Positive solutions for a nonlinear algebraic system with nonnegative coefficient matrix

Due to its numerous applications, existence of positive solutions for the algebraic system x=GFx has been extensively studied, where G is the coefficient matrix and F:Rn→Rn is nonlinear. However, all results require the matrix G to be positive. When G contains a zero-element, positive solutions have not been proved because of the difficulties of cone construction. In the present paper, an existence result is obtained for nonnegative G by introducing a new cone. To show applications of the theorem, two explanatory examples are given. The new result can be naturally extended to some more general systems. In particular, the system can be transformed into an operator equation on a Banach space. Thus, the new method also provides a novel idea for operator equations.