Recently, some authors have proved monotonicity results for delta and nabla fractional differences separately. In this article, we use dual identities relating delta and nabla fractional difference operators to prove shortly the monotonicity properties for the (left Riemann) nabla fractional differences using the corresponding delta type properties. Also, we proved some monotonicity properties for the Caputo fractional differences. Finally, we use the Q−operator dual identities to prove monotonicity results for the right fractional difference operators.Keywords: right (left) delta and nabla fractional sums, right (left) delta and nabla Riemann and Caputo fractional differences, Q-operator, dual identity.

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Recently, Jarad et al. in (Adv. Differ. Equ. 2017:247, 2017) defined a new class of nonlocal generalized fractional derivatives, called conformable fractional derivatives (CFDs), based on conformable derivatives. In this paper, sufficient conditions are established for the oscillation of solutions of generalized fractional differential equations of the formRe(α) ≥ 0 in the Riemann-Liouville setting and a I α,ρ is the left-fractional conformable integral operator. The results are also obtained for CFDs in the Caputo setting. Examples are provided to demonstrate the effectiveness of the main result.

Based on certain mathematical inequalities and Volterra sum equations, we establish oscillation criteria for higher order fractional difference equations with mixed nonlinearities. The problem is addressed for equations involving Riemann-Liouville and Caputo operators. Two examples are constructed to demonstrate the validity of the proposed assumptions. Our results improve those obtained in the previous works.

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