An Existence and Uniqueness Result for a Bending Beam Equation without Growth Restriction

and Applied Analysis 3 that the partial derivatives fu and fv exist, the conjecture means that if for large |u| |v| the pair ( fu t, u, v ,−fv t, u, v ) 1.10 lies in a certain rectangle R α, β;a, b which does not intersect any of the eigenline Lk of LEVP 1.6 ; then BVP 1.1 is solvable. But they could not prove the conjecture. Recently, the present author 11 has partly answered this conjecture and shows that if the rectangle R α, β;a, b is replaced by the circle B ( α, β; r ) {( x, y ) x − α 2 y − β2 ≤ r2 } , 1.11 the conjecture is correct. In other words, the following result is obtained. Theorem B. Assume that f has partial derivatives fu and fv in 0, 1 ×R ×R. If there exists a circle B α, β; r , which does not intersect any of the eigenline Lk of LEVP 1.6 , such that ( fu t, u, v ,−fv t, u, v ) ∈ Bα, β; r 1.12 for large |u| |v|, then the BVP 1.1 has at least one solution. See 11, Theorem 2 and Corollary 2 . Condition 1.12 means that f is linear growth on u and v. If f is not linear growth on u or v, Theorem B is invalid. In this paper, we will extend Theorem B to the case that the circle B α, β; r is replaced by an unbounded domain. Let ε ∈ 0, π6 be a positive constant; then we will use the parabolic sector Dε { ( x, y ) ∈ R2 | y ≤ − x 2 4 π6 − ε } 1.13 to substitute the the circle B α, β; r in Theorem B. Noting thatDε is contained in the parabolic sector D0 { ( x, y ) ∈ R2 | y ≤ − x 2 4π6 } 1.14 and D0 only contacts the first eigenline L1 at 2π4, −π2 , we see that Dε does not intersect any of the eigenline Lk. Our new result is as follows. Theorem 1.1. Assume that f has partial derivatives fu and fv in 0, 1 ×R ×R. If there is a positive constant ε ∈ 0, π6 such that ( fu t, u, v ,−fv t, u, v ) ∈ Dε, 1.15 then the BVP 1.1 has a unique solution. 4 Abstract and Applied Analysis In Theorem 1.1, Condition 1.15 allows f t, u, v to be superlinear in u and v, and an example will be showed at the end of the paper. The proof of Theorem 1.1 is based on LeraySchauder fixed point theorem and a differential inequality, which will be given in the next section. 2. Proof of the Main Results Let I 0, 1 and H L2 I be the usual Hilbert space with the interior product u, v ∫1 0 u t v t dt and the norm ‖u‖2 ∫1 0 |u t |dt 1/2 . Form ∈ N, letWm,2 I be the usual Sobolev space with the norm ‖u‖m,2 ∑m i 0 ‖u i ‖2 . u ∈ Wm,2 I which means that u ∈ Cm−1 I , u m−1 t is absolutely continuous on I and u m ∈ L2 I . Given h ∈ L2 I , we consider the linear fourth-order boundary value problem LBVP u 4 t h t , t ∈ I, u 0 u 1 u′′ 0 u′′ 1 0. 2.1 Let G t, s be Green’s function to the second-order linear boundary value problem −u′′ 0, u 0 u 1 0, 2.2 which is explicitly expressed by G t, s ⎧ ⎨ ⎩ t 1 − s , 0 ≤ t ≤ s ≤ 1, s 1 − t , 0 ≤ s ≤ t ≤ 1. 2.3 For every given h ∈ L2 I , it is easy to verify that the LBVP 2.1 has a unique solution u ∈ W4,2 I in Carathéodory sense, which is given by


Introduction and Main Results
In this paper we deal with the existence of a solution of the fourth-order ordinary differential equation boundary value problem BVP where f : 0, 1 × R × R → R is continuous.This problem models deformations of an elastic beam in the equilibrium state, whose ends are simply supported.Owing to its importance in physics, the solvability of this problem has been studied by many authors; see 1-14 .
In 1 , Aftabizadeh showed the existence of a solution to BVP 1.1 under the restriction that f is a bounded function.In 2, Theorem 1 , Yang extended Aftabizadeh's result and showed the existence for BVP 1.1 under the growth condition of the form where a, b, and c are positive constants such that a/π 4 b/π 2 < 1.

Abstract and Applied Analysis
In 6 , del Pino and Manásevich further extended the result of Yang and obtained the following existence theorem.
and that there are positive constants a, b, and c such that and f satisfies the growth condition Then the BVP 1.1 possesses at least one solution.

1.6
In 6 it was shown that α, β is an eigenvalue pair of LEVP 1.6 if and only if α/ kπ 4 β/ kπ 2 1 for some k ∈ N. Hence, for k ∈ N the straight line is called an eigenline of LEVP 1.6 .Conditions 1.3 -1.4 trivially imply that It is easy to prove that condition 1.8 is equivalent to the fact that the rectangle does not intersect any of the eigenline L k of LEVP 1.
for large |u| |v|, then the BVP 1.1 has at least one solution.
See 11, Theorem 2 and Corollary 2 .Condition 1.12 means that f is linear growth on u and v.If f is not linear growth on u or v, Theorem B is invalid.
In this paper, we will extend Theorem B to the case that the circle B α, β; r is replaced by an unbounded domain.Let ε ∈ 0, π 6 be a positive constant; then we will use the parabolic sector to substitute the the circle B α, β; r in Theorem B. Noting that D ε is contained in the parabolic sector and D 0 only contacts the first eigenline L 1 at 2π 4 , −π 2 , we see that D ε does not intersect any of the eigenline L k .Our new result is as follows.
Theorem 1.1.Assume that f has partial derivatives f u and then the BVP 1.1 has a unique solution.
In Theorem 1.1, Condition 1.15 allows f t, u, v to be superlinear in u and v, and an example will be showed at the end of the paper.The proof of Theorem 1.1 is based on Leray-Schauder fixed point theorem and a differential inequality, which will be given in the next section.

Proof of the Main Results
Let I 0, 1 and H L 2 I be the usual Hilbert space with the interior product u, v . For m ∈ N, let W m,2 I be the usual Sobolev space with the norm u m,2 Given h ∈ L 2 I , we consider the linear fourth-order boundary value problem LBVP

2.1
Let G t, s be Green's function to the second-order linear boundary value problem

2.3
For every given h ∈ L 2 I , it is easy to verify that the LBVP 2.1 has a unique solution u ∈ W 4,2 I in Carathéodory sense, which is given by holds.Let u Sh, then u ∈ W 4,2 I is the unique solution of LBVP 2.1 , and u, u , and u 4 can be expressed by the Fourier series expansion of the sine system.Since u 4 h, by the integral formula of Fourier coefficient, we obtain that

2.8
On the other hand, since cosine system {cos kπt | k 0, 1, 2, . ..} is another complete orthogonal system of L 2 I , every v ∈ L 2 I can be expressed by the cosine series expansion where a k 2 1 0 h s cos kπs ds, k 0, 1, 2, . ... For the above u Sh, by the integral formula of the coefficient of cosine series, we obtain the cosine series expansions of u and u :

2.10
By 2.8 -2.10 and Parseval equality, we have that

2.11
This implies that 2.5 holds.
Proof of Theorem 1.1.We define a mapping F : By the continuity of f, F :

2.13
We need to prove that the set of the solutions of 2.13 is bounded in C 2 I .Let u ∈ C 2 I be a solution of 2.13 for λ ∈ 0, 1 .Set h λF u .Since h ∈ C I , by the definition of S, u Sh ∈ C 4 I is the unique solution of LBVP 2.1 .Hence u satisfies the differential equation

2.14
Set M max t∈I |f t, 0, 0 |.Multiplying the first formula of 2.14 by −u t and by the theorem of differential mean value, we have where ξ θu, η θu for some θ ∈ 0, 1 .In the last step of this estimation we use the inequality which is derived from the inequality 2pq ≤ p 2 q 2 by choosing 2.17 Since Hence, we obtain that

2.21
From this and Lemma 2.1, we obtain that Hence, by the continuity of the Sobolev embedding W

2.26
This implies that u 2 0, and hence we have u 1 u 2 .Thus BVP continuous and it maps every bounded set of C 2 I into a bounded set of C I .Hence, the composite mapping S • F : C 2 I → C 2 I is completely continuous.By the definition of the solution operator S of LBVP 2.1 , the solution of BVP 1.1 is equivalent to the fixed point of S•F.We first use the Leray-Schauder fixed point theorem 15 to show that S • F has a fixed point.For this, we consider the homotopic family of the operator equations s h s ds dτ : Sh t .2.4If h ∈ C I , the solution is in C 4 I , and it is a classical solution.Moreover, the solution operator of LBVP 2.1 , S : L 2 I → W 4,2 I is a linearly bounded operator.By the compactness of the Sobolev embedding W 4,2 I → C 2 I , we see that S : L 2 I → C 2 I is a completely continuous operator.Hence the restriction S : C I → C 2 I is completely continuous.Proof.Since sine system {sin kπt | k ∈ N} is a complete orthogonal system of L 2 I , every h ∈ L 2 I can be expressed by the Fourier series expansion 3,2 I → C 2 I , we have is the Sobolev embedding constant.This means that the set of the solutions of 2.13 is bounded in C 2 I .By the Leray-Schauder fixed point theorem 15 , S • F has a fixed point in C 2 I which is a solution of BVP 1.1 .Now, let u 1 , u 2 ∈ C 4 I be two solutions of BVP 1.1 .Set u u 2 − u 1 and h F u 2 − F u 1 .Then u S F u 2 − F u 2 Sh is the solution of LBVP 2.1 , and it satisfies the equation u 4 t f t, u 2 , u 2 − f t, u 1 , u 2 , t ∈ I. 2.24 Multiplying this equality by − u 2 − u 1 and by the theorem of differential mean value and Condition 1.15 , we have that 1.1 has only one solution.The proof of Theorem 1.1 is completed.on v, one can check that all the known results of 1-14 are not applicable to this equation.But, if max t∈I |a t | < 2π 3 , then ∈ 0, 4π 6 .Hence, Condition 1.15 holds, and by Theorem 1.1, the boundary value problem 2.27 has a unique solution.