## Existence of extremal solutions for quadratic fuzzy equations.(English)Zbl 1102.54004

The authors revise and extend certain fuzzy equations. Applying results such as the well-known fixed point theorem of Tarski, they also present some results regarding the existence of extremal solutions of the fuzzy equation $Ex^2+ Fx+ G= x,\tag{1}$ where $$E$$, $$F$$, $$G$$ and $$x$$ are positive fuzzy numbers satisfying certain conditions. Some of the results are as follows:
i) If $$E,F,G\succeq\chi\{0\}$$ and there exist $$u_0\in E^1$$ with $$u_0\succ\chi\{0\}$$ and $$Eu^2_0+ Fu_0+ G\preceq u_0$$, then (1) has an extremal solution
ii) If $$E,F\succeq\chi\{0\}$$ and if there exist $$\alpha,\beta\in E^1$$ with $$\beta> \alpha\geq \chi\{0\}$$ and $$E\alpha^2+ F\alpha+ G\geq \alpha$$, $$E\beta^2+ F\beta+ G\leq\beta$$, then (1) has extremal solutions in $$[\alpha,\beta]$$. Moreover, if $$\alpha= \beta$$, $$\alpha$$ is a solution of (1).
iii) Let $$F: E^1\to E^1$$ be nondecreasing and suppose that there exist $$\alpha,\beta\in E^1$$ with $$\alpha\leq\beta$$ and $$F(\alpha)\geq \alpha$$, $$F(\beta)\leq\beta$$, then the equation $$F(x)= x$$ has extremal solutions in $$[\alpha,\beta]$$; if $$\alpha=\beta$$, this is a fixed point for $$F$$.

### MSC:

 54A40 Fuzzy topology 54H25 Fixed-point and coincidence theorems (topological aspects)
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