A new nonlinear duffing system with sequential fractional derivatives

Abstract

By considering the Caputo fractional derivative and Riemann-Liouville integral, in the present work, we are concerned with a nonlinear sequential fractional differential system of Duffing oscillator type. The considered system has neither the commutativity nor the semi group properties, since the sum of the two orders of derivatives, of the left hand side of the problem, are outside the interval [0,1]. With the absence of these two properties, we have to find other arguments to obtain the integral representation of the problem, to be able thereafter to present the other main results. Then, using the contraction mapping principle and Scheafer theorem, two main theorems on the uniqueness and existence of solutions are proved. Finally, some examples are given to illustrate the proposed main results.