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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 Hlder 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.
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 [1–8].
Integral inequalities play an important role in researches not only on properties of solutions for various differential and integral equations [9–12], 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  obtained upper bounds for solutions of a class of nonlinear fractional differential systems by a result of two dimensional linear integral inequalities. Ye  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.
2. Main Results
This section is devoted to studying properties of solutions for R-L fractional differential systems with a delay and presents the main result of this paper. First, we give some lemmas on integral inequalities.
Lemma 1 (let and ). Suppose , and , , , , and is a contant. If andthen where ,
Proof. Set , . Then, , , are nondecreasing for . Hence, for , we have Set . (5) and (6) can be rewritten as a matrix form Then we have Hence, That is, ThenWhen , from (1),LetThus, Therefore, (2) is satisfied and the proof is completed.
Lemma 2. Suppose , and , , , , and and are constants. If and then,
(i) when , setand we havewhereand , , , ;
(ii) when , let , and we havewhere and , , , .
Proof. (i) Suppose . By Hlder inequality and , for and , we haveThus,Let , , , . Then, for , we haveSet . For By (26), (27), and Lemma 1, (17) is satisfied.
(ii) Suppose . Set and . Then . By Hlder inequality and , for and , we haveThus, Let , , , , . Then, we haveSet . For ,By (30), (31), and Lemma 1, (21) is satisfied and the proof is completed.
Now, we introduce some definitions on Riemann-Liouville fractional derivative and fractional primitive.
Definition 3 (see ). The fractional derivative of order of a function is given by
Definition 4 (see ). The fractional derivative of order of a function is given by
Consider the following Riemann-Liouville fractional differential system with a delay:where , are constants, and , are continuous functions. Then, a result can be obtained on the solutions of system (34).
Theorem 5. Consider system (34). and and satisfy the following condition:where and . Then, solutions of system (34) satisfy that
(i) when ,where, and , , , ;
(ii) when , where, and , , , .
Proof. Fractional differential system (34) with a delay can be converted to the following integral equation:By condition (36), we haveSet and . Then, according to Lemma 2, the result is obtained and the proof is completed.
3. An Illustrative Example
In this section, we give an illustrative example to show effectiveness of results obtained in this paper.
Example 1. Consider the following fractional differential equation:where , .
It is obvious that , . From (47) and Theorem 5, we obtain , , , . Thus, , , .
Therefore, where for , and for ,
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work is supported by the National Natural Science Foundation of China (G11671227, G61403223, G11701310).
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.View at: MathSciNet
I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.View at: MathSciNet
W. Sun, Y. Wang, and R. Yang, “L2 disturbance attenuation for a class of time delay Hamiltonian systems,” Journal of Systems Science & Complexity, vol. 24, no. 4, pp. 672–682, 2011.View at: Publisher Site | Google Scholar | MathSciNet
W. Sun and L. Peng, “Robust adaptive control of uncertain stochastic Hamiltonian systems with time varying delay,” Asian Journal of Control, vol. 18, no. 2, pp. 642–651, 2016.View at: Publisher Site | Google Scholar | MathSciNet
Y. Wang and L. Liu, “Positive solutions for a class of fractional 3-point boundary value problems at resonance,” Advances in Difference Equations, Paper No. 7, 13 pages, 2017.View at: Publisher Site | Google Scholar | MathSciNet
Y. Wang and L. Liu, “Positive properties of the Green function for two-term fractional differential equations and its application,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 10, no. 4, pp. 2094–2102, 2017.View at: Publisher Site | Google Scholar | MathSciNet
K. M. Zhang, “On a sign-changing solution for some fractional differential equations,” Boundary Value Problems, vol. 2017, no. 59, 8 pages, 2017.View at: Google Scholar | MathSciNet
X. Zhang, L. Liu, Y. Wu, and B. Wiwatanapataphee, “Nontrivial solutions for a fractional advection dispersion equation in anomalous diffusion,” Applied Mathematics Letters, vol. 66, pp. 1–8, 2017.View at: Publisher Site | Google Scholar | MathSciNet
H. Liu and F. Meng, “Some new nonlinear integral inequalities with weakly singular kernel and their applications to FDEs,” Journal of Inequalities and Applications, vol. 2015, no. 209, 2015.View at: Publisher Site | Google Scholar | MathSciNet
L. Li, M. Han, X. Xue, and Y. Liu, “Ψ-stability of nonlinear Volterra integro-differential systems with time delay,” Abstract and Applied Analysis, pp. 1–5, 2013.View at: Google Scholar | MathSciNet
Q.-H. Ma and E.-H. Yang, “Some new Gronwall-Bellman-BIHari type integral inequalities with delay,” Periodica Mathematica Hungarica. Journal of the J\'Anos Bolyai Mathematical Society, vol. 44, no. 2, pp. 225–238, 2002.View at: Publisher Site | Google Scholar | MathSciNet
H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007.View at: Publisher Site | Google Scholar | MathSciNet
Q.-H. Ma and J. Pecaric, “On some qualitative properties for solutions of a certain two-dimensional fractional differential systems,” Computers & Mathematics with Applications. An International Journal, vol. 59, no. 3, pp. 1294–1299, 2010.View at: Publisher Site | Google Scholar | MathSciNet
H. Ye and J. Gao, “Henry-Gronwall type retarded integral inequalities and their applications to fractional differential equations with delay,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4152–4160, 2011.View at: Publisher Site | Google Scholar | MathSciNet