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Existence results for semilinear functional differential inclusions involving Riemann-Liouville fractional derivative. (English) Zbl 1198.34178

The authors are concerned with the existence of mild solutions for the fractional order semilinear functional differential inclusion of the form:

${D}^{\alpha }y\left(t\right)-Ay\left(t\right)\in F\left(t,{y}_{t}\right),\phantom{\rule{0.166667em}{0ex}}t\in J:=\left[0,b\right];\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}y\left(t\right)=\phi \left(t\right),\phantom{\rule{0.166667em}{0ex}}t\in \left[-r,0\right],$

where ${D}^{\alpha }$ is the standard Riemann-Liouville fractional derivative, $F:J×C\left(\left[-r,0\right],E\right)\to 𝒫\left(E\right)$ is a multivalued function, $\left(E,|·|\right)$ is a separable real Banach space and $𝒫\left(E\right)$ is the family of all nonempty subsets of $E$. The operator $A:D\left(A\right)\subset E\to E$, densely defined possibly unbounded), is a generator of a strongly continuous semigroup ${\left\{T\left(t\right)\right\}}_{t\ge 0}$ of bounded linear operators from $E$ to $E$. The function $\phi :\left[-r,0\right]\to E$ is a given continuous function such that $\phi \left(0\right)=0$. For any function $y$ defined on $\left[-r,b\right]$ and any $t\in J$, ${y}_{t}$ denotes the element of $C\left(\left[-r,0\right],E\right)$ defined by

${y}_{t}\left(\theta \right)=y\left(t+\theta \right),\phantom{\rule{1.em}{0ex}}\theta \in \left[-r,0\right]·$

Both cases of convex valued and nonconvex valued right hand side are considered. The proofs are, respectively, achieved by means of Martelli’s fixed point theorem and the contraction multivalued fixed point theorem due to Covitz and Nadler. The compactness of the set of mild solutions defined on $\left[-r,b\right]$ is also proved.

##### MSC:
 34K37 Functional-differential equations with fractional derivatives 34K09 Functional-differential inclusions 34K30 Functional-differential equations in abstract spaces 47N20 Applications of operator theory to differential and integral equations