Existence of solutions of a special class of fuzzy integral equations.

*(English)*Zbl 1140.45006Let \(E^n\) 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.

(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.

Reviewer: Dariusz Bugajewski (Baltimore)

##### Keywords:

Banach contraction principle; existence; uniqueness; continuous solutions; fuzzy integral equations
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\textit{K. Balachandran} and \textit{K. Kanagarajan}, J. Appl. Math. Stochastic Anal. 2006, No. 5, Article ID 52620, 8 p. (2006; Zbl 1140.45006)

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