## Some relations between the Caputo fractional difference operators and integer-order differences.(English)Zbl 1321.39024

Summary: In this article, we are concerned with the relationships between the sign of Caputo fractional differences and integer nabla differences. In particular, we show that if $$N-1<\nu<N$$, $$f:\mathbb{N}_{a-N+1}\to\mathbb{R}$$, $$\nabla^\nu_{a^*}f(t)\geq 0$$, for $$t\in\mathbb{N}_{a+1}$$ and $$\nabla^{N-1}f(a)\geq 0$$, then $$\nabla^{N-1}f(t)\geq 0$$ for $$t\in\mathbb{N}_a$$. Conversely, if $$N-1<\nu<N$$, $$f:\mathbb{N}_{a-N+1}\to\mathbb{R}$$, and $$\nabla^{N}f(t)\geq 0$$ for $$t\in\mathbb{N}_{a+1}$$, then $$\nabla^{\nu}_{a^*}f(t)\geq 0$$, for each $$t\in\mathbb{N}_{a+1}$$. As applications of these two results, we get that if $$1<\nu<2$$, $$f:\mathbb{N}_{a-1}\to\mathbb{R}$$, $$\nabla^\nu_{a^*}f(t)\geq 0$$ for $$t\in\mathbb{N}_{a+1}$$ and $$f(a)\geq f(a-1)$$, then $$f(t)$$ is an increasing function for $$t\in \mathbb{N}_{a-1}$$. Conversely if $$0<\nu<1$$, $$f:\mathbb{N}_{a-1}\to\mathbb{R}$$ and $$f$$ is an increasing function for $$t\in\mathbb{N}_{a}$$, then $$\nabla^\nu_{a^*}f(t)\geq 0$$, for each $$t\in\mathbb{N}_{a+1}$$. We also give a counterexample to show that the above assumption $$f(a)\geq f(a-1)$$ in the last result is essential. These results demonstrate that, in some sense, the positivity of the $$\nu$$-th order Caputo fractional difference has a strong connection to the monotonicity of $$f(t)$$.

### MSC:

 39A70 Difference operators 26A33 Fractional derivatives and integrals 39A22 Growth, boundedness, comparison of solutions to difference equations
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