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Periodic boundary value problems for second order impulsive integrodifferential equations of mixed type in Banach spaces. (English) Zbl 0849.45006
The author considers the existence of minimal and maximal solutions for the periodic boundary value problems of second order impulsive integro-differential equations of mixed type in Banach spaces $- u'' = f(t,u,Tu,Su) \text{ for } t\neq t_k,\;\Delta u |_{t = t_k} = I_k \bigl( u(t_k) \bigr),\;\Delta u' |_{t = t_k} = \overline I_k \bigl( u(t_k) \bigr)\;(k = 1,2, \dots, m)$ $$u(0) = u(2 \pi)$$, $$u'(0) = u'(2 \pi)$$, where $$f \in C[J \times E^3;E]$$, $$J = [0, 2 \pi]$$, $$E$$ is a real Banach space, $$0 < t_1 < \cdots < t_m < 2 \pi$$, $$I_k \in C [E;E]$$, $$\overline I_k \in C[E;E]$$, $$\Delta u |_{t = t_k} = u(t^+_k) - u(t^-_k)$$, $$\Delta u' |_{t = t_k} = u'(t^+_k) - u'(t^-_k)$$ $$(k = 1,2, \dots, m)$$. The operators $$T,S$$ are given by $Tu (t) = \int^t_0 k(t,s) u(s) ds, \quad Su (t) = \int^{2 \pi}_0k_1 (t,s) u(s)ds$ with $$k \in C[D,R]$$, $$D = \{(t,s) \in \mathbb{R}^2 : 0 \leq s \leq t \leq 2 \pi\}$$, $$k_1 \in C[J \times J; \mathbb{R}]$$. The method of proof is based on the monotone iterative technique and cone theory.

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations 45M15 Periodic solutions of integral equations 45G15 Systems of nonlinear integral equations 45L05 Theoretical approximation of solutions to integral equations
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