Multi-term fractional differential equation in reflexive Banach space

Mathematical and Computer Modelling

Volumn 49, Issue 3-4, 2009, Pages 829-834

  Salem, Hussein A H ^a  

^a Alexandria University Faculty Of Science   (Egypt)

Author keywords

Cauchy problem; Fixed point theorem; Fractional calculus

Indexed keywords

BANACH SPACES; TOPOLOGY;

CAUCHY PROBLEM; FIXED POINT THEOREM; FRACTIONAL CALCULUS; FRACTIONAL DIFFERENTIAL EQUATIONS; WEAK TOPOLOGIES;

FUNCTIONS;

EID: 58049173648     PISSN: 08957177     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mcm.2008.02.002     Document Type: Article

References (31)

1. Arino O., Gautier S., and Penot T. A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations. Funkcial Ekvac. 27 (1984) 273-279
2. Babakhani A., and Daftardar-Gejji V. Existence of positive solutions of nonlinear fractional differential equations. J. Math. Anal. Appl. 278 (2003) 434-442
3. Ball J.M. Weak continuity properties of mapping and semi-groups. Proc. Roy. Soc. Edinburgh Sect. A 72 (1973-1974) 275-280
4. Cichoń M. Weak solutions of ordinary differential equations in Banach spaces. Discuss. Math. Differ Incl. 15 (1995) 5-14
5. Cichoń M., El-Sayed A.M., and Salem H.A.H. Existence theorem for nonlinear functional integral equations of fractional orders. Comment. Math. prace Mat. 41 (2001) 59-67
6. Daftardar-Gejji V. Positive solutions of a system of non-autonomous fractional differential equations. J. Math. Anal. Appl. 302 (2005) 56-64
7. Debanth L. Recent applications of fractional calculus to scinence and engineering. Int. J. Math. Appl. Sci. 54 (2003) 3413-3442
8. Diestel J., and Uhl Jr. J.J. Vector measures, math. surveys 15 (1977), Amer. Math. Soc, Providence, R.I.
9. El-Sayed A.M.A. Linear differential equations of fractional order. Appl. Math. Comput. 55 (1993) 1-12
10. El-Sayed A.M.A. Nonlinear functional differential equations of arbitrary order. Nonlinear Anal. 33 2 (1998) 181-186
11. El-Sayed A.M.A., and Ibrahim A.G. Set-valued integral equations of arbitrary (fractional) order. Appl. Math. Comput. 118 (2001) 113-121
12. Geitz R.F. Pettis integration. Proc. Amer. Math. Soc. 82 (1981) 81-86
13. Hadid S.B. Local and global existence theorem on differential equation on non integral order. Math. Z. 7 (1995) 101-105
14. Hille E., and phillips R.S. Functional Analysis and Semi-groups. Amer. Math. Soc. Colloq. Publ. Vol. 31 (1957), Amer. Math. Soc, Providence, R.I
15. Knight W.J. Solutions of differential equations in B-spaces. Duke Math. J. 41 (1974) 437-442
16. Kubiaczyk I., and Szufla S. Kneser's theorem for weak solutions of differential equations in Banach spaces. In: Lakshmikantham V. (Ed). Nonlinear Equations in Abstract Spaces (1978) 387-404
17. Kilbas A.A., and Trujillo J.J. Differential equations of fractional order: methods, results problems, I. Appl. Anal. 78 1-2 (2001) 153-192
18. Kilbas A.A., Srivastava H.M., and Trujillo J.J. Theory and Applications of Fractional Differential Equations (2006), North-Holland Math. Studies (204) Amsterdam, New York
19. Mitchell A.R., and Smith Ch. An existence theorem for weak solutions of differential equations in Banach spaces. In: Lakshmikantham V. (Ed). Nonlinear Equations in Abstract Spaces (1978) 387-404
20. Luszczi Z., and Rzepecki B. An existence theorem for ordinary differential equations of order α ∈ (0, 1]. Demonste. Math. 20 3-4 (1987) 471-475
21. O'Regan D. Fixed point theory for weakly sequentially continuous mapping. Math. Comput. Modeling 27 5 (1998) 1-14
22. Pettis B.J. On integration in vector spaces. Trans. Amer.Math. Soc. 44 (1938) 277-304
23. Podlubny I. Fractional Differential Equation (1999), Acad. Press, San Diego-New York-London
24. Ross B., Samko S.G., and Love E.R. Functions that have no first order derivative might have fractional derivative of all orders less than one. Real Anal. Exchange 20 (1994/95) 140-157
25. Rossikhin Yu., and Shitikova M.V. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50 1 (1997) 15-67
26. Salem H.A.H. Monotonic solutions of multi term fractional differential equations. Comment. Math. prace Mat. 47 1 (2007) 1-7
27. Salem H.A.H., El-Sayed A.M.A., and Moustafa O.L. Continuous solutions of some nonlinear fractional order integral equations. Comment. Math. prace Mat. 42 (2002) 209-220
28. Salem H.A.H., El-Sayed A.M.A., and Moustafa O.L. A note on the fractional calculus in Banach spaces. Studia Sci. Math. Hungar. 42 2 (2005) 115-113
29. Salem H.A.H., and Väth M. An abstract Gronwall lemma and application to global existence results for functional differential and integral equations of fractional order. J. Integral Equations Appl. 16 (2004) 411-439
30. Samko S., Kilbas A., and Marichev O. Fractional Integrals and Drivatives (1993), Gordon and Breach Science Publisher
31. Szep A. Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces. Studia Sci. Math. Hungar. 6 (1971) 197-203