Monotone iterative technique and symmetric positive solutions to fourth-order boundary value problem with integral boundary conditions (Q2321442)

Summary: The purpose of this paper is to investigate the existence of symmetric positive solutions for a class of fourth-order boundary value problem: \(u^{\left(4\right)}(t) + \beta u''(t) = f(t, u(t), u''(t))\), \(0 < t < 1\), \(u(0) = u(1) = \int_0^1 p(s) u(s) d s\), \(u''(0) = u''(1) = \int_0^1 q \left(s\right) u''(s) d s\), where \(p, q \in L^1 [0,1]\), \(f \in C([0,1] \times [0, \infty) \times(- \infty, 0], [0, \infty))\). By using a monotone iterative technique, we prove that the above boundary value problem has symmetric positive solutions under certain conditions. In particular, these solutions are obtained via the iteration procedures.