A Generalized Nonlinear Bagley–Torvik Equation in Distributions

This paper investigates the fractional calculus of distributions supported on R+ in the sense of L. Schwartz, based on distributional convolutions. We further study a generalized Bagley–Torvik equation involving an arbitrary number of fractional derivative terms with orders in the interval (0,2). The existence and uniqueness of solutions for its nonlinear form are established in a space of continuous functions by applying Banach’s contraction principle, the Leray–Schauder fixed-point theorem, inverse operators, and the multivariate Mittag–Leffler function. Finally, several examples are presented, in which the values of multivariate Mittag–Leffler functions are computed to illustrate the main results.