Study On The Existence And Uniqueness For Solutions Of Some Fuzzy Fractional Integral-differential Equations

Fuzzy fractional differential equations are an important branch of fuzzy mathematics,which can be used as an important mathematical tool for modeling real systems with uncertainties,especially in real-world applications such as seismology,fluid mechanics,and measurement of viscoelastic material properties,and so on.There is a huge potential for application in modeling various physical problems.In this paper,by using the monotonic iteration technique and the upper and lower solution methods,we study the existence and uniqueness of solutions to two class of fuzzy fractional delay integral-differential equations with fractional derivatives.On the one hand,we introduce the concept of Caputo type interval-valued fractional conformable calculus and prove the comparison principle with Caputo type interval-valued fractional conformable derivatives.Based on this,we study the initial value problem of a class of fuzzy fractional delay integral-differential equations with Caputo-type interval-valued fractional conformable derivatives.By applying the monotone iterative method with the method of upper and lower solutions,this paper not only proves the existence of the extremum solutions of the above mentioned fuzzy fractional delay integral-differential equations,but also gives a monotone iterative sequence for approximating the extremum solutions.Furthermore,we prove the existence of unique solutions by developing a class of generalized Gronwall inequalities with Caputo type interval valued fractional conformable derivatives.On the other hand,we study a class of fuzzy fractional delay integral-differential equations with generalized ABC fractional derivatives.By using the monotonic iterative method,we not only obtain the existence and uniqueness of the solution to the initial value problem of the fuzzy fractional delay integral-differential equation with generalized ABC fractional derivatives,but also give a monotonic iterative sequence that approximates the unique solution.Finally,it is also proved to the continuous dependence on the initial value.