It is evident that fuzzy arithmetic is different than that of normal arithmetic due to the different classes of functions and symbols. This article is concerned with an appropriate definition of the fractional derivative and integral in a fuzzy sense and a new concept of solutions for a fuzzy Caputo fractional initial value problem (IVP) is presented. Further, under some sufficient conditions on IVP, the existence, uniqueness, and stability results of the solution are established by applying the iterate methods. For the validation of established results, a particular fuzzy fractional Riccati differential equation is presented.
In this article, we consider a class of stochastic fractional differential equations (SFDEs) driven by L'evy noise in the sense of a newly defined OBC-fractional derivative. This is a generalized Caputo type fractional derivative introduced recently by Zaid Odibat and Dumitru Baleanu. Under some suitable sufficient conditions, we have employed fixed point theorem to obtain existence and uniqueness results for the considered equation. We have also presented anexample which illustrates the applicability of our obtained results.
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