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Initial value problems for nonlinear second order impulsive integro-differential equations of mixed type in Banach spaces. (English) Zbl 1093.45006

The authors improve some earlier results on the existence of a unique solution to the second order impulsive integrodifferential equation \[ \begin{aligned} &x'' = f(t,x,x',Tx,Sx), \quad t\in [0,a], \quad t\neq t_k,\\ &\Delta x_{| t=t_k}= I_k(x(t_k),x'(t_k)),\\ &\Delta x'_{| t=t_k}= \overline{I}_k(x(t_k),x'(t_k)),\quad k=1,\ldots,m,\\ &x(0) = x_0, \quad x'(0)=x'_0,\end{aligned} \] in a Banach space, where \((Tx)(t) = \int_0^t k(t,s)x(s)\,ds\) and \((Sx)(t) = \int_0^a h(t,s)x(s)\,ds\). In the proof the equation is solved on the intervals between the impulse points \(t_k\) using assumptions on the Lipschitz continuity of the function \(f\).

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45G10 Other nonlinear integral equations
45J05 Integro-ordinary differential equations
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