Some fixed point theorems for fuzzy mappings. (English) Zbl 0685.54030
Let X be a linear metric space and let W denote the family of normal, convex, upper semicontinuous, compactly supported fuzzy sets defined on X. Suppose F is a fuzzy mapping, i.e. F:X\(\to W\). Then a \(z\in X\) is a fixed point of F if \(\{\) \(z\}\) \(\subset F(z)\), that is \(F(z)(z)=1.\)
The authors extend some fixed point theorems for multivalued mappings into a fuzzy setting. For related results see also R. K. Bose and D. Sahani [Fuzzy Sets Syst. 21, 53-58 (1987; Zbl 0609.54032)], D. Butnariu [Fuzzy Sets Syst. 7, 191-207 (1982; Zbl 0473.90087)] and S. Heilpern [J. Math. Anal. Appl. 83, 566-569 (1981; Zbl 0486.54006)].
Recently A. Chitra and P. V. Subrahmanyam [J. Math. Anal. Appl. 124, 584-590 (1987; Zbl 0628.47035)] introduced an elegant method for proving fuzzy fixed point theorems: define a multivalued mapping \(g(x)=F_ 1(x)\), where \(F_ 1(x)\) denotes the l-level set of F(x), and apply a corresponding fixed point theorem to the multivalued mapping g.
In literature there also appear different kinds of fixed point theorems related to fuzziness. See for instance O. Hadžic [Fuzzy Sets Syst. 29, 115-125 (1989)], O. Kaleva [Fuzzy Sets Syst. 17, 53-65 (1985; Zbl 0584.54004)] and M. D. Weiss [J. Math. Anal. Appl. 50, 142-150 (1975; Zbl 0297.54004)].
The authors extend some fixed point theorems for multivalued mappings into a fuzzy setting. For related results see also R. K. Bose and D. Sahani [Fuzzy Sets Syst. 21, 53-58 (1987; Zbl 0609.54032)], D. Butnariu [Fuzzy Sets Syst. 7, 191-207 (1982; Zbl 0473.90087)] and S. Heilpern [J. Math. Anal. Appl. 83, 566-569 (1981; Zbl 0486.54006)].
Recently A. Chitra and P. V. Subrahmanyam [J. Math. Anal. Appl. 124, 584-590 (1987; Zbl 0628.47035)] introduced an elegant method for proving fuzzy fixed point theorems: define a multivalued mapping \(g(x)=F_ 1(x)\), where \(F_ 1(x)\) denotes the l-level set of F(x), and apply a corresponding fixed point theorem to the multivalued mapping g.
In literature there also appear different kinds of fixed point theorems related to fuzziness. See for instance O. Hadžic [Fuzzy Sets Syst. 29, 115-125 (1989)], O. Kaleva [Fuzzy Sets Syst. 17, 53-65 (1985; Zbl 0584.54004)] and M. D. Weiss [J. Math. Anal. Appl. 50, 142-150 (1975; Zbl 0297.54004)].
Reviewer: O.Kaleva
Citations:
Zbl 0609.54032; Zbl 0473.90087; Zbl 0486.54006; Zbl 0628.47035; Zbl 0584.54004; Zbl 0297.54004References:
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| [2] | Bose, R. K.; Mukherjee, R. N., On fixed points of nonexpansive set-valued mappings, (Proc. Amer. Math. Soc., 72 (1978)), 97-98 · Zbl 0396.47034 |
| [3] | Bose, R. K.; Sahani, D., Fuzzy mappings and fixed point theorems, Fuzzy Sets and Systems, 21, 53-58 (1987) · Zbl 0609.54032 |
| [4] | Dotson, W. G., On fixed points of nonexpansive mappings on nonconvex sets, (Proc. Amer. Math. Soc., 38 (1973)), 155-156 · Zbl 0274.47029 |
| [5] | Heilpern, S., Fuzzy mappings and fixed point theorem, J. Math. Anal. Appl., 83, 566-569 (1981) · Zbl 0486.54006 |
| [6] | Kirk, W. A.; Downing, D., Fixed point theorems for set-valued mappings in metric and Banach spaces, Math. Japonica, 22, 99-112 (1977) · Zbl 0372.47030 |
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