Positivity analysis for mixed order sequential fractional difference operators

Abstract

We consider the positivity of the discrete sequential fractional operators $ \left(^{\rm RL}_{a_{0}+1}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ defined on the set $ \mathscr{D}_{1} $ (see (1.1) and Figure 1) and $ \left(^{\rm RL}_{a_{0}+2}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ of mixed order defined on the set $ \mathscr{D}_{2} $ (see (1.2) and Figure 2) for $ \tau\in\mathbb{N}_{a_{0}} $. By analysing the first sequential operator, we reach that $ \bigl(\nabla {f}\bigr)(\tau)\geqq 0, $ for each $ \tau\in{\mathbb{N}}_{a_{0}+1} $. Besides, we obtain $ \bigl(\nabla {f}\bigr)(3)\geqq 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.

Keywords

• discrete delta riemann-liouville fractional difference
• negative lower bound
• convexity analysis
• analytical and numerical results