Combined Liouville-Caputo Fractional Differential Equation

This paper studies a fractional differential equation combined with a Liouville-Caputo fractional differential operator, namely, (LC) D (beta)(eta),(gamma) Q (t) = lambda upsilon(t, Q (t)), t epsilon [c, d], beta, gamma epsilon (0, 1], eta epsilon [ 0, 1], where Q(c) = q(c) is a bounded and non-negative initial value. The function upsilon : [c, d] * R R is Lipschitz continuous in the second variable, lambda> 0 is a constant and the operator (LC) D-eta(beta),gamma h is a convex combination of the left and the right Liouville-Caputo fractional derivatives. We study the wellposedness using the fixed-point theorem, estimate the growth bounds of the solution and examine the asymptotic behaviours of the solutions. Our findings are illustrated with some analytical and numerical examples. Furthermore, we investigate the effect of noise on the growth behaviour of the solution to the combined Liouville-Caputo fractional differential equation.