The class of problems considered here consists of nonlinear ordinary differential equation two-point boundary value problems of the form u&dprime; ( x ) = f ( x , u ( x ) , u′ ( x ) ) , 0 < x < 1, subject to homogeneous Sturm-Liouville or periodic boundary conditions. This work is one of a bouquet of papers stemming from the work of Schröder and colleagues, including the author. Other problems are dealt with in published or pending companion papers.

The thrust of the endeavor is the following. Suppose that an approximate solution v is given that satisfies the boundary conditions and renders the defect d { v } ( x ) = - v&dprime; ( x ) + f ( x , v ( x ) , v′ ( x ) ) sufficiently small. A suitable local existence theorem is established that guarantees the existence of a solution u close to v provided certain inequalities are satisfied; this serves the dual function of proving existence and bounding the error. The constants determining the requisite inequalities and bounds are then determined by a combination of analytical and numerical means. Linearization of the equation in the neighborhood of v plays an essential role. Any suitable numerical method can be used to generate v : a polynomial collocation method is used in the examples. The constants, however, must be determined so as to guarantee that the inequalities are satisfied: interval arithmetic is used here where computations are required.

The paper is generally well written, but cannot be fully appreciated or understood in its own terms because of extensive cross-referencing of other papers for motivation and key results, only some of which are currently available. Is the least publishable unit syndrome at work?