Abstract:
In the article, we consider a boundary value problem
for a nonlinear ordinary differential equation of even order which, obviously,
has a trivial solution. Sufficient conditions for the existence and uniqueness
of a positive solution to this problem are obtained.
With the help of linear transformations of T. Y. Na [T. Y. Na, Computational Methods in Engineering Boundary Value Problems, Acad. Press, NY, 1979, ch. 7],
the boundary value problem is reduced to the Cauchy problem,
the initial conditions of which make it possible to uniquely
determine the transformation parameter. It is shown that the transformations
of T. Y. Na uniquely determine the solution of the original problem.
In addition, based on the proof of the uniqueness of a positive solution
to the boundary value problem, a sufficiently effective non–iterative
numerical algorithm for constructing such a solution is obtained.
A corresponding example is given.
Keywords:boundary value problem, positive solution, existence, uniqueness.