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Hindawi 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 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 positive solutions of some fourth-order beam equation boundary value 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 Boundary Value 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 boundary value 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 positive solutions for differential equation boundary value problems, see 3, 6. With this fixed point theorem, Bai and Wang 6 discussed the existence, uniqueness, 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 a new fixed point theorem in a cone and concavity of function, we show the existence of positive solutions for 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 positive solutions 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 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 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 for boundary value 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. Boundary Value 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 Boundary Value 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 Boundary Value 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 Boundary Value 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 boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 116, no. 2, pp. 415–426, 1986. 2 R. P. Agarwal, “On fourth order boundary value problems arising in beam analysis,” Differential and Integral Equations, vol. 2, no. 1, pp. 91–110, 1989. 3 R. P. Agarwal, D. O’Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference, and Integral Equations, Kluwer Academic Publishers, Boston, Mass, USA, 1999. 4 Z. B. Bai, “The method of lower and upper solutions for a bending of an elastic beam equation,” Journal of Mathematical Analysis and Applications, vol. 248, no. 1, pp. 195–202, 2000. 5 Z. B. Bai and W. G. Ge, “Existence of positive solutions to fourth order quasilinear boundary value problems,” Acta Mathematica Sinica, vol. 22, no. 6, pp. 1825–1830, 2006. 6 Z. B. Bai and H. Y. Wang, “On positive solutions of some nonlinear fourth-order beam equations,” Journal of Mathematical Analysis and Applications, vol. 270, no. 2, pp. 357–368, 2002. 7 A. Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 302– 320, 1994. 8 C. De Coster and L. Sanchez, “Upper and lower solutions, Ambrosetti-Prodi problem and positive solutions for fourth order O.D.E,” Rivista di Matematica Pura ed Applicata, no. 14, pp. 1129–1138, 1994. 9 M. A. Del Pino and R. F. Man ´ asevich, “Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition,” Proceedings of the American Mathematical Society, vol. 112, no. 1, pp. 81–86, 1991. 10 J. Ehme, P. W. Eloe, and J. Henderson, “Upper and lower solution methods for fully nonlinear boundary value problems,” Journal of Differential Equations, vol. 180, no. 1, pp. 51–64, 2002. 11 C. P. Gupta, “Existence and uniqueness theorems for the bending of an elastic beam equation,” Applicable Analysis, vol. 26, no. 4, pp. 289–304, 1988. 12 C. V. Pao, “On fourth-order elliptic boundary value problems,” Proceedings of the American Mathematical Society, vol. 128, no. 4, pp. 1023–1030, 2000. Boundary Value Problems 9 13 Q. L. Yao, “Existence and multiplicity of positive solutions to a singular elastic beam equation 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 boundary value 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
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