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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
Full Text:
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