ON THE EXISTENCE INTERVAL FOR THE INITIAL VALUE PROBLEM OF A FRACTIONAL DIFFERENTIAL EQUATION

Abstract

ON THE EXISTENCE INTERVAL FOR THE INITIAL VALUE PROBLEM OF A FRACTIONAL DIFFERENTIAL EQUATION

Abstract

We compute via a comparison function technique, a new bound for the existence interval of the initial value problem for a fractional differential equation given by means of Caputo derivatives. We improve in this way the estimate of the existence interval obtained very recently in the literature.

Keywords

• Fractional differential equation,Leray-Schauder alternative,Local existence,Estimate of existence interval,2000 AMS Classification: 34 A 12

References

1. Agrawal, O. P. Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl. 272, 368–379, 2002.
2. Ahlfors, L. V. Complex Analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable(McGraw-Hill, New York, 1979).
3. Atanackovi¸c, T. M., Konjik, S. and Pipilovi¸c, S. Variational problems with fractional deriva- tives: Euler-Lagrange equations, J. Phys. A 41 (9): Art. No. 095201, 2008
4. B˘aleanu, D., Muslih, S. I. and Rabei, E. M. On fractional Euler-Lagrange and Hamilton equations and the fractional generalization of total time derivative, Nonlin. Dyn. 53, 67–74, 2008.
5. B˘aleanu, D. New applications of fractional variational principles, Rep. Math. Phys. 61, 331–335, 2008.
6. B˘aleanu, D. and Trujillo, J. J. On exact solutions of a class of fractional Euler-Lagrange equations, Nonlin. Dyn. 52, 199–206, 2008.
7. Caputo, M. Linear models of dissipation whose Q is almost frequency indepedent, II, Geo- phys. J. Roy. Astronom. Soc. 13, 529–539, 1967.
8. Chen, W. An iterative learning observer for fault detection and accommodation in nonlinear time-delay systems, Internat. J. Robust Nonlin. Control 16 (1) DOI 10.1002/rnc.1033, 2006. [9] Delbosco, D. and Rodino, L. Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204, 609–625, 1996.

9. Deng, W., Li, C. and L¨u, J. Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dyn. 48, 409–416, 2007.
10. Diethelm, K. and Ford, N. J. Analysis of fractional differential equations, J. Math. Anal. Appl. 265, 229–248, 2002.
11. Diethelm, K. The analysis of Fractional Differential Equations (Springer-Verlag, Berlin, 2010).
12. Dugundji, J. and Granas, A. Fixed point theory I (Monogr. Matem. 61, PWN, Warszawa, 1982).
13. Gl¨ocke, W. G. and Nonnenmacher, T. F. A fractional calculus approach to self-similar pro- tein dynamics, Biophys. J. 68, 46–53, 1995.
14. Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. Theory and applications of fractional differential equations(North-Holland Math. Stud. 204, Elsevier, Amsterdam, 2006).
15. Klimek, M. Lagrangian and Hamiltonian fractional sequential mechanics, Czech. J. Phys. 52, 1247–1252, 2002.
16. Lakshmikantham, V. and Vatsala, A. S. Basic theory of fractional differential equations, Nonlinear Anal. TMA 69, 2677–2682, 2008.
17. Machado, J. A. Tenreiro A probabilistic interpretation of the fractional-order differentiation, Frac. Calc. Appl. Anal. 8, 73–80, 2003.
18. Magin, R. L. Fractional calculus in bioengineering (Begell House Publ., Inc., Connecticut, 2006).
19. Magin, R. L., Abdullah, O., B˘aleanu, D. and Xiaohong, J. Z. Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, J. Mag. Res. 190, 255–270, 2008.
20. Mainardi, F., Luchko, Yu. and Pagnini, G. ??????, Frac. Calc. Appl. Anal. 4, 153–161, 2001.
21. Metzler, R., Schick, W., Kilian, H. G. and Nonennmacher, T. F. Relaxation in filled poly- mers: A fractional calculus approach, J. Chem. Phys 103, 7180–7186, 1995.
22. Momani, S. and Odibat, Z. A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor’s formula, J. Comput. Appl. Math. 220(2008), 85–95, 2008.
23. Momani, S. and Odibat, Z. Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A 365, 345–350, 2007.
24. Muslih, S. I. and B˘aleanu, D. Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives, J. Math. Anal. Appl. 304, 599–606, 2005. [26] Mustafa, O. G. and Rogovchenko, Yu. V. Estimates for domains of local invertibility of diffeomorphisms, Proc. Amer. Math. Soc. 135, 69–75, 2007.
25. Mustafa, O. G. On the existence interval in Peano’s theorem, Ann. A.I. Cuza Univ. Ser. Math., Ia¸si, Romania, LI, 55–64, 2005.
26. Podlubny, I. Fractional Differential Equations (Academic Press, San Diego, 1999).
27. Sabatier, J., Agrawal, O. P. and Machado, J. A. Tenreiro (eds.) (Advances in fractional calculus. Theoretical developments and applications in physics and engineering, Springer- Verlag, Dordrecht, 2007).
28. Samko, S. G., Kilbas, A. A. and Marichev, O. I. Fractional Integrals and Derivatives. Theory and Applications(Gordon and Breach, Switzerland, 1993).