New monotonicity conditions in discrete fractional calculus with applications to extremality conditions

摘要

We present some initial results for detecting extrema of a function f : ℕ_a → ℝ when the only a priori information is some limited pointwise data about f as well as information about an appropriate fractional difference of f. In particular, such results provide some analogues of the well-known tests for detecting local extrema using the sign of Δf(t). Certain of these results are consequences of some new monotonicity conditions, which relate conditions on the sign of Δ_a ~νf(t), for 0 < ν < 1, to the monotonicity of f. Finally, we provide some numerical examples to illustrate the results.