Existence of solutions of fuzzy integral equations in Banach spaces (Q1914521)

We consider the fuzzy integral equation \[ \varphi (u) = w_0 + \int^u_{u_0} F \bigl( u,s, \varphi (s) \bigr) ds, \quad \varphi(u_0) = w_0, \tag{1} \] where \(F : J \times J \times T(X) \to T(X)\) is continuous, \(J = [u_0, u_0 + d]\), and \(T(X)\) is a regular fuzzy set. The purpose of this paper is to study the local existence of solutions and the approximate solutions for problem (1), based on the properties of the \(\alpha\)-index of Kuratowski.