Comparison principles for fractional differential equations with the Caputo derivatives

Abstract

In this paper, we deal with comparison principles for fractional differential equations involving the Caputo derivatives of order p with 0 ≤ n -1 < p ≤ n. First, we present comparison results with strict inequalities for fractional differential equations with the Caputo derivatives. Then we investigate local existence and extremal solutions for fractional differential equations with the Caputo derivatives. Finally, we consider comparison results with nonstrict inequalities for fractional differential equations with the Caputo derivatives. MSC: Primary 26A33; 26D10; secondary 34A12; 34A40

A Generalized Uncertain Fractional Forward Difference Equations of Riemann-Liouville Type

“…Also, an existence and uniqueness theorem of solutions to a special type of UFFDEs is presented by Lu, Q., . Meanwhile, have been created the relations between uncertain fractional differential equations and the associated fractional differential equations via the comparison theorems for fractional differential equations of Caputo type by Lu, Z., & Zhu, Y. (2018).…”

A Correlation Between Solutions of Uncertain Fractional Forward Difference Equations and Their Paths

“…Besides the discrete fractional calculus, the uncertain fractional differential and difference equations have been introduced and investigated in order to model the continuous or discrete systems with memory effects and human uncertainty (see for e.g., [24][25][26][27][28]). In Lu and Zhu [27], the relations between uncertain fractional differential equations and the associated fractional differential equations have been created via comparison theorems for fractional differential equations of Caputo type in Lu and Zhu [26]. Lu et al [28] presented analytic solutions to a type of special linear uncertain fractional difference equation (UFDE) by the Picard iteration method.…”

Uncertain fractional forward difference equations for Riemann–Liouville type

“…Recently, UFDEs were extended and applied to European option pricing in [20]. Meanwhile, comparison principles of the fractional differential equations were proposed in [21]. Furthermore, a numerical approach for solving UFDEs was also provided in [22] by the comparison principles in [21].…”

“…Meanwhile, comparison principles of the fractional differential equations were proposed in [21]. Furthermore, a numerical approach for solving UFDEs was also provided in [22] by the comparison principles in [21]. To our knowledge, research on the fractional equations in the uncertain environment is just beginning, and a lot of work needs to be done.…”