Multiple positive solutions of a nonlinear fourth order periodic boundary value problem

The fourth order periodic boundary value problem ${\displaystyle u^{\left(4\right)}-m⁴u+F(t,u)=0}$, 0 < t < 2π, with ${\displaystyle u^{\left(i\right)}\left(0\right)=u^{\left(i\right)}(2\pi )}$, i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of ${\displaystyle \pm {10}^{-7}}$.