Existence and uniqueness of the solution of nonlinear fuzzy Volterra integral equations

Document Type: Research Paper

Authors

1 Department of mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

2 Department of mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran

Abstract

In this paper the fixed point theorem of Schauder is used to prove the existence of a continuous solution of the nonlinear fuzzy Volterra integral equations. Then using some conditions the uniqueness of the solution is investigated.

Keywords


[1] G. A. Anastassiou and S. G. Gal, On a fuzzy trigonometric approximation theorem of
Weierstrass-type, J. Fuzzy Math., 9 (3) (2001), 701-708.
[2] K. Balachandran and K. Kanagarajan, Existence of solutions of general nonlinear fuzzy
Volterra-Fredholm integral equations, J. Appl. Math. Stochast. Anal., 3 (2005), 333-343.
[3] K. Balachandran and P. Prakash, Existence of solutions of nonlinear fuzzy Volterra-Fredholm
integral equations, Indian J. Pure Appl. Math., 33 (3) (2002), 329-343.
[4] K. Balachandran and P. Prakash, Existence of solutions of nonlinear fuzzy integral equations
in Banach spaces, Libertas Math., 21 (2001), 91-97.
[5] J. J. Buckley and T. Feuring, Fuzzy integral equations, J. Fuzzy Math., 10 (2002), 1011-1024.
[6] D. Dubois and H. Prade, Fundamentals of fuzzy sets, Springer Netherlands Publisher, 2000.
[7] M. Friedman, M. Ma and A. Kandel, On fuzzy integral equations, Fundam. Inform., 37
(1999), 89-99.

[8] S. G. Gal, Approximation theory in fuzzy setting, in: G. A. Anastassiou (Ed.), Handbook
of Analytic-Computational Methods in applied Mathematics, Chapman & Hall, CRC Press,
Boca Raton, London, New York, Washington DC, (2000), (Chapter 13).
[9] D. N. Georgiou and I. E. Kougias, On fuzzy Fredholm and Volterra integral equations, J.
Fuzzy Math., 9 (4) (2001), 943-951.
[10] D. N. Georgiou and I. E. Kougias, Bounded solutions for fuzzy integral equation, Int. J. Math.
Math. Sci., 31 (2) (2002), 109-114.
[11] R. Goetschel and W. Voxman, Elementary fuzzy calculus, Fuzzy Sets Syst., 18 (1986), 31-43.
[12] A. Karoui and A. Jawahdou, Existence and approximate Lp and continuous solutions of
nonlinear integral equations of the Hammerstein and Volterra types, Appl. Math. Comput.,
216 (2010), 2077-2091.
[13] J. Mordeson and W. Newman, Fuzzy integral equations, Inform. Sci., 81 (4) (1995), 215-229.
[14] J. J. Nieto and R. Rodriguez-lopez, Bounded solutions for fuzzy di erential and integral
equations, Chaos Solitons & Fractals, 27 (5) (2006), 1376-1386.
[15] J. Y. Park and J. U. Jeong, On the existence and uniquness of solutions of fuzzy Volterra-
Fredholm integral equation, Fuzzy Sets Syst., 115 (2000), 425-431.
[16] J. Y. Park and J. U. Jeong, A note on fuzzy integral equations, Fuzzy Sets Syst., 108 (1999),
193-200.
[17] J. Y. Park, Y. C. Kwun and J. U. Jeong, Existence of solutions of fuzzy integral equations
in Banach spaces, Fuzzy Sets Syst., 72 (1995), 373-378.
[18] J. Y. Park, S. Y. Lee and J. U. Jeong, The approximate solutions of fuzzy functional integral
equation, Fuzzy Sets Syst., 110 (2000), 79-90.
[19] C. Wu and Z. Gong, On Henstock integral of fuzzy-number-valueed functions, Fuzzy Sets
Syst., 120 (2001), 523-532.