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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 {\parindent=9mm \item{(A5)} for each positive number $r$ and $x\in C(J,X)$ with $\|x\|_\infty\le r$, there exists a function $l_r\in L^1(J,\Bbb R_+)$ such that $$\sup\Bigg\{|f|: f(t)\in F\bigg(t,x(t), \int_0^t g(t,s,x(s))\,dx,\ \int_0^b h(t,s,x(s))\,ds \bigg)\Bigg)\le l_r(t)$$ for a.e. $t\in J,$ \par} only depends upon the local properties of multivalued map on a bounded set. An example is also given to illustrate our main results.

93C25Control systems in abstract spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
34A60Differential inclusions
34A37Differential equations with impulses
93B28Operator-theoretic methods in systems theory
Full Text: DOI
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