Analytical and Numerical Monotonicity Analyses for Discrete Delta Fractional Operators

In this paper, first, we intend to determine the relationship between the sign of () c₀ ∆^β y (c₀ + 1), for 1 < β < 2, and (∆y)(c₀ + 1) > 0, in the case we assume that c₀ ∆^β y (c₀ + 1) is negative. After that, by considering the set D_ℓ+1,θ ⊆ D_ℓ,θ, which are (subsets) of (1, 2), we will extend our previous result to make the relationship between the sign of () c₀ ∆^β y (z) and (∆y)(z) > 0 (the monotonicity of y), where c₀ ∆^β y (z) will be assumed to be negative for each z ∈ Nc^T₀:= {c₀, c₀ + 1, c₀ + 2, …, T} and some T ∈ N_c0:= {c₀, c₀ + 1, c₀ + 2, … }. The last part of this work is devoted to see(the possibility) of information reduction regarding the monotonicity of y despite the non-positivity of c₀ ∆^β y (z) by means of numerical simulation.