Necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and solution representation

Abstract

In this paper, we investigate the necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and the solution representation by using the multivariate Mittag-Leffler function. First we convert fuzzy initial value problem into the cut problem (system of fractional differential equations with inequality constraints) and obtain existence results for the solution of the cut problem under (1,1)- differentiability. Next we study the conditions for the solutions of the cut problem to constitute the solution of a fuzzy initial value problem and suggest a necessary and sufficient condition for the (1,1)-solution. Also, some examples are given to verify the effectiveness of our proposed method. The necessary and sufficient condition, solution representation for (1,2)-solution of initial value problem of fuzzy fractional Bagley-Torvik equation are shown in Appendix.