Abstract
Comparison theorems are presented for solutions to general classes of boundary value problems for ordinary differential equations. Upper and lower bounding functions jointly satisfying an extended system of inequalities imply the existence of a solution to the original problem which is bounded by these functions. The theory associates with any differential equation of order n, an extended equation of order 2 n. Variational theory also associates a linear equation, the Jacobian equation, of order n with every differential equation of order n. It transpires that these maps commute.
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