The monotone method for periodic differential equations with the non well-ordered upper and lower solutions

In this paper, we consider the following periodic problem: x ″ ( t ) = g ( t , x ( t ) , x ′ ( t ) ) , a.e.  t ∈ ( a , b ) , x ( a ) = x ( b ) , x ′ ( a ) = x ′ ( b ) . The unique solution is obtained by constructing an auxiliary periodic system with bounded solutions and proving these bounds to be equal, in which case they are the expected unique solution.