Existence of Positive Solutions for a Fourth-Order Periodic Boundary Value Problem

and Applied Analysis 3 The strongly maximum principle implies that the fourth-order linear boundary value problem LBVP L4 u : u 4 − βu′′ αu 0, 0 ≤ t ≤ 1, u i 0 − u i 1 0, i 0, 1, 2, u 3 0 − u 3 1 1 1.7 has a unique positive solution Φ : 0, 1 → 0,∞ , see 3, Lemma 3 . This function has been introduced in 2, Lemma 2.1 and Remark 2.1 . Let I 0, 1 , and set σ mint∈I Φ t maxt∈I Φ t , M maxt∈I |Φ′′ t | mint∈I Φ t . 1.8 Let f : I × R × R → R be continuous. To be convenient, we introduce the notations f0 lim inf u→ 0 min |v|≤Mu,t∈I ( f t, u, v u ) , f0 lim sup u→ 0 max |v|≤Mu,t∈I ( f t, u, v u ) , f∞ lim inf u→ ∞ min |v|≤Mu,t∈I ( f t, u, v u ) , f∞ lim sup u→ ∞ max |v|≤Mu,t∈I ( f t, u, v u ) . 1.9 Our main result is as follows. Theorem 1.1. Let f : 0, 1 × R × R → R be continuous, and let the assumption 1.3 hold. If f satisfies one of the following conditions: (F1) f0 < α, f∞ > α, (F2) f0 > α, f∞ < α, then PBVP 1.1 has at least one positive solution. Clearly, Theorem 1.1 is an extension of Theorem A. Since that α is an eigenvalue of linear eigenvalue problem u 4 − βu′′ αu λu, 1.10 with periodic boundary condition, if one inequality in F1 or F2 of Theorem 1.1 is not true, the existence of solution to PBVP 1.1 cannot be guaranteed. Hence, F1 and F2 are the optimal conditions for the existence of the positive of PBVP 1.1 . In Theorem 1.1, the condition F1 allows that f t, u, v may be superlinear growth on u and v, for example, f t, u, v u2 v2, and the condition F2 allows that f t, u, v may be sublinear growth on u and v, for example, f t, u, v 3 √ u2 v2. 4 Abstract and Applied Analysis The proof of Theorem 1.1 is based on the theory of the fixed point index in cones. Since the nonlinearity f of PBVP 1.1 contains u′′, the argument of Theorem A in 3 is not applicable to Theorem 1.1. We will prove Theorem 1.1 by choosing a proper cone of C2 I in Section 3. Some preliminaries to discuss PBVP 1.1 are presented in Section 2. 2. Preliminaries LetC I be the Banach space of all continuous functions on the unit interval I 0, 1 with the norm ‖u‖C max0≤t≤ω |u t |. Let C I denote the cone of all nonnegative functions in C I . Generaly, for n ∈ N, we use C I to denote the Banach space of the nth-order continuous differentiable functions on I with the norm ‖u‖Cn ∑n k 1 ‖u k ‖C. In C2 I , we define a new norm by ‖u‖C02 ‖u‖C ∥ ∥u′′ ∥ ∥ C. 2.1 Then ‖u‖C02 is equivalent to ‖u‖C2 . In fact, for every u ∈ C2 I , it is clear that ‖u‖C02 ≤ ‖u‖C2 . On the other hand, by the Lagrange mean-value theorem, there exists ξ ∈ 0, 1 such that u 1 − u 0 u′ ξ . For t ∈ I, we have ∣u′ t ∣ ≤ ∣u′ t − u′ ξ ∣ ∣u′ ξ ∣ ∣∣∣∣ ∫ t ξ u′′ s ds ∣∣∣∣ |u 1 − u 0 | ≤ ∫1 0 ∣u′′ s ∣ ds |u 1 | |u 0 | ≤ ∥∥u′′∥∥C 2‖u‖C ≤ 2‖u‖C02 . 2.2 Hence, ‖u‖C ≤ 2‖u‖C02 . By this, we have ‖u‖C2 ‖u‖C ∥u′ ∥ C ∥u′′ ∥ C ‖u‖C02 ∥u′ ∥ C ≤ 3‖u‖C02 . 2.3 Therefore, the norms ‖u‖C02 and ‖u‖C2 are equivalent. Let α, β ∈ R satisfy the assumption 1.3 . For h ∈ C I , we consider the fourth-order linear periodic boundary value problem LPBVP u 4 t − βu′′ t αu t h t , 0 ≤ t ≤ 1, u i 0 u i 1 , i 0, 1, 2, 3. 2.4 Let Φ t be the unique positive solution of LBVP 1.7 , and set G t, s ⎧ ⎨ ⎩ Φ t − s , 0 ≤ s ≤ t ≤ 1, Φ 1 t − s , 0 ≤ t < s ≤ 1. 2.5 By 3, Lemma 1 , we have the following result. Abstract and Applied Analysis 5 Lemma 2.1. Let α, β ∈ R satisfy the assumption 1.3 . Then for every h ∈ C I , LPBVP 2.4 has a unique solution u t which is given byand Applied Analysis 5 Lemma 2.1. Let α, β ∈ R satisfy the assumption 1.3 . Then for every h ∈ C I , LPBVP 2.4 has a unique solution u t which is given by u t ∫1 0 G t, s h s ds : Sh t , t ∈ R. 2.6 Moreover, S : C I → C4 I is a linear bounded operator. Let σ and M be the positive constants given by 1.8 . Choose a cone K in C2 I by K { u ∈ C2 I | u t ≥ σ‖u‖C, ∣ ∣u′′ t ∣ ∣ ≤ M|u t |, t ∈ I } . 2.7 We have the following. Lemma 2.2. Let α, β ∈ R satisfy the assumption 1.3 . Then for every h ∈ C I , the solution of LPBVP 2.4 u Sh ∈ K. Namely, S C I ⊂ K. Proof. Let h ∈ C I , u Sh. For every t ∈ I, from 2.6 it follows that 0 ≤ u t ∫1 0 G t, s h s ds ≤ max t∈I Φ t ∫1 0 h s ds, 2.8


Introduction
This paper concerns the existence of positive solutions for the fourth-order periodic boundary value problem PBVP u 4 t − βu t αu t f t, u t , u t , 0 ≤ t ≤ 1, where α, β ∈ R and f : 0, 1 ×R ×R → R is continuous, R 0, ∞ .PBVP 1.1 describes the deformations of an elastic beam in equilibrium state with periodic boundary condition.In the equation, the u denotes the bending moment term which represents bending effect.Owing to its importance in physics, the existence of solutions to this problem has been studied by some authors, see 1-6 .In practice, only its positive solutions are significant.In this paper, we discuss the existence of positive solutions of PBVP 1.1 .
In 1, 2 , Cabada and Lois obtained the maximum principles for fourth-order operator L 4,α u u 4 αu in periodic boundary condition and then they proved the existence of solutions and the validity of the monotone method in the presence of lower and upper solutions for the periodic boundary problem u 4 t g t, u t , 0 ≤ t ≤ 1, u i 0 u i 1 , i 0, 1, 2, 3.

1.2
In 3 , the present author established a strongly maximum principle for operator L 4 u u 4 − βu αu in periodic boundary condition, and showed that if α, β satisfy the assumption then L 4 is strongly inverse positive in space As an application of this strongly maximum principle, the author considered the existence of positive solutions for the special fourth-order periodic boundary problem and obtained the following result.
Theorem A. Let g : 0, 1 × R → R be continuous and the assumption 1.3 hold.If g satisfies one of the following conditions (G1) then PBVP 1.5 has at least one positive solution.
Based upon this strongly maximum principle, the authors of 4, 5 further consider the existence and multiplicity of positive solutions of PBVP 1.5 .In 6 , Bereanu obtained existence results for PBVP 1.5 by using the method of topological degree.However, all of these works are on the special equation 1.5 , and few people consider the existence of positive solutions of PBVP 1.1 that nonlinearity f contains the bending moment term u .The purpose of this paper is to discuss the existence of positive solutions of PBVP 1.1 .
The strongly maximum principle implies that the fourth-order linear boundary value problem LBVP has a unique positive solution Φ : 0, 1 → 0, ∞ , see Let f : I × R × R → R be continuous.To be convenient, we introduce the notations 1.9 Our main result is as follows.
Theorem 1.1.Let f : 0, 1 × R × R → R be continuous, and let the assumption 1.3 hold.If f satisfies one of the following conditions:

has at least one positive solution.
Clearly, Theorem 1.1 is an extension of Theorem A. Since that α is an eigenvalue of linear eigenvalue problem u 4 − βu αu λu, 1.10 with periodic boundary condition, if one inequality in F1 or F2 of Theorem 1.1 is not true, the existence of solution to PBVP 1.1 cannot be guaranteed.Hence, F1 and F2 are the optimal conditions for the existence of the positive of PBVP 1.1 .
In Theorem 1.1, the condition F1 allows that f t, u, v may be superlinear growth on u and v, for example, f t, u, v u 2 v 2 , and the condition F2 allows that f t, u, v may be sublinear growth on u and v, for example, f t, u, v The proof of Theorem 1.1 is based on the theory of the fixed point index in cones.Since the nonlinearity f of PBVP 1.1 contains u , the argument of Theorem A in 3 is not applicable to Theorem 1.1.We will prove Theorem 1.1 by choosing a proper cone of C 2 I in Section 3. Some preliminaries to discuss PBVP 1.1 are presented in Section 2.

Preliminaries
Let C I be the Banach space of all continuous functions on the unit interval I 0, 1 with the norm u C max 0≤t≤ω |u t |.Let C I denote the cone of all nonnegative functions in C I .Generaly, for n ∈ N, we use C n I to denote the Banach space of the nth-order continuous differentiable functions on I with the norm Then On the other hand, by the Lagrange mean-value theorem, there exists ξ ∈ 0, 1 such that

2.2
Hence, u C ≤ 2 u C 02 .By this, we have Therefore, the norms u C 02 and u C 2 are equivalent.Let α, β ∈ R satisfy the assumption 1.3 .For h ∈ C I , we consider the fourth-order linear periodic boundary value problem LPBVP

2.4
Let Φ t be the unique positive solution of LBVP 1.7 , and set

2.5
By 3, Lemma 1 , we have the following result.Let σ and M be the positive constants given by 1.8 .Choose a cone K in C 2 I by We have the following.

2.13
Therefore, u ∈ K.This means that S C I ⊂ K.
For every u ∈ K, since f :

2.14
We have the following.
By the definition of S and K, the positive solution of PBVP 1.1 is equivalent to the nontrivial fixed point of A. We will find the nonzero fixed point of A by using the fixed point index theory in cones.
We recall some concepts and conclusions on the fixed point index in 7, 8 .Let E be a Banach space, and let K ⊂ E be a closed convex cone in E. Assume Ω is a bounded open subset of E with boundary ∂Ω, and K ∩ Ω / ∅.Let A : K ∩ Ω → K be a completely continuous mapping.If Au / u for any u ∈ K∩∂Ω, then the fixed point index i A, K∩Ω, K has definition.One important fact is that if i A, K∩Ω, K / 0, then A has a fixed point in K∩Ω.The following two lemmas are needed in our argument.

Proof of the Main Result
Proof of Theorem 1.1.Choose the working space E C 2 I with the norm u C 02 .Let K be the closed convex cone in C 2 I defined by 2.7 , and let A : K → K be the operator defined by 2.14 .By Lemma 2.3 and the definition of K, the nonzero fixed of the operator A is the positive solution of PBVP 1.1 .Let 0 < r < R < ∞, and set We show that, if r is small enough and R large enough, the operator A has a fixed point in K ∩ Ω 2 \ Ω 1 in either case that F1 holds or F2 holds.

3.6
By this inequality and 3.3 , we have Integrating this inequality from 0 to 1 and using the periodic boundary condition 3.4 , we obtain that Since 1 0 u 0 t dt ≥ σ u 0 C > 0, form this inequality it follows that α ≤ α − ε, which is a contradiction.Hence, A satisfies the condition of Lemma 2.4 in K ∩ ∂Ω 1 .By Lemma 2.4 we have 3.9 On the other hand, since f ∞ > α, by the definition of f ∞ , there exist ε 1 > 0 and H > 0 such that

3.10
Choose R > max{ 1 M/σ H, δ}, and let e t ≡ 1.Clearly, e ∈ K \ {θ}.We show that A satisfies the condition of Lemma 2.5 in K ∩ ∂Ω 2 ; namely, u − Au / τe, for every u ∈ K ∩ ∂Ω 2 and τ ≥ 0. In fact, if there exist Au 1 , by definition of A and Lemma 2.1, u 1 t ∈ C 4 I satisfies the differential equation and the periodic boundary condition 3.4 .Since u 1 ∈ K ∩ ∂Ω 2 , by the definition of K, we have

3.12
By the second inequality of 3.12 , we have

3.13
Consequently, 3.14 By 3.14 and the first inequality of 3.12 , we have

3.15
From this, the second inequality of 3.12 , and 3.10 , it follows that

3.16
By this and 3.11 , we have

3.17
Integrating this inequality on I and using the periodic boundary condition 3.4 , we get that Now, by the additivity of fixed point index, 3.9 , and 3.19 , we have

3.20
Hence, A has a fixed point in K ∩ Ω 2 \ Ω 1 , which is the positive solution of PBVP 1.1 .
By the assumption of f 0 > α and the definition of f 0 , there exist ε > 0 and δ > 0, such that

3.24
Integrating this inequality on I and using the periodic boundary condition 3.4 , we have

3.29
From this, the second inequality of 3.12 , and 3.27 , it follows that

3.30
By this inequality and 3.28 , we have

3.31
Integrating this inequality on I and using the periodic boundary condition 3.4 , we obtain that

Lemma 2 . 3 .
A : K → K is a completely continuous operator.Proof.Let D ⊂ K be a bounded set in C 2 I .By the continuity of f : I ×R ×R → R , F D is a bounded set in C I .By the boundedness of the operator S : C I → C 4 I , A D S F D is a bounded set in C 4 I .By the compactness of the embedding
form this inequality it follows that α ≤ α − ε 1 , which is a contradiction.This means that A satisfies the condition ofLemma 2.4in K ∩ ∂Ω 2 .By Lemma 2.4, ∩ Ω 2 \ Ω 1 , K i A, K ∩ Ω 2 , K − i A, K ∩ Ω 1 , K 1.3.34Hence,Ahas a fixed point in K ∩ Ω 2 \ Ω 1 , which is the positive solution of PBVP 1.1 .The proof of Theorem 1.1 is completed.Example 3.1.Consider the superlinear fourth-order periodic boundary problemu 4 − u u a 1 t u 2 a 2 t u 2 , 0 ≤ t ≤ 1, , a 2 ∈ C Iand a 1 t , a 2 t > 0 for t ∈ I.It is easy to verify that α 1 and β 1 satisfy the assumption P .f t, u, v a 1 t u 2 a 2 t v 2 satisfies the condition F1 , in which f 0 0 and f ∞ ∞.Hence, by Theorem 1.1, 3.35 has at least one positive solution., b 2 ∈ C I and b 1 t , b 2 t > 0 for t ∈ I.For PBVP 3.36 , it is easy to verify that α 1 and β −1 satisfy the assumption 1.3 , and f t, u, v b 1 t √ u b 2 t |v| satisfies the condition F2 with f 0 ∞ and f ∞ 0. By Theorem 1.1, 3.36 has a positive solution.Since 3.35 and 3.36 have nonlinear terms of u , which are not in the range considered by 1-6 , the existence results in Example 3.1, and Example 3.2 cannot be obtained from 1-6 .