Combined Liouville–Caputo Fractional Differential Equation

This paper studies a fractional differential equation combined with Liouville–Caputo operator, namely, LCDηβ,γQ(t)=λϑ(t,Q(t)),t∈[c,d],β,γ∈(0,1],η∈[0,1], where Q(c)=qc is bounded and non-negative initial value. The function ϑ:[c,d]×R→R Lipschitz continuous in the second variable, λ>0 constant operator LCDηβ,γ convex combination of left right derivatives. We study well-posedness using fixed-point theorem, estimate growth bounds solution examine asymptotic behaviours solutions. Our findings are illustrated some analytical numerical examples. Furthermore, we investigate effect noise on behaviour to equation.