EXISTENCE AND UNIQUENESS THEOREM FOR FUZZY INTEGRAL EQUATION OF FRACTIONAL ORDER

TitleEXISTENCE AND UNIQUENESS THEOREM FOR FUZZY INTEGRAL EQUATION OF FRACTIONAL ORDER
Publication TypeJournal Article
Year of Publication2008
AuthorsBENCHOHRA, MOUFFAK, DARWISH, MOHAMEDABDALLA
Volume12
Issue1
Start Page13
Pagination9
Date Published2008
ISSN1083-2564
AMS99Z00
Abstract

We present an existence and uniqueness theorem for integral equations of fractional order involving fuzzy set valued mappings of a real variable whose values are normal, convex, upper semicontinuous and compactly supported fuzzy sets in ${ \mathbb{R}^n}$. The method of successive approximation is the main tool in our analysis.

URLhttp://www.acadsol.eu/en/articles/12/1/2.pdf
Refereed DesignationRefereed
Full Text

REFERENCES
[1] R.P. Agarwal, M. Benchohra, D. O’Regan and A. Ouahab, Fuzzy solutions for multi-point
boundary value problems, Mem. Differential Equations Math. Phys. 35 (2005), 1–14.
[2] R.P. Agarwal, D. O’Regan, and V. Lakshmikantham, Fuzzy Volterra integral equations: a
stacking theorem approach, Appl. Anal. 83 (5) (2004), 521–532.
[3] A. Arara and M. Benchohra, Fuzzy solutions for boundary value problems with integral boundary
conditions, Acta Math. Univ. Comenianae LXXV (1) (2006), 119–126.
[4] M.A. Darwish, On existence of fuzzy integral equation of Urysohn-Volterra type, Discussiones
Mathematicae Differential Inclusions Control and Optimization 28 (2008).
[5] M.A. Darwish, On maximal and minimal solutions of fuzzy integral equation of Urysohn type,
Intern. J. Math. Anal. (to appear).
[6] M.A. Darwish and A.A. El-Bary, Existence of fractional integral equation with hysteresis,
Appl. Math. Comput. 176 (2006), 684–687.
[7] M.A. Darwish, On quadratic integral equation of fractional orders, J. Math. Anal. Appl. 311
(2005), 112–119.
[8] D. Dubois and H. Prade, Towards fuzzy differential calculus, Part 1. Integraation of fuzzy
mappings, Fuzzy Sets and Systems 8 (1982), 1–17.
[9] D. Dubois and H. Prade, Towards fuzzy differential calculus, Part 2. Integraation of fuzzy
mappings, Fuzzy Sets and Systems 8 (1982), 105–116.
[10] M. Friedman, Ma Ming and A. Kandel, Solutions to fuzzy integral equations with arbitrary
kernels, Internat. J. Approx. Reason. 20 (3) (1999), 249–262.
[11] R. Goetschel and W. Voxman, Elementary Calculus, Fuzzy Sets and Systems 18 (1986), 31–43.
[12] R. Hilfer, Applications of Fractional calculus in Physics, World Scientific, Singapore, 2000.
[13] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987), 301–317.
[14] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991.
[15] V. Lakshmikantham and R. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions,
Taylor & Francis, New York, 2003.
[16] J. Mordeson and W. Newman, Fuzzy integral equations, Information Sciences 87 (1995), 215–229.
[17] S. Nanda, On integration of fuzzy mappings, Fuzzy Sets and Systems 32 (1989), 95–101.
[18] K.B. Oldham and J. Spanier, The Fractional Calculus: Theory and applications of Differentiation
and Integration to Arbitary Order, Academic Press, New York and London, 1974.
[19] J. Y. Park and J.U. Jeong, A note on fuzzy integral equations, Fuzzy Sets and Systems 108
(1999), 193–200.
[20] I. Podlubny, Fractional Differential Equations, Academic Press, New York and London, 1999.
[21] M. L. Puri and D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986), 409–422.
[22] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and
Applications, Gordon and Breach Science Publs., Amsterdam, 1993. [Russian Edition 1987]
[23] P. V. Subrahmanyam and S.K. Sudarsanam, A note on fuzzy Volterra integral equations, Fuzzy
Sets and Systems 81 (1996), 237–240.