The Existence of the Extremal Solution for the Boundary Value Problems of Variable Fractional Order Differential Equation With Causal Operator

Abstract

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results.

On the Existence and Stability of Variable Order Caputo Type Fractional Differential Equations

“…Yufeng et al [17] discussed the existence and uniqueness of variable order FDEs by considering the iterative series with the contraction mapping principle. Jiang et al [18] provide the existence of the solution of a variable order fractional differential equation with two point boundary values. A tempered variable order FDE was studied in [19] for the Mittag-Leffler stability.…”

The Tracking Control of the Variable-Order Fractional Differential Systems by Time-Varying Sliding-Mode Control Approach

“…Fractional-order (FO) calculus has been developed rapidly over the past decades and the FO operators have been considered as an important tool to model the complex phenomena in science and engineering [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. It is mentioned that, compared with the constant-order (CO) fractional-order operators, the variable-order (VO) fractional-order operators possess more complicated properties and are more suitable for describing some complex phenomena.…”