Analytical and Numerical Monotonicity Analyses for Discrete Delta Fractional Operators

In this paper, first, we intend to determine the relationship between the sign of Δ c 0 β y ( c 0 + 1 ) , for 1 < β < 2 , and Δ y ( c 0 + 1 ) > 0 , in the case we assume that Δ c 0 β y ( c 0 + 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 Δ c 0 β y ( z ) and Δ y ( z ) > 0 (the monotonicity of y ), where Δ c 0 β y ( z ) will be assumed to be negative for each z ∈ N c 0 T : = { c 0 , c 0 + 1 , c 0 + 2 , ⋯ , T } and some T ∈ N c 0 : = { c 0 , c 0 + 1 , c 0 + 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 Δ c 0 β y ( z ) by means of numerical simulation.