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On the solvability of periodic boundary value problems with impulse. (English) Zbl 1123.34022

The authors consider an boundary value problem for a first-order differential equation with periodic boundary conditions:
\[ x'= f(t, x),\quad t\in [0,N],\quad t\neq t_1,\tag{1} \]
\[ x(0)= x(N),\quad 0< N\in\mathbb{R},\tag{2} \]
where \(f: [0,N]\times \mathbb{R}^n\to\mathbb{R}^n\) is continuous on \(([0, N]\setminus\{t_1\})\times \mathbb{R}^n\). The impulse at \(t= t_1\) is given by the continuous function \(I_t: \mathbb{R}^n\to\mathbb{R}^n\).
By means of the fixed point theorem due to Schäfer, sufficient conditions for the existence of solutions to (1)–(2) are given.

MSC:

34B37 Boundary value problems with impulses for ordinary differential equations
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