Controllability of mixed Volterra–Fredholm-type integro-differential systems in Banach space

Dhakne and Kendre [On abstract nonlinear mixed Volterra–Fredholm integro-differential equations, Presented Paper in the International Conference at IIT-Bombay, 11–13 December, 2004] has proved the existence of the abstract nonlinear mixed Volterra–Fredholm integro-differential system of the type x ′ ( t ) = f t , x ( t ) , ∫ 0 t k ( t , s , x ( s ) ) d s , ∫ 0 T h ( t , s , x ( s ) ) d s , x ( 0 ) = x 0 ∈ X ; t ∈ J = [ 0 , T ] . s 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 x ′ ( t ) = Ax ( t ) + ( Bu ) ( t ) + f t , x ( t ) , ∫ 0 t g ( t , s , x ( s ) ) d s , ∫ 0 T h ( t , s , x ( s ) ) d s , x ( 0 ) = x 0 , t ∈ J = [ 0 , T ] , 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 . sult 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.