Existence and Uniqueness of Positive Solutions to Nonlinear Systems of Equations With M-Type Functions

This paper considers the existence and uniqueness of positive solutions to a class of nonlinear systems of equations, which has wide applications in fluid mechanics and electrical circuit theory. In this paper, we define a new class of functions, called M-type function, and then, we show some good properties of M-type function as well as the relationship between this kind of functions and other kinds of functions. In particular, by fully exploiting the properties of M-type function, we present that under a reasonable condition, the nonlinear system of equations with an M-type function, which can be seen as a generalization of the multilinear system with a nonsingular M-tensor, and a positive right-hand side vector has a unique positive solution. Meanwhile, for this class of problems, we give a preliminary and simple algorithm framework. What is more, under an additional assumption, we illustrate the feasibility of using Newton step when improving the original iteration algorithm to obtain better convergence. Numerical experiments show the stability and efficiency of the proposed algorithms.