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Hindawi Publishing Corporation Boundary Value Problems Volume 2008, Article ID 574842, 9 pages doi:10.1155/2008/574842 Research Article Multiplicity of Positive Periodic Solutions of Singular Semipositone Third-Order Boundary Value Problems Yigang Sun Department of Applied Mathematics, Hohai University, Nanjing 210098, China Correspondence should be addressed to Yigang Sun, hongyimingsun@163.com Received 2 July 2007; Accepted 13 December 2007 Recommended by Colin Rogers We establish the existence of multiple positive solutions for a singular nonlinear third-order peri- odic boundary value problem. We are mainly interested in the semipositone case. The proof relies on a nonlinear alternative principle of Leray-Schauder, together with a truncation technique. Copyright q 2008 Yigang Sun. This is an open access article distributed under the Creative Commons Attribution License, which p ermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In this paper, we study the existence and multiplicity of positive periodic solutions of the fol- lowing singular nonlinear third-order periodic boundary value problem: u   ρ 3 u  ft, u, 0 ≤ t ≤ 2π, u i 0u i 2π,i 0, 1, 2. 1.1 Here ρ ∈ 0, 1/ √ 3 is a positive constant and ft, u is continuous in t, u and 2π- is periodic in t. We are mainly interested in the case that ft, u may be singular at u  0 and satisfies the following semipositone condition: G 1  There exists a constant L>0 such that Ft, uft, uL ≥ 0 for all t, u ∈ 0, 2π×0, ∞. During the last two decades, singular periodic problems have deserved the attention of many researchers 1–8. Third-order boundary value problems have also been studied in 9–11. For the problem 1.1, we recall the following results. In 12, by using Schauder fixed- point theorem, together with perturbation technique, it was established the existence of at least one positive solution under some suitable conditions of ft, u. One hard restriction in 12 was the monotonicity on ft, u.In13, this restricted condition was removed and the existence 2 Boundary Value Problems of multiple positive solutions was obtained by using the fixed-point index theory. Recently, instead of Schauder fixed-point theorem and fixed-point index theory, Chu and Zhou 10 em- ployed a nonlinear alternative principle of Leray-Schauder and a fixed-point theorem in cones due to Krasnoselskii 14 to study problem 1.1. It was proved that 1.1 has at least two pos- itive solutions for the positone case and has at least one positive solution for the semipositone case. For the convenience of the reader, we recall the following result obtained in 10 for the semipositone case. Theorem 1.1. Suppose that (G 1 ) is satisfied. Furthermore, it is assumed that G 2  there exist continuous nonnegative functions gu and hu on 0, ∞ such that Ft, u ≤ guhu ∀t, u ∈ 0, 2π × 0, ∞1.2 and gu > 0 is nonincreasing and hu/gu is nondecreasing in u; G 3  there exist continuous, nonnegative functions g 1 u and h 1 u on 0, ∞ such that Ft, u ≥ g 1 uh 1 u ∀t, u ∈ 0, 2π × 0, ∞1.3 and g 1 u > 0 is nonincreasing and h 1 u/g 1 u is nondecreasing in u; G 4  there exists r>ρω/σsuch that r gσr/ρ − ω  1   hr/ρ − ω/gr/ρ − ω  > 1 ρ 2 , 1.4 where ω  L/ρ 3 , σ  m/M will be given in Section 2. G 5  There exists R>rsuch that R g 1 R/ρ − ω  1   h 1 σR/ρ − ω/g 1 σR/ρ − ω  ≤ 1 ρ 2 . 1.5 Then problem 1.1 has a positive solution u with ut > 0 for t ∈ 0, 2π and r/ρ < u  ω <R/ρ. The rest of this paper is organized as follows. In Section 2, some preliminary results will be given. In Section 3, w e will state and prove the main results. Furthermore, an illustrating example will be given. 2. Preliminaries In this section, we present some preliminary results. First, as in 13, we transform the problem into an integral equation. For any function u ∈ C0, 2π, we define the operator Jut  2π 0 gt, xuxdx, 2.1 where gt, x ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ e −ρt−x 1 − e −2πρ , 0 ≤ x ≤ t ≤ 2π, e −ρ2πt−x 1 − e −2πρ , 0 ≤ t ≤ x ≤ 2π. 2.2 Yigang Sun 3 By a direct calculation, w e can easily obtain  2π 0 gt, xdx  1 ρ . 2.3 Next we consider the equation u  − ρu   ρ 2 u  F  t, J  ut  − ω  , 0 ≤ t ≤ 2π 2.4 with the following periodic boundary condition: u0u2π,u  0u  2π. 2.5 If ut >L/ρ 2 , for all t ∈ 0, 2π, is a solution of problem 2.4-2.5,itiseasytoverify that ytJut − ω is a positive solution of problem 1.1for more details, see 10. Consequently, we will concentrate our study on problem 2.4-2.5. Lemma 2.1 see 12. The boundary value problem 2.4-2.5 is equivalent to integral equation ut  2π 0 Gt, sF  s, J  us  − ω  ds, 2.6 where Gt, s  ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2e ρ/2t−s  sin √ 3/2ρ2π − t  se −ρπ sin √ 3/2ρt − s  √ 3ρ  e ρπ  e −ρπ − 2cos √ 3ρπ  , 0 ≤ s ≤ t ≤ 2π, 2e ρ/22πt−s  sin √ 3/2ρs − te −ρπ sin √ 3/2ρ2π − s  t  √ 3ρ  e ρπ  e −ρπ − 2cos √ 3ρπ  , 0 ≤ t ≤ s ≤ 2π. 2.7 Moreover, we have the estimates 0 <m 2sin √ 3ρπ √ 3ρ  e ρπ  1  2 ≤ Gt, s ≤ 2 √ 3ρ sin √ 3ρπ  M, ∀s, t ∈ 0, 2π. 2.8 In applications below, we take X  C0, 2π with the supremum norm · and we define an operator T : X → X by Tut  2π 0 Gt, sF  s, Jus  ds, 2.9 where F : 0, 2π × R → 0, ∞ is a continuous function. It is easy to see that T is continuous and completely continuous. 4 Boundary Value Problems 3. Main results In this section, we state and prove the main results of this paper. Theorem 3.1. Suppose that ft, u satisfies (G 1 )-(G 5 ). In addition, suppose that G 6  there exists a nonincreasing positive continuous function g 0 u on 0, ∞ and a constant R 0 such that ft, u ≥ g 0 u for t, u ∈ 0, 2π × 0,R 0 ,whereg 0 u satisfies the strong force condition, that is, lim u→0  g 0 u∞ and lim x→0   R 0 x g 0 udu ∞. Then problem1.1 has at least one positive periodic solution u with ω<u  ω <r/ρ. Proof. We only need to show that problem 2.4-2.5 has a solution ut >L/ρ 2 and L/ρ 2 < u <r, for all t ∈ 0, 2π. To do so, we will use the Leray-Schauder alternative principle, together with a truncation technique. Let N 0  {n 0 ,n 0  1, },wheren 0 ∈ N is chosen such that 1/n 0 <σr− L/ρ 2  and 1 ρ 2 gσr/ρ − ω  1  hr/ρ − ω gr/ρ − ω   1 n 0 <r. 3.1 For λ ∈ 0, 1, consider the family of equations u  − ρu   ρ 2 u  λF n  t, J  ut  − ω   ρ 2 n ,n∈ N 0 , 3.2 where F n t, x ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Ft, x,x≥ 1 nρ , F  t, 1 nρ  ,x≤ 1 nρ . 3.3 Problem 3.2-2.5 is equivalent to the following fixed-point problem in C0, 2π: utλ  2π 0 Gt, sF n  s, J  us  − ω  ds  1 n . 3.4 We claim that any fixed point u of 3.4 must satisfy u / r for all λ ∈ 0, 1. Otherwise, assume that u is a solution of 3.4 for some λ ∈ 0, 1 such that u  r.Wehave ut − 1 n  λ  2π 0 Gt, sF n  s, J  us  − ω  ds ≥ λm  2π 0 F n  s, J  us  − ω  ds  σMλ  2π 0 F n  s, J  us  − ω  ds ≥ σmax t  λ  2π 0 Gt, sF n  s, J  us  − ω  ds   σ     u − 1 n     . 3.5 Yigang Sun 5 By n, n 0 ∈ N 0 , it is evident that 1/n ≤ 1/n 0 <r. Hence, for all t ∈ 0, 2π,wehave ut ≥ σ     x − 1 n      1 n ≥ 1 n , ut ≥ σ     x − 1 n      1 n ≥ σ  x− 1 n   1 n  σ  r − 1 n   1 n ≥ σr; 3.6 thus, by conditions G 2  and G 4 ,wehave utλ  2π 0 Gt, sF n  s, J  us  − ω  ds  1 n ≤  2π 0 Gt, sF  s, J  us  − ω  ds  1 n ≤  2π 0 Gt, sg  J  us  − ω   1  hJus − ω gJus − ω  ds  1 n ≤ 1 ρ 2 gσr/ρ − ω  1  hr/ρ − ω gr/ρ − ω   1 n . 3.7 Therefore, r  u≤ 1 ρ 2 gσr/ρ − ω  1  hr/ρ − ω gr/ρ − ω   1 n 0 3.8 which is a contradiction to the choice of n 0 and the claim is proved. From this claim, the nonlinear alternative of Leray-Schauder guarantees that 3.4 has a fixed point, denoted by u n for n ∈ N 0 with the property u n  <r. In order to pass the solutions u n of the truncation equation 3.2with λ  1 to that of the original problem 1.1,weneedthefactu  n ≤H for some constant H>0 for all n ≥ n 0 . Integrating 3.2 with λ  1 from 0 to 2π,weobtain ρ 2  2π 0 u n tdt   2π 0  F n  t, J  u n t  − ω   ρ 2 n  dt. 3.9 By the periodic boundary condition, u  n t 0 0 for some t 0 ∈ 0, 2π.Then   u  n    max 0≤t≤2π   u  n t    max 0≤t≤2π      t t 0 u  n sds      max 0≤t≤2π      t t 0  F n  s, J  u n s  − ω   ρ 2 n  ρu  n s − ρ 2 u n s  ds     ≤  2π 0  F n  s, J  u n s  − ω   ρ 2 n  ds  ρ 2  2π 0 u n sds  ρ   u n t − u n  t 0     2ρ 2  2π 0 u n sds  ρ   u n t − u n  t 0    < 4πρ 2 r  2ρr : H. 3.10 6 Boundary Value Problems In the next lemma, we will show that there exists a constant δ>0 such that u n t − L ρ 2 ≥ δ, ∀t ∈ 0, 2π 3.11 for n large enough. Since u i n , i  0, 1 are bounded, {u n } n∈N 0 is bounded and equicontinuous family on 0, 2π. Now the Arzela-Ascoli theorem guarantees that {u n } n∈N 0 has a subsequence, {u n } n∈N n k , converging uniformly to a function u ∈ C0, 2πobviously, δ ≤ ut ≤ r. Furthermore, u n k satisfies the integral equation u n k t  2π 0 Gt, sF  s, J  u n k s  − ω  ds  1 n k . 3.12 Letting k →∞,weobtainthat ut  2π 0 Gt, sF  s, J  us  − ω  ds, 3.13 where the uniform continuity of Ft, · on 0, 2π × δ/ρ, r/ρ is used. Hence, ut is a positive periodic solution of 2.4-2.5. Finally, it is not difficult to show that u <r,bynotingthatifu  r, the argument similar to the proof of the first claim will yield a contradiction. Lemma 3.2. There exists a constant δ>0 such that any solution u n t satisfies 3.11 for n large enough. Proof. The conclusion is established using the strong force condition of ft, u. By condition G 3 , there exists R 1 ∈ 0,R 0  and a continuous function g 0 · satisfying the strong force condi- tion such that F  t, J  u n t  − ω  −  ρ 2 J  u n t  − ω  ≥ g 0  J  u n t  − ω  > max  L, ρ 2 r  ρH  , 3.14 for all t, u ∈ 0, 2π × 0,R 1 . Choose n 1 ∈ N 0 such that 1/n 1 <R 1 and let N 1  {n 1 ,n 1  1, }.Forn ∈ N 1 ,let 0 <α n  min t  u n t − L ρ 2  ≤ max t  u n t − L ρ 2   β n . 3.15 First we claim that β n >R 1 for all n ∈ N 1 . Otherwise, it is easy to verify that F n  t, J  u n t  − ω  >ρ 2 r  ρH. 3.16 Infact,if1/n ≤ u n t − L/ρ 2 ≤ R 1 , following from 3.14,wehave F n  t, J  u n t  − ω   F  t, J  u n t  − ω  ≥ ρ 2  J  u n t  − ω   g 0  J  u n t  − ω  ≥ g 0  J  u n t  − ω  >ρ 2 r  ρH 3.17 Yigang Sun 7 and if u n t − L/ρ 2 ≤ 1/n,wehave F n  t, J  u n t  − ω   F  t, 1 nρ  ≥ ρ n  g 0  1 nρ  ≥ g 0  1 nρ  >ρ 2 r  ρH. 3.18 By 3.16 and integrating 3.2with λ  1 from0to2π,weobtainthat 0   2π 0  u  n − ρu  n  ρ 2 u n − F n  t, J  u n t  − ω  − ρ 2 n  dt ≤ ρ 2  2π 0 u n tdt −  2π 0 F n  t, J  u n t  − ω  dt < 0. 3.19 This is a contradiction and thus the claim is proved. Next we claimed that u  n t > 0, for all t ∈ 0, 2π. Suppose α n <R 1 ,thatis, α n  min t  u n t − L ρ 2   u n  a n  − L ρ 2 <R 1 < max t  u n t − L ρ 2   β n . 3.20 So there exists c n ∈ 0, 2πwithout loss of generality, we assume a n <c n  such that u n c n  − L/ρ 2  R 1 and u n t ≤ R 1  L/ρ 2 for t ∈ a n ,c n . It can be checked that F n  t, J  u n t  − ω  >ρ 2 r  ρH. 3.21 By 3.2 with λ  1and3.21, we can easily obtain that u  n t > 0, as u  0 a n 0, u  n t > 0for all t ∈ a n ,c n , and the function y n : u n − L/ρ 2 is strictly increasing on a n ,c n . We use ξ n to denote the inverse function of y n restricted to a n ,c n . In order to obtain 3.14, first we will show that u n t − L ρ 2 ≥ 1 nρ , for some n ∈ N 1 . 3.22 Otherwise, there should exist b n ∈ a n ,c n  such that x n b n  − L/ρ 2  1/n for some n ∈ N 1 and u n t − L ρ 2 ≤ 1 n , ∀a n ≤ t ≤ b n , 1 n ≤ u n t − L ρ 2 ≤ R 1 , ∀b n ≤ t ≤ c n . 3.23 Multiplying 3.2with λ  1 by u  n t and integrating from b n to c n ,weobtain  R 1 1/n F  ξ n y,Jy  dy   c n b n F  t, J  u n t  − ω  u  n tdt   c n b n F n  t, J  u n t  − ω  u  n tdt   c n b n u  n tu  n tdt −  c n b n  ρu  n  ρ 2 u n − ρ 2 n  u  n tdt. 3.24 By the facts u n  <rand u  n  <H, one can easily obtain that the last equation is bounded, that is, there exist a constant η>0 such that  R 1 1/n F  ξ n y,Jy  dy ≤ η. 3.25 8 Boundary Value Problems On the other hand, by G 3  we can choose n 2 ∈ N 1 large enough such that  R 1 1/n F  ξ n y,Jy  dy ≥  R 1 1/n g 0 Jydy > η 3.26 for all n ∈ N 2  {n 2 ,n 2  1, }.So3.22 holds for n ∈ N 2 . As a last step, we will show that 3.14 holds. Multiplying 3.2 by u  n t and integrating from a n to c n ,weobtain  R 1 α n F  ξ n y,Jy  dy   c n a n F  t, J  u n t  − ω  u  n tdt   c n a n F n  t, J  u n t  − ω  u  n tdt   c n a n u  n tu  n tdt −  c n a n  ρu  n  ρ 2 u n − ρ 2 n  u  n tdt. 3.27 In the same way as in the proof of 3.24, one may readily prove that the right-hand side of the above equality is bounded. On the other hand, by G 3  if n ∈ N 2 ,  R 1 α n F  ξ n y,Jy  dy ≥  R 1 α n g 0 Jydy  M  R 1 − α n  −→ ∞,  α n −→ 0   . 3.28 Thus, the claim is confirmed. Combined with Theorems 1.1 and 3.1, we can obtain the following multiplicity result. Theorem 3.3. Suppose that (G 1 )–(G 6 ) are satisfied. Then problem 1.1 has at least two positive peri- odic solutions u, u with ω<u  ω <r/ρ<u  ω <R/ρ. Corollary 3.4. Let the nonlinearity in 1.1 be ft, ubtu −α  μctu β  et, 0 ≤ t ≤ 2π, 3.29 where α>0 and β ≥ 0, bt, ct, et are nonnegative continuous functions and bt > 0, for all t ∈ 0, 2π, μ>0 is a positive parameter. Then i if β<1,problem1.1 has at least one positive periodic solution for each μ>0; ii if β ≥ 1,problem1.1 has at least one positive periodic solution for each 0 <μ<μ ∗ ,whereμ ∗ is some positive constant; iii if β>1,problem1.1 has at least two positive periodic solutions for each 0 <μ<μ ∗ ,hereμ ∗ is the same as in (ii). Proof. Let M  max 0≤t≤2π |et| and gub 0 u −α ,huμc 0 u β  M, g 1 ub 1 u −α ,h 1 uμc 1 u β , 3.30 where b 0  max t bt > 0,c 0  max t ct > 0,b 1  min t bt > 0,c 1  min t ct > 0. 3.31 Yigang Sun 9 Then conditions G 1 –G 3  and G 5  are satisfied. The existence condition G 4  becomes μ< ρ 2 rσr/ρ − ω α − Lr/ρ − ω α − b 0 c 0 r/ρ − ω αβ 3.32 for some r > L/ρ 2 σ. Hence, problem 1.1 has at least one positive periodic solution for 0 <μ<μ ∗ : sup r>0 ρ 2 rσr/ρ − ω α − Lr/ρ − ω α − b 0 c 0 r/ρ − ω αβ . 3.33 Note that μ ∗  ∞ if β<1andμ ∗ < ∞ if β ≥ 1. We have the desired results i and ii. If β>1, then the existence condition G 6  becomes μ ≥ ρ 2 RR/ρ − ω α − b 1 c 1 σR/ρ − ω αβ . 3.34 Since β>1, the right-hand side goes to 0 as R →∞.Hence,foranygiven0<μ<μ ∗ ,it is always possible to find such R  r that 3.34 is satisfied. Thus, 1.1 has an additional periodic solution u such that u >r. This implies that iii holds. References 1 F. Merdivenci Atici and G. Sh. Guseinov, “On the existence of positive solutions for nonlinear differ- ential equations with periodic boundary conditions,” Journal of Computational and Applied Mathematics, vol. 132, no. 2, pp. 341–356, 2001. 2 J. Chu and M. 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Corporation Boundary Value Problems Volume 2008, Article ID 574842, 9 pages doi:10.1155/2008/574842 Research Article Multiplicity of Positive Periodic Solutions of Singular Semipositone Third-Order Boundary Value. this paper, we study the existence and multiplicity of positive periodic solutions of the fol- lowing singular nonlinear third-order periodic boundary value problem: u   ρ 3 u  ft, u,. establish the existence of multiple positive solutions for a singular nonlinear third-order peri- odic boundary value problem. We are mainly interested in the semipositone case. The proof relies on a