Eigenvalue and spectral intervals for a nonlinear algebraic system

This paper studies the nonlinear algebraic system of the form x=λAF(x)x=λAFx, where λ>0,Aλ>0,A is a positive n×nn×n square matrix,x=x1,x2,...,xnT,Fx=fx1,fx2,...,fxnT,the nondecreasing continuous function f   is defined on [0,∞)[0,∞) and f(u)>0fu>0 for u>0u>0.The system covers many problems that arise in applications such as difference equations, boundary value problems and dynamical networks. Letf0=limu→0cfuuandf∞=limu→∞fuu.We classify the system according to six pairs of possible values of f0f0 and f∞f∞ with the assumption that both f0f0 and f∞f∞ exist (including ∞∞). For each case, R+=(0,∞)R+=(0,∞) is divided into intervals for the value of λλ (λλ-intervals) that corresponds to existence, multiplicity and nonexistence of positive solutions of the system respectively. We then obtain intervals that contain only the eigenvalues of the nonlinear operator T=AFT=AF. Also, the resolvent and the spectrum of T are determined. Two examples are given to show the applications of the results.