Periodic boundary value problems for second order impulsive integrodifferential equations of mixed type in Banach spaces (Q1909995)

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.