Eigenvalue characterization of a system of difference equations - Agarwal, R.P.; O'Regan, D.; P. J. Y. Wong (Нелінійні коливання, 2004)
We consider the following system of difference equations: ui(k) = λ∑gi(k, l)Pi(l, u1(l), u2(l), . . . , un(l)), k ∈ {0, 1, . . . }, 1 ≤ i ≤ n, where λ > 0 and T ≥ N ≥ 0. Our aim is to determine those values of λ such that the above system has a constant-sign solution. In addition, explicit intervals for λ will be presented. The generality of the results obtained is illustrated through applications to several well known boundary-value problems. We also extend the above problem to that on {0, 1, . . . }, ui(k) = λ∑gi(k, l)Pi(l, u1(l), u2(l), . . . , un(l)), k ∈ {0, 1, . . . , T}, 1 ≤ i ≤ n, Finally, both systems above are extended to the general case when λ is replaced by λi .