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Existence of solutions of a special class of fuzzy integral equations. (English) Zbl 1140.45006
Let \(E^n\) be a family of functions \(u:\mathbb R^n\to [0,1]\) which satisfy the following conditions:
(i) \(u\) is normal, that is, there exists \(x_0\in\mathbb R^n\) such that \(u(x_0)=1\);
(ii) \(u\) is fuzzy convex;
(iii) \(u\) is upper continuous;
(iv) \([u]^0=cl\{x\in\mathbb R^n:\;u(x)>0\}\) is compact.
The authors investigate the existence and uniqueness of continuous solutions to the fuzzy integral equations of the form
\[ x(t)=\varphi(t)+x(t)\cdot \int_0^tK(t,s)f(s,x(s))\,ds+ \int_0^tg(t,s,x(s))\,ds, \]
where \(\varphi:[0,T]\to E^n\), \(k:[0,T]\times [0,T]\to\mathbb R\), \(f:[0,T]\times E^n\to E^n\) and \(g:[0,T]\times [0,T]\times E^n\to E^n\) are continuous functions (\(T>0\)).
The proof of their result is based on the Banach contraction principle. The paper does not contain any example illustrating the results.
MSC:
45G10 Other nonlinear integral equations
26E50 Fuzzy real analysis
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