Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

Abstract

Based on Guo-Krasnoselskii’s fixed point theorem, the existence of positive solutions for a class of nonlinear algebraic systems of the formx=GFxis studied firstly, whereGis a positiven×nsquare matrix,x=col⁡(x1,x2,…,xn), andF(x)=col⁡(f(x1),f(x2),…,f(xn)), where,F(x)is not required to be satisfied sublinear or superlinear at zero point and infinite point. In addition, a new cone is constructed inRn. Secondly, the obtained results can be extended to some more general nonlinear algebraic systems, where the coefficient matrixGand the nonlinear term are depended on the variablex. Corresponding examples are given to illustrate these results.

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

“…. (11) In this case, Theorems 1 and 2 are valid for (10) or (11). To the best of our knowledge, such a system cannot be handled by the previous results.…”

“…f : R → R is continuous, and G = (g ij ) n×n is an n × n square matrix. The existence of positive solutions for system (1) has been extensively studied in the literature; see [2,3,5,6,10,11,13,15,17,20,21,24,[27][28][29] and the references therein. However, to the best of our knowledge, almost all obtained results require that the coefficient matrix G ≥ 0 or G > 0, where G ≥ 0 if g ij ≥ 0 and G > 0 if g ij > 0 for (i, j) ∈ [1, n] An n × n square matrix G is called a sign-changing coefficient matrix if its elements change the sign.…”

Algebraic Systems with Positive Coefficients and Positive Solutions

“…Related results can be found in the papers in the bibliography. General methods for solving systems are described in [4][5][6][7][8][9][10][11][12][13][14][15][16]. For applications to discrete inclusions, see [17].…”