Fuzzy integro-differential equations with compactness type conditions (Q2877963)

The authors consider a fuzzy integro-differential equation of the type NEWLINE\[NEWLINEx'(t)=F(t,x(t),(Vx(t))),\, x(0)=x_0,\,t \in [0, T],NEWLINE\]NEWLINE where NEWLINE\[NEWLINE(Vx(t))= \int _{0} ^{t} K(t,s) x(s) ds,NEWLINE\]NEWLINE \(F:I \times E^n \times E^n \rightarrow E^n\) and \( E^n\) is a complete fuzzy metric space consisting of all mappings \(x: \mathbb{R^n} \rightarrow [0, T]\) satisfy certain properties. Assuming that \(F\) is strongly measurable on \(E^n \times E^n \) and continuous for almost all \(t \in I,\) using the Kuratowski measure of noncompactness and the Hausdorff distance they obtain criteria under which the solution of the considered fuzzy integro-differential equation exists.