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In this paper, we study the spectrum structure of second-order difference operators with sign-changing weight. We apply the Sylvester inertia theorem to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. We also show that the eigenfunction corresponding to the j -th positive/negative eigenvalue changes its sign exactly j-1 times.

Let T > 1 be an integer, T = {1, 2, ..., T}. This article is concerned with the global structure of the set of positive solutions to the discrete second-order boundary value problemswhere r ≠ 0 is a parameter, m : T → R changes its sign, m(t) ≠ 0 for t ∈ T and f : ℝ ℝ is continuous. Also, we obtain the existence of two principal eigenvalues of the corresponding linear eigenvalue problems. MSC (2010): 39A12; 34B18.

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