A Solution of Nonlinear Boundary Value Problem of System With Rectangular Coefficients

Abstract

This paper discusses a nonlinear boundary value problem of system with rectangular coefficients of the form  with boundary conditions of the form  A(t)x' + B(t)x = f(t,x) and  which is  is a real  matrix with  whose entries are continuous on the form B₁x(t_o)=a  and B₂x(T)=b which is A(t) is a real m  n matri with m > n matrix with m > n whose entries  are continuous on J = [t_o,T] and f E C[J x Rⁿ, Rⁿ]. B₁, B₂  are nonsingular matrices such that  and  are constant vectors, especially about the proof of the uniqueness of its solution. To prove it, we use Moore-Penrose generalized inverse and method of variation of parameters to find its solution. Then we show the uniqueness of it by using fixed point theorem of contraction mapping. As the result, under a certain condition, the boundary value problem has a unique  solution.