Parametrization for non-linear problems with integral boundary conditions

摘要

We consider the integral boundary-value problem for a certain class of non-linear system of ordinary differential equations of the form egin{equation*} rac{dxleft(tight)}{dt} =fleft(t,xleft(tight)ight), tin left[0,Tight],: xin mathbb{R}^{n}, end{equation*} egin{equation*} Ax(0)+int_0^T{B(s)x(s)ds}+Cx(T)=d, end{equation*} where $f:left[0,Tight]imes Do mathbb{R}^{n}$, $Dsubset mathbb{R}^{n}$ is a closed and bounded domain, $A$ and $C$ are some $n imes n$ singular matrixes. We give a new approach for studying this problem, namely by using an appropriate parametrization technique the given problem is reduced to the equivalent parametrized two-point boundary-value problem with linear boundary conditions without integral term. To study the transformed problem we use a method based upon a special type of successive approximations, which are constructed analytically. We establish a sufficient conditions for the uniformly convergence of this sequence and introduce a certain finite-dimensional 'determining' system of algebraic or transcendental equations whose solutions give all the initial values of the solutions of the given boundary--value problem. Based upon the properties of the functions of the constructed sequence and of the determining equations, using the Brower degree, we give efficient conditions for the solvability of the original integral boundary-value problem.