On the existence of solutions of fractional integro-differential equations

Fractional Calculus and Applied Analysis

Volumn 15, Issue 1, 2012, Pages 44-69

  Aghajani, Asadollah ^a   Jalilian, Yaghoub ^b   Trujillo, Juan J ^c  

^a IRAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Iran)

^b RAZI UNIVERSITY   (Iran)

^c UNIVERSIDAD DE LA LAGUNA   (Spain)

Author keywords

Caputo fractional derivative; Fractional integral inequality; Fractional integro differential equation; Nonlinear alternative Leray Schauder

Indexed keywords

EID: 84862666375     PISSN: 13110454     EISSN: 13142444     Source Type: Journal    
DOI: 10.2478/s13540-012-0005-4     Document Type: Article

References (42)

1. R.P. Agarwal and B. Ahmad, Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62 (2011), 1200-1214.
2. 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. Appl. Math. 109 (2010), 973-1033.
3. A. Aghajani, J. Bánas, and Y. Jalilian, Existence of solutions for a class of nonlinear Volterra singular integral equations. Comput. Math. Appl. 62 (2011), 1215-1227.
4. A. Anguraj, P. Karthikeyan, and J. J. Trujillo, Existence of solutions to fractional mixed integrodifferential equations with nonlocal initial condition. Adv. Differ. Equ. 2011, Art. ID 690653, 12 pp.
5. K. Balachandran, and S. Kiruthika, Existence results for fractional integrodifferential equations with nonlocal condition via resolvent operators. Comput. Math. Appl., 62 (2011), 1350-1358.
6. K. Balachandran, S. Kiruthika, and J.J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces. Commun. Nonlin. Sci. Numer. Sim. 16 (2011), 1970-1977.
7. K. Balachandran and J. J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Analysis: Th. Meth. Appl. 72 (2010), 4587-4593.
8. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods. World Scientific Publishing Company, 2012.
9. D. Baleanu and O. G. Mustafa, On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59 (2010), 1835-1841.
10. J. Caballero, J. Harjani, and K. Sadarangani, On existence and uniqueness of positive solutions to a class of fractional boundary value problems. Bound. Val. Probl. 2011: 25; http://www.boundaryvalueproblems.com/content/2011/1/25
11. Ph. Clément, G. Gripenberg, and S-O. Londen, Schauder estimates for equations with fractional derivatives. Trans. Amer. Math. Soc. 352 (2000), 2239-2260.
12. S. Das, Functional Fractional Calculus for System Identification and Controls. Springer-Verlag, Berlin, Heidelberg, 2008.
13. S. Das, Functional Fractional Calculus. Springer-Verlag, Berlin, Heidelberg, 2011.
14. Z. Hu and W. Liu, Solvability for fractional order boundary value problems at resonance. Bound. Val. Probl. 2011: 20; http://www. boundaryvalueproblems.com/content/2011/1/20
15. K. Diethelm and N. J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002), 229-248.
16. D. Idczak and R. Kamocki, On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in Rn. Fract. Calc. Appl. Anal. 14, No 4 (2011), 538-553; DOI: 10.2478/s13540-011- 0033-5, http://www. springerlink.com/content/1311-0454/14/4/
17. A.A. Kilbas and S.A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously, differentiable functions. Diff. Eq. 45 (2005), 84-89. (Pubitemid 40551100)
18. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.
19. J. Klafter, S.C. Lim, and R. Metzler, Fractional Dynamics in Physics: Recent Advances. World Scientific, Singapore, 2011.
20. A.N. Kochubei, Fractional differential equations: α-entire solutions, regular and irregular singularities. Fract. Calc. Appl. Anal. 12, No 2 (2009), 135-158, http://www.math.bas.bg/∼fcaa
21. N. Kosmatov, Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Analysis: Th. Meth. Appl. 70 (2009), 2521-2529.
22. M. Kostić, Abstract time-fractional equations: existence and growth of solutions, Fract. Cal. Appl. Anal. 14, No 2 (2011), 301-316; DOI: 10.2478/s13540-011-0018-4, http://www.springerlink.com/content/1311-0454/14/2/
23. C. Li, Z. Zhao, and Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl. 62 (2011), 855-875.
24. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London, 2010.
25. R. Magin, Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl., 59 (2010), 1586-1593.
26. R. Magin, M. Ortigueira, I. Podlubny, and J. J. Trujillo, On the fractional signals and systems. Signal Process., 91 (2011), 350-371.
27. F.C. Merala, T. J. Roystona, and R. Magin, Fractional calculus in viscoelasticity: An experimental study, Commun. Nonlin. Sci. Numer. Sim. 15 (2010), 939-945.
28. B.G. Pachpatte, Inequalities for Differential and Integral Equations. Academic Press, 1998.
29. I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Beijing and Springer-Verlag, Berlin - Heidelberg, 2011.
30. S. Pilipović and M. Stojanović, Fractional differential equations through Laguerre expansions in abstract spaces: Error estimates. Integr. Transf. Spec. Funct. 17 (2006), 877-887. (Pubitemid 44851413)
31. I. Podlubny, Fractional Differential Equations.Academic Press New York, 1999.
32. W. Rui, Existence of solutions of nonlinear fractional differential equations at resonance. Elec. J. Qualit. Th. Diff. Eq. 66 (2011), 1-112.
33. J. Sabatier, O.P. Agrawal, and J.A. Tenreiro-Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, 2007.
34. J. Sabatier, H.C. Nguyen, C. Farges, J.-Y. Deletage, X. Moreau, F. Guillemard, and B. Bavoux, Fractional models for thermal modeling and temperature estimation of a transistor junction. Adv. Differ. Eq. 2011, Article # 687363.
35. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon, 1993.
36. V.E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer-Verlag, Berlin - Heidelberg, 2010.
37. J. A. Tenreiro-Machado, V. Kiryakova, and F. Mainardi, Recent history of fractional calculus. Commun. Nonlin. Sci. Numer. Sim. 16 (2011), 1140-1153; doi:10.1016/j.cnsns.2010.05.027, http://www.sciencedirect.com/science/article/ pii/S1007570410003205
38. J. A. Tenreiro-Machado, Time-delay and fractional derivatives, Adv. Differ. Eq. (2011), Article ID 934094.
39. Y. Tian and Z. Bai, Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comput. Math. Appl. 59 (2010), 2601-2609.
40. Z. Wei, Q. Li, and J. Che, Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. and Appl. 367 (2010), 260-272.
41. S.B. Yuste and L. Acedo, An explicit finite difference method and a new von neumann-type stability analysis for fractional diffusion equations, SIAM J. on Numerical Analysis 42 (2005), 1862-1874. (Pubitemid 41634613)
42. E. Zeidler, Nonlinear Functional Analysis and Applications, I: Fixed Point Theorems. Springer-Verlag, New York, 1986.