Random fuzzy fractional integral equations – theoretical foundations

This paper presents mathematical foundations for studies of random fuzzy fractional integral equations which involve a fuzzy integral of fractional order. We consider two different kinds of such equations. Their solutions have different geometrical properties. The equations of the first kind possess solutions with trajectories of nondecreasing diameter of their consecutive values. On the other hand, the solutions to equations of the second kind have trajectories with nonincreasing diameter of their consecutive values. Firstly, the existence and uniqueness of solutions is investigated. This is showed by using a method of successive approximations. An estimation of error of nth approximation is given. Also a boundedness of the solution is indicated. To show well-posedness of the considered theory, we prove that solutions depend continuously on the data of the equations. Some concrete examples of random fuzzy fractional integral equations are solved explicitly.