Some New Delay Integral Inequalities Based on Modified Riemann-Liouville Fractional Derivative and Their Applications
Zhimin Zhao, Run Xu*

Abstract

By using the properties of modified Riemann-Liouville fractional derivative, some new delay integral inequalities have been studied. First, we offered explicit bounds for the unknown functions, then we applied the results to the research concerning the boundness, uniqueness and continuous dependence on the initial for solutions to certain fractional differential equations.

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Zhao, Z. and Xu, R. (2015) Some New Delay Integral Inequalities Based on Modified Riemann-Liouville Fractional Derivative and Their Applications. Journal of Applied Mathematics and Physics, 3, 465-477. doi: 10.4236/jamp.2015.35059.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Jiang, F.C. and Meng, F.W. (2007) Explicit Bounds on Some New Nonlinear Integral Inequalities with Delay. Journal of Computational and Applied Mathematics, 205, 479-486. http://dx.doi.org/10.1016/j.cam.2006.05.038 [2] Zhang, H.X. and Meng, F.W. (2008) Integral Inequalities in Two Independent Variables for Retarded Volterra Equations. Applied Mathematics and Computation, 199, 90-98. http://dx.doi.org/10.1016/j.amc.2007.09.026 [3] Yuan, Z.L., Yuan, X.W., Meng, F.W. and Zhang, H.X. (2008) Some New Delay Integral Inequalities and Their Applications. Journal of Computational and Applied Mathematics, 180, 191-200. [4] Zheng, B. (2013) Some New Gronwall-Bellman-Type Inequalities Based on the Modified Riemann-Liouville Fractional Derivative. Hindawi Publishing Corporation Journal of Applied Mathematics, 2013, Article ID: 341706.http://dx.doi.org/10.1155/2013/341706 [5] Zheng, B. (2014) Explicit Bounds Derived by Some New Inequalities and Applications in Fractional Integral Equations. Journal of Inequalities and Applications, 2014, 4.http://www.journalofinequalitiesandapplications.com/content/2014/1/4 http://dx.doi.org/10.1186/1029-242X-2014-4 [6] Zheng, B. and Feng, Q.H. (2013) New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations. Hindawi Publishing Corporation Abstract and Applied Analysis, 2013, Article ID: 705126. http://dx.doi.org/10.1155/2013/705126 [7] Denton, Z. and Vatsala, A.S. (2010) Fractional Integral Inequalities and Applications. Computers and Mathematics with Applications, 59, 1087-1094. http://dx.doi.org/10.1016/j.camwa.2009.05.012 [8] Ye, H.P., Gao, J.M. and Ding, Y.S. (2007) A Generalized Gronwall Inequality and Its Application to a Fractional Differential Equation. Journal of Mathematical Analysis and Applications, 328, 1075-1081.http://dx.doi.org/10.1016/j.jmaa.2006.05.061 [9] Khalil, R., Al Horani, M., Yousef, A. and Sababheh, M. (2014) A New Definition of Fractional Derivative. Journal of Computational and Applied Mathematics, 264, 65-70. http://dx.doi.org/10.1016/j.cam.2014.01.002 [10] Jessada, T., Ntouyas, S.K. and Weerawat, S. (2014) Some New Riemann-Liouville Fractional Integral Inequalities. Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences, 2014, Article ID: 869434. http://dx.doi.org/10.1155/2014/869434 [11] Jalilian, Y. and Jalilian, R. (2013) Existence of Solution for Delay Fractional Differential Equations. Mediterranean Journal of Mathematics, 10, 1731-1747.