Research Article Fixed Points and Stability in Nonlinear Equations with Variable Delays

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1 Hindawi Publihing Corporation Fixed Point Theory and Application Volume 21, Article ID , 14 page doi:1.1155/21/ Reearch Article Fixed Point and Stability in Nonlinear Equation with Variable Delay Liming Ding, 1, 2 Xiang Li, 1 and Zhixiang Li 1 1 Department of Mathematic and Sytem Science, College of Science, National Univerity of Defence Technology, Changha 4173, China 2 Air Force Radar Academy, Wuhan 431, China Correpondence hould be addreed to Liming Ding, limingding@ohu.com Received 9 July 21; Accepted 18 October 21 Academic Editor: Hichem Ben-El-Mechaiekh Copyright q 21 Liming Ding et al. 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. We conider two nonlinear calar delay differential equation with variable delay and give ome new condition for the boundedne and tability by mean of the contraction mapping principle. We obtain the difference of the two equation about the tability of the zero olution. Previou reult are improved and generalized. An example i given to illutrate our theory. 1. Introduction Fixed point theory ha been ued to deal with tability problem for everal year. It ha conquered many difficultie which Liapunov method cannot. While Liapunov direct method uually require pointwie condition, fixed point theory need average condition. In thi paper, we conider the nonlinear delay differential equation x t atxt r 1 t btgxt r 2 t, x t atfxt r 1 t btgxt r 2 t, where r 1 t,r 2 t :,,, r max{r 1,r 2 }, a, b :, R, f, g : R R are continuou function. We aume the following: A1 r 1 t i differentiable, A2 the function t r 1 t,t r 2 t :, r, i trictly increaing, A3 t r 1 t,t r 2 t a t.

2 2 Fixed Point Theory and Application Many author have invetigated the pecial cae of 1.1 and 1.2. Since Burton 1 ued fixed point theory to invetigate the tability of the zero olution of the equation x t atxt r, many cholar continued hi idea. For example, Zhang 2 ha tudied the equation x t atxt rt, 1.3 Becker and Burton 3 have tudied the equation x t atfxt rt, 1.4 Jin and Luo 4 have tudied the equation x t atxt r 1 t btx 1/3 t r 2 t. 1.5 Burton 5 and Zhang 6 have alo tudied imilar problem. Their main reult are the following. Theorem 1.1 Burton 1. Suppoe that rt r, a contant, and there exit a contant α<1 uch that a r d t r a r e aurdu r du d α, 1.6 r au for all t and ad. Then, for every continuou initial function ψ : r, R, the olution xt xt,,ψ of 1.3 i bounded and tend to zero a t. Theorem 1.2 Zhang 2. Suppoe that r i differentiable, the invere function ht of t rt exit, and there exit a contant α, 1 uch that for t i lim inf t ah >, 1.7 ii ah d t rt e ahudu ah ahv dv d θ, r 1.8 where θt e ahudu a r d. Then, the zero olution of aymptotically table if and only if iii 1.3 i ahd, a t. 1.9

3 Fixed Point Theory and Application 3 Theorem 1.3 Burton 7. Suppoe that rt r, a contant. Let f be odd, increaing on,l, and atifie a Lipchitz condition, and let x fx be nondecreaing on,l. Suppoe alo that for each L 1,L, one ha L1 fl 1 up t fl 1 up t e aurdu a r d fl 1 up t au r du t r e aurdu a r au r du d < L 1, r 1.1 and there exit J> uch that Then, the zero olution of 1.4 i table. a rd J for t Theorem 1.4 Becker and Burton 3. Suppoe f i odd, trictly increaing, and atifie a Lipchitz condition on an interval l, l and that x fx i nondecreaing on,l. If up audu < 1 t t 1 t rt 2, 1.12 where t 1 i the unique olution of t rt, and if a continuou function ã :, R exit uch that at ãt ( 1 r t ), 1.13 on,, then the zero olution of 1.5 i table at t. Furthermore, if f i continuouly differentiable on l, l with f / and audu a t, 1.14 then the zero olution of 1.4 i aymptotically table. In the preent paper, we adopt the contraction mapping principle to tudy the boundedne and tability of 1.1 and 1.2. That mean we invetigate how the tability property will be when 1.3 and 1.4 are added to the perturbed term btgxt r 2 t. We obtain their difference about the tability of the zero olution, and we alo improve and generalize the pecial cae r 1 t r 1. Finally, we give an example to illutrate our theory. 2. Main Reult From exitence theory, we can conclude that for each continuou initial function ψ : r, R there i a continuou olution xt,,ψ on an interval,t for ome T > and

4 4 Fixed Point Theory and Application xt,,ψψt on r,.letcs 1,S 2 denote the et of all continuou function φ : S 1 S 2 and ψ max{ ψt : t r, }. Stability definition can be found in 8. Theorem 2.1. Suppoe that the following condition are atified: i g, and there exit a contant L> o that if x, y L,then gx g ( y ) x y, 2.1 ii there exit a contant α, 1 and a continuou function h : r, R uch that h d e hudu h hu du d r 1 e hudu[ h r 1 ( 1 r 1 ) a b ] d α, 2.2 iii lim inf t hd >. 2.3 Then, the zero olution of 1.1 i aymptotically table if and only if iv hd, a t. 2.4 Proof. Firt, uppoe that iv hold. We et J up t { } hd. 2.5 Let S {φ φ C r,,r, φ up t r φt < }, then S i a Banach pace. Multiply both ide of 1.1 by e hd, and then integrate from to t to obtain xt x e hd e hudu hxd e hudu ax r 1 d e hudu bgx r 2 d. 2.6

5 Fixed Point Theory and Application 5 By performing an integration by part, we have xt x e hd e hudu ( r 1 huxudu) d e hudu[ h r 1 ( 1 r 1 ) a ] x r 1 d e hudu bgx r 2 d, 2.7 or xt x e hd e hd hxd hxd r 1 e hudu h huxudu d r 1 e hudu[ h r 1 ( 1 r 1 ) a ] x r 1 d e hudu bgx r 2 d. 2.8 Let { M φ φ S, upφt } L, φt ψt for t r,, φt at. 2.9 t r Then, M i a complete metric pace with metric up t φt ηt for φ, η M. For all φ M, define the mapping P ( Pφ ) t ψt, t r,, ( ) Pφ t ψe hd e hd hψd hφd r 1 t r 1 t e hudu h huφudu d r e hudu[ h r 1 ( 1 r 1 ) a ] φ r 1 d e hudu bg ( φ r 2 ) d, t.

6 6 Fixed Point Theory and Application By i and g, ( Pφ ) t ] ψ [1 h d e hd r 1 [ L h d e hudu h hu du d r 1 e hudu[ h r1 ( 1 r 1 ) a b ] d ] ] ψ [1 h d e J αl. r Thu, when ψ δ 1 αl/1 r 1 h dej, Pφt L. We now how that Pφt at. Since φt andt r 1 t a t, for each ε>, there exit a T 1 > uch that t>t 1 implie φt r 1 t <ε.thu,for t T 1, I 1 hφd ε h d αε. t r 1 t 2.12 Hence, I 1 at. And I 2 e hudu h huφudu d r 1 T1 e hudu h hu φudu d r 1 e hudu h hu φu du d T 1 r 1 T1 L e hudu h hu du d ε r 1 T 1 e hudu h r 1 hu du d, 2.13 By ii and iv, there exit T 2 >T 1 uch that t T 2 implie T1 L e hudu h hu du d < ε. r Apply ii to obtain I 2 <ε 2ε <2ε.Thu,I 2 at. Similarly, we can how that the ret term in 2.1 approache zero a t. Thi yield Pφt at, and hence Pφ M.

7 Fixed Point Theory and Application 7 Alo, by ii, P i a contraction mapping with contraction contant α. By the contraction mapping principle, P ha a unique fixed point x in M which i a olution of 1.1 with x ψ on r, and xt at. In order to prove tability at t, let ε>begiven. Then, chooe m>othat m<min{ε, L}. Replacing L with m in M, we ee there i a δ>uch that ψ <δimplie that the unique continuou olution x agreeing with ψ on r, atifie xt m<εfor all t r. Thi how that the zero olution of 1.1 i aymptotically table if iv hold. Converely, uppoe iv fail. Then, by iii, there exit a equence {t n }, t n a n n uch that lim n hd l for ome l R. We may chooe a poitive contant N atifying N n hd N, 2.15 for all n 1. To implify the expreion, we define ω h r1 ( 1 r 1 ) a b h r 1 hu du, 2.16 for all. By ii, we have n e n hudu ωd α Thi yield n e hudu ωd αe n hudu αe N The equence { n hudu ωd} i bounded, o there exit a convergent ubequence. For brevity of notation, we may aume that e n lim e hudu ωd γ, 2.19 n for ome γ R and chooe a poitive integer k o large that n t k e hudu ωd < δ 4N, 2.2 for all n k, where δ > atifie2δ Je N α<1.

8 8 Fixed Point Theory and Application By iii, J in 2.5 i well defined. We now conider the olution xt xt, t k,ψ of 1.1 with ψt k δ and ψ δ for t k. We may chooe ψ o that xt L for t t k and ψ ( t k ) k t k r 1t k hψd 1 2 δ It follow from 2.1 with xt Pxt that for n t k, xt n n t n r 1 t n hxd 1 2 δ e n n t hudu k t k e n hudu ωd 1 2 δ e n t k hudu e n hudu n e tn t hudu( 1 k 2 δ e t tn k hudu t k e hudu ωd t k ) e hudu ωd e tn t hudu( 1 tn ) k 2 δ N e hudu ωd t k 1 4 δ e n t k hudu 1 4 δ e 2N > On the other hand, if the olution of 1.1 xt xt, t k,ψ at,incet n r 1 t n a n and ii hold, we have n xt n hxd a n, 2.23 t n r 1t n which contradict Hence, condition iv i neceary for the aymptotically tability of the zero olution of 1.1. The proof i complete. When r 1 t r 1, a contant, ht at r 1, we can get the following. Corollary 2.2. Suppoe that the following condition are atified: i g, and there exit a contant L> o that if x, y L,then gx g ( y ) x y, 2.24

9 Fixed Point Theory and Application 9 ii there exit a contant α, 1 uch that for all t, one ha t r 1 a r 1 d e aur 1du b d α, e aur1du a r 1 au r 1 du d r iii lim inf t a r 1 d > Then, the zero olution of 1.1 i aymptotically table if and only if iv a r 1 d, a t Remark 2.3. We can alo obtain the reult that xt i bounded by L on r,. Ourreult generalize Theorem 1.1 and 1.2. Theorem 2.4. Suppoe that a continuou function ã :, R exit uch that at ãt1 r 1 t and that the invere function ht of t r 1t exit. Suppoe alo that the following condition are atified: i there exit a contant J>uch that up t { ãhd} <J, ii there exit a contant L>uch that fx,x fx,gx atify a Lipchitz condition with contant K>on an interval L, L, iii f and g are odd, increaing on,l. x fx i nondecreaing on,l, iv for each L 1,L, one ha L 1 fl 1 up t gl 1 up t e ãhudu ãh d fl 1 up e ãhudu b d < L 1. t ãh d 2.28 Then, the zero olution of 1.2 i table.

10 1 Fixed Point Theory and Application Proof. By iv, there exit α, 1 uch that L fl up t glup t e ãhudu ãh d flup e ãhudu b d αl. t ãh d 2.29 Let S be the pace of all continuou function φ : r, R uch that φk : up {e dk2 ãh b d } φt : t r, <, 2.3 where d > 3 i a contant. Then, S, K i a Banach pace, which can be verified with Cauchy criterion for uniform convergence. The equation 1.2 can be tranformed a x t ãhtfxt d dt ãhtxt ãht [ xt fxt ] ãhfxd btgxt r 2 t 2.31 d dt ãhfxd btgxt r 2 t. By the variation of parameter formula, we have xt x e ãhd e ãhd r 1 ãhfxd e ãhudu[ x fx ] d ãhfxd e ãhudu bgx r 2 d Let { M φ φ S, up φt L, φt ψt, t r, }, 2.33 t r

11 Fixed Point Theory and Application 11 then M i a complete metric pace with metric φ η K for φ, η M. For all φ M, define the mapping P ( Pφ ) t ψt, t r,, ( Pφ ) t ψe ãhd e ãhd r 1 ãhf ( ψ ) d e ãhudu[ φ f ( φ )] d ãhf ( φ ) d e ãhudu bg ( φ r 2 ) d By i, iii, and2.29, we have ( Pφ ) t ψ e J e J f ( ψ ) ãh d L fl up r 1 t flup t ãh d glup t ψ e J e J f ( ψ ) r 1 ãh d αl. e ãhudu b d e r 1 ãhudu ãhd 2.35 Thu, there exit δ,l uch that e J 1Kδ r 1 ãh d < 1 αl and Pφt L. Hence, Pφ M. We now how that P i a contraction mapping in M. For all φ, η M, ( Pφ ) t ( Pη ) t e ãhudu ãh ( ) ( ) φ f φ η f η d ãh ( ) ( ) f φ f η d 2.36 e ãhudu b g ( φ r2 ) g ( η r 2 ) d.

12 12 Fixed Point Theory and Application Since e dk2 t ãh b d e ãhudu ãh ( ) ( ) φ f φ η f η d e ãhudu ãh K φ η e dk2 ãhu bu du e dk2 ãhu bu du d e ãhudu ãh Ke dk2 ãhu bu du d φ η K 1 φ ηk, d e dk2 ãh b d ãh f ( φ ) f ( η ) d ãh K φ ηe dk2 ãhu bu du e dk2 ãh Ke dk2 ãhu bu du d φ ηk 1 φ ηk, d ãhu bu du d e dk2 t ãh b d e ãhudu b ( g φ r2 ) g ( η r 2 ) d e ãhudu b K φ r2 η r 2 e dk2 r 2 ãhu bu du e dk2 r 2 ãhu bu du e dk2 ãhu bu du d e ãhudu b Ke dk2 ãhu bu du d φ η K 1 φ ηk, d 2.37 we have e dk2 ãh b d Pφt Pηt 3/d φ η K. That mean Pφ Pη 3/d φ η K. Hence, P i a contraction mapping in M with contant 3/d. By the contraction mapping principle, P ha a unique fixed point x in M, which i a olution of 1.2 with x ψ on r, and up t r xt L. In order to prove tability at t, let ε> be given. Then, chooe m>othat m<min{ε, L}. Replacing L with m in M, we ee there i a δ> uch that ψ <δimplie that the unique continuou olution x agreeing with ψ on r, atifie xt m<εfor all t r. Thi how that the zero olution of 1.2 i table. That complete the proof.

13 Fixed Point Theory and Application 13 When r 1 t r 1, a contant, we have the following. Corollary 2.5. Suppoe that the following condition are atified: i there exit a contant J>uch that up t { a r 1d} <J, ii there exit a contant L>uch that fx,x fx,gx atify a Lipchitz condition with contant K>on an interval L, L, iii f and g are odd, increaing on,l. x fx i nondecreaing on,l, iv for each L 1,L, one ha L1 fl 1 up t gl 1 up t e aur 1du a r 1 d fl 1 up e aur 1du b d < L 1. t t r 1 au r 1 du 2.38 Then, the zero olution of the equation x t atfxt r 1 btgxt r 2 t 2.39 i table. Corollary 2.6. Suppoe that the following condition are atified: i there exit a contant J>uch that up t { ad} <J, ii there exit a contant L>uch that fx, x fx, gx atify a Lipchitz condition with contant K>on an interval L, L, iii f and g are odd, increaing on,l. x fx i nondecreaing on,l, iv for each L 1,L, one ha L1 fl 1 up t e audu a d gl 1 up t e audu b d < L Then, the zero olution of x t atfxt btgxt rt 2.41 i table. Remark 2.7. The zero olution of 1.2 i not a aymptotically table a that of 1.1. Thekey i that M i not complete under the weighted metric when added the condition to M that φt at. Remark 2.8. Theorem 2.4 make ue of the technique of Theorem 1.3 and 1.4.

14 14 Fixed Point Theory and Application 3. An Example We ue an example to illutrate our theory. Conider the following differential equation: x t atxt r 1 t btgxt r 2 t. 3.1 where r 1 t.281t, r 2 CR,R, gx x 3, at 1/.719t 1, andbt μ in t/t 1, μ>. Thi equation come from 4. Chooing ht 1.2/t 1, we have h d.719t d 1.2 ln t 1.719t 1 <.396, e hudu h r1 ( 1 r 1 ) a d t e hudu h r 1 hu du d <.396, 1.2/u1du d e < e 1.2/u1du 1.2 d <.1592, 1 e hudu b d μ Let α : μ/1.2, when μ i ufficiently mall, α<1. Then, the condition ii of Theorem 2.1 i atified. Let L 3/3, then the condition i of Theorem 2.1 i atified. And hd 1.2/ 1d 1.2 lnt 1, then the condition iii and iv of Theorem 2.1 are atified. According to Theorem 2.1, the zero olution of 3.1 i aymptotically table. Reference 1 T. A. Burton, Stability by fixed point theory or Liapunov theory: a comparion, Fixed Point Theory, vol. 4, no. 1, pp , B. Zhang, Fixed point and tability in differential equation with variable delay, Nonlinear Analyi: Theory, Method and Application, vol. 63, no. 5 7, pp. e233 e242, L. C. Becker and T. A. Burton, Stability, fixed point and invere of delay, Proceeding of the Royal Society of Edinburgh. Section A, vol. 136, no. 2, pp , C. Jin and J. Luo, Stability in functional differential equation etablihed uing fixed point theory, Nonlinear Analyi: Theory, Method & Application, vol. 68, no. 11, pp , T. A. Burton, Stability and fixed point: addition of term, Dynamic Sytem and Application, vol. 13, no. 3-4, pp , B. Zhang, Contraction mapping and tability in a delay-differential equation, in Dynamic Sytem and Application, vol. 4, pp , Dynamic, Atlanta, Ga, USA, T. A. Burton, Stability by fixed point method for highly nonlinear delay equation, Fixed Point Theory, vol. 5, no. 1, pp. 3 2, J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equation, Springer, New York, NY, USA, 1993.