We demonstrate that there exist sequences u n n = 0 ∞ ⊆ (0, + ∞) \ u₍ \₍ = ₀^ (0, +) such that u n + 1 > u n u₍ + ₁> u₍ for each n ∈ N 0 n N₀, but yet the fractional difference of the sequence is uniformly negative. This demonstrates a pathological property of nonlocal discrete operators.
Chang et al. (Wed,) studied this question.