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

Nieto, Juan J.; Rodríguez-López, Rosana

Fixed Point Theory Appl. 2005, No. 3, 321-342 (2005).

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 \(F\).

Reviewer: S. Ganguly (Kolkata)

MSC:

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

Keywords:

fixed point theorem of Tarski; Lipschitzian fuzzy number