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.


34B37 Boundary value problems with impulses for ordinary differential equations
Full Text: DOI


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