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Zbl 1167.93007
Chang, Y.-K.; Chalishajar, D.N.
Controllability of mixed Volterra-Fredholm-type integro-differential inclusions in Banach spaces. (English)
[J] J. Franklin Inst. 345, No. 5, 499-507 (2008). ISSN 0016-0032; ISSN 1879-2693/e

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.

MSC 2000:: *93B05 Controllability
93C25 Control systems in abstract spaces
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
34A60 ODE with multivalued right-hand sides
34A37 Differential equations with impulses
93B28 Operator-theoretic methods in systems theory

Keywords: controllability; mixed Volterra-Fredholm-type integro-differential inclusions; Bohnenblust-Karlin's fixed point theorem

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