## 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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### References:

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