Remarks on the lower and upper solutions method for second- and third-order periodic boundary value problems

摘要

We study the solvability and the approximation of the solutions by monotone iteration of the second- and third-order periodic boundary value problems (BVPs) -u″+cu′ = ⨍(t,u), u(0) = u(2π), u′(0) = u′(2π), and u‴+au″+bu′ = ⨍(t,u), u(0) = u(2π), u′(0) = u′(2π), u″(0) = u″(2π), in the presence of lower and upper solutions, which may not satisfy the usual ordering condition. To this end some maximum and antimaximum principles are stated and proved for certain linear differential operators naturally associated with the previously stated problems.