Positivity analysis for mixed order sequential fractional difference operators

Abstract

We consider the positivity of the discrete sequential fractional operators ((RL)(a0+1) del(v1) (RL)(a0) del(v2) f) (tau) defined on the set D-1 (see (1.1) and Figure 1) and (RL)(a0+2) del(v1) (RL)(a0) del(v2) f) (tau) of mixed order defined on the set D-2 (see (1.2) and Figure 2) for tau is an element of N-a0. By analysing the first sequential operator, we reach that (del f(tau) >= 0; for each tau is an element of Na0+1. Besides, we obtain (del f(tau) >= 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.