A Cantilever Beam Problem with Small Deflections and Perturbed Boundary Data

The boundary value problem of a fourth-order beam equation u 4 = λ f x , u , u ′ , u ″ , u ′ ′ ′ , 0 ≤ x ≤ 1 is investigated. We formulate a nonclassical cantilever beam problem with perturbed ends. By determining appropriate values of λ and estimates for perturbation measurements on the boundary data, we establish an existence theorem for the problem under integral boundary conditions u 0 = u ′ 0 = ∫ 0 1 p x u x d x , u ″ 1 = u ′ ′ ′ 1 = ∫ 0 1 q x u ″ x d x , where p , q ∈ L 1 0 , 1 , and f is continuous on 0 , 1 × 0 , ∞ × 0 , ∞ × − ∞ , 0 × − ∞ , 0 .