Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix

Abstract

Existence of positive solutions for the nonlinear algebraic system $x=\lambda GF ( x ) $ x = λ G F ( x ) has been extensively studied when the $n\times n$ n × n coefficient matrix G is positive or nonnegative. However, to the best of our knowledge, few results have been obtained when the coefficient matrix changes sign. In this case, some commonly applied analysis methods such as the cone theory, the Krein–Rutman theorem, the monotone iterative techniques, and so on cannot be directly applied. In this note, we prove the existence of positive solutions for the above nonlinear algebraic system with sign-changing coefficient matrix taking the advantages of the classical Brouwer fixed point theorem combined with a decomposition condition on the coefficient matrix. We provide an example in solving a second-order difference equation with periodic boundary conditions to illustrate the applications of the results.

Algebraic Systems with Positive Coefficients and Positive Solutions

“…In [19], applications to extremum problems are presented, while [20] describes applications to parameter estimation. Linear systems with positive coefficients and positive solutions are studied in [21,22], while special classes of systems are investigated in [23][24][25]. Applications to boundary value problems can be found in [26,27] and the references therein.…”