Positivity analysis for mixed order sequential fractional difference operators

Abstract

<abstract><p>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 <xref ref-type="fig" rid="Figure1">Figure 1</xref>) 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 <xref ref-type="fig" rid="Figure2">Figure 2</xref>) 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.</p> </abstract>