Some Properties for Solutions of Riemann-Liouville Fractional Differential Systems with a Delay

In this paper, we study properties for solutions of Riemann-Liouville (R-L) fractional differential systems with a delay. Some results on integral inequalities are first presented by H ̈ older inequality. Then we investigate properties on solutions for R-L fractional systems with a delay by using the obtained inequalities and obtain upper bound of solutions. Finally, an illustrative example is considered to support our new results.


Introduction
Fractional differential equations have been studied for several centuries. At first the researches were only on the pure theoretical aspect. In the last few years, more and more fractional differential equations have been applied to described some actual researches, such as mechanics, aerodynamics, chemistry, and the electrodynamics of complex mediums [1][2][3][4][5][6][7][8].
Integral inequalities play an important role in researches not only on properties of solutions for various differential and integral equations [9][10][11][12], but also on some fractional differential equations. Recently, some results are obtained on properties of solutions for a fractional differential equation with or without delays. For example, Ma [13] obtained upper bounds for solutions of a class of nonlinear fractional differential systems by a result of two dimensional linear integral inequalities. Ye [14] studied the Henry-Gronwall type retarded integral inequalities and then obtained a certain properties of fractional differential equations with delay. This paper studies some properties for solutions of R-L fractional differential systems with a delay. First, we obtain some results on the integral inequalities by Hlder inequality. Then, using the obtained inequalities, properties are investigated on solutions for R-L fractional systems with a delay, and upper bound of solutions is obtained. Moreover, an illustrative example is studied to show that new results presented in this paper work very well.

An Illustrative Example
In this section, we give an illustrative example to show effectiveness of results obtained in this paper. ] . (51)

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.