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.