- About this Journal
- Abstracting and Indexing
- 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
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 398632, 9 pages
A Class of Fractional -Laplacian Integrodifferential Equations in Banach Spaces
1Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis and College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, China
2College of Sciences, Hezhou University, Hezhou, Guangxi 542899, China
Received 17 April 2013; Accepted 26 June 2013
Academic Editor: Paul Eloe
Copyright © 2013 Yiliang Liu and Liang Lu. 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.
- G. Q. Chai, “Positive solutions for boundary value problem of fractional differential equation with -Laplacian operator,” Boundary Value Problems, p. 2012, article 18, 2012.
- T. Y. Chen, W. Liu, and Z. G. Hu, “A boundary value problem for fractional differential equation with -Laplacian operator at resonance,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 6, pp. 3210–3217, 2012.
- T. Y. Chen and W. B. Liu, “An anti-periodic boundary value problem for the fractional differential equation with a -Laplacian operator,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1671–1675, 2012.
- A. A. Kilbsa, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
- X. Y. Liu and Z. H. Liu, “Existence results for fractional differential inclusions with multivalued term depending on lower-order derivative,” Abstract and Applied Analysis, vol. 2012, Article ID 423796, 24 pages, 2012.
- Z. H. Liu and X. W. Li, “On the controllability of impulsive fractional evolution inclusions in Banach spaces,” Journal of Optimization Theory and Applications, vol. 156, no. 1, pp. 167–182, 2013.
- Z. H. Liu and D. Motreanu, “A class of variational-hemivariational inequalities of elliptic type,” Nonlinearity, vol. 23, no. 7, pp. 1741–1752, 2010.
- Z. H. Liu and J. H. Sun, “Nonlinear boundary value problems of fractional differential systems,” Computers & Mathematics with Applications, vol. 64, no. 4, pp. 463–475, 2012.
- Z. H. Liu and J. H. Sun, “Nonlinear boundary value problems of fractional functional integro-differential equations,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3228–3234, 2012.
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
- F. Wang, Z.-H. Liu, and J. Li, “Complete controllability of fractional neutral differential systems in abstract space,” Abstract and Applied Analysis, vol. 2013, Article ID 529025, 11 pages, 2013.
- H. Okochi, “On the existence of periodic solutions to nonlinear abstract parabolic equations,” Journal of the Mathematical Society of Japan, vol. 40, no. 3, pp. 541–553, 1988.
- A. Alsaedi, “Existence of solutions for integrodifferential equations of fractional order with antiperiodic boundary conditions,” International Journal of Differential Equations, vol. 2009, Article ID 417606, 9 pages, 2009.
- J. B. Liu and Z. H. Liu, “On the existence of anti-periodic solutions for implicit differential equations,” Acta Mathematica Hungarica, vol. 132, no. 3, pp. 294–305, 2011.
- Z. H. Liu, “Anti-periodic solutions to nonlinear evolution equations,” Journal of Functional Analysis, vol. 258, no. 6, pp. 2026–2033, 2010.
- E. Kaslik and S. Sivasundaram, “Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions,” Nonlinear Analysis. Real World Applications, vol. 13, no. 3, pp. 1489–1497, 2012.
- M. S. Tavazoei and M. Haeri, “A proof for non existence of periodic solutions in time invariant fractional order systems,” Automatica, vol. 45, no. 8, pp. 1886–1890, 2009.
- M. S. Tavazoei, “A note on fractional-order derivatives of periodic functions,” Automatica, vol. 46, no. 5, pp. 945–948, 2010.
- M. S. Tavazoei and M. Haeri, “Simplification in the proof presented for non existence of periodic solutions in time invariant fractional order systems,” In press, http://arxiv.org/abs/1202.5878.
- S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Physics Letters A, vol. 372, no. 42, pp. 6309–6320, 2008.
- S. C. Lim and L. P. Teo, “The fractional oscillator process with two indices,” Journal of Physics A, vol. 42, no. 6, Article ID 065208, 34 pages, 2009.
- B. Ahmad and J. J. Nieto, “Sequential fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3046–3052, 2012.
- B. Ahmad, J. J. Nieto, and A. Alsaedi, “A nonlocal three-point inclusion problem of Langevin equation with two different fractional orders,” Advances in Difference Equations, vol. 2012, article 54, 2012.
- B. Ahmad, J. J. Nieto, A. Alsaedi, and M. El-Shahed, “A study of nonlinear Langevin equation involving two fractional orders in different intervals,” Nonlinear Analysis. Real World Applications, vol. 13, no. 2, pp. 599–606, 2012.
- B. Ahmad and J. J. Nieto, “Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions,” International Journal of Differential Equations, vol. 2010, Article ID 649486, 10 pages, 2010.
- A. P. Chen and Y. Chen, “Existence of solutions to nonlinear Langevin equation involving two fractional orders with boundary value conditions,” Boundary Value Problems, vol. 2011, Article ID 516481, 17 pages, 2011.
- D. R. Smart, Fixed Point Theorems, Cambridge University Press, London, UK, 1980.