Extremal Positive Solutions For The Distributed Order Fractional Hybrid Differential Equations

In this article, we prove the existence of extremal positive solution for the distributed order fractional hybrid differential equation $$int_{0}^{1}b(q)D^{q}[frac{x(t)}{f(t,x(t))}]dq=g(t,x(t)),$$ using a fixed point theorem in the Banach algebras. This proof is given in two cases of the continuous and discontinuous function $g$, under the generalized Lipschitz and Caratheodory conditions.