On the fourth-order nonlinear beam equation of a small deflection with nonlocal conditions

$ {equation*} u^{(4)}+A(x)u = \lambda f (x, \ u, \ u''), \ 0<x<1 {equation*} $ subject to the integral boundary conditions: $ {equation*} u(0) = u(1) = \int_{0}^{1}p(x)u(x)dx, \ u''(0) = u''(1) = \int_{0}^{1}q(x)u''(x)dx, {equation*} $ where $ A\in \mathbb{C}[0, 1], $ $ \lambda > 0 $ is a parameter and $ p, q \in \mathbb{L}^{1}[0, 1].