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