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Periodic boundary value problems for first order impulsive integro-differential equations of mixed type. (English) Zbl 1057.45002

The authors consider the periodic boundary value problem for first-order impulsive integro-differential equations of mixed type: \[ \begin{gathered} x'(t)= f(t,x(t),[Tx](t),[Sx](t)),\;t\in I,\;t\neq t_k\quad (k= 1,2,\dots, p),\\ \Delta x(t_k)= I_k(x(t_k)),\\ x(0)= x(2n),\end{gathered} \] where \[ \begin{gathered} f\in C(I\times R^1\times R^1\times R^1; R^1),\;I= [0,2n],\;I_k\in C(R^1; R^1),\\ \Delta x(t_k)= x(t^+_k)- x(t^-_k),\quad 0< t_1< t_2<\cdots< t_p< 2n,\\ [Sx])t_= \int^t_0 K(t,s)x(s)\,ds,\;[Tx](t)= \int^{2n}_0 H(t,s) x(s)\,ds.\end{gathered} \] Applying Lakshmikantham’s monotone iterative technique and the method of upper and lower solutions they obtain extremal solutions of the above problem.

MSC:

45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
45M15 Periodic solutions of integral equations
45L05 Theoretical approximation of solutions to integral equations
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