Research Article Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations

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1 International Scholarly Reearch Network ISRN Mathematical Analyi Volume 20, Article ID 85203, 9 page doi:0.502/20/85203 Reearch Article Exitence for Nonocillatory Solution of Higher-Order Nonlinear Differential Equation Yazhou Tian, 2 and Fanwei Meng 2 Department of Baic Coure, Qingdao Technological Univerity Linyi), Feixian 27300, Shandong, China 2 Department of Mathematic, Qufu Normal Univerity, Qufu 27365, Shandong, China Correpondence hould be addreed to Yazhou Tian, tianyazhou369@63.com Received 8 Augut 20; Accepted 22 September 20 Academic Editor: Z. Dola Copyright q 20 Y. Tian and F. Meng. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. The exitence of nonocillatory olution of the higher-order nonlinear differential equation rtxtptxt τ n m Q itf i xt σ i 0, t t 0,wherem,n 2areinteger, τ>0, σ i 0, r,p,q i Ct 0,,R, f i CR, R i, 2,...,m, i tudied. Some new ufficient condition for the exitence of a nonocillatory olution of above equation are obtained for general Q i t i, 2,...,m which mean that we allow ocillatory Q i t i, 2,...,m. In particular, our reult improve eentially and extend ome known reult in the recent reference.. Introduction Conider the higher-order nonlinear neutral differential equation [rtxt Ptxt τ n ] Q i tf i xt σ i 0, t t 0.. With repect to., throughout, we hall aume the following: i m, n 2 are integer, τ>0, σ i 0, ii r, P, Q i Ct 0,,R, rt > 0, f i CR, R, i, 2,...,m. Let ρ max im {τ, σ i }. By a olution of., wemeanafunctionxt Ct ρ,,r for ome t t 0 which ha the property that xtptxt τ C n t,,r and rtxtptxt τ n C t,,r and atifie. on t,. A nontrivial olution of. i called ocillatory if it ha arbitrarily large zero, and, otherwie, it i nonocillatory.

2 2 ISRN Mathematical Analyi The exitence of nonocillatory olution of higher-order nonlinear neutral differential equation received much le attention, which i due mainly to the technical difficultie ariing in it analyi. In 998, Kulenovic and Hadziomerpahic invetigated the exitence of nonocillatory olution of econd-order nonlinear neutral differential equation xt cxt τ Q txt σ Q 2 txt σ 2 0, t t 0, E 0 where c i a contant. In 2006, Zhang and Wang 2 invetigated the econd neutral delay differential equation with poitive and negative coefficient: [ rtxt Ptxt τ ] Q tfxt σ Q 2 tgxt σ 2 0, t t 0, E where τ>0, σ i 0,Q,Q 2 Ct 0,,R,f,g CR, R, xfx > 0, xgx > 0, x / 0. By uing Banach contraction mapping principle, they proved the following theorem which extend the reult in. Theorem A 2, Theorem 2.3. Aume that H f and g atify local Lipchitz condition and xfx > 0, xgx > 0, forx/ 0; H 2 Q i t 0, i, 2, aq t Q 2 t i eventually nonnegative for every a>0; H 3 t t 0 t 0 Q i t/d dt <, i, 2 hold if one of the following two condition i atified: H Pt > eventually, and 0 <P 2 P <P 2 2 <, H 5 Pt < eventually, and <P 2 P <, where P lim up t Pt, P 2 lim inf t Pt,then. ha a nonocillatory olution. In 2007, Zhou 3 tudie the exitence of nonocillatory olution of the following econd-order nonlinear differential equation. [ rtxt Ptxt τ ] Q i tf i xt σ i 0, t t 0, E where f i CR, R i, 2,...,m. By uing Kranoelkii fixed point theorem, they proved the following theorem. Theorem B 3, Theorem. Aume that there exit nonnegative contant c and c 2 uch that c c 2 <, c 2 Pt c. Further, aume that t t 0 t 0 Q i t d dt <, i, 2,...,m..2 Then. ha a bounded nonocillatory olution.

3 ISRN Mathematical Analyi 3 In thi paper, by uing Kranoelkii fixed point theorem and ome new technique, we obtain ome ufficient condition for the exitence of a nonocillatory olution of. for general Q i t i, 2,...,m which mean that we allow ocillatory Q i t i, 2,...,m. Meanwhile, we extend the main reult of 2, Main Reult The following fixed point theorem will be ued to prove the main reult in thi ection. Lemma 2. ee 3, Kranoelkii fixed point theorem. Let X be a Banach pace, let Ω be a bounded cloed convex ubet of X, and let S, S 2 be map of Ω into X uch that S x S 2 y Ω for every pair x, y Ω.IfS i a contraction and S 2 i completely continuou, then the equation S x S 2 x x 2. ha a olution in Ω. Theorem 2.2. Aume that there exit nonnegative contant c and c 2 uch that c c 2 <, < c 2 Pt c <. Further, aume that t t 0 t 0 n 2 Q i t d dt <, i, 2,...,m. 2.2 Then. ha a bounded nonocillatory olution. Proof. By interchanging the order of integral, we note that 2.2 i equivalent to t 0 n 2 Q i t d dt <, i, 2,..., m. 2.3 By 2.3, we chooe T>t 0 ufficiently large uch that T n 2 M Q i u du d < c c 2, 2. where M max c c 2 /2x{ f i x :i m}. Let Ct 0,,R be the et of all continuou function with the norm x up t t0 xt <. Then Ct 0,,R i a Banach pace. We define a bounded, cloed, and convex ubet Ω of Ct 0,,R a follow: Ω { x xt Ct 0,,R : c c 2 xt, t t 0 }

4 ISRN Mathematical Analyi Define two map S and S 2 : Ω Ct 0,,R a follow: 3 c 3c 2 Ptxt τ, t T, S xt S xt, t 0 t T, n t n 2 Q i uf i xu σ i du d, t T, S 2 xt t S 2 xt, t 0 t T. 2.6 i We hall how that for any x, y Ω, S x S 2 y Ω. In fact, x, y Ω,andt T,weget S xt S 2 y ) t 3 c 3c 2 Ptxt τ 3 c 3c 2 t t n 2 c 2 T Q i uf i yu σi ) ) du d n 2 M Q i u du d c 3c 2 c 2 c c 2. Furthermore, we have S xt S 2 y ) t 3 c 3c 2 Ptxt τ n 2 Qi uf i yu σi ) ) du d t 3 c 3c 2 c n 2 M Q i u du d T c 3c 2 c c c 2 c c 2. 2 Hence, c c 2 2 S xt S 2 y ) t, for t t Thu, we have proved that S x S 2 y Ω for any x, y Ω. ii WehallhowthatS i a contraction mapping on Ω. In fact, for x, y Ω and t T, we have S xt S y ) t Pt xt τ yt τ c0 x y, 2.0

5 ISRN Mathematical Analyi 5 where c 0 max{c,c 2 }. Thi implie that S x S y c0 x y. 2. Since 0 <c 0 <, we conclude that S i a contraction mapping on Ω. iii We now how that S 2 i completely continuou. Firt, we will how that S 2 i continuou. Let x k x k t Ω be uch that x k t xt a k. Becaue Ω i cloed, x xt Ω. For t T, we have S 2 x k t S 2 xt t T n 2 n 2 Q i u fi x k u σ i f i xu σ i ) du d Q i u f i x k u σ i f i xu σ i ) du d. 2.2 Since f i x k t σ i f i xt σ i 0ak for i, 2,...,m, by applying the Lebegue dominated convergence theorem, we conclude that lim k S 2 x k t S 2 xt 0. Thi mean that S 2 i continuou. Next, we how that S 2 Ω i relatively compact. It uffice to how that the family of function {S 2 x : x Ω} i uniformly bounded and equicontinuou on t 0,. The uniform boundedne i obviou. For the equicontinuity, according to Levitan reult, weonly need to how that, for any given ε>0, T, can be decompoed into finite ubinterval in uch a way that on each ubinterval all function of the family have change of amplitude le than ε.by2.3, for any ε>0, take T T large enough o that n 2 T M Q i u du d < ε Then, for x Ω, t 2 t T, S 2 xt 2 S 2 xt t 2 n 2 Q i u fi xu σ i ) du d t n 2 n 2 t 2 t n 2 Q i u fi xu σ i M Q i u du d M Q i u du d ) du d 2. < ε 2 ε ε. 2

6 6 ISRN Mathematical Analyi For x Ω, T t <t 2 T, S 2 xt 2 S 2 xt t2 t n 2 Q i uf i xu σ i du d t [ t 2 n 2 t n 2] t 2 t2 n 2 M Q i u du d t n 3! t 2 t ξ n 3 M Q i u du d t 2 t2 n 2 M Q i u du d t n 3! t 2 t n 2 M Q i u du d, T Q i uf i xu σ i du d 2.5 where t <ξ<t 2. Then there exit δ>0 uch that S 2 xt 2 S 2 xt <ε, if 0 <t 2 t <δ. 2.6 For any x Ω, t 0 t <t 2 T, itieaytoeethat S 2 xt 2 S 2 xt 0 <ε. 2.7 Therefore, {S 2 x : x Ω} i uniformly bounded and equicontinuou on t 0,, and hence S 2 Ω i relatively compact. By Lemma 2., there i x 0 Ω uch that S x 0 S 2 x 0 x 0.Itieay to ee that x 0 t i a nonocillatory olution of.. The proof i complete. Theorem 2.3. Aume that < c Pt c 2 < and 2.2 hold. Then. ha a bounded nonocillatory olution. Proof. We chooe poitive contant M,M 2,αuch that c M <α< c 2 M 2. c min{α M c c 2 /c, c 2 M 2 α}. Chooing T>t 0 ufficiently large uch that T n 2 M Q i u du d < c, 2.8 where M max M xm 2 { f i x : i m}.

7 ISRN Mathematical Analyi 7 Let Ct 0,,R be the et a in the proof of Theorem 2.2. We define a bounded, cloed, and convex ubet Ω of Ct 0,,R a follow: Ω{x xt Ct 0,,R : M xt M 2,t t 0 }. 2.9 Define two map S and S 2 : Ω Ct 0,,R a follow: n S 2 xt n 2! α xt τ S xt Pt τ Pt τ, t T, S xt, t 0 t T, Ptτ Q i uf i xu σ i ) du d, t T, tτ t τ n 2 S 2 xt, t 0 t T i We hall how that for any x, y Ω, S x S 2 y Ω. In fact, for every x, y Ω,andt T,weget S xt S 2 y ) t α c c c 2 M, S xt S 2 y ) t α c 2 M 2 c 2 c c 2 M Thu, we have proved that S x S 2 y Ω. Since <c Pt c 2 <, we get that S i a contraction mapping. We alo can prove that {S 2 x : x Ω} i uniformly bounded and equicontinuou on t 0,, and hence S 2 Ω i relatively compact. So by Lemma 2., there i x 0 Ω uch that S x 0 S 2 x 0 x 0.Thati, α x 0 t Pt τ x 0t τ n Pt τ Pt τ t τ n 2 Q i uf i x 0 u σ i du d. tτ 2.22 It i eay to ee that x 0 t i a bounded nonocillatory olution of.. The proof i complete. Theorem 2.. Aume that <c Pt c 2 < and 2.2 hold. Then. ha a bounded nonocillatory olution. Proof. We chooe poitive contant M 3, M, αuch that M c 2 M 3 <α<c M. c min{α M c 2 M 3, c M α}. Chooing T>t 0 ufficiently large uch that

8 8 ISRN Mathematical Analyi T n 2 M Q i u du d < c, 2.23 where M max M3 xm { f i x : i m}. Let Ct 0,,R be the et a in the proof of Theorem 2.2. We define a bounded, cloed, and convex ubet Ω of Ct 0,,R a follow: Ω{x xt Ct 0,,R : M 3 xt M,t t 0 }. 2.2 Define two map S and S 2 : Ω Ct 0,,R a follow: n S 2 xt n 2! α xt τ S xt Pt τ Pt τ, t T, S xt, t 0 t T, Ptτ Q i uf i xu σ i du d, t T, tτ t τ n 2 S 2 xt, t 0 t T i We hall how that for any x, y Ω, S x S 2 y Ω. In fact, for every x, y Ω and t T,weget S xt S 2 y ) t c 2 α M c M 3, S xt S 2 y ) t α c c c M Thu, we have proved that S x S 2 y Ω. Since <c Pt c 2 <, wegets i a contraction mapping. We alo can prove that {S 2 x : x Ω} i uniformly bounded and equicontinuou on t 0,, and, hence, S 2 Ω i relatively compact. So by Lemma 2., there i x 0 Ω uch that S x 0 S 2 x 0 x 0.Thati, α x 0 t Pt τ x 0t τ n Pt τ Pt τ t τ n 2 Q i uf i x 0 u σ i du d. tτ 2.27 It i eay to ee that x 0 t i a bounded nonocillatory olution of.. The proof i complete. Remark 2.5. If we let n 2inTheorem 2.2, we get the Theorem in 3. In the cae where n 2, rt, Theorem 2.2 improve eentially Theorem 2.2 in 5. Remark 2.6. The condition of Theorem 2. relaxing the hypothee H of Theorem 3 in 2.

9 ISRN Mathematical Analyi 9 Remark 2.7. Theorem 2.3 and 2. improve eentially Theorem 3 in 2, we allow that Q i t i, 2,...,m are ocillatory. Acknowledgment Thi reearch wa upported by Natural Science Foundation of Shandong Province of China ZR2009AM0 and ZR2009AQ00 and Doctor of Minitry of Education Reference M. R. S. Kulenovic and S. Hadziomerpahic, Exitence of nonocillatory olution of econd order linear neutral delay equation, Mathematical Analyi and Application, vol. 228, no. 2, pp. 36 8, Z. Y. Zhang and X. X. Wang, The ocillatory and nonocillatory criteria of econd nonlinear neutral equation, Sytem Science and Mathematical Science, vol. 26, no. 3, pp , 2006 Chinee. 3 Y. Zhou, Exitence for nonocillatory olution of econd-order nonlinear differential equation, Mathematical Analyi and Application, vol. 33, no., pp. 9 96, B. M. Levitan, Some quetion of the theory of almot periodic function I, Upekhi Matematichekikh Nauk, vol. 2, no. 5, pp , 97 Ruian. 5 X. Y. Lin, Ocillation of econd-order nonlinear neutral differential equation, Mathematical Analyi and Application, vol. 309, no. 2, pp. 2 52, 2005.

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