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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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