Extremal Positive Solutions For The Distributed Order Fractional Hybrid Differential Equations

Abstract

In this article, we prove the existence of extremal positive solution for the distributed order fractional hybrid differential equation<br />$$int_{0}^{1}b(q)D^{q}[frac{x(t)}{f(t,x(t))}]dq=g(t,x(t)),$$<br />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.