B Baculíková

C Zhang

GS Ladde

Haixia Wang

I Győri

J Zhong

JK Hale

JSW Wong

L Erbe

L Ye

M Hasanbulli

MA El-Sayed

P Wang

S Sun

SR Grace

T Li

Tongxing Li

W Shi

Ying Jiang

Youliang Fu

Z Zhang

Advances in Difference Equations

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Jiang et al. Advances in Diﬀerence Equations  (2015) 2015:223 DOI 10.1186/s13662-015-0556-xRESEARCH Open AccessOscillation of forced second-order neutraldelay differential equationsYing Jiang1, Youliang Fu1, Haixia Wang2 and Tongxing Li3,4**Correspondence:litongx2007@163.com3LinDa Institute of ShandongProvincial Key Laboratory ofNetwork Based IntelligentComputing, Linyi University, Linyi,Shandong 276005, P.R. China4School of Informatics, LinyiUniversity, Linyi, Shandong 276005,P.R. ChinaFull list of author information isavailable at the end of the articleAbstractThe objective of this paper is to study oscillation of a forced second-order neutraldiﬀerential equation. By using the generalized Riccati substitution and integraltechnique, a new suﬃcient condition is obtained which insures that all solutions tothe studied equation are oscillatory. An illustrative example is included.MSC: 34C10; 34K11Keywords: oscillation; second-order; forced term; neutral diﬀerential equation1 IntroductionIn this paper, we are concerned with the oscillation of a forced second-order nonlinearneutral diﬀerential equation(r(t)[x(t) + P(t)x(τ (t))]′)′ +m∑i=Qi(t)fi(x(t))+l∑j=Rj(t)gj(x(τ (t)))= F(t), (.)where t ≥ t > ,m≥ , and l ≥  are integers.We suppose that the following assumptionsare satisﬁed:(A) r ∈ C([t,∞), (,∞)), P,Qi,Rj ∈ C([t,∞), [,∞)), fi, gj ∈ C(R,R), yfi(y) > , andygj(y) >  for y = , i = , , . . . ,m, and j = , , . . . , l;(A) τ ∈ C([t,∞),R), τ (t)≤ t, and limt→∞ τ (t) =∞;(A) there exist constants αi >  and βj >  such that fi(y)/y ≥ αi and gj(y)/y ≥ βj for y = ,i = , , . . . ,m, and j = , , . . . , l;(A) for any T ≥ t, there exist T ≤ s < t ≤ s < t such thatF(t){≤ , t ∈ [s, t],≥ , t ∈ [s, t],andl∑j=βjRj(t)≥m∑i=αiQi(t)P(t), t ∈ [s, t]∪ [s, t]. (.)© 2015 Jiang et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.Jiang et al. Advances in Diﬀerence Equations  (2015) 2015:223 Page 2 of 5Throughout the paper, we deﬁnez(t) := x(t) + P(t)x(τ (t)). (.)By a solution of (.) wemean a function x ∈ C([Tx,∞),R),Tx ≥ t, which has the propertyrz′ ∈ C([Tx,∞),R) and satisﬁes (.) on [Tx,∞). We consider only those solutions x of(.) which satisfy condition sup{|x(t)| : t ≥ T} >  for all T ≥ Tx. We assume that (.)possesses such solutions. A solution of (.) is called oscillatory if it has arbitrarily largezeros on the interval [Tx,∞); otherwise, it is termed nonoscillatory.As is well known, the study of qualitative theory of diﬀerential equations is of impor-tance both in theory and applications. For instance, the problems of oscillatory behav-ior of neutral diﬀerential equations have a number of practical applications in the studyof distributed networks containing lossless transmission lines which arise in high-speedcomputers where the lossless transmission lines are used to interconnect switching cir-cuits. For some related contributions on oscillation of various classes of neutral diﬀerentialequations, we refer the reader to [–] and the references cited therein.In what follows, we provide some background details that motivated our study. El-Sayed[] and Wong [] investigated the second-order forced linear diﬀerential equation(p(t)x′)′ + q(t)x = f (t).Zhang et al. [] studied a second-order neutral diﬀerential equation(r(t)[x(t) + p(t)x(t – τ )]′)′ +Q(t)f(x(t))+Q(t)g(x(t – τ ))=H(t), (.)where Q and Q are nonnegative functions. Equation (.) is a special case of (.). Inthe sequel, using a generalized Riccati substitution which diﬀers from those exploited in[, , ], a new oscillation criterion for (.) is presented. Furthermore, an illustrativeexample is provided.2 Main resultsTheorem . Assume that conditions (A)-(A) are satisﬁed and let Bk = {u ∈ C[sk , tk] :u(t) ≡ ,u(sk) = u(tk) = }, k = , . If there exist functions u ∈ Bk , ρ ∈ C([t,∞), (,∞)),and σ ∈ C([t,∞),R) such that, for k = , ,Jk(u,ρ,σ )=∫ tksk{ρ[u( m∑i=αiQi + rσ  – (rσ )′)– r(u′ + uρ′ρ + uσ)]}(t) dt > , (.)then every solution of (.) is oscillatory.Proof Suppose that x is a nonoscillatory solution of (.) which is eventually positive. Thenz deﬁned by (.) is also eventually positive. Using (A), for any T ≥ t, there exist t > s ≥T such that F(t)≤  for t ∈ [s, t]. From (A), (.), (.), and (.), we haveJiang et al. Advances in Diﬀerence Equations  (2015) 2015:223 Page 3 of 5(rz′)′(t) = F(t) –m∑i=Qi(t)fi(x(t))–l∑j=Rj(t)gj(x(τ (t)))≤ –m∑i=αiQi(t)x(t) –l∑j=βjRj(t)x(τ (t))≤ –[ m∑i=αiQi(t)x(t) +m∑i=αiQi(t)P(t)x(τ (t))]= –m∑i=αiQi(t)z(t). (.)For t ≥ T , we deﬁne a generalized Riccati substitution byV (t) := –ρ(t)[ r(t)z′(t)z(t) + r(t)σ (t)]. (.)Then we haveV ′ = –ρ ′( rz′z + rσ)– ρ( rz′z + rσ)′= ρ′ρV – ρ( rz′z)′– ρ(rσ )′= ρ′ρV – ρ (rz′)′z + ρr(z′)z – ρ(rσ )′. (.)By virtue of (.), we obtain(z′z)=( V–ρr – σ)=( Vρr)+ σ  + Vσρr . (.)For t ∈ [s, t], substituting (.) and (.) into (.), we conclude thatV ′ = ρ′ρV – ρ (rz′)′z + ρr( V ρr + σ + Vσρr)– ρ(rσ )′= –ρ (rz′)′z + ρ[rσ  – (rσ )′]+(ρ ′ρ+ σ)V + Vρr≥ ρ[ m∑i=αiQi + rσ  – (rσ )′]+(ρ ′ρ+ σ)V + Vρr . (.)Let u ∈ B be given as in the hypothesis. Multiplying (.) by u and integrating the result-ing inequality from s to t, we have∫ tsuV ′ dt ≥∫ tsuρ[ m∑i=αiQi + rσ  – (rσ )′]dt +∫ ts(ρ ′ρ+ σ)Vu dt+∫ tsV ρr u dt. (.)Jiang et al. Advances in Diﬀerence Equations  (2015) 2015:223 Page 4 of 5Integrating (.) by parts and using the fact that u(s) = u(t) = , we deduce that–∫ tsuu′V dt ≥∫ tsuρ[ m∑i=αiQi + rσ  – (rσ )′]dt+∫ ts(ρ ′ρ+ σ)Vu dt +∫ tsV ρr u dt.That is,∫ ts[uV ρr + uV(u′ + u(ρ ′ρ + σ))]dt +∫ tsuρ[ m∑i=αiQi + rσ  – (rσ )′]dt ≤ .Hence,∫ ts[ uV√ρr +√ρr(u′ + uρ′ρ + uσ)]dt+∫ ts[uρ( m∑i=αiQi + rσ  – (rσ )′)– ρr(u′ + uρ′ρ + uσ)]dt ≤ ,which is equivalent to∫ ts[ uV√ρr +√ρr(u′ + uρ′ρ + uσ)]dt + J(u,ρ,σ )≤ , (.)where J(u,ρ,σ ) is as in (.). Since J(u,ρ,σ ) > , inequality (.) yields∫ ts[ uV√ρr +√ρr(u′ + uρ′ρ + uσ)]dt ≤ –J(u,ρ,σ ) < ,which is a contradiction. This contradiction proves that x is not eventually positive.When x is eventually negative, we use u ∈ B and F(t)≥  on [s, t] to arrive at a similarcontradiction. The proof is complete. Example . For t ≥ , consider the forced second-order neutral delay diﬀerential equa-tion(x(t) + x( t))′′+ x(t) + tx( t)= sin t. (.)Let r(t) = , P(t) = /, τ (t) = t/, m = l = , Q(t) = , R(t) = t, f(y) = g(y) = y,α = β = , u = sin t, ρ(t) = , and σ (t) = . Set s = (n+ )π , t = (n+ )π , s = (n+ )π ,and t = (n + )π . ThenJ(u,ρ,σ ) =∫ ts{ρ[u( m∑i=αiQi + rσ  – (rσ )′)– r(u′ + uρ′ρ + uσ)]}(t) dt=∫ (n+)π(n+)π( sin t – cos t)dt = π .Similarly, J(u,ρ,σ ) = π/. Hence, by Theorem ., every solution of (.) is oscillatory.Jiang et al. Advances in Diﬀerence Equations  (2015) 2015:223 Page 5 of 5Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll four authors contributed equally to this work. They all read and approved the ﬁnal version of the manuscript.Author details1Qingdao Technological University, Feixian, Shandong 273400, P.R. China. 2College of Economics, Ocean University ofChina, Qingdao, Shandong 266100, P.R. China. 3LinDa Institute of Shandong Provincial Key Laboratory of Network BasedIntelligent Computing, Linyi University, Linyi, Shandong 276005, P.R. China. 4School of Informatics, Linyi University, Linyi,Shandong 276005, P.R. China.AcknowledgementsThe authors are grateful to the editors and two anonymous referees for a very thorough reading of the manuscript andfor kindly prompting improvements in presentation. This research is supported by NNSF of P.R. China (GrantNo. 61403061), NSF of Shandong Province (Grant No. ZR2012FL06), and the AMEP of Linyi University, P.R. China.Received: 9 December 2014 Accepted: 26 June 2015References1. Baculíková, B, Džurina, J: Oscillation theorems for second-order nonlinear neutral diﬀerential equations. Comput.Math. 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Oscillation of forced second-order neutral delay differential equations

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Oscillation of forced second-order neutral delay differential equations

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