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Controllability of mixed Volterra-Fredholm-type integro-differential inclusions in Banach spaces. (English) Zbl 1167.93007

Summary: The paper establishes a sufficient condition for the controllability of semilinear mixed Volterra-Fredholm-type integro-differential inclusions in Banach spaces. We use Bohnenblust-Karlin’s fixed point theorem combined with a strongly continuous operator semigroup. Our main condition

for each positive number $r$ and $x\in C\left(J,X\right)$ with ${\parallel x\parallel }_{\infty }\le r$, there exists a function ${l}_{r}\in {L}^{1}\left(J,{ℝ}_{+}\right)$ such that

$sup\left\{|f|:f\left(t\right)\in F\left(t,x\left(t\right),{\int }_{0}^{t}g\left(t,s,x\left(s\right)\right)\phantom{\rule{0.166667em}{0ex}}dx,\phantom{\rule{4pt}{0ex}}{\int }_{0}^{b}h\left(t,s,x\left(s\right)\right)\phantom{\rule{0.166667em}{0ex}}ds\right)\right\)\le {l}_{r}\left(t\right)$

for a.e. $t\in J,$

only depends upon the local properties of multivalued map on a bounded set. An example is also given to illustrate our main results.

##### MSC:
 93B05 Controllability 93C25 Control systems in abstract spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 34A60 Differential inclusions 34A37 Differential equations with impulses 93B28 Operator-theoretic methods in systems theory
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