- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
International Journal of Differential Equations
Volume 2014 (2014), Article ID 780636, 10 pages
Existence Results for an Impulsive Neutral Fractional Integrodifferential Equation with Infinite Delay
Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand Pin-247667, India
Received 24 December 2013; Revised 6 March 2014; Accepted 3 April 2014; Published 4 May 2014
Academic Editor: Kanishka Perera
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.
- I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1993.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
- J. K. Hale and J. Kato, “Phase space for retarded equations with infinite delay,” Funkcialaj Ekvacioj, vol. 21, no. 1, pp. 11–41, 1978.
- Y. Hino, S. Murakami, and T. Naito, Functional-Differential Equations with Infinite Delay, vol. 1473 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1991.
- E. M. Hernández, 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.
- E. Hernández and H. R. Henríquez, “Existence results for partial neutral functional-differential equations with unbounded delay,” Journal of Mathematical Analysis and Applications, vol. 221, no. 2, pp. 452–475, 1998.
- F. Li, “An existence result for fractional differential equations of neutral type with infinite delay,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 52, pp. 1–15, 2011.
- F. Li and J. Zhang, “Existence of mild solutions to fractional integrodifferential equations of neutral type with infinite delay,” Advances in Difference Equations, vol. 2011, Article ID 963463, 15 pages, 2011.
- R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010.
- J. Dabas and A. Chauhan, “Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay,” Mathematical and Computer Modelling, vol. 57, no. 3-4, pp. 754–763, 2013.
- J. Wang, M. Fečkan, and Y. Zhou, “On the new concept of solutions and existence results for impulsive fractional evolution equations,” Dynamics of Partial Differential Equations, vol. 8, no. 4, pp. 345–361, 2011.
- M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of Contemporary Mathematics and Its Applications, Hindawi, New York, NY, USA, 2006.
- M. Fečkan, Y. Zhou, and J. Wang, “On the concept and existence of solution for impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 3050–3060, 2012.
- 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.
- Y.-K. Chang, A. Anguraj, and M. M. Arjunan, “Existence results for impulsive neutral functional differential equations with infinite delay,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 1, pp. 209–218, 2008.
- 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.
- 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.
- R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Birkhäuser, Berlin, Germany, 1992.
- S. Ji and G. Li, “A unified approach to nonlocal impulsive differential equations with the measure of noncompactness,” Advances in Difference Equations, vol. 2012, article 182, 14 pages, 2012.
- P. Chen and Y. Li, “Nonlocal problem for fractional evolution equations of mixed type with the measure of noncompactness,” Abstract and Applied Analysis, vol. 2013, Article ID 784816, 12 pages, 2013.
- 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.
- X. Xue, “Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 7, pp. 2593–2601, 2009.
- Z. Fan and G. Mophou, “Nonlocal problems for fractional differential equations via resolvent operators,” International Journal of Differential Equations, vol. 2013, Article ID 490673, 9 pages, 2013.
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, NY, USA, 1983.
- E. Bazhlekova, Fractional evolution equations in Banach spaces [Ph.D. thesis], Eindhoven University of Technology, 2001.
- R. P. Agarwal, M. Meehan, and D. O'Regan, Fixed Point Theory and Applications, Cambridge Tracts in Mathematics, Cambridge University Press, New York, NY, USA, 2001.
- R. Ye, “Existence of solutions for impulsive partial neutral functional differential equation with infinite delay,” Nonlinear Analysis: Theory, Methods and Applications, vol. 73, no. 1, pp. 155–162, 2010.