Fuzzy fractional integral equations under compactness type condition

Fractional Calculus and Applied Analysis

Volumn 15, Issue 4, 2012, Pages 572-590

  Agarwal, Ravi P ^a   Arshad, Sadia ^b   O'Regan, Donal ^c   Lupulescu, Vasile ^b,d  

^a TEXAS A AND M UNIVERSITY KINGSVILLE   (United States)

^b GC UNIVERSITY   (Pakistan)

^c NATIONAL UNIVERSITY OF IRELAND   (Ireland)

^d Universitatea Constantin Brancusi din Targu Jiu   (Romania)

Author keywords

Existence of solution; Fractional calculus; Fuzzy fractional integral equation; Measure of noncompactness

Indexed keywords

EID: 84869161109     PISSN: 13110454     EISSN: 13142444     Source Type: Journal    
DOI: 10.2478/s13540-012-0040-1     Document Type: Article

References (37)

1. R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 74 (2010), 2859-2862.
2. R.P. Agarwal, M. Belmekki, M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative. Adv. Difference Equ. (2009) Article ID 981728, 47pp.
3. R.P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundry value problems for nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109 (2010), 973-1033.
4. R.P. Agarwal, Y. Zhou, J. Wang, X. Luo, Fractional functional differential equations with causal operators in Banach spaces. Mathematical and Compututer Modelling 54 (2011), 1440-1452.
5. R. Alikhani, F. Bahrami, A. Jabbari, Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations. Nonlinear Anal. 75 (2012), 1810-1821.
6. T. Allahviranloo, S. Salahshour, S. Abbasbandy, Explicit solutions of fractional differential equations with uncertainty. Soft Computing 16, No 2 (2012), 297-302.
7. S. Arshad, V. Lupulescu, On the fractional differential equations with uncertainty. Nonlinear Anal. 74 (2011), 3685-3693.
8. S. Arshad, V. Lupulescu, Fractional differential equation with fuzzy initial condition. Electronic J. of Differential Equations 2011 (2011), 1-8.
9. M. Benchohra, M. A. Darwish, Existence and uniqueness theorem for fuzzy integral equations of fractional order. Communications in Applied Analysis 12 (2008), 13-22.
10. W. Congxin, S. Shiji, Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-Type conditions. Inform. Sci. 108 (1998), 123-134.
11. K. Deimling, Nonlinear Functional Analysis. Springer-Verlag, Berlin-Heidelberg, 1985.
12. P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets. World Scientific, Singapore, 1994.
13. K. Diethelm, An efficient parallel algorithm for the numerical solution of fractional differential equations. Fract. Calc. and Appl. Anal. 14, No 3 (2011), 475-490; DOI: 10.2478/s13540-011-0029-1; at http://www.springerlink.com/ content/1311-0454/14/3/.
14. K. Diethelm, The Analysis of Fractional Differential Equations. Springer, 2004.
15. K. Diethelm, The mean value theorems and a Nagumo-Type uniqueness theorem for Caputo's fractional calculus. Fract. Calc. and Appl. Anal. 15, No 2 (2012), 304-313; DOI: 10.2478/s13540-012-0022-3; at http://www.springerlink.com/ content/1311-0454/15/2/.
16. T. Donchev, A. Nosheen, On the solution set of fuzzy systems. In: Nonlinear Anal., 2012.
17. A.M.A. El-Sayed, A.-G. Ibrahim, Set-valued integral equations of fractional-orders. Applied Mathematics and Computation 118 (2001), 113-121.
18. S.R. Grace, R.P. Agarwal, P.J.Y. Wong and A. Zafer, On the oscillation of fractional differential equations. Fract. Calc. and Appl. Anal. 15, No 2 (2012), 222-231; DOI: 10.2478/s13540-012-0016-1; at http://www.springerlink.com/ content/1311-0454/15/2/.
19. S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory. Kluwer, Dordrecht, 1997.
20. O. Kaleva, The Cauchy problem for fuzzy differential equations. Fuzzy Sets and Systems 35 (1990), 389-396.
21. D. Kandilakis, N.S. Papageorgiou, On the properties of the Aumann integral with applications to differential inclusions and control systems. Czech. Math. Journ. 39 (1989), 1-15.
22. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. vol. 204 of North-Holland Mathematics Studies, Elsevier, New York, 2006.
23. M. Kisielewicz, Multivalued differential equations in separable Banach spaces. J. Opt. Theory Appl. 37 (1982), 231-249.
24. V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations. Nonlinear Anal. 69 (2008), 2677-2682.
25. V. Lakshmikantham, R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions. Taylor & Francis, London, 2003.
26. V. Lakshmikantham, S. Leela, Nonlinear Differential Equations in Abstract Spaces. Pergamon Press, New York, 1969.
27. V. Lupulescu, Causal functional differential equations in Banach spaces. Nonlinear Anal. 69 (2008), 4787-4795.
28. M.T. Malinowski, Random fuzzy differential equations under generalized Lipschitz condition. Nonlinear Analysis: Real World Applications 13, No 2 (2012), 860-881.
29. K.S. Miller, B. Ross, An introduction to Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993.
30. C.V. Negoita, D. Ralescu, Applications of Fuzzy Sets to Systems Analysis. Wiley, New York, 1975.
31. K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Application of Differentiation and Integration to an arbitrary order. Academic Press, New York-London, 1974.
32. N.S. Papageorgiou, Existence of solutions for integrodifferential inclusions in Banach spaces. Commentationes Mathematicae Universitatis Carolinae 32, No 4 (1991), 687-696.
33. J.Y. Park, H. K. Han, Existence and uniqueness theorem for a solution of fuzzy Volterra integral equations. Fuzzy Sets and Systems 105 (1999), 481-488.
34. M. Puri, D. Ralescu, Fuzzy random variables. J. Math. Anal. Appl. 114 (1986), 409-422.
35. S. Salahshour, T. Allahviranloo, S. Abbasbandy, Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun. Nonlinear Sci. Numer. Simulat. 17 (2012), 1372-1381.
36. S. Song, Q. Liu, Q. Xu, Existence and comparison theorems to Volterra fuzzy integral equations in (En,D). Fuzzy Sets and Systems 104 (1999), 315-321.
37. H. Wang, Y. Liu, Existence results for fuzzy integral equations of fractional order. Int. Journal of Math. Analysis 5 (2011), 811-818.