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Discrete Dynamics in Nature and Society
Volume 2017, Article ID 5129051, 10 pages
https://doi.org/10.1155/2017/5129051
Research Article

A Generalization of Linear and Nonlinear Retarded Integral Inequalities in Two Independent Variables

School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

Correspondence should be addressed to Guojing Xing; nc.ude.uds@udsjgx

Received 26 August 2017; Accepted 5 December 2017; Published 25 December 2017

Academic Editor: Eric R. Kaufmann

Copyright © 2017 Ying Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Y. Sun, “Stabilization of switched systems with nonlinear impulse effects and disturbances,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 56, no. 11, pp. 2739–2743, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. Y. Zhang, M. Wang, H. Xu, and K. L. Teo, “Global stabilization of switched control systems with time delay,” Nonlinear Analysis: Hybrid Systems, vol. 14, pp. 86–98, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. É. Gyurkovics, G. Szabó-Varga, and K. Kiss, “Stability analysis of linear systems with interval time-varying delays utilizing multiple integral inequalities,” Applied Mathematics and Computation, vol. 311, pp. 164–177, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  4. Z. Li, Y. Bai, C. Huang, and H. Yan, “Further results on stabilization for interval time-delay systems via new integral inequality approach,” ISA Transactions®, vol. 68, pp. 170–180, 2017. View at Publisher · View at Google Scholar · View at Scopus
  5. B. Yang, J. Wang, and J. Wang, “Stability analysis of delayed neural networks via a new integral inequality,” Neural Networks, vol. 88, pp. 49–57, 2017. View at Publisher · View at Google Scholar · View at Scopus
  6. T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics, vol. 20, no. 4, pp. 292–296, 1919. View at Publisher · View at Google Scholar · View at MathSciNet
  7. R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol. 10, pp. 643–647, 1943. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. I. Bihari, “A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations,” Acta Mathematica Hungarica, vol. 7, pp. 81–94, 1956. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. B. G. Pachpatte, Inequalities for Differential and Integral Equations, vol. 197, Academic Press, New York, NY, USA, 1998. View at MathSciNet
  10. B. G. Pachpatte, “Explicit bounds on certain integral inequalities,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 48–61, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. A. Abdeldaim and A. A. El-Deeb, “On generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro-differential equations,” Applied Mathematics and Computation, vol. 256, pp. 375–380, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. A. Abdeldaim and A. A. El-Deeb, “On some generalizations of certain retarded nonlinear integral inequalities with iterated integrals and an application in retarded differential equation,” Journal of the Egyptian Mathematical Society, vol. 23, no. 3, pp. 470–475, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  13. H. Zhang and F. Meng, “Integral inequalities in two independent variables for retarded Volterra equations,” Applied Mathematics and Computation, vol. 199, no. 1, pp. 90–98, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. H. Zhang and F. Meng, “On certain integral inequalities in two independent variables for retarded equations,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 608–616, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599–612, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Y. Tian, M. Fan, and F. Meng, “A generalization of retarded integral inequalities in two independent variables and their applications,” Applied Mathematics and Computation, vol. 221, pp. 239–248, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. Y. Tian, Y. Cai, L. Li, and T. Li, “Some dynamic integral inequalities with mixed nonlinearities on time scales,” Journal of Inequalities and Applications, vol. 2015, no. 1, pp. 1–10, 2015. View at Publisher · View at Google Scholar · View at Scopus
  18. X. Liu, L. Zhang, P. Agarwal, and G. Wang, “On some new integral inequalities of Gronwall-Bellman-Bihari type with delay for discontinuous functions and their applications,” Indagationes Mathematicae, vol. 27, no. 1, pp. 1–10, 2016. View at Publisher · View at Google Scholar · View at Scopus
  19. L. Li, F. Meng, and L. He, “Some generalized integral inequalities and their applications,” Journal of Mathematical Analysis and Applications, vol. 372, no. 1, pp. 339–349, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. R. Xu and F. Meng, “Some new weakly singular integral inequalities and their applications to fractional differential equations,” Journal of Inequalities and Applications, vol. 2016, no. 1, article no. 78, pp. 1–16, 2016. View at Publisher · View at Google Scholar · View at Scopus