Existence and uniqueness theorems for nonlinear fractional differential equations

Demonstratio Mathematica

Volumn 33, Issue 3, 2000, Pages 583-602

  Kilbas, A A ^a   Bonilla, B ^b   Trujillo, J J ^b  

^a BELARUSIAN STATE UNIVERSITY   (Belarus)

^b UNIVERSIDAD DE LA LAGUNA   (Spain)

Author keywords

Differential equation of fractional order; Existence and uniqueness theorem; Riemann Liouville fractional derivative

Indexed keywords

EID: 0012415112     PISSN: 04201213     EISSN: 23914661     Source Type: Journal    
DOI: 10.1515/dema-2000-0315     Document Type: Article

References (53)

1. A. Z. Al-Abedeen, Existence theorem on differential equations of generalized order, Rafidain J. Sci. Mosul. Univ. Iraq 1 (1976), 95-104.
2. A. Z. Al-Abedeen and H. L. Arora, A global existence and uniqueness theorem for ordinary differential equation of generalized order, Canad. Math. Bull. 21, No. 3 (1978), 267-271.
3. M. A. Al-Bassam, Some existence theorems on differential equation of generalized order, J. Reine Angew. Math. 218 (1965), 70-78.
4. M. A. Al-Bassam, Existence of series solution of a type of differential equations of generalized order, Bull. College Sci. (Baghdad) 9 (1966), 175-180.
5. M. A. Al-Bassam, On the existence of series solution of differential equations of generalized order, Portugal Math. 29 (1970), 5-11.
6. H. L. Arora and J. G. Alshamani, Stability of differential equation of noninteger order through fixed point in the large, Indian J. Pure Appi. Math. 11, No. 3 (1980), 307-313.
7. J. H. Barrett, Differential equations of non-integer order, Canad. J. Math. 6, No. 4 (1954), 529-541.
8. L. Blank, Numerical Solution of Differential Equations of Fractional Order, Numerical Analysis Report of Dep. Of Math., Manchester Center of Computational Math., March 1996, No. 287, pp. 1-16.
9. L. M. B. C. Campos, On the solution of some simple fractional differential equations, Internat. J. Math. Math. Sci. 13, No. 3 (1990), 481-496.
10. D. Delbosco, Fractional calculus and function spaces, J. Fract. Cale. 6 (1994), 45-53.
11. D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appi. 204, No. 2 (1996), 609-625.
12. K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal. 5 (1997), Mar., pp. 1-6 (electronic).
13. K. Diethelm and G. Walz, Numerical solution of fractional order differential equations by extrapolation, Numer. Algorithms 16, No. 3-4 (1997), 231-253 (1998).
14. M. M. Dzhrbashyan, A boundary value problem for a Sturm-Liouville type differential operator of fractional order (Russian), Izv. Akad. Nauk Armyan. SSR, Ser. Mat. 5, No. 2 (1970), 71-96.
15. M. M. Dzhrbashyan, Harmonic Analysis and Boundary Value Problems in the Complex Domain, Operator theory: Advances and Applications, 65, Birkhauser Verlag, Basel, 1993.
16. M. M. Dzhrbashyan and A. B. Nersesyan, Fractional derivatives and the Cauchy problem for differential equations of fractional order (Russian), Izv. Akad. Nauk Armyan. SSR, Ser. Mat. 3, No. 1 (1968), 3-29.
17. A. M. A. El-Sayed, Fractional differential equations, Kyungpook Math. J. 28, No. 2 (1988), 119-122.
18. A. M. A. El-Sayed, On the fractional differential equations, Appi. Math. Comput. 49, No. 2-3 (1992), 205-213.
19. A. M. A. El-Sayed, Linear differential equations of fractional order, Appi. Math. Comput. 55, no. 1 (1993), 1-12.
20. A. M. A. El-Sayed and F. M. Gaafar, Sturm-Liouville equation of fractional order, Kyungpook Math. J. 37, No. 1 (1997), 9-17.
21. A. M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional differential equations, Appi. Math. Comput. 68, No. 1 (1995), 15-25.
22. A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Runctions, Vol. Ill, McGraw-Hill Boo. Coop., New York, 1954
23. Reprinted Krieger, Melbourne, Florida, 1981.
24. R. G or enfio and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, Fractal and Fractional Calculus in Continuum Mechanics (Udine, 1996), 223-276, CISM Courses and Lectures, 378, Springer, Vienna, 1997.
25. S. B. Hadid, Local and global existence theorems on differential equations of non-integer order, J. Frac. Cale. 7 (1995), 101-105.
26. S. B. Hadid and J. G. Alshamanii, Liapunov stability of differential equations of noninteger order, Arab. J. Math. 7, No. 1-2 (1986), 5-17.
27. S. B. Hadid, B. Masaedeh and S. Momani, On the existence of maximal and minimal solutions of differential equations of non-integer order, J. Fract. Cale. 9 (1996), 41-44.
28. S. B. Hadid, A. A. Ta'ani and S. M. Momani, Some existence theorems on differential equation of generalized order through a fixed-point theorem, J. Fract. Cale. 9 (1996), 45-49.
29. N. Hayek, J. J. Trujillo, M. Rivero, B. Bonilla and J. C. Moreno, An extension of Picard-Lindeloff theorem to fractional differential equations, Appi. Anal. 70, No. 3-4 (1999), 347-361.
30. A. A. Kilbas, B. Bonilla, J. Rodriguez, J. J. Trujillo and M. Rivero, Compositions of fractional integrals and derivatives with Bessel-type function and solution of differential and integral equations (Russian). Doki. Akad. Nauk Belarusi 42, No. 2 (1998), 25-29.
31. A. A. Kilbas and M. Saigo, On solution in closed form of nonlinear integral and differential equations of fractional order. Univalent functions and the Briot-Bouquet differential equations (Japanese) (Kyoto, 1996). Surikaisekikenkyusho Kokyuroku, No. 963 (1996), 39-50.
32. A. A. Kilbas and M. Saigo, Solution in closed form of a class of linear differential equations of of fractional order (Russian), Differ. Uravn. 33, No. 2 (1997), 195-204;
33. translation in Differential Equations 33, No. 2 (1997), 194-204.
34. A. A. Kilbas and M. Saigo, Application of Fractional Calculus to Solve Abel-Volterra Nonlinear and Linear Integral Equations, Transforms methods and special functions, Varna '96, 236-263, Bulgarian Acad. Sci., Sofia, 1998.
35. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Russian), Nauka, Moscow, 1989.
36. I. P. Leskovskii, On the theory of fractional differentiation (Russian), Differ. Uravn. 13, No. 1 (1977), 170-173.
37. Y. Luchko and R. Gorenflo, The initial value problem for some fractional differential equations with the Caputo derivatives, Fachbereich Math, und Inf., Freie Univ. Berlin, No. A8, (1998), pp. 1-23.
38. Y. Luchko and H. M. Srivastava, The exact solution of certain differential equations of fractional order by using operational calculus, Comput. Math. Appi. 29, No. 8 (1995), 73-85.
39. Z. Luszczki and B. Rzepecki, An existence theorem for ordinary differential equations of order a e (0,1], Demonstratio Math. 20, No. 3-4 (1987), 471-475.
40. F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, Fractal and fractional calculus in continuum mechanics (Udine, 1996), 291-348, CISM Courses and Lectures, 378, Springer, Vienna, 1997.
41. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Lohn Wiley & Sons, New York, 1993.
42. A. M. Nakhushev, The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms (Russian), Dokl. Akad. Nauk SSSR 234, No. 2 (1977), 308-311.
43. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
44. E. Pitcher and W. E. Sewell, Existence theorem for solutions of differential equations of non-integral order, Bull. Amer. Math. Soc. 44, No. 2 (1938), 100-107 (A correction in No. 12, p. 888).
45. I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solutions and Some of Their Applications. Mathematics in Sciences and Engineering. 198, Academic Press, San-Diego, 1999.
46. M. Saigo and A. A. Kilbas, On Mittag-Leffler type function and applications, Integral Transform. Spec. Funct. 7, No. 1-2 (1998), 97-112.
47. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, New York, 1993.
48. N. P. Semenchuk, A class of differential equations of non-integer order (Russian), Differ. Uravn. 18, No. 10 (1982), 1831-1833.
49. M. Kh. Shkhanukov, On the convergence of difference schemes for diferential equations with a fractional derivative (Russian), Doki. Akad. Nauk 348, No. 6 (1996), 746-748.
50. H. M. Srivastava, S. Owaand K. Nishimoto, Some fractional differintegral equations, J. Math. Anal. Appi. 106, No. 2 (1985), 360-366.
51. A. Z.-A. M. Tazali, Local existence theorems for ordinary differential equations of fractional order, Lecture Notes in Math. 964 (1982), 652-665.
52. A. Z. M. Tazali and I. F. Karim, Continuous dependence and a parameter and continuation of solutions in the extended sense in ordinary differential equations of fractional order, Dirâsât Ser. B Pure Appi. Sci. 21, No. 1 (1994), 270-280.
53. K. Wiener, On solutions of a differential equation of nonintegral order that occurs in the theory of polarography (German), Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 32, No. 1 (1983), 41-46.