Positive integrable solutions for nonlinear integral and differential inclusions of fractional-orders

In this paper we study the global existence of positive integrable solution for the nonlinear integral inclusion of fractional order $$x(t)\in p(t)+I^{\alpha }{F}_1(t,I^{\beta }{f}_2(t,x(\phi (t)))),t\in (0,1).$$ As an application the global existence of the solution for the initial-value problem of the arbitrary (fractional) orders differential inclusion $$\frac{dx(t)}{dt}\in p(t)+I_1^{\text{alphaF}}(t,D^{\text{gammax}}(t))),\text{a.e.}tgt0$$ will be studied.