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Periodic boundary value problems of second-order impulsive differential equations. (English) Zbl 1176.34032

Sufficient conditions are obtained for the existence of solutions to periodic boundary value problem for second-order impulsive differential equations. The problems have the form
\[ x''(t)= f(t,x(t), x'(t)), \]
\[ \Delta x(t_k)= I_k(x(t_k)),\;\Delta x'(t_k)= I^*_k(x'(t_k)), \]
\[ x(0)= x(T),\quad x'(0)= x'(T). \]
Here, \(k= 1,2,\dots, p\), the functions \(I_k\), \(I^*_k\) are continuous, \(\Delta x(t_k)= x(t^+_k)- x(t_k)\),
\[ \Delta x'(t_k)= x'(t^+_k)- x'(t_k), \]
the right-hand side \(f\) satisfies the Carathéodory conditions. Monotone iterative technique is employed.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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References:

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