On the parametrization of three-point nonlinear boundary-value problems

Abstract

A three-point boundary-value problem for a system of nonlinear differential equations is reduced to a family of two-point problems, whose solutions are investigated by using the numerical-analytic method.

A new approach to non-local boundary value problems for ordinary differential systems

“…(13), where u 0 ðÁ; n; gÞ is given by (8). By Theorem 3, the function u 1 ðÁ; n; gÞ, which is the uniform limit of the successive approximations (9), is the only solution of (13).…”

“…where u 0 is given by (8). Version 2 (Polynomial interpolation) Formula (52) is modified so that the polynomial approximations of the integrands are used, i. e., instead of (9), one uses the formula v mþ1 ðt; n; gÞ :¼ u 0 ðt; n; gÞ þ where l is fixed and p l y stands for the polynomial of degree l interpolating the function y at l suitably chosen nodes.…”

“…The idea that we are going to employ is based on the reduction to a family of simpler auxiliary boundary problems obtained by ''freezing'' certain values of the solution sought for (see, e.g., [8][9][10]). In our case, the auxiliary problems will have two-point linear separated conditions at a and b: uðaÞ ¼ n; uðbÞ ¼ g;…”

“…The vectors n and g in (8) and (9) are treated as unknown parameters. Considering formulae (8) and (9), one arrives immediately at the following Proposition 1. If, for a fixed pair ðn; gÞ 2 D 0  D 1 , the sequence fu m ðÁ; n; gÞ : m P 0g converges to a function u 1 ðÁ; n; gÞ uniformly on ½a; b, then:…”

Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions

“…Later, appropriate versions of this method were developed for handling more general types of nonlinear boundary value problems for ordinary and functional-differential equations. We refer, e.g., to the books [3][4][5], the articles [6][7][8][9][10][11][12], and the series of survey articles [13] for the related references.…”