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(to)=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 = [to,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.