Table of Contents
International Journal of Partial Differential Equations
Volume 2014 (2014), Article ID 640931, 8 pages
http://dx.doi.org/10.1155/2014/640931
Research Article

Existence of the Mild Solution for Impulsive Semilinear Differential Equation

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand 247667, India

Received 26 December 2013; Revised 15 April 2014; Accepted 18 April 2014; Published 18 May 2014

Academic Editor: Yuri N. Skiba

Copyright © 2014 Alka Chadha and Dwijendra N. Pandey. 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. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, World Scientific Publishing, Teaneck, NJ, USA, 1989.
  2. L. Zhu and G. Li, “Existence results of semilinear differential equations with nonlocal initial conditions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 5133–5140, 2011. View at Publisher · View at Google Scholar
  3. J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.
  4. M. Benchohra, J. Henderson, and S. Ntouyas, IImpulsive Differential Equations and Inclusions, vol. 2, Hindawi Publishing Corporation, New York, NY, USA, 2006.
  5. Y.-K. Chang, V. Kavitha, and M. M. Arjunan, “Existence results for impulsive neutral differential and integrodifferential equations with nonlocal conditions via fractional operators,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 32–43, 2010. View at Publisher · View at Google Scholar
  6. E. Hernández and D. O'Regan, “On a new class of abstract impulsive differential equations,” Proceedings of the American Mathematical Society, vol. 141, no. 5, pp. 1641–1649, 2013. View at Publisher · View at Google Scholar
  7. M. Pierri, D. O'Regan, and V. Rolnik, “Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses,” Applied Mathematics and Computation, vol. 219, no. 12, pp. 6743–6749, 2013. View at Publisher · View at Google Scholar
  8. Z. Yan, “Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2252–2262, 2011. View at Publisher · View at Google Scholar
  9. E. Hernández M., M. Rabello, and H. R. Henríquez, “Existence of solutions for impulsive partial neutral functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 1135–1158, 2007. View at Publisher · View at Google Scholar
  10. J. H. Liu, “Nonlinear impulsive evolution equations,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 6, no. 1, pp. 77–85, 1999. View at Google Scholar
  11. R. H. Martin, Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Robert E. Krieger Publishing, Malabar, Fla, USA, 1987.
  12. J. J. Nieto and R. Rodríguez-López, “Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 593–610, 2006. View at Publisher · View at Google Scholar
  13. X. Xue, “Nonlinear differential equations with nonlocal conditions in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 4, pp. 575–586, 2005. View at Publisher · View at Google Scholar
  14. X. Xue, “Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 7, pp. 2593–2601, 2009. View at Publisher · View at Google Scholar
  15. R. P. Agarwal, M. Benchohra, and D. Seba, “On the application of measure of noncompactness to the existence of solutions for fractional differential equations,” Results in Mathematics, vol. 55, no. 3-4, pp. 221–230, 2009. View at Publisher · View at Google Scholar
  16. Z. Fan and G. Li, “Existence results for semilinear differential equations with nonlocal and impulsive conditions,” Journal of Functional Analysis, vol. 258, no. 5, pp. 1709–1727, 2010. View at Publisher · View at Google Scholar
  17. J. X. Sun and X. Y. Zhang, “A fixed point theorem for convex-power condensing operators and its applications to abstract semilinear evolution equations,” Acta Mathematica Sinica. Chinese Series, vol. 48, no. 3, pp. 439–446, 2005. View at Google Scholar · View at MathSciNet
  18. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1983.
  19. H.-B. Shi, W.-T. Li, and H.-R. Sun, “Existence of mild solutions for abstract mixed type semilinear evolution equations,” Turkish Journal of Mathematics, vol. 35, no. 3, pp. 457–472, 2011. View at Google Scholar
  20. Y. X. Li, “Existence of solutions to initial value problems for abstract semilinear evolution equations,” Acta Mathematica Sinica. Chinese Series, vol. 48, no. 6, pp. 1089–1094, 2005. View at Google Scholar
  21. K. Li, J. Peng, and J. Gao, “Nonlocal fractional semilinear differential equations in separable Banach spaces,” Electronic Journal of Differential Equations, vol. 2013, no. 7, pp. 1–7, 2013. View at Google Scholar