Monotonicity results for sequential fractional differences of mixed orders with negative lower bound. Issue 11 (2nd November 2021) - Dahal, Rajendra; Goodrich, Christopher S.; Lyons, Benjamin
Abstract : We investigate the relationship between the sign of the discrete fractional sequential difference ( Δ 1 + a − μ ν Δ a μ f ) ( t ) and the monotonicity of the function t ↦ f ( t ) in the case where the orders of the fractional difference are mixed – i.e. μ ∈ ( 0, 1 ) but ν ∈ ( 1, 2 ) with μ + ν ∈ ( 1, 2 ) . We consider the special case in which the bound 1 ( Δ 1 + a − μ ν Δ a μ f ) ( t ) ≥ − ε f ( a ), for some ε > 0 and where f ( a ) ≥ 0, is satisfied. Even though the fractional difference is negative, we demonstrate that there still exists a relationship between this quantity and the monotonicity of f . We augment our analytical results with numerical simulations.