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Controllability of mixed Volterra-Fredholm-type integro-differential systems in Banach space. (English) Zbl 1119.93016
Summary: {\it M. B. Dhakne} and {\it S. D. Kendre} [Commun. Appl. Nonlinear Anal. 13, No. 4, 101--112 (2006; Zbl 1119.45005)] have proved the existence of the abstract nonlinear mixed Volterra-Fredholm integro-differential system of the type \align x'(t) &=f\left( t,x(t), \quad \int_0^t k(t,s,x(s)) \text{d}s, \quad \int_0^T h(t,s,x(s)) \text{d}s\right), \\ x(0) &= x_0, \quad t \in J = [0,T].\endalign In this short article, we have studied sufficient conditions for controllability of semi-linear mixed Volterra-Fredholm-type integro-differential systems in Banach space of the type\left. \aligned x'(t) &= Ax(t) + (Bu)(t) + f\left( t,x(t), \quad \int_0^t g(t,s,x(s)) \text{d}s, \quad \int_0^T h(t,s,x(s)) \text{d}s\right), \\ x(0) &= x_0, \quad t \in J = [0,T], \endaligned\right\} where the state $x(.)$ takes values in a Banach space $X$ and the control function $u(.)$ is given in $L^{2}(J,U)$, with $U$ as a Banach space. Here $A$ is the infinitesimal generator of a strongly continuous semigroup in a Banach space $X$. $B$ is a bounded linear operator from $U$ into $X$. The result is obtained by using the application of the topological transversality theorem known as Leray-Schauder alternative and rely on a priori bounds of solutions. An example is provided to illustrate the theory.

##### MSC:
 93B05 Controllability 93C23 Systems governed by functional-differential equations 45J05 Integro-ordinary differential equations 45N05 Abstract integral equations, integral equations in abstract spaces
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##### References:
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