On the Hadamard Type Fractional Differential System

A Gronwall inequality for a general Caputo fractional operator

“…[6]). Recently, the Gronwall inequality has been generalized for the study of fractional differential equations, with dependence on the Riemann-Liouville fractional derivative [31] and for the Hadamard fractional derivative [11]. Here we present a more general form, valid for the Katugampola fractional derivative.…”

Positive Solutions for a System of Hadamard-Type Fractional Differential Equations with Semipositone Nonlinearities

“…Coupled systems of fractional-order differential equations have also been investigated by many authors (see [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][25][26][27][28][29][30][31][32][33][34][35][36] and the references therein). In [7], the authors used coincidence degree theory to establish an existence result for a coupled system of nonlinear fractional differential equations:…”

“…ere are a few results in the literature on Hadamardtype fractional differential equations (see [23][24][25][26][27][28][29][30][31][32][33][34][35][36]). In [23], the authors used fixed-point methods to obtain some existence theorems for Hadamard-type fractional boundary value problems:…”

Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative

“…On one hand, the Hadamard fractional derivative involves an integral kernel of logarithmic function with an arbitrary exponent, which could be more effective in describing the ultra-slow diffusion processes. On the other hand, the t d dt in its definition has shown to be invariant on the half-axis in concerns of dilation [13]. For more knowledge of the Hadamard fractional integral and derivative, we refer the reader to [5,23].…”