Monotonicity results for delta and nabla caputo and Riemann fractional differences via dual identities

Abstract

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.

A solution to nonlinear Volterra integro-dynamic equations via fixed point theory

“…We refer the readers to some very recent studies related with these two types of derivatives of fractional order [2,3]. In this study, we only deal with the delta derivative.…”

Monotonicity results for h-discrete fractional operators and application

“…Remark 5.1 If we let h = 1 in Theorem 5.1, then we reobtain the results in [34] via using the dual identities presented in [30,31], or else we refer to [37].…”

“…It is worth mentioning that Atıcı et al in [34] studied the monotonicity properties of delta fractional differences and obtained a delta-fractional difference version of the mean-value theorem. In [37], interesting monotonicity results were provided using the dual identities related to delta and nabla fractional operators. In [38], the authors studied the relationship between the discrete sequential fractional operators and monotonicity.…”

On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations

“…In particular, the fractional calculus has been used in many research works related to biological, biomechanics, magnetic fields, echanics of micro/nano structures, and physical problems (see [1][2][3][4][5][6][7]). We can find fractional delta difference calculus and fractional nabla difference calculus in [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and [25][26][27][28][29][30][31][32][33][34][35][36], respectively. Definitions and properties of fractional difference calculus are presented in the book [37].…”