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