An abstract Gronwall lemma and applications to global existence results for functional differential and integral equations of fractional order

Journal of Integral Equations and Applications

Volumn 16, Issue 4, 2004, Pages 411-439

  Salem, Hussein A H ^a   V̈ath, Martin ^b  

^a Alexandria University Faculty Of Science   (Egypt)

^b UNIVERSITY OF WÜRZBURG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 20444421082     PISSN: 08973962     EISSN: None     Source Type: Journal    
DOI: 10.1216/jiea/1181075299     Document Type: Article

References (31)

1. A.Z. Al-Abedeen and H.L. Arora, A global existence and uniqueness theorem for ordinary differential equations of generalized order, Canad. Math. Bull. 21 (1978), 267-271.
2. M.A. Al-Bassam, Some existence theorems on differential equations of generalized order, J. Reine Angew. Math. 218 (1965), 70-78.
3. J. Appell, E. de Pascale, H.T. Nguyêñ and P.P. Zabreiko, Multi-valued superpositions, Diss. Math. (Rozprawy Mat.) 345, Polish Scientific Publ.,Warszawa, 1995.
4. J. Appell, H.T. Nguyêñ and P.P. Zabreiko, Multivalued superposition operators in ideal spaces of vector functions II, Indag. Math. 2 (1991), 397-409.
5. J. Appell and M. V̈ath, The space of Carath'eodory functions, Trudy Inst. Mat. NAN Belarus, Minsk 2 (1999), 39-43.
6. J. Appell, Misure di non compattezza di insiemi ed operatori integrali in spazi di funzioni misurabili, Rend. Mat. Appl. 21 (2001), 159-190.
7. M. Cicho'n, A.MA. El-Sayed and H.A.H. Salem, Existence theorem for nonlinear functional integral equations of fractional order, Comment. Math. Prace Mat. 41 (2001), 59-67.
8. K. Deimling, Nonlinear functional analysis, Springer, Berlin, 1985.
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, W.G. El-Sayed and O.L. Moustafa, On some fractional functional equations, Pure Math. Appl. 6 (1995), 321-332.
11. A.M.A. El-Sayed and F.M. Gaafar, Exact solution of linear differential equations of arbitrary order, Kyungpook Math. J. 37 (1997), 181-189.
12. A.M.A. El-Sayed and A.G. Ibrahim, Set-valued integral equations of arbitrary (fractional) order, Appl. Math. Comput. 118 (2001), 113-121.
13. R. Gorenflo and S. Vessella, Abel integral equations, Lecture Notes in Math., vol. 1461, Springer, Berlin, 1991.
14. S.B. Hadid, Local and global existence theorem on differential equations of non-integer order, Math. Z. 7 (1995), 101-105.
15. G.H. Hardy and J.E. Littlewood, Some properties of fractional integrals, Math. Z. 27 (1928), 565-606.
16. C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72.
17. Y. Katznelson, An introduction to harmonic analysis, John Wiley & Sons, New York, 1968.
18. M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik and P.E. Sobolevskii, Integral operators in spaces of summable functions, Nauka, Moscow, 1958 (in Russian); Noordhoff, Leyden 1976 (in English).
19. K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. S'er. Sci. Math. 13 (1965), 397-403.
20. W.A.J. Luxemburg and A.C. Zaanen, Compactness of integral operators in Banach function spaces, Math. Ann. 149 (1963), 150-180.
21. E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 361-381.
22. I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
23. I. Podlubny and A.M.A. El-Sayed, On two definitions of fractional calculus, Preprint uef-03-96 (isbn 80-7099-252-2), Slovak Academy of Sciences, Institute of Experimental Phy., 1996.
24. W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, Singapore, 1987.
25. H.A.H. Salem, A.M.A. El-Sayed and O.L. Moustafa, Continuous solutions of some nonlinear fractional order integral equations, Comment. Math. Prace Mat. 42 (2002), 209-220.
26. M. V̈ath, Ideal spaces, Lecture Notes in Math. 1664, Springer, Berlin, 1997.
27. M. V̈ath, Abstract Volterra equations of the second kind, J. Integral Equations Appl. 10 (1998), 319-362.
28. M. V̈ath, Complete continuity of some nonlinear Volterra operator in Banach spaces, Z. Anal. Anwendungen 17 (1998), 23-35.
29. M. V̈ath, Linear and nonlinear abstract Volterra equations, Functional Differential Equations 5 (1998), 499-512.
30. M. V̈ath, Volterra and integral equations of vector functions, Marcel Dekker, New York, 2000.
31. M. V̈ath, Integration theory. A second course, World Scientific Publishers, Singapore, 2002.