A nonlocal boundary value problem for a nonlinear fractional differential equation with two indices

Communications on Applied Nonlinear Analysis

Volumn 17, Issue 3, 2010, Pages 69-80

  Ahmad, Bashir ^a   Eloe, Paul ^b  

^a KING ABDULAZIZ UNIVERSITY   (Saudi Arabia)

^b UNIVERSITY OF DAYTON   (United States)

Author keywords

Differential equations of fractional order; Existence of solutions; Fixed point theorem; Nonlocal boundary conditions

Indexed keywords

EID: 78649793054     PISSN: 1074133X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (27)

1. B. Ahmad, Existence of solutions for irregular boundary value problems involving nonlinear fractional differential equations, Appl. Math. Lett. 23 (2010), 390-394.
2. B. Ahmad, Existence of solutions for fractional differential equations of order q ⋯ (2,3] with anti-periodic boundary conditions, J. Appl. Math. Comput. (in press) doi: 10.1007/s12190-009-0328-4.
3. B. Ahmad, John R. Graef, Coupled systems of nonlinear fractional differential equations with nonlocal boundary conditions, Panamer. Math. J. 19 (2009), 29-39.
4. B. Ahmad, J.J. Nieto, Existence of solutions for nonlocal boundary value problems of higher order nonlinear fractional differential equations, Abstr. Appl. Anal. (2009), Art. ID 494720, 9 pp.
5. B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 58 (2009), 1838-1843.
6. B. Ahmad, J.J. Nieto, Solvability of nonlinear Langevin equation involving two fractional orders with Dirichlet boundary conditions, Int. J. Differ. Equ. (2010), Art. ID 649486, 10 pp.
7. B. Ahmad, S. Sivasundaram, Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations, Nonlinear Anal. Hybrid Syst. 3 (2009), 251-258.
8. A. Arara, M. Benchohra, N. Hamidi, J.J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Anal. 72 (2010), 580-586.
9. F.M. Atici, P.W. Eloe, Fractional q-calculus on a time scale, J. Nonlinear Math. Phys. 14 (2007), 333-344.
10. F.M. Atici, P.W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), 981-989.
11. D. Baleanu, A.K. Golmankhaneh, A. K. Golmankhaneh, Fractional Nambu mechanics, Int. J. Theor. Phys. 48 (2009), 1044-1052.
12. S. Bhalekar, V. Daftardar-Gejji Boundary value problems for multi-term fractional differential equations, J. Math. Anal. Appl. 345 (2008), 754-765.
13. B. Bonilla, M. Rivero, L. Rodriguez-Germa, J.J. Trujillo, Fractional differential equations as alternative models to nonlinear differential equations, Appl. Math. Comput. 187 (2007), 79-88.
14. W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation, second ed. World Scientific, Singapore, 2004.
15. B. Datsko, V. Gafiychuk, V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J. Comput. Appl. Math. 220 (2008), 215-225.
16. J.V. Devi, V. Lakshmikantham, S. Leela, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.
17. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
18. V. Lakshmikantham, A.S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett. 21 (2008), 828-834.
19. M. P. Lazarevic, A. M. Spasic, Finite-time stability analysis of fractional order time delay systems: Gronwall's approach, Math. Comput. Modelling 49 (2009), 475-481.
20. S.C. Lim, M. Li, L.P. Teo, Langevin equation with two fractional orders, Phys. Lett. A 372 (2008), 6309-6320.
21. S.C. Lim, L.P. Teo, The fractional oscillator process with two indices, J. Physics A: Math. Theor. 42 (2009), 065208 (34 pages).
22. G.M. Mophou, Existence and uniqueness of mild solutions to impulsive fractional differential equations, Nonlinear Anal. 72 (2010), 1604-1615.
23. G. M. N'Guerekata, A Cauchy problem for some fractional abstract differential equation with non local conditions, Nonlinear Anal. 70 (2009), 1873-1876.
24. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
25. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
26. D.R. Smart, Fixed Point Theorems, Cambridge University Press, 1980.
27. M.R. Ubriaco, Entropies based on fractional calculus, Phys. Lett. A 373 (2009), 25162519.