On two fractional differential inclusions.

Dumitru Baleanu,Maysaa’ Mohamed Al Qurashi,Vahid Hedayati,Shahram Rezapour

doi:10.1186/s40064-016-2564-z

Dumitru Baleanu, Maysaa’ Mohamed Al Qurashi + Show 2 more

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https://doi.org/10.1186/s40064-016-2564-z

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Abstract

We investigate in this manuscript the existence of solution for two fractional differential inclusions. At first we discuss the existence of solution of a class of fractional hybrid differential inclusions. To illustrate our results we present an illustrative example. We study the existence and dimension of the solution set for some fractional differential inclusions.

Highlights

As you know, fractional dynamical systems be used in modeling of some real processes and there are many published works about the existence of solutions for many fractional differential equations and inclusions
We investigate in this manuscript the existence of solution for two fractional differential inclusions
At first we discuss the existence of solution of a class of fractional hybrid differential inclusions

Summary

Background

Fractional dynamical systems be used in modeling of some real processes and there are many published works about the existence of solutions for many fractional differential equations (see for example Baleanu et al 2013a, b, c; Chai 2013 and the references therein) and inclusions Theorem 3 (Dhage 2006) Suppose that X is a Banach algebra space, S ∈ Pbd,cl,cv(X ) and A : S → Pcl,cv,bd(X ) and B : S → Pcp,cv(X ) two multifunctions satisfying the following conditions. Theorem 5 (Agarwal et al 2013) Suppose that C is a nonempty closed convex subset of Banach space X. This implies that, B has a closed graph and the operator B is upper semi-continuous. We review existence and dimension of the solution set of the fractional differential inclusion problem cDαy(s) ∈ G s, y(s), (φy)(s), (ψy)(s), cDβ1 y(s), . 0s(s − t)α−βi−1|v1(t) − v2(t)| ≤ i2 x − y and so − y = l x − y This implies that the multifunction N is a contraction via closed values. If Lebesgue measure of the set s : dim G(s, y1, y2, . . . , y2k+3) < 1 for some y1, y2, . . . , y2k+3 ∈ R is zero and l < 1, the set of all solutions of the problem (2) is infinite dimensional, where l is defined in Theorem 9

Proof Define the operator Υ by

Conclusions