Positivity analysis for mixed order sequential fractional difference operators

Abstract

We consider the positivity of the discrete sequential fractional operators( RL a0 +1∇ν1 defined on the set D1 (see (1.1) and Figure 1) and( RL a0 +2∇ν1 RL a0 ∇ν2 f) (τ) RL a0 ∇ν2 f) (τ) of mixed order defined on the set D2 (see (1.2) and Figure 2) for τ ∈ Na0 . By analysing the first sequential operator, we reach that (∇f )(τ)≧ 0, for each τ∈ Na0 +1. Besides, we obtain(∇ f)(3) ≧ 0 by analysing the second sequential operator. Furthermore, some conditions to obtain the proposed monotonicity results are summarized. Finally, two practical applications are provided to illustrate the efficiency of the main theorems.