Hindawi Publishing Corporation BoundaryValue Problems Volume 2009, Article ID 393259, 9 pages doi:10.1155/2009/393259 ResearchArticlePositiveSolutionsforSomeBeamEquationBoundaryValue Problems Jinhui Liu 1, 2 and Weiya Xu 3 1 Department of Civil Engineering, Hohai University, Nanjing 210098, China 2 Zaozhuang Coal Mining Group Co., Ltd, Jining 277605, China 3 Graduate School, Hohai University, Nanjing 210098, China Correspondence should be addressed to Jinhui Liu, jinhuiliu88@163.com Received 2 September 2009; Accepted 1 November 2009 Recommended by Wenming Zou A new fixed point theorem in a cone is applied to obtain the existence of positivesolutions of some fourth-order beamequationboundaryvalue problems with dependence on the first-order derivative u iυ tft, ut,u t, 0 <t<1,u0u1u 0u 10, where f : 0, 1 × 0, ∞ × R → 0, ∞ is continuous. Copyright q 2009 J. Liu and W. Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction It is well known that beam is one of the basic structures in architecture. It is greatly used in the designing of bridge and construction. Recently, scientists bring forward the theory of combined beams. That is to say, we can bind up some stratified structure copings into one global combined beam with rock bolts. The deformations of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by following equation of deflection curve: d 2 dx 2 EI z d 2 v dx 2 q x , 1.1 where E is Yang’s modulus constant, I z is moment of inertia with respect to z axes, determined completely by the beam’s shape cross-section. Specially, I z bh 3 /12 if the cross- section is a rectangle with a height of h and a width of b. Also, qx is loading at x.Ifthe 2 BoundaryValue Problems loading of beam considered is in relation to deflection and rate of change of deflection, we need to research the more general equation u 4 x f x, u x ,u x . 1.2 According to the forms of supporting, various boundary conditions should be considered. Solving corresponding boundaryvalue problems, one can obtain the expression of deflection curve. It is the key in design of constants of beams and rock bolts. Owing to its importance in physics and engineering, the existence of solutions to this problem has been studied by many authors, see 1–10. However, in practice, only its positive solution is significant. In 1, 9, 11, 12, Aftabizadeh, Del Pino and Man ´ asevich, Gupta, and Pao showed the existence of positive solution for u iv t f t, u t ,u t 1.3 under some growth conditions of f and a nonresonance condition involving a two-parameter linear eigenvalue problem. All of these results are based on the Leray-Schauder continuation method and topological degree. The lower and upper solution method has been studied for the fourth-order problem by several authors 2, 3, 7, 8, 13, 14. However, all of these authors consider only an equation of the form u iv t f t, u t , 1.4 with diverse kind of boundary conditions. In 10, Ehme et al. gave some sufficient conditions for the existence of a solution of u iv t f t, u t ,u t ,u t ,u t 1.5 with some quite general nonlinear boundary conditions by using the lower and upper solution method. The conditions assume the existence of a strong upper and lower solution pair. Recently, Krasnosel’skii’s fixed point theorem in a cone has much application in studying the existence and multiplicity of positivesolutionsfor differential equationboundaryvalue problems, see 3, 6. With this fixed point theorem, Bai and Wang 6 discussed the existence, uniqueness, multiplicity, and infinitely many positivesolutionsfor the equation of the form u iv t λf t, u t , 1.6 where λ>0 is a constant. BoundaryValue Problems 3 In this paper, via a new fixed point theorem in a cone and concavity of function, we show the existence of positivesolutionsfor the following problem: u iv t f t, u t ,u t , 0 <t<1, u 0 u 1 u 0 u 1 0, 1.7 where f : 0, 1 × 0, ∞ × R → 0, ∞ is continuous. We point out that positivesolutions of 1.7 are concave and this concavity provides lower bounds on positive concave functions of their maximum, which can be used in defining a cone on which a positive operator is defined, to which a new fixed point theorem in a cone due to Bai and Ge 5 can be applied to obtain positive solutions. 2. Fixed Point Theorem in a Cone Let X be a Banach space and P ⊂ X a cone. Suppose α, β : X → R are two continuous nonnegative functionals satisfying α λx ≤ | λ | α x ,β λx ≤ | λ | β x , for x ∈ X, λ ∈ 0, 1 , M 1 max α x ,β x ≤ x ≤ M 2 max α x ,β x , for x ∈ X, 2.1 where M 1 ,M 2 are two positive constants. Lemma 2.1 see 5. Let r 2 >r 1 > 0,L 2 >L 1 > 0 are constants and Ω i x ∈ X | α x <r i ,β x <L i ,i 1, 2 2.2 are two open subsets in X such that θ ∈ Ω 1 ⊂ Ω 1 ⊂ Ω 2 . In addition, let C i x ∈ X | α x r i ,β x ≤ L i ,i 1, 2; D i x ∈ X | α x ≤ r i ,β x L i ,i 1, 2. 2.3 Assume T : P → P is a completely continuous operator satisfying S 1 αTx ≤ r 1 ,x∈ C 1 ∩ P ; βTx ≤ L 1 ,x ∈ D 1 ∩ P ; αTx ≥ r 2 ,x ∈ C 2 ∩ P ; βTx ≥ L 2 ,x ∈ D 2 ∩ P; or S 2 αTx ≥ r 1 ,x ∈ C 1 ∩ P; βTx ≥ L 1 ,x ∈ D 1 ∩ PαTx ≤ r 2 ,x ∈ C 2 ∩ P; βTx ≤ L 2 ,x ∈ D 2 ∩ P, then T has at least one fixed point in Ω 2 \ Ω 1 ∩ P. 4 BoundaryValue Problems 3. Existence of PositiveSolutions In this section, we are concerned with the existence of positivesolutionsfor the fourth-order two-point boundaryvalue problem 1.7. Let X C 1 0, 1 with u max{max 0≤t≤1 |ut|, max 0≤t≤1 |u t|} be a Banach space, P {u ∈ X | ut ≥ 0,u is concave on 0, 1}⊂X a cone. Define functionals α u max 0≤t≤1 | u t | ,β u max 0≤t≤1 u t , for u ∈ X, 3.1 then α, β : X → R are two continuous nonnegative functionals such that u max α u ,β u 3.2 and 2.1 hold. Denote by Gt, s Green’s function forboundaryvalue problem −y t 0, 0 <t<1, y 0 y 1 0. 3.3 Then Gt, s ≥ 0, for 0 ≤ t, s ≤ 1, and G t, s ⎧ ⎨ ⎩ t 1 − s , 0 ≤ t ≤ s ≤ 1, s 1 − t , 0 ≤ s ≤ t ≤ 1. 3.4 Let M max 0≤t≤1 1 0 G t, s G s, x dx ds, N max 0≤t≤1 1 0 3/4 1/4 G t, s G s, x dx ds, A max 1 0 1 − s G s, x dx ds, 1 0 sG s, x dx ds , B max 1 0 1−h h 1 − s G s, x dx ds, 1 0 1−h h sG s, x dx ds . 3.5 However, 1.7 has a solution u ut if and only if u solves the operator equation u t Tu t : 1 0 1 0 G t, s G s, x f x, u x ,u x dx ds. 3.6 It is well know that T : P → P is completely continuous. BoundaryValue Problems 5 Theorem 3.1. Suppose there are four constants r 2 >r 1 > 0,L 2 >L 1 > 0 such that max{r 1 ,L 1 }≤ min{r 2 ,L 2 } and the following assumptions hold: A 1 ft, x 1 ,x 2 ≥ max{r 1 /M, L 1 /A}, for t, x 1 ,x 2 ∈ 0, 1 × 0,r 1 × −L 1 ,L 1 ; A 2 ft, x 1 ,x 2 ≤ min{r 2 /M, L 2 /A}, for t, x 1 ,x 2 ∈ 0, 1 × 0,r 2 × −L 2 ,L 2 . Then, 1.7 has at least one positive solution ut such that r 1 ≤ max 0≤t≤1 u t ≤ r 2 or L 1 ≤ max 0≤t≤1 u t ≤ L 2 . 3.7 Proof. Let Ω i u ∈ X | α u <r i ,β u <L i ,i 1, 2, 3.8 be two bounded open subsets in X. In addition, let C i u ∈ X | α u r i ,β u ≤ L i ,i 1, 2; D i u ∈ X | α u ≤ r i ,β u L i ,i 1, 2. 3.9 For u ∈ C 1 ∩ P,byA 1 , there is α Tu max t∈ 0,1 1 0 G t, s G s, x f x, u x ,u x dx ds ≥ r 1 M · max t∈ 0,1 1 0 G t, s G s, x dx ds r 1 . 3.10 For u ∈ P , because T : P → P,soTu ∈ P,thatistosayTu concave on 0, 1, it follows that max t∈ 0,1 Tu t max Tu 0 , Tu 1 . 3.11 6 BoundaryValue Problems Combined with A 1 and f ≥ 0, for u ∈ D 1 ∩ P, there is β Tu max t∈ 0,1 Tu t max t∈ 0,1 − t 0 s 1 0 G s, x f x, u x ,u x dx ds 1 t 1 − s 1 0 G s, x f x, u x ,u x dx ds max 1 0 1 − s 1 0 G s, x f x, u x ,u x dx ds, 1 0 s 1 0 G s, x f x, u x ,u x dx ds ≥ L 1 A · max 1 0 1 − s G s, x dx ds, 1 0 sG s, x dx ds L 1 A · A L 1 . 3.12 For u ∈ C 2 ∩ P,byA 2 , there is α Tu max t∈ 0,1 1 0 G t, s G s, x f x, u x ,u x dx ds ≤ max t∈ 0,1 1 0 G t, s G s, x · r 2 M dx ds r 2 M · max t∈ 0,1 1 0 G t, s G s, x dx ds r 2 . 3.13 For u ∈ D 2 ∩ P,byA 2 , there is β Tu max 1 0 1 − s 1 0 G s, x f x, u x ,u x dx ds, 1 0 s 1 0 G s, x f x, u x ,u x dx ds ≤ L 2 A · max 1 0 1 − s G s, x dx ds, 1 0 sG s, x dx ds L 2 A · A L 2 . 3.14 BoundaryValue Problems 7 Now, Lemma 2.1 implies there exists u ∈ Ω 2 \ Ω 1 ∩ P such that u Tu, namely, 1.7 has at least one positive solution ut such that r 1 ≤ α u ≤ r 2 or L 1 ≤ β u ≤ L 2 , 3.15 that is, r 1 ≤ max 0≤t≤1 u t ≤ r 2 or L 1 ≤ max 0≤t≤1 u t ≤ L 2 . 3.16 The proof is complete. Theorem 3.2. Suppose there are five constants 0 <r 1 <r 2 , 0 <L 1 <L 2 , 0 ≤ h<1/2 such that max{r 1 /N, L 1 /B}≤min{r 2 /M, L 2 /A}, and the following assumptions hold A 3 ft, x 1 ,x 2 ≥ r 1 /N, for t, x 1 ,x 2 ∈ 1/4, 3/4 × r 1 /4,r 1 × −L 1 ,L 1 ; A 4 ft, x 1 ,x 2 ≥ L 1 /B, for t, x 1 ,x 2 ∈ h, 1 − h × 0,r 1 × −L 1 ,L 1 ; A 5 ft, x 1 ,x 2 ≤ min{r 2 /M, L 2 /A}, for t, x 1 ,x 2 ∈ 0, 1 × 0,r 2 × −L 2 ,L 2 . Then, 1.7 has at least one positive solution ut such that r 1 ≤ max 0≤t≤1 u t ≤ r 2 or L 1 ≤ max 0≤t≤1 u t ≤ L 2 . 3.17 Proof. We just need notice the following difference to the proof of Theorem 3.1. For u ∈ C 1 ∩P, the concavity of u implies that ut ≥ 1/4αur 1 /4fort ∈ 1/4, 3/4. By A 3 , there is α Tu max t∈ 0,1 1 0 G t, s G s, x f x, u x ,u x dx ds ≥ max t∈ 0,1 1 0 3/4 1/4 G t, s G s, x f x, u x ,u x dx ds ≥ max t∈ 0,1 1 0 3/4 1/4 G t, s G s, x · r 1 N dx ds r 1 N · max t∈ 0,1 1 0 3/4 1/4 G t, s G s, x dx ds r 1 . 3.18 8 BoundaryValue Problems For u ∈ D 1 ∩ P,byA 4 , there is β Tu max 1 0 1 − s 1 0 G s, x f x, u x ,u x dx ds, 1 0 s 1 0 G s, x f x, u x ,u x dx ds ≥ max 1 0 1 − s 1−h h G s, x f x, u x ,u x dx ds, 1 0 s 1−h h G s, x f x, u x ,u x dx ds ≥ L 1 B · max 1 0 1−h h 1 − s G s, x dx ds, 1 0 1−h h sG s, x dx ds L 1 B · B L 1 3.19 The rest of the proof is similar to Theorem 3.1 and the proof is complete. References 1 A. R. Aftabizadeh, “Existence and uniqueness theorems for fourth-order boundaryvalue problems,” Journal of Mathematical Analysis and Applications, vol. 116, no. 2, pp. 415–426, 1986. 2 R. P. Agarwal, “On fourth order boundaryvalue problems arising in beam analysis,” Differential and Integral Equations, vol. 2, no. 1, pp. 91–110, 1989. 3 R. 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Yao, “Existence and multiplicity of positivesolutions to a singular elastic beamequation rigidly fixed at both ends,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2683–2694, 2008. 14 J. Schr ¨ oder, “Fourth order two-point boundaryvalue problems; estimates by two-sided bounds,” Nonlinear Analysis: Theory, Methods & Applications, vol. 8, no. 2, pp. 107–114, 1984. . Publishing Corporation Boundary Value Problems Volume 2009, Article ID 393259, 9 pages doi:10.1155/2009/393259 Research Article Positive Solutions for Some Beam Equation Boundary Value Problems Jinhui. Ω 1 ∩ P. 4 Boundary Value Problems 3. Existence of Positive Solutions In this section, we are concerned with the existence of positive solutions for the fourth-order two-point boundary value problem. multiplicity, and infinitely many positive solutions for the equation of the form u iv t λf t, u t , 1.6 where λ>0 is a constant. Boundary Value Problems 3 In this paper, via