| Title | PERTURBED FUNCTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL ORDER |
| Publication Type | Journal Article |
| Year of Publication | 2007 |
| Authors | BELARBI, ABDELKADER, BENCHOHRA, MOUFFAK, HAMANI, SAMIRA, NTOUYAS, SOTIRISK |
| Secondary Title | Communications in Applied Analysis |
| Volume | 11 |
| Issue | 4 |
| Start Page | 429 |
| Pagination | 440 |
| Date Published | 12/2007 |
| Type of Work | scientific: mathematics |
| ISSN | 1083–2564 |
| AMS | 26A33, 34K05 |
| Abstract | In this paper, we shall establish sufficient conditions for the existence of solutions, as well as extremal solutions, for perturbed fractional functional differential equations.
|
| URL | http://www.acadsol.eu/en/articles/11/4/1.pdf |
| Short Title | PERTURBED FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATIONS |
| Refereed Designation | Refereed |
| Full Text | REFERENCES[1] T. A. Burton and C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr. 189 (1998), 23–31.
[2] D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609–625. [3] B. C. Dhage and J. Henderson, Existence theory for nonlinear functional boundary value problems, Electron. J. Qual. Theory Differ. Equ. 2004, No. 1, 15 pp. [4] K. Diethelm and A.D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in “Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties” (F. Keil, W. Mackens, H. Voss, and J. Werther, Eds), pp 217–224, Springer-Verlag, Heidelberg, 1999. [5] K. Diethelm and N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002), 229–248.
[6] K. Diethelm and G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms 16 (1997), 231–253. [7] A. M. A. El-Sayed, Fractional order evolution equations, J. Fract. Calc. 7 (1995), 89–100. [8] A. M. A. El-Sayed, Fractional order diffusion-wave equations, Intern. J. Theo. Physics 35 (1996), 311–322. [9] A. M. A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal. 33 (1998), 181–186. [10] A. M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional differential equations, Appl. Math. Comput. 68 (1995), 15–25. [11] L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (1991), 81–88. [12] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. [13] W. G. Glockle and T. F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995), 46–53. [14] J. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. [15] S. Heikkila and V. Lakshmikantham, Monotone Iterative Technique for Nonlinear Discontinuous Differential Equations, Marcel Dekker Inc., New York, 1994. [16] D. Henry, Geometric Theory of Semilinear Parabolic Partial Differential Equations, Springer-Verlag, Berlin/New York, 1989. [17] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. [18] A. A. Kilbas and S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differential Equations 41 (2005), 84–89. [19] A. A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. [20] V. Kolmanovskii, and A. Myshkis, Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and its Applications, 463. Kluwer Academic Publishers, Dordrecht, 1999. [21] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in “Fractals and Fractional Calculus in Continuum Mechanics” (A. Carpinteri and F. Mainardi, Eds), pp. 291–348, Springer-Verlag, Wien, 1997. [22] F. Metzler, W. Schick, H. G. Kilian and T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995), 7180–7186. [23] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. [24] S. M. Momani and S. B. Hadid, Some comparison results for integro-fractional differential inequalities. J. Fract. Calc. 24 (2003), 37–44. [25] S. M. Momani, S. B. Hadid and Z. M. Alawenh, Some analytical properties of solutions of differential equations of noninteger order, Int. J. Math. Math. Sci. 2004 (2004), 697–701. [26] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974. [27] I. Podlubny, Fractional Differential Equations, Academic Press, Singapore, 1999.
[28] I. Podlubny, I. Petraˇs, B. M. Vinagre, P. O’Leary and L. Dorˇcak, Analogue realizations of fractional-order controllers. Fractional order calculus and its applications, Nonlinear Dynam. 29 (2002), 281–296. [29] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993. [30] C. Yu and G. Gao, Existence of fractional differential equations, J. Math. Anal. Appl. 310 (2005), 26–29. |