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On the existence and uniqueness of solutions of fuzzy Volterra-Fredholm integral equations. (English) Zbl 0966.45001
Fuzzy integral equations were introduced by {\it H. Y. Chen} [J. Math. Anal. Appl. 80, 19-30 (1981; Zbl 0506.45014)] as the method of solution of fuzzy differential equations [cf. also {\it D. Dubois} and {\it H. Prade}, Fuzzy Sets Syst. 8, 105-116 (1982; Zbl 0493.28003); {\it R. Goetschel jun.} and {\it W. Voxman}, ibid. 18, 31-43 (1986; Zbl 0626.26014); {\it O. Kaleva}, ibid. 24, 301-317 (1987; Zbl 0646.34019)]. This paper contains the existence and uniqueness theorem for the fuzzy version of Volterra-Fredholm integral equation in $\bbfR^n$ [cf. {\it P. V. Subrahmanyam} and {\it S. K. Sudarsanam}, ibid. 81, No. 2, 237-240 (1996; Zbl 0884.45002); {\it M. Friedman, M. Ming} and {\it A. Kandel}, Fundam. Inform. 37, No. 1-2, 89-99 (1999; Zbl 0934.45011)]. A verification of its assumptions is difficult (two pages of assumptions without examples).

MSC:
45B05Fredholm integral equations
45D05Volterra integral equations
46S40Fuzzy functional analysis
03E72Fuzzy set theory
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Full Text: DOI
References:
[1] Dubois, D.; Prade, H.: Towards fuzzy differential calculus. Part 1. Fuzzy sets and systems 8, 1-17 (1982) · Zbl 0493.28002
[2] Dubois, D.; Prade, H.: Towards fuzzy differential calculus. Part 2. Fuzzy sets and systems 8, 105-116 (1982) · Zbl 0493.28003
[3] Kaleva, O.: Fuzzy differential equations. Fuzzy sets and systems 24, 301-317 (1987) · Zbl 0646.34019
[4] Puri, M. L.; Ralescu, D. A.: Differentials of fuzzy functions. J. math. Anal. appl. 91, 552-558 (1983) · Zbl 0528.54009
[5] Puri, M. L.; Relescu, D. A.: Fuzzy random variables. J. math. Anal. appl. 114, 409-422 (1986) · Zbl 0592.60004
[6] Seikkala, S.: On the fuzzy initial value problem. Fuzzy sets and systems 24, 319-330 (1987) · Zbl 0643.34005