (2.6) ) α r /α (u)duds τ(t), (2.7) r /α (u)du

7 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), Proof. Assume tht (.3) hs nonoscilltory solution x. Without loss of generlity, we my suppose tht there exists t 0 such tht x(t) > 0, xτ(t) > 0 for t, nd xg(t, ξ) > 0 for (t, ξ), ), b. As in the proof of Theorem 2., we hve (2.4), (2.7), nd two possible cses (I) nd (II) (s those in the proof of Theorem 2.) for z. Assume first tht cse (I) holds. It follows from g(t, ξ) g(t, ) τ(t), z > 0, nd z > 0 tht Define Riccti trnsformtion ω by (r(t)(z (t)) α ) + pα 0 (rτ(t)(z τ(t)) α ) zα τ(t) 2 α Then ω > 0. Applying (2.4) nd τ(t) t yields By differentiting (2.9), we conclude tht Q(t, ξ)dσ(ξ). (2.8) ω(t) := ρ(t) r(t)(z (t)) α (z τ(t)) α, t. (2.9) z τ(t) ( ) r(t) /α z (t). rτ(t) ω (t) = ρ (t) ρ(t) ω(t) + (t)) α ) ρ(t)(r(t)(z (z τ(t)) α αr(t)ρ(t)τ (t) (z (t)) α z τ(t) (z τ(t)) α+ ρ +(t) ρ(t) ω(t) + ρ(t)(r(t)(z (t)) α ) (z τ(t)) α αρ(t)τ (t) r(α+)/α (t) r /α τ(t) ( z (t) z τ(t) ) α+ = ρ +(t) ρ(t) ω(t) + (t)) α ) ατ (t) ρ(t)(r(t)(z (z τ(t)) α (ρ(t)rτ(t)) /α ω(α+)/α (t). (2.20) Similrly, define nother Riccti trnsformtion ν by Clerly, ν > 0 nd ν(t) := ρ(t) rτ(t)(z τ(t)) α (z τ(t)) α, t. ν (t) = ρ (t) ρ(t) ν(t) + τ(t)) α ) ( z ρ(t)(rτ(t)(z (z τ(t)) α αρ(t)τ ) τ(t) α+ (t)rτ(t) z τ(t) ρ +(t) ρ(t) ν(t) + τ(t)) α ) ατ (t) ρ(t)(rτ(t)(z (z τ(t)) α (ρ(t)rτ(t)) /α ν(α+)/α (t). (2.2) In view of (2.20) nd (2.2), we get ω (t) + pα 0 (r(t)(z ν (t)) α ) (t) ρ(t) (z τ(t)) α + pα 0 (rτ(t)(z τ(t)) α ) (z τ(t)) α + ρ +(t) ρ(t) ω(t) + pα 0 ρ + (t) ρ(t) ν(t) ατ (t) (ρ(t)rτ(t)) /α ω(α+)/α (t) ατ (t) (ρ(t)rτ(t)) /α ν(α+)/α (t). Set C := ρ +(t) ρ(t) nd D := ατ (t) (ρ(t)rτ(t)) /α.

8 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), By virtue of (2.3) nd (2.8), we hve ω (t) + pα 0 ν (t) ρ(t) ( ) zτ(t) α ( ) 2 α z Q(t, ξ)dσ(ξ) + τ(t) (α + ) α+ + pα 0 rτ(t)(ρ + (t)) α+ (ρ(t)τ (t)) α. Similrly, s in the proof of Theorem 2., we obtin (2.5), nd hence ( ) zτ(t) α z G 2 (t), τ(t) where G 2 is defined s in (2.7). Therefore, ω (t) + pα 0 ν (t) ρ(t)g 2(t) 2 α t 3 Q(t, ξ)dσ(ξ) + ( ) (α + ) α+ + pα 0 rτ(t)(ρ + (t)) α+ ( ρ(t)) α. By integrting the ltter inequlity from t 3 (t 3 > ) to t, we hve ρ(s)g2 (s) b 2 α Q(s, ξ)dσ(ξ) + pα 0 / rτ(s)(ρ +(s)) α+ (α + ) α+ ( ρ(s)) α ds ω(t 3 ) + pα 0 ν(t 3 ), which contrdicts (2.6). Assume now tht cse (II) holds. As in the proof of Cse (II) in Theorem 2., we rrive t the desired conclusion. The proof is complete. 3. Oscilltion criteri for the cse (.5) In this section, we estblish some oscilltion criteri for (.3) under the ssumption tht (.5) holds. Similrly, s in Section 2, we begin with the cse when g(t, ) τ(t) holds. Theorem 3.. Let conditions (A )-(A 5 ), (.), (.5), (2.2), nd α be stisfied. Suppose tht g(t, ) C (t 0, ), R), g (t, ) > 0, nd g(t, ) τ(t) t for t t 0. Assume further tht there exists function ρ C (t 0, ), (0, )) such tht (2.) holds for ll sufficiently lrge t 0 nd for some >. If there exists function ζ C (t 0, ), R) such tht, ζ(t) t, ζ(t) g(t, ), ζ (t) > 0 for t t 0, nd for ll sufficiently lrge t 0, where 2 α ϑ α (t)g 3 (t) ( ) α α+ ( ) Q(t, ξ)dσ(ξ) + pα 0 ζ (t) α + ϑ(t)r /α dt =, (3.) ζ(t) G 3 (t) := (g(t, ) ) α, (3.2) then every solution x of (.3) is either oscilltory or stisfies lim t x(t) = 0. Proof. Assume tht (.3) hs nonoscilltory solution x. Without loss of generlity, we my suppose tht there exists t 0 such tht x(t) > 0, xτ(t) > 0 for t, nd xg(t, ξ) > 0 for (t, ξ), ), b. Then we hve z > 0. Bsed on condition (.5), there exist three possible cses (I), (II) (s those in the proof of Theorem 2.), nd (III) z > 0, z > 0, z < 0, nd (r z α z ) 0. Assume tht cse (I) nd cse (II) hold. By using the proof of Theorem 2., we get the conclusion of Theorem 3.. Assume now tht cse (III) holds. In view of g(t, ξ) g(t, ), z > 0, nd z < 0, inequlity (2.7) reduces to ( r(t)( z (t)) α ) + pα 0 ( rτ(t)( z τ(t)) α ) zα g(t, ) b 2 α Q(t, ξ)dσ(ξ). (3.3)

9 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), From (r z α z ) 0, we hve (r( z ) α ) 0, which shows tht r( z ) α is nondecresing. Thus, we obtin z (s) r/α (t) r /α (s) z (t), s t. An integrtion from ζ(t) to l yields By pssing to the limit s l, we get tht is, Define function ϕ by Clerly, ϕ < 0 nd Similrly, we define nother function φ by l z (l) z ζ(t) + r /α (t)z (t) r /α (s)ds. ζ(t) 0 z ζ(t) + r /α (t)z (t)ϑ(t), ϑ(t) r/α (t)z (t) z ζ(t). ϕ(t) := r(t)( z (t)) α (z ζ(t)) α, t. (3.4) ϑ α (t)ϕ(t). (3.5) φ(t) := rτ(t)( z τ(t)) α (z ζ(t)) α, t. (3.6) Then φ < 0. From the monotonicity of r( z ) α nd τ(t) t, we obtin rτ(t)( z τ(t)) α r(t)( z (t)) α. Hence, 0 < φ(t) < ϕ(t). By virtue of (3.5), we hve Now, by differentiting (3.4), we rrive t ϑ α (t)φ(t). (3.7) ϕ (t) = ( r(t)( z (t)) α ) (z ζ(t)) α + αr(t)ζ (t)( z (t)) α z ζ(t) (z ζ(t)) α+. By virtue of ζ(t) t nd the fct tht r( z ) α is nondecresing, we get nd so Similrly, by differentiting (3.6), we hve z ζ(t) r/α (t) r /α ζ(t) z (t), ϕ (t) ( r(t)( z (t)) α ) (z ζ(t)) α αζ (t) r /α ζ(t) ( ϕ(t))(α+)/α. (3.8) φ (t) = ( rτ(t)( z τ(t)) α ) (z ζ(t)) α + αζ (t)rτ(t)( z τ(t)) α z ζ(t) (z ζ(t)) α+.

10 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), By tking into ccount tht r( z ) α is nondecresing nd ζ(t) τ(t), we conclude tht nd hence It follows from (3.3), (3.8), nd (3.9) tht z ζ(t) r/α τ(t) r /α ζ(t) z τ(t), φ (t) ( rτ(t)( z τ(t)) α ) (z ζ(t)) α αζ (t) r /α ζ(t) ( φ(t))(α+)/α. (3.9) ϕ (t) + pα 0 φ (t) ( r(t)( z (t)) α ) (z ζ(t)) α + pα 0 ( rτ(t)( z τ(t)) α ) (z ζ(t)) α Applying z > 0 nd z < 0 implies tht Hence, we hve where G 3 is s in (3.2). Then, we hve αζ (t) r /α ζ(t) ( ϕ(t))(α+)/α pα 0 αζ (t) r /α ζ(t) ( φ(t))(α+)/α ( ) zg(t, ) α 2 α z Q(t, ξ)dσ(ξ) ζ(t) αζ (t) r /α ζ(t) ( ϕ(t))(α+)/α pα 0 αζ (t) r /α ζ(t) ( φ(t))(α+)/α. z(t) = z( ) + ( ) zg(t, ) α ( zg(t, ) z = ζ(t) z g(t, ) ϕ (t) + pα 0 φ (t) 2 α G 3 (t) αpα 0 z (s)ds (t )z (t). z ) g(t, ) α z G 3 (t), ζ(t) Q(t, ξ)dσ(ξ) ζ (t) r /α ζ(t) ( φ(t))(α+)/α. αζ (t) r /α ζ(t) ( ϕ(t))(α+)/α By multiplying the ltter inequlity by ϑ α (t) nd integrting the resulting inequlity from ( > ) to t, we get ϑ ϑ α (t)ϕ(t) ϑ α α (s)ζ (s) ( )ϕ( ) + α r /α ϕ(s) + ϑα (s)ζ (s) ζ(s) r /α ζ(s) ( ϕ(s))(α+)/α ds + pα 0 (ϑ α (t)φ(t) ϑ α ( )φ( )) + αpα 0 ϑ α (s)ζ (s) r /α φ(s) + ϑα (s)ζ (s) ζ(s) r /α ζ(s) ( φ(s))(α+)/α ds Set + A := 2 α ϑ α (s)g 3 (s) Q(s, ξ)dσ(ξ)ds 0. ϑ α (s)ζ (s) α/(α+) r /α ϕ(s) nd B := ζ(s) By using the inequlity (see 2) α α + ϑ α (s)ζ (s) r /α ζ(s) ϑ α (s)ζ (s) α/(α+) α r /α. ζ(s) α + α AB/α A (α+)/α α B(α+)/α, for A 0 nd B 0, (3.0)

11 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), we obtin On the other hnd, define A := ϑ α (s)ζ (s) r /α ϕ(s) + ϑα (s)ζ (s) ζ(s) r /α ζ(s) ( ϕ(s))(α+)/α ( ) α α+ ζ (s) α α + ϑ(s)r /α ζ(s). ϑ α (s)ζ (s) α/(α+) r /α φ(s) nd B := ζ(s) α α + ϑ α (s)ζ (s) r /α ζ(s) ϑ α (s)ζ (s) α/(α+) α r /α. ζ(s) By virtue of (3.0), we hve ϑ α (s)ζ (s) r /α φ(s) + ϑα (s)ζ (s) ζ(s) r /α ζ(s) ( φ(s))(α+)/α ( ) α α+ ζ (s) α α + ϑ(s)r /α ζ(s). By using (3.5) nd (3.7), we conclude tht 2 α ϑ α (s)g 3 (s) ( ) α α+ ( ) Q(s, ξ)dσ(ξ) + pα 0 α + ϑ α ( )ϕ( ) + pα ϑ α ( )φ( ) + + pα 0, which contrdicts (3.). This completes the proof. ζ (s) ϑ(s)r /α ds ζ(s) With proof similr to the proof of Theorems 2.2 nd 3., we cn obtin the following criterion for (.3) ssuming tht g(t, ) τ(t). Theorem 3.2. Let conditions (A )-(A 5 ), (.), (.5), (2.2), nd α be stisfied. Suppose tht τ(t) t nd g(t, ) τ(t) for t t 0. Assume lso tht there exists function ρ C (t 0, ), (0, )) such tht (2.6) holds for ll sufficiently lrge t 0 nd for some >. If there exists function ζ C (t 0, ), R) such tht ζ(t) t, ζ(t) g(t, ), ζ (t) > 0 for t t 0, nd (3.) holds for ll sufficiently lrge t 0, then the conclusion of Theorem 3. remins intct. 4. Exmples Similr results cn be obtined under the ssumption tht 0 < α. In this cse, utilizing 5, Lemm 2, one hs to replce Q(t, ξ) := min{q(t, ξ), q(τ(t), ξ)} with Q(t, ξ) := 2 α min{q(t, ξ), q(τ(t), ξ)} nd proceed s bove. In this section, we illustrte possible pplictions with two exmples. Exmple 4.. For t, consider third-order neutrl differentil eqution x(t) + x(t 2π) + π 4π xt + ξdξ = 0. (4.) Let α =, = 4π, b = π, r(t) =, p(t) = p 0 =, τ(t) = t 2π, q(t, ξ) =, g(t, ξ) = t + ξ, nd σ(ξ) = ξ. Note tht Q(t, ξ) = min{q(t, ξ), q(τ(t), ξ)} =, g (t, ) = > 0, g(t, ) = t 4π < t, nd g(t, ) < τ(t). Moreover, let = nd ρ(t) =, then nd G (t) = 4π 4π 2 α ρ(t)g (t) s r /α (u)duds r /α (u)du = t2 /2 (4π + )t + β, β = 8π 2 t2 2 t (4π + ) 2 + 4π +, Q(t, ξ)dσ(ξ) ( (α + ) α+ + pα 0 ) rg(t, )(ρ + (t)) α+ (ρ(t)g (t, )) α dt

12 C. M. Jing, T. X. Li, J. Nonliner Sci. Appl. 9 (206), = 5π G (t)dt = 5π 2 2(4π + )t + 2β dt =. t (4π + ) Hence, by Theorem 2., every solution x of (4.) is either oscilltory or stisfies lim t x(t) = 0. As mtter of fct, x(t) = sin t is n oscilltory solution to (4.). Exmple 4.2. For t, consider third-order neutrl differentil eqution (x(t) + p(t)x(t γ)) + (ξ + )xt + ξdξ = 0, (4.2) where 0 < p(t) p 0, p 0 nd γ re positive constnts. Let α =, = 0, b =, r(t) =, τ(t) = t γ, q(t, ξ) = ξ+, g(t, ξ) = t+ξ, nd σ(ξ) = ξ. Note tht Q(t, ξ) = min{q(t, ξ), q(τ(t), ξ)} = ξ+, τ(t) = t γ t, nd g(t, ) = t τ(t). Moreover, let =, ρ(t) =, nd ζ(t) = t +, then we hve ϑ(t) = /(t + ), 0 G 2 (t) = γ γ s r /α (u)duds r /α (u)du = γ γ s u 2 duds u 2 du = (t γ)2 + ( ln )(t γ) (t γ)ln(t γ) t γ, nd nd thus 2 α ρ(t)g 2 (t) = 3 2 G 2 (t)dt = 3 2 G 3 (t) = (g(t, ) ) α = t, ( ) Q(t, ξ)dσ(ξ) (α + ) α+ + pα 0 rτ(t)(ρ + (t)) α+ ( ρ(t)) α dt (t γ) 2 + ( ln )(t γ) (t γ)ln(t γ) dt =, t γ nd 2 α ϑ α (t)g 3 (t) = ( α Q(t, ξ)dσ(ξ) 3(t t ) 2(t + ) + p 0 4(t + ) α + dt =. ) α+ ( ) + pα 0 ζ (t) ϑ(t)r /α dt ζ(t) Therefore, by Theorem 3.2, every solution x of (4.2) is either oscilltory or stisfies lim t x(t) = 0. Acknowledgment This reserch is supported by NNSF of P. R. Chin (Grnt Nos nd ), CPSF (Grnt No. 205M58209), nd NSF of Shndong Province (Grnt Nos. ZR206JL02 nd ZR202FL06), DSRF of Linyi University (Grnt No. LYDX205BS00), nd the AMEP of Linyi University, P. R. Chin. References R. P. Agrwl, M. Bohner, W.-T. Li, Nonoscilltion nd oscilltion: theory for functionl differentil equtions, Monogrphs nd Textbooks in Pure nd Applied Mthemtics, Mrcel Dekker, Inc., New York, (2004). 2 B. Bculíková, J. Džurin, On the symptotic behvior of clss of third order nonliner neutrl differentil equtions, Cent. Eur. J. Mth., 8 (200), , 3 B. Bculíková, J. Džurin, Oscilltion of third-order neutrl differentil equtions, Mth. Comput. Modelling, 52 (200), B. Bculíková, J. Džurin, Oscilltion theorems for second order neutrl differentil equtions, Comput. Mth. Appl., 6 (20), B. Bculíková, J. Džurin, Oscilltion theorems for second-order nonliner neutrl differentil equtions, Comput. Mth. Appl., 62 (20), , 2, 4

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