On the nonlinear system of fourth-order beam equations with integral boundary conditions

The purpose of this paper is to establish an existence theorem for a system of nonlinear fourth-order differential equations with two parameters $ \begin{eqnarray*} \left\{ \begin{array}{rcl} u^{(4)}+A(x)u& = &\lambda f(x, u, v, u'', v''), \ 0<x<1, \\ v^{(4)}+B(x)v& = &\mu g(x, u, v, u'', v''), \ 0<x<1 \end{array} \right. \end{eqnarray*} $ subject to the coupled integral boundary conditions: $ \begin{eqnarray*} \left\{ \begin{array}{rcl} u(0) = u'(1) = u'''(1) = 0, \ u''(0)& = & \int_{0}^{1}p(x)v''(x)dx, \\ v(0) = v'(1) = v'''(1) = 0, \ v''(0)& = & \int_{0}^{1}q(x)u''(x)dx, \end{array} \right. \end{eqnarray*} $ where $ A, \ B \in C[0, 1], $ $ p, q\in L^{1}[0, 1], $ $ \lambda > 0, \mu > 0 $ are two parameters and $ f, g: [0, 1]\times[0, \infty)\times[0, \infty)\times(-\infty, 0)\times(-\infty, 0) \rightarrow \mathbb{R} $ are two continuous functions satisfy the growth conditions