Research Article Existence and Uniqueness of Periodic Solution for Nonlinear Second-Order Ordinary Differential Equations

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1 Hindawi Publishing Corporaion Boundary Value Problems Volume 11, Aricle ID 19156, 11 pages doi:1.1155/11/19156 Research Aricle Exisence and Uniqueness of Periodic Soluion for Nonlinear Second-Order Ordinary Differenial Equaions Jian Zu College of Mahemaics, Jilin Universiy, Changchun 131, China Correspondence should be addressed o Jian Zu, zujian1984@gmail.com Received May 1; Acceped 6 March 11 Academic Edior: Kanishka Perera Copyrigh q 11 Jian Zu. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. We sudy periodic soluions for nonlinear second-order ordinary differenial problem x f, x, x. By consrucing upper and lower boundaries and using Leray-Schauder degree heory, we presen a resul abou he exisence and uniqueness of a periodic soluion for secondorder ordinary differenial equaions wih some assumpion. 1. Inroducion The sudy on periodic soluions for ordinary differenial equaions is a very imporan branch in he differenial equaion heory. Many resuls abou he exisence of periodic soluions for second-order differenial equaions have been obained by combining he classical mehod of lower and upper soluions and he mehod of alernaive problems The Lyapunov-Schmid mehod as discussed by many auhors 1 1. In 11, he auhor gives a simple mehod o discuss he exisence and uniqueness of nonlinear wo-poin boundary value problems. In his paper, we will exend his mehod o he periodic problem. We consider he second-order ordinary differenial equaion x f, x, x. 1.1 Throughou his paper, we will sudy he exisence of periodic soluions of 1.1 wih he following assumpions: H 1 f, f x, and f x are coninuous in R R R, and f, x x f π, x, x, 1.

2 Boundary Value Problems H N <α γ sin π < 4N [ γ< 4 N 1 π 4 β< N 1, 1 γ 4α ] β 1, N 1 if N>, 1.3 where N is some posiive ineger, α inf R 3 fx, β sup fx, γ sup fx. R 3 R The following is our main resul. Theorem 1.1. Assume ha H 1 and H hold, hen 1.1 has a unique π-periodic soluion.. Basic Lemmas The following resuls will be used laer. Lemma.1 see 1. Le x C 1,h, R h> wih x x h, x > for,h,.1 hen h x x h d 4 h x d,. and he consan h/4 is opimal. Lemma. see 1. Le x C 1 a, b, R a, b R, a < b wih he boundary value condiions x a x b, hen b a x d b a π b a x d..3 Consider he periodic boundary value problem x p x q x, x x π, x x π..4

3 Boundary Value Problems 3 Lemma.3. Suppose ha p, q are L -inegrable π-periodic funcion, where p, q saisfy he condiion H, wih α inf,π q, β sup q,,π γ sup p,,π.5 hen.4 has only he rivial π-periodic soluion x. Proof. If on he conrary,.4 has a nonzero π-periodic soluion x, hen using.4, we have e p s ds x e p s ds q x,.6 where, π is undeermined. Firsly, we prove ha x has a leas one zero in, π. Ifx /, we may assume x >. Since x is a π-periodic soluion, here exiss a, π wih x x π. Then, π e p s ds x d π e p s ds q xd <,.7 we could ge a conradicion. Wihou loss of generaliy, we may assume ha x x π, x x π A>; hen here exiss a sufficienly small δ> such ha x δ/ >, x π δ/ <. Since x is a coninuous funcion, here mus exis a δ/, π δ/ wih x. Secondly, we prove ha x has a leas N zeros on, π. Considering he iniial value problem ϕ γϕ αϕ, ϕ, ϕ A..8 Obviously, A ϕ e 4α γ/ sin γ.9 is he soluion of.8 and α ϕ A eγ/ sin θ,.1

4 4 Boundary Value Problems where θ,π/ wih sin θ /4α. Since N< <N 1.11 holds under he assumpions of H, here is a,π, such ha θ π, i.e., π <π..1 Now, le N>. By he condiions H,.11,and.1, we have sin sin θ 4α π < π <π. 4N Since sin is decreasing in π/,π, we have < <π/n. Therefore, > sin π 4N, ϕ >, ϕ >, for,,ϕ..15 We also consider he iniial value problem ψ γψ αψ, ψ ϕ, ψ..16 Clearly, α ψ 4α γ ϕ e γ / sin θ.17 is he soluion of.16, where θ is he same as he previous one, and ψ α ϕ e 4α γ / sin. γ.18 Hence, here exiss a 1, π wih 1,π, such ha 1 θ π..19

5 Boundary Value Problems 5 Then, ψ 1.. From.1 and.19, i follows ha 1 π θ, 4 i.e., π 1 <π. 4.1 By H and.1, we have sin 4 1 sin θ Since sin is decreasing on π/,π, we have < 1 <π/n,and 4α > sin π 4N.. ψ <, ψ >, for, 1..3 We now prove ha x has a zero poin in, 1. If on he conrary x > for, 1, hen we would have he following inequaliies: x ϕ, for,,.4 x ψ, for, 1..5 In fac, from.4,.8, and.15, we have ϕ x ϕ x ϕ x ϕ x ϕ x ϕ x γϕ αϕ x ϕ p x q x.6 γ p ϕ x p ϕ x ϕ x q α ϕ x p ϕ x ϕ x, wih,. Seing y ϕ x ϕ x, and since y p y,.7 we obain ye p s ds,,..8

6 6 Boundary Value Problems Noice ha ϕ x, which implies y, ye p s ds,,..9 So, we have ϕ x ϕ x,,, i.e., ϕ,,..3 x Inegraing from o,,weobain ϕ s ds ϕ x s x lim ϕ ϕ x x ϕ x..31 Therefore, ϕ x 1,,,.3 which implies.4. By a similar argumen, we have.5. Therefore, <x 1 ψ 1, a conradicion, which shows ha x has a leas one zero in, 1,wih 1 <π/n. We le x 1, 1, 1.If 1 1 < π, hen from a similar argumen, here is a 1, 1 1, such ha x and so on. So, we obain ha x has a leas N zeros on, π. Thirdly, we prove ha x has a leas N 3 zeros on, π. If, on he conrary, we assume ha x only has N zeros on, π, we wrie hem as < 1 < < N 1 π..33 Obviously, x i /, i, 1,...,N Wihou loss of generaliy, we may assume ha x >. Since x i x i 1 <, i, 1,...,N,.35 we obain x N 1 <, which conradics x N 1 x >. Therefore, x has a leas N 3 zeros on, π.

7 Boundary Value Problems 7 Finally, we prove Lemma.3. Since x has a leas N 3 zeros on, π, here are wo zeros and ξ wih <ξ π/ N 1. By Lemmas.1 and., we have ξ x d ξ x x d ξ p x x d [ γ 4 ξ β ] ξ π ξ x d. ξ q x d.36 From H, i follows ha γ 4 ξ β π ξ πγ 4 N 1 β N 1 < Hence, ξ x d,.38 which implies x for,ξ.alsox. Therefore, x for, π, a conradicion. The proof is complee. 3. Proof of Theorem 1.1 Firsly, we prove he exisence of he soluion. Consider he homoopy equaion x αx λ f, x, x αx λf, x, x, 3.1 where λ, 1 and α inf R 3 f x. When λ 1, i holds 1.1. We assume ha Φ is he fundamenal soluion marix of x αx wihφ I. Equaion 3.1 can be ransformed ino he inegral equaion x x Φ x x From H 1, x is a π-periodic soluion of 3., hen Φ 1 s ds. 3. λf s, x s,x s x I Φ π x π Φ π Φ 1 s ds. λf s, x s,x s For I Φ π is inverible, x π I Φ π 1 Φ π Φ 1 s ds. x λf s, x s,x s

8 8 Boundary Value Problems We subsiue 3.4 ino 3., x Φ I Φ π 1 Φ π x Φ Φ 1 s π λf s, x s,x s Φ 1 s ds λf s, x s,x s ds. 3.5 Define an operaor P λ : C 1, π C 1, π, 3.6 such ha [ x π P λ ] Φ I Φ π 1 Φ π Φ 1 s ds x λf s, x s,x s Φ Φ 1 s ds. λf s, x s,x s 3.7 Clearly, P λ is a compleely coninuous operaor in C 1, π. There exiss B>, such ha every possible periodic soluion x saisfies x B denoe he usual normal in C 1, π. If no, here exiss λ k λ and he soluion x k wih x k k. We can rewrie 3.1 in he following form: x k αx k λ k f x, xk,θx k dθx k λ k f x, θx k, dθx k λ k f,, λ k αx k. 3.8 Le y k x k / x k R, obviously y k 1 k 1,,.... I saisfies he following problem: y k αy k λ k f x, xk,θx k dθy k λ k f x, θx k, dθy k λ k f,, / x k λ k αy k, 3.9 in which we have f,, x k k. 3.1 Since {y k }, {y } are uniformly bounded and equiconinuous, here exiss coninuous funcion k u, v and a subsequence of {k} 1 denoe i again by {k} 1, such ha lim k y k u, lim k y k v uniformly in R. Using H 1 and H, { f x, θx k, dθ} and 1

9 Boundary Value Problems 9 { f x, x k,θx k dθ} are uniformly bounded. By he Hahn-Banach heorem, here exiss 1 L -inegrable funcion p, q, and a subsequence of {k} 1 denoe i again by {k} 1, such ha f x, θx k, dθ ω q, f x, xk,θx ω k dθ p, 3.11 where ω denoes weakly converges o in L, π. As a consequence, we have u αu λ p u λ q u λ αu, 3.1 ha is, u λ p u λ q 1 λ α u Denoe ha p λ p, q λ q 1 λ α, hen we ge p λ p γ, λ α 1 λ α q λ β 1 λ α, 3.14 which also saisfy he condiion H.Noiceha p and q are L -inegrable on, π,so u saisfies Lemma.3. Hence, we have u for, π, which conradics u 1. Therefore, PC 1, π is bounded. Denoe Ω { } x C 1, π, x <B 1, h λ x x P λ x Because / h λ Ω for λ, 1, by Leray-Schauder degree heory, we have deg x Px,Ω, deg h 1 x, Ω, deg h x, Ω, / So, we conclude ha P has a leas one fixed poin in Ω,hais, 1.1 has a leas one soluion. Finally, we prove he uniqueness of he equaion when he condiion H 1 and H holds. Le x 1 and x be wo π-periodic soluions of he problem. Denoe x x 1 x,, π, hen x is a soluion of he following problem: x f x, x x,x θx dθx x x π, x x π. f x, x θx,x dθx, 3.17 By Lemma.3, we have x for, π.

10 1 Boundary Value Problems Le x kπ x,, π, k Z. We have x kπ x f, x, x f, x, x f kπ, x, x, 3.18 wih, π, k Z. Denoe x kπ, π by x R.So,x is he soluion of he problem 1.1. The proof is complee. 4. An Example Consider he sysem x 3 sin x 6x cos x p, 4.1 where p p π is a coninuous funcion. Obviously, α inf fx inf 6 sin x 5, R 3 R 3 β sup R 3 fx sup 6 sin x 7, R 3 γ sup f x sup R 3 R 3 3 sin 3 4. saisfy Theorem 1.1, hen here is a unique π-periodic soluion in his sysem. Acknowledgmens The auhor expresses sincere hanks o Professor Yong Li for useful discussion. He would like o hank he reviewers for helpful commens on an earlier draf of his paper. References 1 C. Bereanu and J. Mawhin, Exisence and mulipliciy resuls for some nonlinear problems wih singular φ-laplacian, Journal of Differenial Equaions, vol. 43, no., pp , 7. J. Ehme, P. W. Eloe, and J. Henderson, Upper and lower soluion mehods for fully nonlinear boundary value problems, Journal of Differenial Equaions, vol. 18, no. 1, pp ,. 3 R. Kannan and V. Lakshmikanham, Exisence of periodic soluions of nonlinear boundary value problems and he mehod of upper and lower soluions, Applicable Analysis, vol. 17, no., pp , H.-W. Knobloch, On he exisence of periodic soluions for second order vecor differenial equaions, Journal of Differenial Equaions, vol. 9, pp , H. W. Knobloch and K. Schmi, Non-linear boundary value problems for sysems of differenial equaions, Proceedings of he Royal Sociey of Edinburgh. Secion A, vol. 78, no. 1-, pp , Y. Liu and W. Ge, Posiive periodic soluions of nonlinear Duffing equaions wih delay and variable coefficiens, Tamsui Oxford Journal of Mahemaical Sciences, vol., no., pp , 4. 7 R. Orega and M. Tarallo, Almos periodic upper and lower soluions, Journal of Differenial Equaions, vol. 193, no., pp , 3.

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