The complementarity problem with a nonlinear continuous mapping f from the nonnegative orthant Rnab21Å+ of Rn into Rn can be written as the system of equations F(x,y)=0 and (x,y)∈RÅ+ab452n, where F denotes the mapping from the nonnegative orthant RÅ+ab452n of R2>n into Rnab21Å+×Rn defined by F(x,y)=(x1y1,...,xnyn,f1(x)-y1,...,fn(x)-yn) for every (x,y)∈RÅ+ab452n. Under the assumption that f is a uniform P-function, this paper establishes that the mapping F is a homeomorphism of RÅ+ab452nonto Rnab21Å+×Rn. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equations F(x,y)=tF(x0,y0) and (x,y)∈RÅ+ab452n from an arbitrary initial point (x0,y0)∈RÅ+ab452n with t=1 until the parameter t attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.
